research papers
Layer groups: Brillouinzone and crystallographic databases on the Bilbao Crystallographic Server
^{a}Institute of Applied Geosciences, Karlsruhe Institute of Technology, Karlsruhe, Germany, ^{b}Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, The Netherlands, and ^{c}Departamento de Física, Universidad del País Vasco UPV/EHU, Spain
^{*}Correspondence email: gemma.delaflor@kit.edu
The section of the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) dedicated to subperiodic groups contains crystallographic and Brillouinzone databases for the layer groups. The crystallographic databases include the generators/general positions (GENPOS), Wyckoff positions (WYCKPOS) and maximal subgroups (MAXSUB). The Brillouinzone database (LKVEC) offers kvector tables and Brillouinzone figures of all 80 layer groups which form the background of the classification of their irreducible representations. The symmetry properties of the wavevectors are described applying the socalled reciprocalspacegroup approach and this classification scheme is compared with that of Litvin & Wike [(1991), Character Tables and Compatibility Relations of the Eighty Layer Groups and Seventeen Plane Groups. New York: Plenum Press]. The specification of independent parameter ranges of k vectors in the representation domains of the provides a solution to the problems of uniqueness and completeness of layergroup representations. The Brillouinzone figures and kvector tables are described in detail and illustrated by several examples.
1. Introduction
The Bilbao Crystallographic Server (https://www.cryst.ehu.es/) (Aroyo et al., 2006, 2011; Tasci et al., 2012) is a free web site that grants access to specialized databases and tools for the resolution of different types of crystallographic, structural chemistry and solidstate physics problems. The server has been operating for more than 20 years, and is in constant improvement and development, offering freeofcharge tools to study an increasing number of crystallographic systems (Elcoro et al., 2017; Gallego et al., 2019; de la Flor et al., 2019). The programs on the server are organized into different sections depending on their degree of complexity, in such a way that the more complex tools make use of the results obtained by the simpler ones. The Bilbao Crystallographic Server (hereafter referred to as BCS) is built on a core of databases that include data from International Tables for Crystallography, Vol. A, Spacegroup Symmetry (Aroyo, 2016; henceforth abbreviated as ITA), Vol. A1, Symmetry Relations between Space Groups (Wondratschek & Müller, 2010), and Vol. E, Subperiodic Groups (Kopský & Litvin, 2010; henceforth referred to as ITE). A kvector database with Brillouinzone figures and classification tables of the wavevectors for all 230 space groups (Aroyo et al., 2014) is also available in the server, together with the magnetic and doublespacegroup databases. The magnetic and the incommensurate structure databases are also hosted on the server.
Aside from the subgroups of space groups with threedimensional lattices which are again space groups, there also exist subgroups called subperiodic groups with translation lattices of dimensions one or two. These groups are essential to describe polymers, nanotubes, nanowires, layered and multilayered materials. The interest in materials with subperiodic symmetry is constantly growing due to their outstanding properties and possible technological applications. There are three types of subperiodic groups: frieze groups (twodimensional groups with onedimensional translation lattices), rod groups (threedimensional groups with onedimensional translation lattices) and layer groups (threedimensional groups with twodimensional translation lattices). The crystallographic data for subperiodic groups are compiled in ITE and also now offered online in the BCS section dedicated to subperiodic groups. This section includes programs which give access to the generators/general positions (GENPOS), Wyckoff positions (WYCKPOS) and maximal subgroups (MAXSUB) databases.
In addition, we have developed the Brillouinzone database of layer groups which contains kvector tables and Brillouinzone figures that form the background of the classification of the irreducible representations of layer groups. The Brillouinzone figures and the wavevector data for space groups are well established and many authors have contributed to the standardization of the data (see e.g. Miller & Love, 1967; Bradley & Cracknell, 1972; Cracknell et al., 1979; Stokes & Hatch, 1988). For layer groups, however, the Brillouinzone and wavevector descriptions proposed by several authors (Ipatova & Kitaev, 1985; Hatch & Stokes, 1986; Milošević et al., 2008) are incomplete and difficult to compare due to the lack of standards in the classification and nomenclature of the k vectors. To the best of our knowledge, the only complete compilation of layergroup k vectors together with the Brillouinzone diagrams is found in the work of Litvin & Wike (1991) (in the following, referred to as LW). In this description the k vectors are labelled following the classification scheme of spacegroup k vectors used by Cracknell et al. (1979). Based on the group–subgroup relations between space and layer groups and using the socalled reciprocalspacegroup approach (cf. Wintgen, 1941; Aroyo & Wondratschek, 1995; Aroyo et al., 2014), we have derived the kvector data and Brillouinzone figures for all 80 layer groups and compared them with the classification of LW. An auxiliary tool for the complete characterization of wavevectors is also available: given the wavevector coefficients referred to a primitive or conventional the program assigns the corresponding wavevector symmetry type, specifies its LW label, determines the layer little cogroup of the wavevector and generates the arms of the wavevector star. The BCS Brillouinzone database of layer groups is accessed by the retrieval tool LKVEC.
