 1. Introduction
 2. Group versus group type
 3. Chirality and handedness. Sohncke, affine and crystallographic spacegroup types
 4. Symmorphic types of space groups and arithmetic crystal classes
 5. Bravais types of lattices, Bravais arithmetic classes and Bravais classes
 6. Geometric crystal classes, holohedries and Laue classes
 7. Crystal systems
 8. Lattice systems
 9. Crystal families
 10. Conclusion
 References
 1. Introduction
 2. Group versus group type
 3. Chirality and handedness. Sohncke, affine and crystallographic spacegroup types
 4. Symmorphic types of space groups and arithmetic crystal classes
 5. Bravais types of lattices, Bravais arithmetic classes and Bravais classes
 6. Geometric crystal classes, holohedries and Laue classes
 7. Crystal systems
 8. Lattice systems
 9. Crystal families
 10. Conclusion
 References
teaching and education
Crystallographic shelves: spacegroup hierarchy explained
^{a}Université de Lorraine, CNRS, CRM2, Nancy, France, ^{b}Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, Bilbao 48080, Spain, and ^{c}Institute for Mathematics, Astrophysics and Particle Physics, Faculty of Science, Mathematics and Computing Science, Radboud University Nijmegen, Postbus 9010, Nijmegen 6500 GL, The Netherlands
^{*}Correspondence email: massimo.nespolo@univlorraine.fr
Space groups are classified, according to different criteria, into types, classes, systems and families. Depending on the specific research topic, some of these concepts will be more relevant to the everyday crystallographer than others. Unfortunately, incorrect use of the classification terms often leads to misunderstandings. This article presents the rationale behind the different classification levels.
Keywords: spacegroup classification; crystal classes; crystal systems; crystal families; Bravais classes; lattice systems.
1. Introduction
A French proverb states that hierarchy is like shelves: the higher they are, the less useful they are. (`La hiérarchie c'est comme une étagère, plus c'est haut, plus c'est inutile.') While this may well apply to social sciences, in exact sciences hierarchy is behind fundamental concepts like taxonomy and phylogenetics. Crystallography is no exception: crystal structures 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 misunderstandings. In this article we present a brief panoramic overview 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 reference 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 W, w), where the (3 × 3) matrix W is called the linear or matrix part and the (3 × 1) column w is the translation or column part. A translation 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 operation (screw rotation or glide reflection), and a location part, which is nonzero when the rotation axis or reflection plane does not pass through the origin. For example, the matrix–column pair
is represented by a matrix–column pair (represents an operation mapping a point with coordinates x, y, z to a point with coordinates + ½, y + ½, + ½. This is found to be a 180° screw rotation about a line ¼, y, ¼, parallel to the b axis but passing through x = ¼, z = ¼, with an intrinsic (screw) component ½ parallel to b. Depending on whether their intrinsic part is zero or not, symmetry operations are of finite or infinite order. In the former case, the operation is a rotation, reflection or rotoinversion and has at least one fixed point, while in the latter case it is a screw rotation or glide reflection and does not leave any point fixed.
A symmetry operation of an object is an (congruence) which maps the object onto itself. Except for the identity and for translations, a geometric element is attached to each which is closely related to the set of fixed points of the operation. Thus, for (glide) reflections and (screw) rotations the geometric elements are planes and lines, respectively. For a rotoinversion, the is the line of the corresponding rotation axis together with the unique on the axis fixed by the rotoinversion. Finally, in the case of an inversion, the inversion centre serves as A symmetry element is defined as the combination of a geometric element with the set of symmetry operations having this in common (the socalled element set) (de Wolff et al., 1989, 1992; Flack et al., 2000). Among the symmetry operations sharing the same the simplest one [more precisely, the one with the smallest positive (possibly zero) intrinsic translation part] is called the defining operation of the It specifies the name (mirror plane, glide plane, rotation axis, screw axis) and the symbol (alphanumeric and graphic) of the For example, consider the x, y, 0 (a plane) with an element set composed of an infinite number of glide reflections g with glide vectors (p + ½, q, 0), where p and q are integers. The defining operation corresponds to p = q = 0 and the symbol g (½, 0, 0) x,y,0 is replaced by the special symbol a x,y,0.
