research papers
Report of a Subcommittee on the Nomenclature of nDimensional Crystallography. II. Symbols for Bravais classes and space groups†
^{a}Institute of Theoretical Physics, University of Nijmegen, NL6525ED Nijmegen, The Netherlands, ^{b}Department of Physics, City University of New York, New York, NY 10031, USA, ^{c}Laboratoire de Physique des Solides, Batiment 510, Université ParisSud, 91405 Orsay CEDEX, France, ^{d}Institute of Physics, Moscow State University, Moscow 119899, Russia, ^{e}UFR Mathématique, Institut Fourier, BP 74, St Martin d'Hères, F38402 France, ^{f}Laboratoire de ChimiePhysique du Solide, Ecole Centrale Paris, Chatenay Malabry, F92295 France, ^{g}Advanced Materials Laboratory, NIMS, Tsukuba, Ibaraki, 3050044 Japan, ^{h}Physics Department, Southern Oregon University, Ashland, OR 97520, USA, and ^{i}Institute of Physics, Czech Academy of Science, Na Slovance 2, Prague 8, 18040 Czech Republic
^{*}Correspondence email: ted@sci.kun.nl
The Second Report of the Subcommittee on the Nomenclature of nDimensional Crystallography recommends specific symbols for Rirreducible groups in 4 and higher dimensions (nD), for centrings, for Bravais classes, for and for space groups (spacegroup types). The relation with higherdimensional crystallographic groups used for the description of aperiodic crystals is briefly discussed. The Introduction discusses the general definitions used in the Report.
Keywords: Commission Report; nomenclature; arithmetic crystal classes; space groups; Bravais classes.
1. Introduction
Recommended symbols for symmetry operations, lattice systems, families and et al., 1999). Similar recommendations are given in this Report for the symbols of crystallographic elements in nD, in particular for Bravais classes, centrings, and spacegroup types. We abbreviate the term `spacegroup type' to `space group', unless there is a reason to do otherwise. Some material that belongs to the topics of the first Report but was left out there will also be dealt with, in particular the question of symbols for (R)irreducible point groups.
in 4D, 5D and 6D have been presented in Report I (Janssen, BirmanIn principle, the crystallographic groups, arithmetic and via the program CARAT, developed in Aachen by W. Plesken and collaborators. It can be obtained via their web site http://wwwb.math.rwthaachen.de/carat/index.html . The notation and presentation given there is not easily adapted for crystallographic use, but all the information is present. According to that source the number of various classes in dimensions up to six is
and space groups, are available in dimensions 1–6The numbers of families and Bravais classes have been published in Plesken & Schulz (2000) and Opgenorth et al. (1998), respectively. The numbers increase rapidly with the dimension. A natural consequence is that the symbols have to be more and more complex. Also, it is clear that tables of these higherdimensional groups are impossible. Fortunately, the number of higherdimensional space groups needed for the description of aperiodic crystals is much lower. For example, there are only 370 nonisomorphic space groups of this type in 4D. As in 3D, where there are 11 enantiomorphic pairs of space groups, enantiomorphism occurs in higher dimensions as well. The number of enantiomorphic pairs, however, is not given in the table above.
A short review of the terminology used, and the definitions of the main notions, is presented first. Definitions may also be found in Vol. A of International Tables for Crystallography (Hahn, 2002), referred to hereafter as ITA. The concepts are clarified by examples in the third section. Nomenclature and symbols are discussed in §§4–8. Recommendations are given in §9. Evidently, the choice of a symbol is a compromise between information richness and conciseness. It is, for that reason, also a matter of taste. Sometimes alternatives will be given. The practice should make it clear what is the best choice.
Precise definitions are often rather dry. To be precise, a mathematical formulation is needed, but the notions have a , a mathematical formulation is given and, in §3, the same notions are introduced in a more descriptive way, with examples. The reader could choose to skip §2 and go directly to §3. To make the correspondence clearer, we have inserted a number of cross references.
and can be described in a more heuristic way. Although we have avoided as much as possible a mathematical symbolism, the section on definitions will not be easy to digest for everybody. Therefore, we have chosen a twotrack approach. In §22. Definitions and symbols
A distance exists in a Euclidean space that is invariant under rigidmotion transformations. These form the Euclidean group E(n) in the ndimensional Euclidean space. A of E(n) is the group of translations T(n). If an origin is chosen, then the of E(n) leaving this origin invariant is the orthogonal group in n dimensions, denoted by O(n). The of E(n) leaving an arbitrary point invariant is conjugate to O(n). The elements of E(n) are pairs of an orthogonal transformation R and a translation t. The product of two elements is given by
This is the definition of a semidirect product. Similarly, the affine group is the semidirect product of the group of translations and the group of nonsingular linear transformations.
Definition 1: Crystallographic space group. An ndimensional crystallographic G is a of E(n) such that its of translations is generated by n independent translations.
Comment: A does not have a fixed point. The translations belonging to the form a called the (translation) lattice. The points obtained from the origin by the translation lattice is called the (point) lattice of the The set of orthogonal transformations R for all elements in the leaves the origin fixed and forms a group of transformations that leaves the lattice of the invariant. This group is the associated with the space group.