The aim of this contribution is to present the crystallographic databases and the Brillouinzone database for layer groups available in the BCS. In the following, the retrieval tools GENPOS, WYCKPOS, MAXSUB and LKVEC, and the procedure to derive and classify the k vectors of layer groups are described in detail. The utility of the programs is demonstrated by several illustrative examples.
2. Crystallographic databases for layer groups
The BCS section Subperiodic Groups: Layer, Rod and Frieze Groups hosts the layergroup crystallographic databases. The structure of these databases is similar to that of the space groups – they include information on generators, general positions, Wyckoff positions and maximal subgroups for the 80 layer groups. Apart from the data shown in ITE the server offers additional information and computer tools that allow the generation of data not available in ITE. The BCS programs and databases use the socalled standard or default settings of the layer groups. These are the specific settings of layer groups that coincide with the conventional layergroup descriptions found in ITE. For layer groups with more than one description in ITE, the following settings are chosen as standard: (i) cellchoice 1 description for the two monoclinic/oblique layer groups p11a (No. 5) and p112/a (No. 7) described with respect to three cell choices in ITE, and (ii) origin choice 2 descriptions (i.e. origin at an inversion centre) for the three layer groups p4/n (No. 52), p4/nbm (No. 62) and p4/nmm (No. 64) listed with respect to two origins in ITE.
Note that, following the conventions of ITE, the ab plane is the plane of periodicity for the layer groups and therefore the translation vectors are of the form
2.1. Generators and general positions
The BCS database of layer groups includes the list of generators/general positions of each layer group. These data can be retrieved using the program GENPOS, by specifying the sequential ITE layergroup number (which, if unknown, can be determined by choosing the corresponding Hermann–Mauguin symbol from a table with the layergroup symbols). The generators and/or general positions of layer groups are specified by their coordinate triplets, the matrixcolumn representations of the corresponding symmetry operations and their geometric interpretation:
(a) The list of coordinate triplets (x, y, z) reproduces the data of the General Positions blocks of layer groups found in ITE. The coordinate triplets may also be interpreted as shorthand descriptions of the matrix presentation of the corresponding symmetry operations.
(b) Matrixcolumn presentation of symmetry operations. With reference to a coordinate system, consisting of an origin O and a basis (a_{1}, a_{2}, a_{3}), the symmetry operations of layer groups are described by (3 × 4) matrixcolumn pairs.
(c) Geometric interpretation. The geometric interpretation of symmetry operations is given (i) following the conventions in ITE [including the symbol of the its glide or screw component (if relevant) and the location of the related symmetry element], and (ii) using the Seitz notation. It is worth pointing out that, in contrast to the online and printed editions of ITE, the programs of BCS use the standard International Union of Crystallography Seitz notation. For example, a twofold rotation around the c axis is denoted by 2_{001} instead of 2_{z} (for details, cf. Litvin & Kopský, 2014).
Fig. 1 shows the general position for the layer group p112/a (No. 7) in the default setting (cellchoice 1).
The program GENPOS lists the generators and/or general positions of layer groups in the standard/default setting as well as in conventional nonstandard settings of monoclinic/rectangular and orthorhombic/rectangular layer groups described in Table 1.2.6.1 of ITE (option ITE settings). In addition, the program can produce the data in any nonconventional setting if the transformation relating the nonconventional setting to the standard one is specified (option Nonconventional setting). The matrixcolumn pair (P, p) of the transformation relating the two settings consists of two parts: a linear part P defined by a (3 × 3) matrix, which describes the change of direction and/or length of the basis vectors (a_{1}, a_{2}, a_{3})_{nonconv} = (a_{1}, a_{2}, a_{3})_{stan}P, and an origin shift p = (p_{1},p_{2},p_{3}) defined as a (3 × 1) column, whose coefficients describe the position of the nonconventional origin with respect to the standard one.