The translational symmetry is captured by the conventional cell. This is a that satisfies the three conditions below (ITA1.3.2):
(i) Its basis vectors a, b, c define a righthanded axial setting.
(ii) Its edges are along symmetry directions of the lattice.
(iii) It is the smallest cell compatible with the above conditions.
The metric properties of the a, b, c, α, β, γ and are reflected by the metric tensor, a square symmetric matrix G whose elements are the scalar products of the basis vectors:
and thus also of the translational are determined by the cell parameters2. Group versus group type
The first and probably most common confusion that occurs in the literature is the use of the term `group' for a type of group. A symmetry group is a group (in the algebraic sense) formed by the set of symmetry operations of a given object. In crystallography, the objects are typically atoms, molecules and crystal structures.
A
is usually described as an idealized periodic pattern of atoms in threedimensional space using the corresponding coordinates with respect to the chosen coordinate system. Conventionally, these coordinate systems are adapted to the symmetry properties of the For example, the basis vectors are oriented along symmetry directions and display the periodicity of the structure.The symmetry of two objects is described by the same group if the symmetry operations of the first object are also symmetry operations of the second and vice versa. However, for different crystal structures this is almost never the case, since a translation bringing one of these structures to overlap with itself will not be a of the other, due to their different cell parameters. On the other hand, after choosing suitable coordinate systems for both structures, the symmetry operations may well be represented by the same matrix–column pairs when expressed in the respective coordinate system. In this case, the space groups of the two crystal structures are said to belong to the same type. The difference between space groups and spacegroup types becomes clear when comparing the symmetry groups of different structures which belong to the same spacegroup type. For example, jadeite, NaAlSi_{2}O_{6}, and thoreaulite, SnTa_{2}O_{7}, are two monoclinic minerals differing in their chemistry, structure, properties and formation environment. To specify the symmetry groups, we also need the cell parameters of the which determine the coordinate system. These are a = 9.418, b = 8.562, c = 5.219 Å and β = 107.58° for jadeite (Prewitt & Burnham, 1966), and a = 17.140, b = 4.865, c = 5.548 Å and β = 91.0° (Mumme, 1970) for thoreaulite. Clearly, applying the translations of one mineral to the other mineral would not bring the structure to coincide with itself and they would thus not be symmetry operations of the second mineral, showing that these two minerals have different space groups. However, when the space groups are expressed with respect to the conventional coordinate systems with basis vectors along the a, b and c axes and with lengths given by the cell parameters, the matrix–column pairs of the symmetry operations become the same and the space groups therefore belong to the same type: C2/c (No. 15). The Hermann–Mauguin symbol [see a recent discussion by Nespolo & Aroyo (2016)] identifies the type of not the itself. In everyday laboratory jargon, the word `type' is often dropped and this does not normally impede the transmission of information. In other cases, like the classifications we are going to discuss below, or in the context of group–subgroup relations, the difference is of paramount importance. For example, a single may have various different subgroups of the same type which can correspond to different phase transitions, or to different domain states arising in a In any case, one should bear in mind that the number of spacegroup types is finite, whereas that of space groups is infinite, due to the infinitely many possible values for the cell parameters and for the orientation and location in space of the geometric elements.