There are other spacegroup types than crystallographic. An arbitrary E(n) that leaves no point invariant (it is fixedpoint free) is called a and the crystallographic space groups are just special cases. In general, the translation is not generated by n independent translations. This is the case if the translation does not span the whole space, as for band and frieze groups. It also occurs if the translation is not generated by n independent translations but by more. For example, the translation group in 2D generated by five vectors making an angle of with each other is generated by at least four translations. Then the generated point set becomes dense and the is not crystallographic. When there is no ambiguity, `crystallographic may be abbreviated to `space group'.
ofDefinition 2: Crystallographic point group. A of O(n) is a crystallographic if it leaves an ndimensional lattice invariant.
Comment: The translation of a G is an invariant and the is isomorphic to the K. Moreover, the leaves the lattice invariant and is for that reason a crystallographic A crystallographic is finite.
Definition 3: Geometric crystal class. Two point groups are called equivalent if they are conjugate subgroups of the orthogonal group O(n). The equivalence classes for the crystallographic point groups are called the geometric crystal classes.
Definition 4: Equivalence of space groups. Two (crystallographic) space groups are called equivalent if they are conjugate subgroups of the affine group. By Bieberbach's theorem (Bieberbach, 1911), this is equivalent to stating that two crystallographic space groups are equivalent if they are isomorphic. The equivalence class of a is called the spacegroup type.
Definition 5: Holohedry. The of a lattice is the consisting of all orthogonal transformations leaving the lattice invariant.
Definition 6: Lattice system. Two lattices belong to the same if their holohedries are geometrically equivalent.
A lattice is spanned by n translation vectors . The lattice is then characterized by the scalar products.
Definition 7: Metric tensor. The of a lattice is the tensor with elements .
Under an element of a K, the basis transforms to a basis and the transforms to the tensor with elements
Here the superscript T denotes the transpose.
Because point groups are subgroups of O(n), they can be presented as groups of orthogonal matrices, on an orthonormal basis. The group of all orthogonal matrices will also be denoted by O(n). A crystallographic leaves an ndimensional lattice invariant. Therefore, on a basis of such an invariant lattice, they are presented as groups of (invertible) integer matrices. The group of all ndimensional invertible integer matrices is denoted by GL().
Definition 8: Arithmetic point group. An arithmetic is a finite group of nD integer matrices.
Comment: The group of matrices presenting the nD crystallographic K depends on the choice of basis for the invariant lattice. Another basis is obtained from the first by an element of GL(). Therefore, the K corresponds to the set of all arithmetic groups that are conjugate in GL() to the first. This leads to a new equivalence.
Definition 9: Arithmetic crystal class. Two arithmetic point groups belong to the same if they are conjugate subgroups of GL().
Two crystallographic point groups in the same
correspond after a choice of invariant lattice basis to two arithmetic point groups that are conjugate by a rational matrix. If there is a conjugating integer matrix, then the two groups are arithmetically equivalent, otherwise at least geometrically equivalent. This means that we can define the of an arithmetic as the set of all arithmetic point groups conjugate to the first by a rational matrix. This implies that each consists of complete In other words, if two arithmetic point groups are arithmetically equivalent they are also geometrically equivalent.The Par abus de langage, space groups may be considered to belong to a well determined arithmetic and geometric crystal class.
of a becomes an arithmetic on a basis that is a basis for the lattice of the Any other lattice basis will lead to an arithmetically equivalent arithmetic group. Therefore, a determines an and consequently also aIn the following, we shall make no distinction between point groups as groups of transformations or as groups of matrices. Because a spacegroup element is a pair of an orthogonal transformation and a translation, which correspond to an ndimensional matrix and an ndimensional vector, respectively, the spacegroup element may also be associated with a pair of an integer matrix and a (real) vector. This is sometimes conveniently presented as an (n+1)dimensional matrix
Definition 10: Arithmetic holohedry. The arithmetic of a lattice with g is the group of invertible integer matrices D(R) such that .
Comment: Because the metric is left invariant, the group is equivalent to a group of orthogonal matrices. Because the group leaves the lattice invariant, it is a finite of GL().
Definition 11: Bravais class. Two lattices belong to the same if their arithmetic holohedries are arithmetically equivalent.
Comment: Because two arithmetically equivalent (arithmetic) point groups are also geometrically equivalent, each consists of whole Bravais classes.
Definition 12: Bravais group. The set of tensors left invariant by an arithmetic D(K) is denoted by S_{K} and defined as the set of metric tensors g for which
for every R in K. Then the of D(K) is the group of all integer matrices D(R) leaving all tensors in S_{K} invariant:
Definition 13: Bravais class of an arithmetic point group. The of an arithmetic is an arithmetic Two arithmetic point groups belong to the same if their Bravais groups are arithmetically equivalent.
In this way, arithmetic point groups (and therefore crystallographic space groups) may be grouped together. A coarser subdivision is given by the following.
Definition 14: Pointgroup system. Two belong to the same pointgroup system if there are two representatives of these classes with geometrically equivalent Bravais groups.
Definition 15: Family. A family of point groups (and of space groups) is the smallest union of pointgroup systems and Bravais classes of point groups such that with each crystallographic both its pointgroup system and its belong to the union.
Comment: Since each determines an and the of its lattice, space groups and lattices can also be assigned to a well defined family.
Definition 16: Conventional lattice. In each family, one lattice is chosen such that the arithmetic of every lattice belonging to the family is equivalent by a rational matrix with the arithmetic of the chosen lattice, or one of its subgroups.