The URL of the program GENPOS is https://www.cryst.ehu.es/subperiodic/get_sub_gen.html.
2.2. Wyckoff positions
The BCS Wyckoffpositions database for layer groups is accessible under the WYCKPOS program. The data on Wyckoff positions can be retrieved by specifying the ITE number of the layergroup type. As a result, the program WYCKPOS shows a table with the Wyckoff positions [see Fig. 2(a)]. Following ITE, each is characterized by its multiplicity, Wyckoff letter, sitesymmetry group and a set of coordinate triplets of the Wyckoffposition points in the For centred subperiodic groups, the centring translations are listed above the coordinate triplets. The sitesymmetry groups [see column three of Fig. 2(a)] are described by oriented symbols which display the same sequence of symmetry directions as the layergroup symbols (cf. Table 1.2.4.1 of ITE). An explicit listing of the symmetry operations of the sitesymmetry group of a point is obtained by clicking directly on its coordinate triplet. A recently implemented auxiliary tool permits the identification of the and the sitesymmetry operations of a point specified by its coordinate triplet [see Fig. 2(b)]. The program accepts as input relative point coordinates in fractions, decimals or variable parameters (indicating a generic value). Fig. 3 shows the list of the symmetry operations of the sitesymmetry group of the layer group cmmm (No. 47) for the points (5/4,3/4,z). The points belong to the 8h with sitesymmetry group ..2.
Apart from the standard/default setting option the program is also able to calculate the Wyckoff positions in different ITE (conventional) settings (option ITE settings) or with respect to a nonconventional setting if the corresponding coordinate transformation (P, p) is defined (option Nonconventional setting).
The URL of the program WYCKPOS is https://www.cryst.ehu.es/subperiodic/get_sub_wp.html.
2.3. Maximal subgroups
The listing of maximal subgroups of layer groups available in ITE is incomplete and lacks additional information, such as, for example, possible unitcell transformations and/or origin shifts involved. In contrast, the BCS database of maximal subgroups of layer groups provides the complete listing (not just by type but individually) of (i) all maximal nonisotypic subgroups for each layer group, and (ii) all maximal isotypic subgroups of indices 2, 3 and 4. The list of maximal subgroups is retrieved by the access tool MAXSUB. Each is specified by its ITE number, Hermann–Mauguin symbol, index, type (t for translationengleiche or k for klassengleiche) and transformation matrixcolumn pair (P, p) that relates the standard setting of the group with that of the (see Fig. 4). The different maximal subgroups are distributed into conjugacy classes. The identification of the symmetry operations as a subset of the elements of the group is achieved by an optional tool of MAXSUB that transforms the generalposition representatives of the to the coordinate system of the group.
The URL of the program MAXSUB is https://www.cryst.ehu.es/subperiodic/get_sub_maxsub.html.
3. Classification of wavevectors
A layer group is defined as a threedimensional crystallographic group with periodicity restricted to a twodimensional subspace. Just as for space groups, the general strategy for determining the irreducible representations, called irreps for short, of such a group is to exploit the fact that it contains the translation reciprocal space spanned by the reciprocal basis defined by .
as a normal abelian and that irreps of abelian groups are onedimensional. The elements of have matrixcolumn pairs of the form where lies in a twodimensional lattice with basis . Each onedimensional irrep of is then of the form for a vector in theStarting from an irrep of , an irrep of the full layer group is obtained by first extending to the 2×2 diagonal blocks of the linear parts of the layer group .
of under the conjugation action of and then inducing to the full group . When the induced irrep of is restricted to , it is the sum of those onedimensional irreps of which lie in the orbit of under the conjugation action of and these are precisely the irreps of that yield irreps of equivalent to the one obtained from . For the action of on the vectors only the part acting on the plane of periodicity of is relevant and the restriction of to this plane gives rise to a plane group associated to . In the convention of ITE, the matrixcolumn pairs of are expressed with respect to a basis such that the first two basis vectors span the plane of periodicity. Therefore, the matrices of the linear parts of the plane group are simply the upperAn explicit computation shows that the in the orbit of are of the form for an element of the reciprocal lattice of , i.e. for with k_{1},k_{2} integers. The set of elements for which is called the little cogroup of . The k vector is called general if contains only the identity element of , i.e. = {I}; otherwise > {I} and k is special (Bradley & Cracknell, 1972; Dresselhaus et al., 2008). If {} is the set of representatives of the decomposition of with respect to , then the set of vectors {} is called the star of while the vectors are called the arms of the star. Using the relation between the elements of the layer group and the plane group associated to , in analogy to , one defines the little layer cogroup which is essential for the derivation of the irreps of .