The type of a lattice is characterized by a number of free parameters, which correspond to the symmetryunrestricted cell parameters and are represented by the independent nonzero elements of the ). It is often overlooked that, while the presence of certain symmetry operations of the can restrict the possible ratios of cell lengths or angular values, their absence cannot impose any restriction. Although this should be selfevident, textbooks often define a type of by imposing that some cell parameters are not identical to each other or to a certain value. For example, a tetragonal lattice is often defined as having a ≠ c, while a and c, being symmetry unrestricted, can take any value, including the special case a = c. This equality is realized within the of the experiment and holds within a certain interval of temperature and pressure [see a detailed discussion by Nespolo (2015b)]. In that case, one speaks of metric specialization (or specialized metric), a phenomenon more frequent than is commonly thought (Janner, 2004a,b) that may severely hinder the structure solution and strategy and which increases the frequency of occurrence of (Nespolo & Ferraris, 2000). prevents the direct classification of space groups based merely on the cell parameters. In fact, two space groups of the same type, one with and the other without have lattices of different symmetry (cubic and tetragonal in the example above). It would, however, be unreasonable to assign these two space groups to different categories (classes, systems, families), because the is often an `accident', not a feature of the structure crystallizing in that group. The higher symmetry of the translation (and thus of the unit cell) is broken by the contents of the which allows only a symmetry group of the lower type. Thus, to apply a consistent classification scheme, we have to abstract from any For this reason, the classification scheme is applied to spacegroup types, not to space groups (ITA1.3.4). We further discuss the consequences of in §§5 and 7.
(Table 1The same distinction between groups and types of groups must also be made for point groups. The common statement that there are 32 crystallographic point groups (in threedimensional space) is again, strictly speaking, incorrect because it actually applies to pointgroup types. As already seen in the case of space groups, point groups too are infinite in number, because of the infinitely many possible orientations in space of the geometric elements. Point groups are represented by matrices with respect to a chosen basis of the underlying a). A finer classification into 136 classes of oriented point groups takes into account the orientation of the symmetry elements with respect to a reference coordinate system and is obtained by considering the subgroups of the cubic and hexagonal holohedries (Nespolo & Souvignier, 2009; indicates the full symmetry of a lattice: see §6 for a precise definition): these classes are of particular importance in the study of domain structures obtained following a For example, a cubic crystal with of type () that undergoes a to 4/m2/m2/m (4/mmm) retains one of the three fourfold axes of the cubic parent phase. It can be oriented along the cubic a, b or c axes, leading to three differently oriented but isomorphic groups of type 4/mmm.
and, similar to space groups, two point groups belong to the same type if their matrices can be made to coincide by choosing suitable bases. Again, the bases are usually chosen in a symmetryadapted way and the transformation between two bases captures the relative orientation of the two point groups. The relative orientation of point groups is crucial to the analysis of general domain structures, in particular for growth twins (Nespolo, 2015The difference between point groups and pointgroup types is the key to the classification of space groups into
The term `space group' (not hyphenated) should be used with reference to a given while the term `spacegroup type' (hyphenated) should instead be used when speaking in general of the type of symmetry common to all crystal structures whose space groups only differ in the values of their cell parameters.3. and handedness. Sohncke, affine and crystallographic spacegroup types
The symmetry group of a i.e. its space group) can be regarded as an object: if we look at a spacegroup diagram as a geometric figure, we can observe how its symmetry elements are distributed in space and find isometries that map those symmetry elements onto each other. In other words, we can find the symmetry group of that drawing. The symmetry operations of an object are isometries (Euclidean mappings), i.e. transformations that do not deform the object and form a group: the symmetry group of that object. The symmetry group depends on the type of object under consideration: for molecules, it is a and for crystal structures, it is a For a spacegroup symmetryelement diagram regarded as a geometric figure, the symmetry is described by the Euclidean (or Cheshire group); it is also termed the `symmetry of the symmetry' because it maps the symmetry elements of the onto themselves (possibly permuting them) (Koch & Fischer, 2006).
(When considering symmetry operations, one distinguishes whether an operation maps a righthanded coordinate system to a righthanded one or a lefthanded one. The former, rotations, screw rotations and translations, are called orientationpreserving, or operations of the first kind: they are characterized by the fact that their linear part has determinant +1. In contrast, reflections, glide reflections, rotoinversions and inversions are called orientationreversing or operations of the second kind: their linear parts have determinant −1. Only operations of the first kind can be applied to real objects in physical space. Every operation of the second kind can be obtained as an operation of the first kind followed by an inversion.