Comment: Usually, but not always, a lattice with a of maximal order is chosen. The rational matrix in the definition is the centring matrix. The determinant of its inverse is the number of points of the lattice inside the conventional This number is called the index of the in the conventional lattice. The arithmetic is usually chosen to show the reducibility of the and to have the `simplest' form. Especially the last criterion means that the choice is sometimes not unique. A basis for the conventional lattice is a conventional basis.
Definition 17: Reducibility. An arithmetic is reducible if there is an invariant of lower dimension. An arithmetic is Rreducible if there is a proper invariant subspace. An arithmetic is  (or R)irreducible if it is not  (or R)reducible.
Comment: One may distinguish between reducibility, decomposability and full reducibility. An arithmetic is reducible if by a basis transformation from GL() all elements may be brought to the form
simultaneously, with the same dimensions of A,B and D for all elements. The group is decomposable if by such a basis transformation the elements may be brought into the form
simultaneously. The group is fully reducible if the elements may be transformed by an element of GL() to a direct sum of irreducible components. Similarly, the Q and Rreducibility, Q and Rdecomposability and full Q and Rreducibility are defined if the basis transformations are from GL(n,Q) and GL(n,R), respectively. reducibility implies Qreducibility, and the latter implies Rreducibility. On the other hand, Rirreducibility implies Qirreducibility, and the latter implies irreducibility.
Definition 18: Reducibility pattern of a crystallographic point group. The reducibility pattern is the space dimension n written as the sum of the dimensions of the irreducible components.
Comment: Generally, the reducibility pattern is different for the reducibility and for the Rreducibility. The Rirreducible subspaces carry a real irreducible representation of the The numbers in the reducibility pattern indicating the dimension of equivalent representations are enclosed in parentheses.
3. Explanation of the definitions
The definitions given above are somewhat mathematical in character, in order to be precise. Since their meanings, however, should be quite clear for crystallographers using them, some examples are now offered in which the definitions are discussed less formally.
Rigid motions in n dimensions (nD) are pairs of nD orthogonal transformations and nD translations. After the choice of an origin and a basis of the nD space, these rigid motions correspond to n×n matrices and nD vectors.
An equivalent description, as used in ITA (Hahn, 2002), gives the transforms of an arbitrary point in the under the spacegroup elements, modulo the lattice translation vectors. For example, the groups Pma2 and P2cm are given as follows.
This is a shorter way to give the transformation if there are many zeros in the matrix. A (crystallographic) Definition 1]. The corresponding orthogonal transformations form the [Definition 2], the translations form an nD lattice. On a basis of this lattice, the matrices of the have integer entries and are, generally, not orthogonal. By a change of basis S and of origin a, the matrices R and the vectors t change according to
is a group of rigid motions [The 3D space groups Pma2 and P2cm are equivalent via a basis transformation interchanging x and z.
Two space groups are considered to be identical if there is an origin shift and/or basis transformation that brings the matrices and translation vectors of the first into the same form as those of the second [Definition 4]. All space groups that are equivalent in this sense form an equivalence class. If there is a real basis transformation bringing the matrices of one in the same form as those of another, then the point groups belong to the same [Definition 3]. A crystallographic with respect to an invariant lattice is a group of integer matrices. Two such groups are arithmetically equivalent if there is a basis transformation for one of the lattices such that the integer matrices become the same [Definition 9]. Because such lattice basis transformations are given by integer matrices, arithmetic equivalence is stronger than geometric equivalence. contain complete In 3D, there are 32 and 73 The 2/m contains two 2/mP and 2/mC.
The other definitions relate to the equivalence of lattices and the classification of space groups, and to the ordering of space and point groups in hierarchical structures. Lattices are characterized by the matrix of n^{2} scalar products of the n basis vectors. This is the [Definition 7]. For example, the for a 3D monoclinic lattice is
The Definition 5]. For the lattice with given above, the is just a of the geometric class 2/m. Two lattices are considered to be equivalent if their symmetry groups are different settings of the same which happens if the holohedries are in the same All lattices equivalent to a certain lattice form a [Definition 6]. All 3D cubic lattices (whether they are primitive, b.c.c. or f.c.c.) belong to one lattice system.
of the orthogonal group that leaves the lattice invariant (each lattice point is transformed into a lattice point), the symmetry of the lattice, is the of the lattice [Because the Definition 10]. The adjective `arithmetic' is used for groups of integer matrices and their crystal classes. The term `arithmetic is standard. Here we use the term arithmetic for any finite group of integer matrices [Definition 8] and arithmetic as well. Because a change of basis gives an arithmetic in the same one may introduce a finer classification of lattices. Two lattices are equivalent if their arithmetic holohedries are in the same The corresponding equivalence class of lattices is the [Definition 11]. For example, there are 7 lattice systems but 14 Bravais classes in three dimensions.
leaves the lattice invariant, its matrices are integer if the chosen basis is a basis of the invariant lattice. As a group of integer matrices, it is called the arithmetic [For each arithmetic Definition 12]. The term `Bravais' is used for the equivalence of lattices or for the of a lattice invariant under a given [Definition 13]. Every lattice left invariant under the 3D mP is also invariant under 2/mP. At the same time, this is the largest group that leaves invariant all lattices invariant under mP. Therefore, 2/mP is the of mP. It is the arithmetic of every primitive monoclinic lattice in 3D.
