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For additional information about this publication click this link. http://hdl.handle.net/2066/196037 Please be advised that this information was generated on 2021-09-29 and may be subject to change. teaching and education Crystallographic shelves: space-group hierarchy explained ISSN 1600-5767 Massimo Nespolo,a* Mois Ilia Aroyob and Bernd Souvignierc aUniversite´ de Lorraine, CNRS, CRM2, Nancy, France, bFı´sica de la Materia Condensada, Facultad de Ciencia y Tecnologı´a, Universidad del Paı´s Vasco, Apartado 644, Bilbao 48080, Spain, and cInstitute for Mathematics, Astrophysics and Particle Physics, Faculty of Science, Mathematics and Computing Science, Radboud University Nijmegen, Postbus Received 19 May 2018 9010, Nijmegen 6500 GL, The Netherlands. *Correspondence e-mail: [email protected] Accepted 7 September 2018 Space groups are classified, according to different criteria, into types, classes, Edited by G. J. McIntyre, Australian Centre for systems and families. Depending on the specific research topic, some of these Neutron Scattering, Australia concepts will be more relevant to the everyday crystallographer than others. Unfortunately, incorrect use of the classification terms often leads to Keywords: space-group classification; crystal misunderstandings. This article presents the rationale behind the different classes; crystal systems; crystal families; Bravais classification levels. classes; lattice systems. 1. Introduction A French proverb states that hierarchy is like shelves: the higher they are, the less useful they are. (‘La hie´rarchie c’est comme une e´tage`re, plus c’est haut, plus c’est inutile.’) While this may well apply to social sciences, in exact sciences hier- archy is behind fundamental concepts like taxonomy and phylogenetics. Crystallography is no exception: crystal struc- tures and their symmetry groups are arranged in a hierarchical way into classes, systems and families that emphasize common features used as classification criteria. Unfortunately, incorrect definitions and sloppy terminology are not rare in textbooks and scientific manuscripts, and frequently lead to misunder- standings. In this article we present a brief panoramic over- view of well known and less well known crystallographic terms in a practical and concrete approach; our aim is also to help the less theoretically inclined crystallographer to understand and apply the concepts expressed by these terms. Our refer- ence is Volume A of International Tables for Crystallography (Aroyo, 2016), whose chapters are indicated henceforth as ITAX where X is the number of the chapter. For the following discussion we need to remind the reader that, with respect to a coordinate system, a symmetry opera- tion is represented by a matrix–column pair (W, w), where the (3 Â 3) matrix W is called the linear or matrix part and the (3Â 1) column w is the translation or column part. A trans- lation is represented as (I, w), where I is the identity matrix. A rotation, reflection or rotoinversion about the origin is represented by (W, 0), where 0 is the zero column. The column w can be decomposed into two components: an intrinsic part, which represents the screw or glide component of the opera- tion (screw rotation or glide reflection), and a location part, which is non-zero when the rotation axis or reflection plane does not pass through the origin. For example, the matrix– column pair 0 1 1 1002 @ 1 A 0102 1 001 2 # 2018 International Union of Crystallography J. Appl. Cryst. (2018). 51, 1481–1491 https://doi.org/10.1107/S1600576718012724 1481 teaching and education represents an operation mapping a point with coordinates x, y, 2. Group versus group type 1 1 1 z to a point with coordinates x + 2 , y + 2 , z + 2 . This is found to The first and probably most common confusion that occurs in 1 1 be a 180 screw rotation about a line 4, y, 4 , parallel to the b the literature is the use of the term ‘group’ for a type of group. 1 1 axis but passing through x = 4 , z = 4 , with an intrinsic (screw) A symmetry group is a group (in the algebraic sense) formed 1 component 2 parallel to b. Depending on whether their by the set of symmetry operations of a given object. In crys- intrinsic part is zero or not, symmetry operations are of finite tallography, the objects are typically atoms, molecules and or infinite order. In the former case, the operation is a rota- crystal structures. tion, reflection or rotoinversion and has at least one fixed A crystal structure is usually described as an idealized point, while in the latter case it is a screw rotation or glide periodic pattern of atoms in three-dimensional space using the reflection and does not leave any point fixed. corresponding coordinates with respect to the chosen coor- A symmetry operation of an object