An object is said to be chiral if it cannot be superimposed on its mirror image by an operation of the first kind; the two nonsuperimposable mirror images of a chiral object are said to possess opposite handedness (left or right). The symmetry group of a chiral object contains only operations of the first kind; if it contained any operation of the second kind, the composition of this operation with a reflection mapping the object to its mirror image would result in an operation of the first kind mapping it to its mirror image, thus making the object achiral. If the chiral object under consideration is a the two variants with opposite handedness are called enantiomorphs and their space groups belong to one of the 65 types containing only operations of the first kind, called Sohncke spacegroup types [after the German mathematician Leonhard Sohncke (1842–1897)].
Sohncke groups are often incorrectly called `chiral groups', so the difference between these two concepts needs to be clarified. are groups representing the possible symmetry of chiral objects, but the groups themselves are not necessarily chiral. The difference between the of an object and the of a symmetry group is, unfortunately, sometimes overlooked (Flack, 2003). There is, indeed, a certain similarity between the two concepts. As we have seen, for a chiral object the symmetry group of the object as a whole is required to consist only of symmetry operations of the first kind. Similarly, a group is called chiral if its Euclidean contains only operations of the first kind. In this case it has a counterpart of opposite handedness which belongs to a different spacegroup type; the two groups then form an enantiomorphic pair and differ by the presence of screw rotations of the same type but turning in opposite directions, like P3_{1} and P3_{2}. Twentytwo types of space groups are chiral and form 11 pairs of enantiomorphic spacegroup types; the other 208 types are achiral. Chiral spacegroup types are Sohncke types because a is contained in its Euclidean but the opposite is not true: 43 of the 65 Sohncke types are not chiral. For example, spacegroup types P4_{1} (No. 76) and P4_{3} (No. 78) are chiral and form an enantiomophic pair; they obviously belong to the Sohncke type of groups. On the other hand, the Sohncke type P4_{2} (No. 77) is not chiral: it does not have an enantiomorphic counterpart because its Euclidean also contains operations of the second kind (e.g. the inversion at the origin).
Summarizing, to distinguish between achiral and chiral types of space groups it is sufficient to check whether their Euclidean normalizers do or do not contain operations of the second kind, in exactly the same way as we distinguish between achiral and chiral crystal structures by checking whether their space groups do or do not contain operations of the second kind.
When considered from a more abstract algebraic viewpoint, spacegroup types belonging to an enantiomorphic pair are not distinguished: they have the same group structure and the symmetry operations in these groups are either identical or differ only for the direction of a screw rotation, like in the case of 3_{1} and 3_{2}. Depending on whether enantiomorphic spacegroup types are considered as different (as crystallographers usually do) or not (as mathematicians usually do), we obtain 230 crystallographic or 219 affine types of space groups. Fig. 1 summarizes the classification of spacegroup types in terms of the of the groups and of the structures crystallizing in them.
4. Symmorphic types of space groups and arithmetic crystal classes
A i.e. its its translation lattice, represented by the and the translational parts of its generators, which reflect the interplay between the and the translation lattice. In order to proceed to the higher shelves in the classification hierarchy of space groups, part of the information on point groups, translation lattices and their interplay is ignored and groups are collected together when they coincide only on a part of these aspects. Keeping the information on the and translation lattice, but neglecting their interplay, leads to the concept of symmorphism [see a recent analysis by Nespolo (2017)] and to the classification into Before proceeding, we need to recall the concept of a sitesymmetry group, which is a of the G containing all the symmetry operations in G that leave a point (the `site') fixed.