of and an element from theThe preceding discussion indicates the importance of the socalled reciprocal plane group , defined as the symmorphic plane group having the as translation vectors and as It is crucial that the operations in act on i.e. they act on rows, as opposed to which acts on columns. In order to identify with one of the symmorphic plane groups in their standard setting, the action on (on rows) has to be transformed into one on (on columns).
3.1. Transformation between and direct space
After fixing a basis for 2×2 matrices acting on columns. We now have to relate the action on the to that on . Defining
the action of the of on is given bythe basis is mapped by to the new basis . The corresponding matrix for the action on
(on rows) must then map the reciprocal basis to the reciprocal basis of . Fromit follows that must be the identity matrix . We thus have , i.e. the action on the rows of is given by the inverse matrices acting on the columns of Since is a group, it contains with every matrix of course also the inverse matrix ; hence the set of matrices remains the same, but they now act from the right on rows.
As an example, we take the layer group (No. 78). Restricting the action to the plane of periodicity gives the plane group p3m1 (No. 14). If we look at the vector , applying the threefold rotation with matrix
to the row gives and applying it again gives . On the other hand, applying the reflection with reflection line 2x,x and matrix
we see that the row is mapped to , which is equal to up to a reciprocallattice vector. The star of consisting of the vectors in the orbit under is therefore .
As we have just demonstrated, it is natural to consider the action of the reciprocal plane group on rows, but on the other hand is by construction isomorphic to a symmorphic plane group . In order to identify with the proper symmorphic plane group, the action on rows has to be transformed to an action on columns. This is of course achieved by simply transposing the matrices of the
, since and the transpose of a row is a column. The symmorphic plane group isomorphic to is therefore the groupThe only problem that may occur is that is not given in its standard setting. This can be seen from different (strongly interrelated) perspectives:
(i) The reciprocal basis may not be a conventional basis of the lattice . For example, the
of a hexagonal lattice is also a hexagonal lattice, but while the vectors in the conventional basis have an angle of 120°, the reciprocal basis vectors enclose an angle of 60° and are therefore not a conventional basis of a hexagonal lattice.(ii) The matrices of the i.e. one has . The transposed matrices fix the inverse as can be seen from inverting the above equation: = = . This corresponds to the fact that the has . But may not be the of a lattice with respect to its conventional basis. For example, a hexagonal lattice has a of the form
fix the of the lattice ,which has inverse
This is the i.e. in a nonconventional setting.
of a hexagonal lattice with basis vectors having an angle of 60°,(iii) The transposed matrices simply do not occur as matrices of one of the point groups of plane groups in their standard setting. For example, the transposed matrix
of the matrix
of a threefold rotation 3^{+} does not belong to the of any of the plane groups.
For plane groups, the hexagonal lattice is actually the only case in which the transposed matrices do not belong to a
in its standard setting. For all other lattices, the inverse of the still belongs to a conventional basis or, in other words, the set of matrices of the does not change by transposing.In order to identify the symmorphic plane group in the case of a hexagonal lattice, the group has to be transformed to a basis in the conventional setting, i.e. which has a equal to a multiple of
Such a transformation is
(or its compositions with powers of a sixfold rotation) and the transposed matrices of the p6mm), those with normals along the two basis vectors and their sum and those with reflection lines along the basis vectors (and their sum). For example, for
have to be conjugated by this. This interchanges the two types of reflections (inthe reflection with reflection line 2x,x, one gets
which is the reflection with reflection line along the a axis.
As a result, for a layer group with plane group of type p3m1, the symmorphic plane group isomorphic to the reciprocal plane group is of type p31m and vice versa.
Based on the identification of the reciprocal plane group with a symmorphic plane group , the special vectors for a layer group can be directly read off from the Wyckoff positions of .