there is a unique arithmetic as It is called the of the (arithmetic) [The lattice of a via Bravais groups to Bravais classes. Both branches come together in the most general classes: the families. (See Fig. 1 of Report I.)
is invariant under its Therefore, a choice of basis for the lattice gives a group of integer matrices, an arithmetic A change of lattice basis gives an arithmetically equivalent group of matrices. Thus the determines a unique The classification hereafter branches, one branch going from to and on to pointgroup systems, the other going fromAs an example, the 3D I4_{1}/a determines the 4/mI, which is contained in the 4/m. This belongs to the tetragonal system with 4/mmm. On the other hand, the arithmetic 4/mI has 4/mmmI as its This is the arithmetic of the bodycentred tetragonal In this case, the family has lattices with 4/mmm. It is the tetragonal family.
The notions of system and family [Definitions 14–15] are exemplified by the case of the point groups in the hexagonal family in 3D. All lattices in this family have two free parameters. However, there are two systems, those with the rhombohedral lattice with and those with a hexagonal lattice with 6/mmm.
The matrices of the Definition 16].
on a lattice basis do not, in general, clearly show the character of the transformation. In many cases, this becomes clearer if a basis is chosen. The spacegroup lattice is obtained from the basis by the addition of vectors in the of the latter. This is the centring. The matrices and vectors corresponding to the orthogonal transformations and translations of the rigid motions then are given with respect to a conveniently chosen basis, the conventional basis [In the 3D cubic family (), there are three Bravais classes: P, I and F. The reason for choosing P as the conventional lattice is to make the pointgroup matrices orthogonal, with each row and each column having precisely one nonzero entry. Both I and Flattices have a of the Bases for the two lattices are obtained from the conventional basis by (rational) basis transformations with determinant 1/2 and 1/4 (indices 2 and 4), respectively.
The centring may be given by the basis transformation or by the lattice vectors inside the conventional 0,0,0; and 0,0,0; ; ; . The index of the Ilattice is two, that of the Flattice is four.
For the examples, these areExamples of the different types of (ir)reducibility [Definition 17] are the following. The group 2/mP in 3D with generating matrices
is reducible, decomposable and fully reducible (see the comments after Definition 17). Therefore, it is also fully Q and Rreducible. The group mc in 2D with generating matrix
is reducible, but not decomposable. It is Qdecomposable and fully Qreducible, and hence also fully Rreducible. The [8]P_{4} in 4D generated by
is  and Qirreducible, but fully Rreducible. The cubic group is Rirreducible, and hence also Q and irreducible.
Examples of the Rreducibility patterns [Definition 18] in 3D are the following:
In each symbol, the dimensions of the irreducible representations appearing in the n. Equivalent irreducible components are put in parentheses.
are given, adding up toThe distinction between  and Rreducibility patterns is illustrated with the following examples in 4D:
The first
may be brought into reduced form by integer matrices, and hence also by real matrices. The second only by a real matrix, the third not by a real matrix and hence neither by an integer matrix.4. revisited
Symbols for Rreducible geometric classes were recommended in Report I, based on the symbols for in lower dimensions but not a notation for the Rirreducible cases. As the proposed symbols were generalizations of the Hermann–Mauguin symbols, it is logical to use this same approach for the Rirreducible classes. The Hermann–Mauguin symbols are based on the choice of a set of generators for a in the class. These generators are given by the corresponding symbols for orthogonal transformations, as presented for arbitrary dimensions in Report I.
The symbols should be unique, in the sense that a given symbol should correspond only to one i.e. the mutual orientation of axes and invariant subspaces is clear for Rreducible point groups in 3D, but is not clear for Rirreducible cubic groups. The orientation of the mirror planes in the symbol , with respect to the threefold rotation axis, is not directly specified. This same problem occurs often in higherdimensional spaces.
Moreover, the symbol should give as much information about the structure of the group as is compatible with conciseness. The latter condition tends to smaller sets of generators. The mutual orientation of the symmetry operators,More information about a ×' when the product is a or by a dot `.' for the general case. When the is the of two subgroups acting in mutually perpendicular subspaces, the is indicated by the symbol . The Hermann–Mauguin symbol may be generalized by composing the symbol for the whole group from the symbols for these generating subgroups, some of which can be cyclic, in which case their symbol is just the symbol for a generator. A complicating factor now is that the symbols for orthogonal transformations often consist of more than one character. For example, the symbol 32 represents both a fourdimensional rotation (a threefold rotation in a twodimensional subspace and a twofold rotation in a perpendicular twodimensional subspace) and a threedimensional of order six in the hexagonal family. It is important to clarify which of the two meanings the symbol represents when used for a higherdimensional We recommend placing the symbol for a in parentheses if it forms one of the generating point groups for the case under consideration, unless no confusion is possible. The symbol 3m only occurs for a 3D of order six, not as a symbol for an orthogonal transformation. In that case, parentheses are not necessary.
may be given by using the property that a may often be constructed as the product of some of its subgroups. The product may be indicated by `An example of a ×44 obtained by taking the of the 3D 32 and the 4D cyclic group generated by an orthogonal transformation 44; 32×44 is the of two cyclic groups, one with the sixfold rotation 32 as generator, the other with the fourfold rotation 44.
that requires parentheses is (32)The order of the group is not necessarily the product of the orders of the generating subgroups. Although it is not always possible to choose subgroups generating the full group in such a way that the product of their orders is the order of the full group, it is convenient to make such a choice, whenever possible, because this immediately gives information about the group. In Table 1, recommendations are made for symbols of the Rirreducible point groups in four dimensions, belonging to the families 21–23 (Table 4 in Report I). This supplements Table 3 in Report I. For comparison, in addition to the recommended symbol the number in Brown et al. (1978) is given (BBNWZ No.). In Table 2, some examples of Rirreducible point groups are given in five and six dimensions.