is characterized by its group of linear parts,A i.e. the group of linear parts occurring in the For a nonsymmorphic the order of the sitesymmetry group of each point (or the corresponding Wyckoff position) is a proper divisor of the order of the point group.
is said to be symmorphic if a coordinate system can be chosen such that all the nontranslation generators have zero translational part. As a consequence, the sitesymmetry group of the origin of this coordinate system is isomorphic to theA symmorphic spacegroup type is easily recognized from its Hermann–Mauguin symbol by the fact that, apart from the letter indicating the centring mode, it coincides with a pointgroup symbol. In particular, it only contains the symbols 1, 2, 3, 4, 6 for rotations, , , , for rotoinversions, and m for reflections. This does not necessarily mean that a symmorphic group does not include glide planes or screw axes, as is sometimes incorrectly stated in the literature. For example, a symmorphic of type C222 (No. 21) contains screw axes parallel to the a and b axes, which are obtained by composing rotations along the a and b axes with the centring translation.
The role of ; Aroyo & Wondratschek, 1995), but for our purposes the symmorphic types of space groups form the basis for further classification of space groups.
in the description of the symmetry of is well established (Wintgen, 1941Each spacegroup type is associated with a unique symmorphic type. Starting with nontranslation generators of the e.g. 3_{1} by 3) and every glideplane symbol a, b, c, d, e, n by the mirrorplane symbol m. All spacegroup types corresponding to the same symmorphic type are gathered into the same arithmetic crystal class, which is identified by the symbol of the corresponding symmorphic type. To avoid any possible confusion, the symbol for the is a modified version of that for the symmorphic spacegroup type, with the letter indicating the centring type moved from the first position to the last. For example, the corresponding to the of type C222 is denoted by 222C. Because there are 73 symmorphic types of space groups, there are also 73 arithmetic crystal classes.
one simply changes their translational parts to zero, and the generated by these modified generators (and the translations of the original group) is then symmorphic by definition. Moreover, the Hermann–Mauguin symbol of the associated symmorphic type can immediately be read off from that of the given spacegroup type: replace every screwaxis symbol by the corresponding rotationaxis symbol (Here the conceptual difference between a group and a class has to be emphasized. A symmetry group is a set of isometries that expresses the symmetry of an object. A class is a set of objects having a common feature with respect to a classification criterion. In the case of ).
we gather into the same class those objects (space groups, spacegroup types, crystal structures) which correspond to the same type of symmorphic A class can be imagined as a shelf where objects (groups, crystal structures, molecules) sharing a common feature are stored. The same conceptual difference occurs again between and types of point groups (see §6Characterizing i.e. space groups with no special Wyckoff positions. These groups are called Bieberbach groups [after the German mathematician Ludwig Georg Elias Moses Bieberbach (1886–1982)] (or fixedpointfree space groups or torsionfree space groups) and they contain (except for the identity) only operations of infinite order: screw rotations, glide reflections and translations. Of the 230 crystallographic spacegroup types, 13 are Bieberbach types of groups.
by the property that they contain sitesymmetry groups which are isomorphic to their point groups, one can look at the other extreme, namely space groups in which all sitesymmetry groups are trivial,5. Bravais types of lattices, Bravais arithmetic classes and Bravais classes
The symmetry of a structure built by atoms occupying the nodes of a lattice is an example of a symmorphic spacegroup symmetry: the Bravais arithmetic crystal class and, by construction, these classes are in onetoone correspondence with the 14 Bravais types of lattices. If a symmorphic does not belong to a Bravais its is not the full symmetry group of a lattice. In that case, the can be extended to a unique of minimal index expressing the full symmetry of the lattice (holohedral supergroup) by adding missing lattice symmetries such that the enlarged symmorphic group belongs to a Bravais For example, a symmorphic of type (No. 115) has a of type which is not the full symmetry group of a lattice, but which is contained (as a of index 2) in a of type 4/mmm, the full symmetry group of a tetragonal lattice. Extending the to 4/mmm gives rise to the symmorphic of type P4/mmm (No. 123) that does belong to the Bravais with symbol 4/mmmP. Note that by choosing a minimal holohedral of the the problem of is circumvented. It does not matter whether the translation lattice of the symmorphic group of type is tetragonal or, accidentally, cubic; since the pointgroup type 4/mmm is already a full symmetry group of a lattice one does not consider possible higher symmetries of a cubic lattice. In the way just described, every symmorphic spacegroup type, and thus also every is assigned to a Bravais and thus at the same time to a Bravais type of lattice.