3.2. Crystallographic conventions in the classification of layergroup irreps
The isomorphism between a reciprocal plane group and a symmorphic plane group allows the application of crystallographic conventions in the classification of the wavevectors (and henceforth of the irreps) of the layer groups :
(i) The unit cells of the symmorphic plane groups listed in ITA can replace the representation domain. It is defined as a simply connected part of the (first) (a of the reciprocal space) which contains exactly one k vector of each orbit of k. The asymmetric units of plane groups can serve as representation domains. The advantage of choosing the crystallographic unit cells and their asymmetric units becomes evident in layer groups where the may belong to different topological types depending on the ratios of the lattice parameters. Lines on the may appear or disappear or change their relative sizes depending on the lattice parameters. In contrast to that, the unit cells and their asymmetric units of ITA are independent of the ratios of the lattice parameters.
as unit cells of the To find all irreps of , it is necessary to consider only the wavevectors of the socalled(ii) The action of the reciprocal plane group on the wavevectors results in their distribution into orbits of symmetryequivalent k vectors with respect to . Thanks to the isomorphism of () ^{*} with the symmorphic plane group , the different types of k vectors correspond to the different kinds of point orbits (Wyckoff positions) of . In this way, a complete list of the special sites in the of () ^{*} is provided by the Wyckoff positions of found in ITA. The sitesymmetry groups of ITA correspond to the little cogroups of the wavevectors and the number of arms of the star of a wavevector follows from the multiplicity of the The Wyckoff positions with zero, one and two variable parameters correspond to special kvector points, special kvector lines and special (or general) kvector planes, respectively. A kvector type, i.e. the set of all k vectors corresponding to a consists of complete orbits of k vectors and thus of full stars of k vectors. The different orbits (and stars) of a kvector type are obtained by varying the free parameters. Correspondingly, the irreps of k vectors of a kvector type are interrelated by parameter variation and are said to belong to the same type of irreps (Boyle, 1986). In this way all wavevector stars giving rise to the same type of irreps are related to the same and designated by the same Wyckoff letter.
(iii) A complete set of irreps of is derived by considering exactly one kvector representative per kvector orbit. To achieve that, it is necessary to specify the exact parameter ranges of the independent kvector regions within the representation domain (or the asymmetric unit). While such data are not available in the literature, the Brillouinzone database of BCS offers the listing of the exact parameter ranges for the k vectors which are absolutely necessary for the solution of the problems of uniqueness and completeness of layergroup irreps. For this purpose it is advantageous to describe the different kvector stars belonging to a applying the socalled uniarm description. Two k vectors of a are called uniarm if one can be obtained from the other by parameter variation. The description of kvector stars of a is called uniarm if the k vectors representing these stars are uniarm. Frequently, in order to achieve a uniarm description, it is necessary to transform k vectors to equivalent ones. In addition, to enable a uniarm description, symmetry lines outside the may be selected as orbit representatives. Such a segment of a line is called a flagpole. Examples of flagpoles are displayed in Fig. 9, and Figs. 10 and 11 for the layer group cm2m (No. 35) [cf. Section 5.2 for a detailed account of the use of flagpoles in the uniarm description of kvector types that belong to 2a of the reciprocal plane group (c1m1)^{*}].
4. The Brillouinzone database for layer groups
The kvector data of the Brillouinzone database of the BCS are accessed by the retrieval tool LKVEC which uses as input the ITE number of the layer group. The output consists essentially of wavevector tables and figures. There are several sets of figures and tables for the same layer group when its Brillouinzone shape depends on the lattice parameters of the The kvector data are the same for layer groups of the same arithmetic crystal class.
In the kvector tables, the wavevector data of LW are compared with the Wyckoffposition data of ITA. In the figures, the and the representation domains of LW, and the asymmetric units, chosen often in analogy to those of ITA, are displayed.
LW describe the monoclinic/rectangular layer groups with respect to a setting that is different from the conventional one found in ITE. Using the relationship between the two settings, we have transformed the kvector data of LW to the conventional setting of ITE for all six monoclinic/rectangular 211p, 211c, m11p, m11c, 2/m11p and 2/m11c. The transformed special kvector points, lines and planes keep the LW labels of the k vectors from which they were derived.
The URL of the program LKVEC is https://www.cryst.ehu.es/subperiodic/get_layer_kvec.html.