is an isometry dinate system. Conventionally, these coordinate systems are (congruence) which maps the object onto itself. Except for the adapted to the symmetry properties of the crystal structure. identity and for translations, a geometric element is attached For example, the basis vectors are oriented along symmetry to each symmetry operation, which is closely related to the set directions and display the periodicity of the structure. of fixed points of the operation. Thus, for (glide) reflections The symmetry of two objects is described by the same group and (screw) rotations the geometric elements are planes and if the symmetry operations of the first object are also lines, respectively. For a rotoinversion, the geometric element symmetry operations of the second and vice versa. However, is the line of the corresponding rotation axis together with the for different crystal structures this is almost never the case, unique inversion point on the axis fixed by the rotoinversion. since a translation bringing one of these structures to overlap Finally, in the case of an inversion, the inversion centre serves with itself will not be a symmetry operation of the other, due as geometric element. A symmetry element is defined as the to their different cell parameters. On the other hand, after combination of a geometric element with the set of symmetry choosing suitable coordinate systems for both structures, the operations having this geometric element in common (the so- symmetry operations may well be represented by the same called element set) (de Wolff et al., 1989, 1992; Flack et al., matrix–column pairs when expressed in the respective coor- 2000). Among the symmetry operations sharing the same dinate system. In this case, the space groups of the two crystal geometric element, the simplest one [more precisely, the one structures are said to belong to the same type. The difference with the smallest positive (possibly zero) intrinsic translation between space groups and space-group types becomes clear part] is called the defining operation of the symmetry element. when comparing the symmetry groups of different structures It specifies the name (mirror plane, glide plane, rotation axis, which belong to the same space-group type. For example, screw axis) and the symbol (alphanumeric and graphic) of the jadeite, NaAlSi2O6, and thoreaulite, SnTa2O7, are two mono- symmetry element. For example, consider the geometric clinic minerals differing in their chemistry, structure, proper- element x, y, 0 (a plane) with an element set composed of an ties and formation environment. To specify the symmetry 1 infinite number of glide reflections g with glide vectors (p + 2 , groups, we also need the cell parameters of the conventional q, 0), where p and q are integers. The defining operation cell, which determine the coordinate system. These are a = 1 ˚ corresponds to p = q = 0 and the symbol g (2,0,0) x,y,0 is 9.418, b = 8.562, c = 5.219 A and = 107.58 for jadeite replaced by the special symbol ax,y,0. (Prewitt & Burnham, 1966), and a = 17.140, b = 4.865, c = The translational symmetry is captured by the conventional 5.548 A˚ and = 91.0 (Mumme, 1970) for thoreaulite. Clearly, cell. This is a unit cell that satisfies the three conditions below applying the translations of one mineral to the other mineral (ITA1.3.2): would not bring the structure to coincide with itself and they (i) Its basis vectors a, b, c define a right-handed axial setting. would thus not be symmetry operations of the second mineral, (ii) Its edges are along symmetry directions of the lattice. showing that these two minerals have different space groups. (iii) It is the smallest cell compatible with the above However, when the space groups are expressed with respect to conditions. the conventional coordinate systems with basis vectors along The metric properties of the conventional cell, and thus also the a, b and c axes and with lengths given by the cell para- of the translational subgroup, are determined by the cell meters, the matrix–column pairs of the symmetry operations parameters a, b, c, , , and are reflected by the metric become the same and the space groups therefore belong to the tensor, a square symmetric matrix G whose elements are the same type: C2/c (No. 15). The Hermann–Mauguin symbol [see scalar products of the basis vectors: a recent discussion by Nespolo & Aroyo (2016)] identifies the 0 1 type of space group, not the space group itself. In everyday a Á aaÁ baÁ c laboratory jargon, the word ‘type’ is often dropped and this B C G ¼ @ b Á abÁ bbÁ c A does not normally impede the transmission of information. In other cases, like the classifications we are going to discuss c Á acÁ bcÁ c 0 1 below, or in the context of group–subgroup relations, the a2 ab cos ac cos difference is of paramount importance.