has not only the given lattice as its translation as usual, but also the full symmetry group of the lattice as its The containing such a symmorphic group is called aRecalling how we collected spacegroup types into Bravais class, and one uses the same names and symbols for the Bravais classes as for the Bravais types of lattices. For example, the spacegroup types , , , (Nos. 115–118) all belong to the same and are, as explained above, assigned to the Bravais 4/mmmP and thus belong to the of primitive tetragonal type. The other belonging to this are 4P, , 4/mP, 422P, 4mmP, and 4/mmmP, and thus all spacegroup types belonging to any of these are associated with the primitive tetragonal lattice type. As a synonym for Bravais classes, the term Bravais flocks is occasionally used (for example, ch. 8.2 of the 5th edition of International Tables of Crystallography, Vol. A. Usually, it is applied to the matrix groups representing the point groups rather than to the space groups.
in the previous section, and combining this with the assignment of an to a Bravais described above, we arrive at an assignment of spacegroup types to a unique Bravais and thus also to a unique Bravais type of lattice. Two spacegroup types corresponding to the same Bravais type of lattice are also said to belong to the sameFigs. 2–4 present a detailed classification of in Bravais and Bravais classes.
6. holohedries and Laue classes
We have seen that by grouping together all the spacegroup types that correspond to the same symmorphic type we get a set of spacegroup types called an geometric crystal class. Because there are 32 types of point groups, by gathering spacegroup types according to their pointgroup type we obtain 32 These classes are denoted by the same Hermann–Mauguin symbol as the corresponding pointgroup types, which is sufficient, since the translation is ignored and thus is not required to be represented by a symbol as is the case for In particular, spacegroup types differing only by the centring mode of their belong to the same The double use of the Hermann–Mauguin symbol for pointgroup types and does not give rise to misunderstandings, because from the context it will always be clear which of the two is meant.
Going one step further and fully neglecting the information on the translation by collecting all spacegroup types for which the point groups are of the same type, we arrive at aThe 11 ) because they are compatible with chiral crystal structures, which can occur in two enantiomorphous modifications. As we have seen in §3, space groups that belong to these classes are known as it would be a logical extension to use the terms Sohncke crystal classes and Sohncke types of point groups, but to the best of our knowledge the name Sohncke has so far only been used for space groups.
corresponding to pointgroup types that contain only operations of the first kind (1, 2, 3, 4, 6, 222, 422, 32, 622, 23, 432) have been called `enantiomorphous classes' (Shuvalov, 1988Those i.e. spacegroup types) show the full symmetry of a lattice are called holohedries. A is therefore not a group, but a class. A holohedral group is a (point or space) group which belongs to a holohedral class; the others are called merohedral. Those whose members (groups) are centrosymmetric are called Laue classes. A contains all the centrosymmetric spacegroup types that correspond to the same pointgroup type. For example, the 2/m contains six spacegroup types: P2/m, P2_{1}/m, P2/c, P2_{1}/c, C2/m and C2/c.
whose members (Pointgroup types that are compatible with the existence of polar or axial vectors are of particular importance for the study of the physical properties of crystals. A vector is characterized by its magnitude and its direction, and the direction is in turn characterized by an orientation and a sense. The orientation is specified by the relationship between the vector and the reference. The sense is specified by the order of two points either on a line parallel to the vector (polar vector or true vector) or on a loop perpendicular to the vector (axial vector or pseudovector). A polar vector has symmetry ∞m and can be represented by a stationary cone or arrow, whereas an axial vector has symmetry ∞/m and can be represented by a cylinder rotating about its axis.