4.1. Guide to the tables
Each kvector table is headed by the corresponding Hermann–Mauguin symbol of the layer group, its ITE number and the symbol of the to which the layer group belongs. If there is more than one table for an then these tables refer to different geometric conditions for the lattice parameters that are indicated after the symbol of the The set of layer groups of the are also indicated in the headline block. They are followed by the symbol of the corresponding reciprocal planegroup type together with the conditions for the lattice parameters of the if any (asterisks denote reciprocalspace quantities). From the kvector table there is a link to the corresponding Brillouinzone figure.
The kvector tables consist of two parts: (i) `Litvin & Wike' description and (ii) `Planegroup description'. The first three columns under the heading `Litvin & Wike' refer to the description of the kvectors found in Tables 24 and 25 of LW. It consists of labels of kvectors (column 1), their parameter descriptions (column 2) and their layer little cogroup (column 3 for primitive lattices and column 4 for ccentred). Note that LW substitutes the Greekcharacter labels for the symmetry points and lines inside the by a symbol consisting of two Roman characters, e.g. GM instead of Γ, LD instead of Λ etc. In order to enable the uniarm description new kvector types, equivalent to those of LW, are added to the kvector lists. Equivalent k vectors (related by the sign ∼) are designated by the same labels; additional indices distinguish the new k vectors.
Different k vectors with the same LW label always belong to the same kvector type, i.e. they correspond to the same k Vectors with different LW labels may either belong to the same or to different types of k vectors. When k vectors with different LW labels belong to the same kvector type, the corresponding parameter descriptions are followed by the letters `ex' (from Latin, with the meaning of `from' or `out of'). Symmetry points or lines of symmetry of LW, related to the same are grouped together in a block. In the kvector tables, neighbouring Wyckoffposition blocks are distinguished by a slight difference in the background colour. The parameter description of the uniarm region of a kvector type is shown in the last row of the corresponding Wyckoffposition block.
The wavevector coefficients of LW (column 2 of the kvector tables) refer always to a irrespective of whether the conventional description of the group in ITE is with respect to a centred or For that reason, for layer groups with the wavevector coefficients with respect to the usual conventional reciprocal basis, i.e. dual to the conventional centred basis, are listed in the column under the heading `Conventional' of the kvector tables. The relations between the conventional coefficients (k_{1}, k_{2}) and the primitive coefficients (, ) are summarized in Table 1. (For layer groups with primitive lattices, the wavevector coefficients referred to a coincide with those referred to the basis dual to the conventional one of ITA.)

The layer little cogroup data of each k vector are listed under the heading `Layer little cogroup' of the kvector tables. The layer little cogroups are subgroups of the of the layer group and are described by oriented pointgroup symbols (as is customary for sitesymmetry groups of Wyckoff positions).
The data for the crystallographic classification scheme of the wavevectors are listed under the heading `Planegroup description' in the kvector tables. The columns `Wyckoff positions' show the `multiplicity', `Wyckoff letter' and `sitesymmetry' of the Wyckoff positions of the corresponding symmorphic plane group of ITA which is isomorphic to the reciprocal plane group () ^{*}. The multiplicity of a divided by the number of lattice points in the conventional of ITA is equal to the number of arms of the star of the corresponding k vector. The alphabetical sequence of the Wyckoff positions determines the sequence of the LW labels. Unlike in ITA, the tables start with the Wyckoff letter `a' for the of the highest Site symmetries are described by means of oriented pointgroup symbols which are also links to more detailed information on the symmetry operations of the sitesymmetry group. Besides the shorthand description, the matrixcolumn representation and the geometric representation of the symmetry operations of the sitesymmetry group, the program also provides a table with the relationship between the symmetry operations of the sitesymmetry group and the layer little cogroup (cf. Fig. 5 and Section 3.1 for detailed explanations).
The parameter description of the Wyckoff positions is shown in the last column of the wavevector tables under the heading `Coordinates'. It consists of a representative coordinate doublet of the and algebraic statements for the description of the independent parameter range. In some cases, the algebraic expressions are substituted by the designation of the parameter region in order to avoid clumsy notation. Because of the dependence of the shape of the on the lattice parameter relations there may be vertices of the with a variable coordinate. If such a point is displayed and designated in the tables and figures by an uppercase letter, then the label of its variable coefficient used in the parameterrange descriptions is the same letter but typed in lower case.