There are several infinite series of Rirreducible groups with very similar structure in various dimensions. The use of similar symbols is recommended in these cases. The first series is that of hypercubic groups, generated by the permutations of the n axes and the n mirrors perpendicular to the axes. Their order is hence 2^{n}.n!. The first members of the series are the 2D group 4mm and the 3D group . A second series is that of the symmetry groups of the generalized (n+1)D rhombohedral lattices, of order 2.(n+1)!. The lattice is the projection of a generalized rhombohedral lattice on a hyperplane perpendicular to the diagonal of the The first members are 6mm and . The third series is that of the symmetry groups of lattices that are the direct sum of a number of identical lowerdimensional lattices. An example is related to the group , the symmetry group of the sum of two 2D hexagonal lattices, of order 144. In case the two hexagonal lattices have the same lattice constant, there is a in 4D that exchanges the two lattices. The full symmetry group then is a group of order 288, with of index two. Subgroups in this pointgroup system all have a of index 2 that is the subdirect product of two subgroups of 6mm. This is (2+2)reducible, and has been assigned a symbol in Report I. In addition, there is one additional generator, exchanging the two invariant subspaces.
The hypercubic lattice in 4D has the special property that there is a
for which the is of higher order than the hypercubic lattice. There is a threefold rotation permuting the 3 centring axes that belongs to the of the but not to that of the hypercubic lattice. This threefold rotation is an additional generator which raises the order of the from 384 to 1152. Because the matrices of the group of order 384 on an orthonormal basis of the hypercubic lattice are simpler, the latter is chosen as the conventional lattice.Proposals for the symbols of all Rirreducible geometric classes in 4D and for some in 5D and 6D are given in Tables 1 and 2. These are based on the considerations given above. For an alternative view, cf. Weigel et al. (2001) and, specifically for 5D, Veysseyre & Veysseyre (2002) and Veysseyre et al. (2002).
First example. The third group of system 21_1 in Table 1(a) has order 60. There are 12 group elements that, on the chosen basis, correspond to orthogonal matrices. These 12 form a generated by
with relations A^{3} = B^{2} = (AB)^{3} = E. They form a 3D tetrahedral group 23 in the space perpendicular to the invariant vector (1,1,1,1). In addition, the group has a generator
This element generates a cyclic
[5]. The two subgroups generate the full 4D with symbol [5].(23). Because the product of the orders of the two groups (5 and 12) is equal to the order of the group, the latter does not have to be given explicitly.Second example. The second in the system 22_2 in Table 1(b) has order 24. There is an Rreducible generated by the matrices
of order 12. It is the group 6m(6m). In addition, there is a generator
The last matrix generates a cyclic group of order 4. It is the group 44. The symbol for the full group then is 6m(6m).44 but, because the product of the orders of the two subgroups is 48, the order of the group may be indicated explicitly: 6m(6m).44_{[24]}.
5. Bravais classes and centrings
The symbol for a et al., 1985, 1989, 1992). This consists of the symbol for the geometric class followed by a symbol for the centring, the latter being the basis transformation from a standard basis for the to a of a lattice from the Bravais class.
is the symbol for the arithmetic of the (de WolffCentrings are given by the basis transformation from a conventional basis for the family to a S, then the number of lattice translations in a of the conventional lattice is called the index of the centring, which is equal to the determinant of S. Conventionally, the basis transformation is given by a lowercase letter for 2D and an uppercase letter for 3D. We also recommend using one or more capital letters in higher dimensions.
of the If the basis transformation isThe basis transformation can be specified either by the matrix S or by the lattice vectors inside a conventional There are four cases that occur similarly for every dimension. One has index 1 and is indicated by P, another has index 2 and has, in addition to the origin, also the centre of the (conventional) it is indicated by I. The third has three lattice translations along the diagonal of the and has symbol R, and the fourth has a lattice translation in the middle of each pair of conventional basis vectors. It has index 2^{n1} in nD space and is indicated by F. If necessary to give the dimension explicitly, it may be added as a subindex: P_{n}, I_{n}, R_{n} and F_{n}. Centrings similar to R have m lattice translations along the diagonal and index m. We recommend that the centring be indicated by K for m = 4 and by Q for m = 5, or by K_{n} and Q_{n}, respectively.
Centrings of an mD such as the C centring in a 3D orthorhombic lattice with centring translation , are given by the same symbol as in lower dimensions but the centred must be indicated. We recommend using a subindex indicating the axes involved (). Thus the C centring of the orthorhombic 3D lattice is given the symbol I_{xy}.
Finally, some centrings may be regarded as the centring of a centring. The basis transformation S is then given by the product of two basis transformations S_{1} and S_{2}. The index of S is the product of the indices of S_{1} and S_{2}. There is a centring with index 16 in 4D, for which the centring matrix can be viewed as the product of the centring matrix for I with that for F.