Point groups compatible with the existence of one polar vector belong to ten pyroelectric (or polar) because crystals in these classes can show pyroelectricity. The ten pyroelectric are 1, 2, 3, 4, 6, m, mm2, 3m, 4mm and 6mm. These correspond to the ten pointgroup types (because of the 1:1 correspondence between pointgroup types and geometric crystal classes) which are subgroups of the group ∞m. The number of crystallographic spacegroup types that belong to these pyroelectric is 68, which includes four pairs of enantiomorphic spacegroup types. The presence of a polar vector along a fixed direction, or parallel to a plane (point group of type m) or arbitrarily oriented (point group 1), implies that the choice of the origin is not fixed by symmetry but can be taken anywhere along the polar direction. When performing the of a structure crystallizing in a pyroelectric the origin should be fixed by the user [by restraining a centre of mass or fixing the coordinate(s) of one atom along the polar direction], otherwise a large number of correlations result, precisely because of the infinitely many possible choices for the origin.
In an analogous way, ferromagnetic materials can crystallize in pointgroup types and spacegroup types that are compatible with the existence of one axial vector. These are the crystallographic subgroups of ∞/m, i.e. 1, 2, 3, 4, 6, , m, 2/m, , , 4/m, and 6/m. The number of crystallographic spacegroup types that belong to the corresponding 13 is 44, which includes four pairs of enantiomorphic spacegroup types.
7. Crystal systems
Spacegroup types are gathered into the same i.e. leave invariant) the same types of Bravais lattices. If H and G are point groups and H is a of G, then H acts on the same types of Bravais lattices as G, but possibly also on other lattice types with more free parameters, whereas the opposite is not necessarily true. For example, 4/mmm acts on tetragonal and cubic lattices, and mmm, which is a of 4/mmm, acts on tetragonal and cubic lattices as well. However, mmm also acts on orthorhombic lattices, on which 4/mmm does not act. The action on different types of lattices is the criterion to classify pointgroup types (and therefore also spacegroup types) into crystal systems.
when their point groups act on (Table 2 shows the lattices on which the pointgroup types act. A is the shelf on which we can gather all the pointgroup types (and the corresponding spacegroup types) which act on the same types of lattices, i.e. which have an identical intersection of rows and columns in Table 2. This criterion of acting on the same types of lattices classifies spacegroup types into seven crystal systems: triclinic (or anorthic), monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic.

It is a common misunderstanding to establish a direct relation between the cell parameters of the i.e. a crystal with a belonging to the orthorhombic crystal system), three of the six cell parameters vary independently and, in a certain interval of temperature and pressure, they might take values that correspond to a more symmetric lattice, within the of the experiment; as discussed above, this is a case of Table 2 shows that pointgroup types 222, mm2 and mmm act not only on orthorhombic but also on tetragonal, hexagonal and cubic lattices. If one adopts the cell parameters as a criterion to estimate the structural symmetry, in the case of one might be tempted to assign the sample to the tetragonal, hexagonal or cubic because the lattice of that sample is tetragonal, hexagonal or cubic. However, the unambiguously shows that the structure is still orthorhombic.
and the Although the cell parameters are a useful indicator, this relation works in general only in the opposite direction: symmetry imposes restrictions on the cell parameters, but absence of symmetry does not. For example, in the case of an orthorhombic crystal (8. Lattice systems
A P23, and F432 are all assigned to the and belong to the cubic Similarly, space groups of types P32 and P622 correspond to the 6/mmm and belong to the hexagonal On the other hand, space groups of types R32 and P622 correspond to the different holohedries and 6/mmm, respectively; a group of type R32 belongs to the rhombohedral whereas P622 belongs to the hexagonal lattice system.