The symbol is then IF or I_{4}F_{4}. The basis vectors , , and span a lattice with 16 basis vectors in the conventional (see Table 3).

Products of (0,0,0,0), , and is indicated by the centring symbol I_{xy,yzu}. This is, of course, the same as I_{xy,xzu}. As an example, all the centring symbols for 4D lattices are given in Table 3. Recommendations for the notation of the Bravais classes in 4D, using these centring symbols, are given in Table 4. Some examples of centring symbols for higherdimensional (5D and 6D) spaces are given in Table 5.
centrings are treated in the same way. If the centrings have the same symbol, but possibly different orientations, then the subindices are combined. A centring in 4D with four lattice translations in the conventional given by


6. Arithmetic crystal classes
Two arithmetic point groups that are geometrically equivalent can be obtained from each other by a conjugation with a rational matrix. In other words, they are related by a centring matrix. As in the case for m1p and 31mp of order 6. The difference between the two groups is the orientation of the mirror planes (m). In the first case, these are perpendicular to the crystal axes, in the second these are along the crystal axes. There are hence two ways to indicate point groups in the same that are arithmetically different: either by the centring symbol or by indicating the orientation in the Bravais group.
in one, two and three dimensions, the symbol for an is the symbol followed by a symbol for the centring of the lattice. However, a may contain several subgroups that are geometrically equivalent but not arithmetically. A well known example in two dimensions is given by the pair of arithmetic groups 3Both approaches are used in 2D and 3D. In 3D, the point groups in the m1P, 31mP and 3mR are geometrically equivalent. We recommend using the same system in higher dimensions, with the introduction of symbols for the centrings to distinguish between Bravais classes. The same symbols are used for the In principle, there are more symbols than strictly needed for the Bravais classes. In 3D, the 14 Bravais classes would have the proposed general notation symbols , 2/mP_{3}, 2/mI_{xz}, mmmP_{3}, mmmI_{3}, mmmF_{3}, mmmI_{xy}, , 6/mmmP_{3}, 4/mmmP_{3}, 4/mmmI_{3}, , , . Symbols such as I_{yz} are available for distinguishing between arithmetic crystal classes.
3Different orientations of subgroups of the Bravais groups are indicated by the invariant spaces of the generators. This may be achieved by giving the axes in the invariant space of the element as superscripts.
A special case occurs if a generator of an Rreducible is the sum of mirrors in the various invariant subspaces. If the group is reducible with components of dimension less than 4, the orientation of the mirrors with respect to the crystal axes may be given. If the mirror plane is perpendicular to a crystal axis, the mirror is indicated by a dot (), otherwise by a double dot (. This corresponds to the notation m1 and 1m, respectively. The positional notation cannot be used here because combinations of a mirror of the first type in one subspace can be combined with one of the other type in another subspace. If the group is irreducible, but Rreducible, the same notation can be used with respect to the projection of the crystal axes on the invariant subspace. The case of twofold rotations instead of mirrors can be treated in the same way.
Example 1. The five in the are distinguished by the centrings. They are denoted as , , , and .
Example 2. In the geometric class 3m(3m), there are three If the elements 33 and 2( = mm) are given by
then three point groups, one from each class, are generated by A, B_{1}, or by A, B_{2}, or by A, B_{3}. The centring is primitive in all cases. Therefore, the symbols are, respectively, , , and . Correspondingly, the two arithmetic classes in the 6m(6m) are 6m(6m)P_{4} and .
Example 3. The 4D geometric class has 11 arithmetic classes. These correspond to the centrings P_{4}, I_{4}, I_{zu}, I_{xyu}, I_{xyz,xyu} and N_{4}. For each of these centrings, except the last, there are two different orientations of the with respect to the crystal axes. One is generated by
and the other by A, B and
The two arithmetic classes with Pcentring are and . The other arithmetic classes are similar, with other centrings, except the Ncentring. In the latter case, only the second orientation occurs.
Example 4. The [5].2 = 5m(5^{2}m) contains two arithmetic classes. A representative is generated by
and another by A and C = B. The groups are irreducible, but Rreducible, and arithmetically nonequivalent. B and C are composed of two mirrors, one in each Rirreducible subspace. The mirrors in both subspaces go either through the projection of basis vectors of the lattice basis or they are both perpendicular. Therefore, the two arithmetic groups can be given the symbols 51m(5^{2}1m) and 5m1(5^{2}m1), or () and ().
Example 5. All for the two 4D families and mmmm are given in Table 6 with their recommended symbols.

7. Space groups
Spacegroup symbols are conveniently chosen as the symbols for the e.g. P2_{1}), and for a glide reflection (e.g. P2/a) the mirror symbol `m' is replaced by a letter that corresponds to the translation. With the de Wolff et al. (1992) nomenclature, the centring symbol for the is placed at the beginning of the spacegroup symbol.
corresponding to the with additional information on the translation parts of the spacegroup elements. The intrinsic part of the corresponding translation in the spacegroup element is given in the symbol for every generator of the This is the translation component that is invariant under origin shifts. In the symbols used up to the 3D case, a subindex represents the intrinsic part of the translation for a screw axis (The same scheme can be adopted in higher dimensions. Each character in the symbol for a e.g. the 4D rotation 43 of order 12, or the element 6(6) in a 4D reducible The intrinsic translations only appear in invariant subspaces. Therefore, the examples given [43 and 6(6)] hence have intrinsic translations only in spaces of dimension higher than 4. We recommend appending an index to the symbol to indicate the intrinsic translation. The position of the index is:
stands for a generator of the or a component of a generator. The latter occurs if an orthogonal transformation is indicated by more than one digit, as– for series of subsequent characters indicating an orthogonal transformation (such as 3 or 43), directly after the last character;
– for characters separated by parentheses, directly after the last character of the corresponding component, followed by an eventual component in other directions.
The intrinsic translation of a spacegroup element is always a rational fraction of a lattice translation in the invariant subspace. It can be given by that fraction and the lattice translation. Suppose the fraction is p/q and the order of the orthogonal transformation is n. Then np/q corresponds to a lattice translation. We take the order of the element (n) as the denominator. With respect to a np/q would be an integer, but, because translations are given with respect to the conventional basis, np/q still may be a fraction. This fraction is placed before the lattice translation vector indicated by , where is the ith basis vector.
It is sufficient to indicate the translation part for a set of generators because the other spacegroup elements including their translation parts are the result of multiplications. However, the intrinsic parts are not always sufficient. There are nonsymmorphic space groups for which a set of generators may be chosen without an intrinsic translation part. An example is the 3D group I2_{1}2_{1}2_{1}. For every element of the the associated translation in the spacegroup element may be changed by adding a lattice translation or by a change of origin. In the case of I2_{1}2_{1}2_{1}, there is an origin for which the rotation along the z axis has a translation () and the spacegroup element is a screw rotation. However, because () is a lattice translation, the translation in the screw rotation may be changed to (), which may be changed to zero by an origin shift. This holds for all three twofold rotations. However, the nonprimitive translations cannot simultaneously be eliminated. The choice of generators of the without intrinsic translations would lead to the symbol I222, which is already the symbol for the symmorphic group. Therefore, the symbol I2_{1}2_{1}2_{1} is used. We recommend choosing in similar situations translation components for which the intrinsic part is nonzero.
A similar situation occurs for the group . Strictly speaking, the elements 5(5^{2}) = [5] and generate the However, this choice leads to a notational problem. There is a nonsymmorphic with generators and . Both translation parts may be eliminated by an origin shift. However, their product m(m) has a translation part (), which is intrinsic. Therefore, it is advisable to use the symbol . The nonsymmorphic character can only be made evident by choosing m(m) = 2 as a (superfluous) generator of the We recommend using more generators for the than strictly necessary, if that is a way to indicate the character of the nonsymmorphic space group.
1: The
with generators(and the lattice translations) has arithmetic
and symbol .2. The
with the lattice translations andas generators has arithmetic m(4m)P. Because the order of the first rotation is four, the fraction np/q = 2 and four times the translation part is . Although the translation part of the second generator is the same, the order of the second rotation is two and np/q = 1. Therefore, the symbol is P4_{2v}m_{v}(4m).
4Lists of space groups may be found in Brown et al. (1978) for 4D (a complete list), in Martinais (1987) for 6D and in Janssen (1988) for 5D and 6D.
This is not the place to give full lists of the recommended symbols for all space groups in dimensions higher than three. As examples, all space groups are given for a selected number of ), 5D (Table 8) and 6D (Tables 9 and 10). Generators for the space groups are given either by generating matrices for the arithmetic together with the associated translations (Table 8) or by the action of the generators on a point (x,y,z,u,v) in 5D or (x,y,z,u,v,w) in 6D (Tables 7, 9 and 10). For example, one generator of the P5_{x}32(5^{2}32) can be written as
in 4D (Table 7or as A, with




8. Symmetries of aperiodic crystals
Higherdimensional i.e. the aperiodic crystals for which the diffraction spots can be labelled by a finite set of integer indices. There are at least three different, but partly overlapping, classes among these aperiodic crystals: the incommensurate displacively/occupationally modulated phases, the incommensurate composites and the quasicrystals.
groups have been adopted in specifying the symmetry of quasiperiodic crystals,The symmetry groups of these structures are (3+d)dimensional space groups, called groups. These are space groups in a space that is the sum of the 3D physical space and a dD internal or perpendicular space. The dimension d is determined by the number n = 3+d of basis vectors of the Fourier module of the structure. This is the set of reciprocallattice vectors spanned with integer coefficients by the positions of the diffraction spots of the aperiodic crystal.
The point groups of these Rreducible in 3D and dD components, both possibly Rreducible themselves. Affine conjugation or isomorphism may be used as coarsest equivalence relation of groups. For modulated structures, this relation may be refined. For these structures, there is a basis such that the matrices of the belong to the group , the of the general group with matrices of the form
groups arewhere A is a 3×3, v a d×3 and B a d×d matrix. Two groups for modulated structures are equivalent if they are conjugate subgroups in the of the affine group that is the semidirect product of the translation group and GL(n,d,R).