can be assigned to a unique by looking for the of minimal order that contains the of the and contains the full symmetry of the lattice type of the Grouping together all space groups that are associated with the same in this way gives rise to the classification of spacegroup types into lattice systems. For example, space groups of typesUsing the classification into Bravais classes as an intermediate step, the lattice systems have a fairly simple description: two Bravais classes belong to the same m), orthorhombic (mmm), tetragonal (4/mmm), rhombohedral (), hexagonal (6/mmm) and cubic ().
if the corresponding Bravais arithmetic classes belong to the same This criterion classifies spacegroup types into seven lattice systems (formerly known as Bravais systems): triclinic (or anorthic) (holohedry ), monoclinic (2/As in the case of crystal systems, a
can affect the lattice of a particular sample but not its lattice system.9. Crystal families
The highest shelf in our classification is that of crystal families. Spacegroup types are gathered into the same ) and their types of Bravais lattices have the same number of free parameters. For example, spacegroup types with holohedries 4/mmm and 6/mmm have lattices with two free parameters (a and c; see Table 1). However, 4/mmm and 6/mmm are not in a group–subgroup relation and thus the corresponding spacegroup types belong to different crystal families (tetragonal and hexagonal, respectively). On the other hand, spacegroup types with holohedries and 6/mmm have lattices with two free parameters (a and c in hexagonal axes) and a of type 6/mmm contains a of type . Therefore, the two spacegroup types belong to the same (hexagonal).
when they correspond to holohedries that are in a group–subgroup relation (Fig. 5This criterion classifies spacegroup types into six crystal families, indicated with a lowercase letter as follows: a (anorthic = triclinic), m (monoclinic), o (orthorhombic), t (tetragonal), h (hexagonal) and c (cubic). The Bravais types of lattices are then indicated by the symbol of the followed by the centring symbol of the in upper case (aP, mP, mC, oP, oC, oI, oF, tP, tI, hR, hP, cP, cI, cF; S for sideface centred is also used to indicate centring of one face, when one abstracts on whether it is A, B or C).
For five of the six crystal families, the . Spacegroup types corresponding to the hexagonal have an hP lattice, whereas spacegroup types corresponding to the trigonal may have an hR or hP lattice. The term trigonal indicates that the groups belonging to this act on both hR and hP lattice types and has no meaning with reference to lattices. Thus, the term `trigonal lattice', used in various textbooks, is incorrect. Similarly, the term `rhombohedral' can not be used with reference to crystal systems, since it applies only to a lattice: the term `rhombohedral is incorrect, despite its widespread use in the French literature, where, for example, αquartz is described as `rhombohedral', despite the fact that the lattice is hexagonal. This confusion has a historical origin. In the XIX century, German and French crystallographers used the same term `crystal system' (Kristallsystem; système cristallin) to indicate, respectively, what are today known as crystal system and lattice system.
coincides with a and a However, the hexagonal is subdivided into two lattice systems (rhombohedral and hexagonal) and into two crystal systems (trigonal and hexagonal). The distribution of spacegroup types in the hexagonal over crystal systems and lattice systems is shown in Table 3

Fig. 6 shows the full hierarchy of the classification of space groups discussed throughout this article, with the crystal families as the highest level.
10. Conclusion
We hope that the explanation and illustration of the various classification criteria for space groups given in this article will help others to apply these concepts correctly and will contribute to reducing misunderstandings due to imprecise terminology. As a quick reference to the reader, we have summarized in Table 4 common errors and why and how they should be corrected.

Acknowledgements
Critical remarks by two anonymous reviewers are gratefully acknowledged for their usefulness in improving the clarity of this article. MIA acknowledges the hospitality and support of CNRSNancy.
Funding information
The work of MIA was supported by the Government of the Basque Country (project No. IT77913) and the Spanish Ministry of Economy and Competitiveness and FEDER funds (project No. MAT201566441P).
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