The nD space groups, but two nonequivalent groups under this equivalence relation may still be isomorphic, and thus equivalent as nD space groups. The (3+1)D groups for modulated crystals are given in Janssen, Janner et al. (1999). In Table 11, the spacegroup symbols are given for the symmorphic groups corresponding to the Bravais groups of the (3+1)D space. groups for 3+d dimensions (d = 1,2,3) can be found on the internet at the web site http://quasi.nims.go.jp/yamamoto/index.html . The Bravais classes for modulated structures in 3+d dimensions are given in Janner et al. (1983).
groups form a subset of all the

A brief introduction to the notation for S. In the diffraction spots can be distinguished as main reflections and satellites. A basis set of satellite vectors may be obtained from a subset by the action of the Finally, each pointgroup element present in the symbol S may have translation components in the internal (or perpendicular) space symbolized by alphanumerical characters a. The symbol for the then is S(. For more details, see IT Vol. C (Janssen, Janner et al., 1999). An example of such a symbol is , the symbol for a (3+1)D with component Pcmn in physical space and satellites . The mirror m in Pcmn accompanies an internal translation given by the symbol `s'. The glide planes `c' and `n' do not have an associated internal translation. This is given by `1'. Generators for this spacegroup transform (x,y,z,u) into (), () and (). This group would be denoted in the recommended notation for nD as .
groups follows. The components of pointgroup elements and translations in physical space form a 3D with symbolFor all quasiperiodic crystals, the nD is an Rreducible group with an invariant subspace that has the dimension of the physical space. However, the equivalence of groups may depend on the type of quasiperiodic crystal. Conjugation in the semidirect product of T(n) and GL(n,d,R) has been adopted as the equivalence relation for displacively modulated crystals. In the case of composites and quasicrystals, the coarser equivalence of the conjugation in E(n) is used conventionally. In each case, it is evident from the notation that the is Rreducible. This notation differs sometimes from that recommended in the present Report for nD space groups. A number of examples are given in Table 12 of groups that are nonequivalent as groups but equivalent as space groups, together with the recommended nD notation.

9. Conclusions and recommendations
Symbols and notation for Rirreducible for for Bravais classes and for space groups are discussed in §§2–7, the relation between nD space groups and the symmetry groups for aperiodic crystals in §8. The conclusions can be summarized in the following recommendations.
(I) Rirreducible receive a symbol that is composed of symbols for generating subgroups.
(a) The full is the product of the subgroups. A is given by `×', a general product by a dot `.'.
(b) The symbols for the subgroups are those of a generator for a cyclic group, the conventional symbols in 2D and 3D, and the symbols given in Report I.
(c) When the symbol of a is identical to that of an orthogonal transformation but the is not the corresponding cyclic group, then the symbol is placed in parentheses.
(d) By preference, the subgroups are chosen such that the product of their orders is the order of the otherwise, the order is indicated by a subindex between brackets at the end.
(II) Bravais classes are indicated by the symbol of the
of the followed by a symbol for the centring.(a) Symbols for nD P, I, F and Rcentrings are P_{n}, I_{n}, F_{n} and R_{n}, respectively.
(b) I, F and R centrings of sublattices are given by the same letters, but with the axes of the involved as a subindex.
(c) When I, F or R centrings occur in more than one the various sublattices are indicated by the corresponding sets of axes, separated by a comma.
(d) When the centring can be considered as the centring of a it is given as a series of centring symbols.
(e) Symbols for 4D are given in Table 4.
(III) The symbol for an arithmetic class is the symbol for the geometric class followed by the centring symbol.
(a) When the orientation of the with respect to the centred basis is relevant, the orientation of the centring is explicitly given.
(b) Different orientations of the with respect to the lattice are given by indicating the invariant spaces for those group operators for which the orientation matters as a superscript for the corresponding symbol.
(c) Mirror operations in an invariant space are given by when the inversion operation is along the projection of a lattice basis vector, by when it is perpendicular; for 3D components, is conventionally denoted as Nm1 and N by N1m.
(IV) A
(type) is given by a centring symbol, the symbol for the (but with the centring symbol removed) and by an indication for the intrinsic part of the translation parts of the generators corresponding to the components of the symbol.(a) The intrinsic translation is given as a subindex at the corresponding symbol for the pointgroup generator.
(b) The translation is given with respect to a conventional basis.
(c) When N is the order of the orthogonal part, the translations are given as sums of 1/N of the basis vectors; the latter are indicated by .
(d) To the symbol for the geometrical are added symbols for new generators if the intrinsic parts associated with the original generators are zero or do not give enough information about the spacegroup type; for these additional generators, elements are chosen with nonzero intrinsic translation.
(e) If necessary, to avoid ambiguity the translation parts of the generators are chosen in such a way that the intrinsic parts are nonzero; this may be done by adding lattice translations of the centring type.
(V) Superspacegroup symbols [IT Vol. C (Janssen, Janner et al., 1999)] may be used for the symmetry description of incommensurate modulated structures. The full ndimensional symbol is recommended for incommensurate composites and quasicrystals.
Correction. In Table 3 of I, the symbol in 19_2 should be .
Footnotes
†Subcommittee renewed by the IUCr Commission on Crystallographic Nomenclature 18 March 1999 with all present coauthors as members. Original version of the Report received by the Commission 22 April 2002, accepted 19 July 2002.
‡Chairman.
§Ex officio, IUCr Commissionon Crystallographic Nomenclature.
Acknowledgements
We thank Subcommittee advisors G. Chapuis, T. Phan, and R. Veysseyre for very useful comments. One of us (DW) acknowledges the computational support by H. Veysseyre.
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