Because of the isomorphism between and () ^{*} the coordinate doublets of the Wyckoff positions of can be interpreted as kvector coefficients (, ) determined with respect to the conventional ITA basis of . The relation between the ITA coefficients (, ) and the conventional coefficients ( k_{1}, k_{2}) is shown in Table 2.

At the bottom of the web page with the kvector table one finds an auxiliary tool which allows the complete characterization of a wavevector of the (not restricted to the first Brillouin zone): given the kvector coefficients referred either to a primitive (LW) or to a conventional basis, the program assigns the k vector to the corresponding wavevector symmetry type, specifies its LW label, and calculates the layer little cogroup and the arms of the kvector star. Consider again the example of the k vector with coefficients for the layer group (No. 78), cf. Fig. 6 and Fig. 8. It is a vector outside the first and its coefficients do not correspond to any of the parameter descriptions of the kvector representatives listed in Fig. 6. The output of the auxiliary tool indicates that k is a point of a special kvector line of type SM and belongs to the Wyckoffposition block 3c. As already commented in Section 3.1, its star consists of three k vectors, = {(−1.21, 1), (1, 0.21), (0.21, −1.21)}. The sitesymmetry group ..m is generated by a reflection plane that can be identified by direct inspection among the symmetry operations of (p31m) ^{*}. The layer little cogroup, however, is mm2, due to the additional reflection in the layer plane.
4.2. Guide to the figures
The headline of each Brillouinzone figure includes the same information as the kvector tables: the Hermann–Mauguin symbol of the layer group, the ITE number and the symbol of the to which the layer group belongs. Different figures for the same are distinguished by the geometric conditions for the lattice. The corresponding conditions for the lattice parameters of the are indicated after the symbol of the reciprocal plane group.
The k_{x} and k_{y}, and the origin with coefficients (0, 0) always coincides with the centre of the and is called Γ (indicated as GM in the kvector tables). In the Brillouinzone figures the representation domains of LW are compared with the asymmetric units of ITA. A statement of whether the representation domain of LW and the are identical or not is given below the kvector table. The asymmetric units are often not fully contained in the but protrude from it, in particular by flagpoles.
are twodimensional objects in the The coordinate axes are designated byThe representatives of the orbits of kvector symmetry points or kvector symmetry lines, as well as the edges of the representation domains of LW and of the asymmetric units are brought out in colour (see Fig. 7):
(a) Symmetry points. A representative point of each orbit of special kvector points is designated by a circle filled in red with its label also in red. Note that a point is coloured red only if it is really a special point, i.e. a point whose layer little cogroup is a of the little cogroups of the points in its neighbourhood. In the figures, a point is marked by its label and an empty circle if it is listed in the corresponding kvector table but is not a point of special symmetry. For example, points listed by LW are not coloured if they form part of a symmetry line or a symmetry plane. The same designation is used for the auxiliary points that have been added in order to facilitate the comparison between the traditional and the reciprocal planegroup descriptions of the kvector types.
(b) Symmetry lines. A line is coloured in red with its label also in red only if it is a special kvector line, i.e. the layer little cogroups of the points on the line are supergroups of the little cogroups of the points in its neighbourhood. The colour of the line is pink for an edge of the which is not a symmetry line. The colour of the line is brown with the name in red for a line which is a symmetry line as well as an edge of the The edges of the representation domains are coloured light blue if the representation domain of LW does not coincide with the Edges of the representation domain and their labels are coloured dark blue if they are symmetry lines. Flagpoles are always coloured in red. Coordinate axes, edges of the or auxiliary lines are displayed by thin solid black lines.
5. Examples
The relation between the traditional and the reciprocal group descriptions of the wavevector types is illustrated by the following examples. The figures and tables included here form part of the output of the access tool LKVEC.
5.1. kVector table and for the layer group p62m (No. 78)
The kvector table and the Brillouinzone diagram of the hexagonal layer group (No. 78) are shown in Fig. 6 and Fig. 8, respectively. The of a hexagonal p lattice is also a hexagonal p lattice and the is a hexagon. The conventional basis for the has while the ITA description of hexagonal layer groups is based on a basis a_{H}, b_{H} with . In the Brillouinzone diagrams, the axis k_{x} is taken along a_{H} while k_{y} points out in the direction of a_{H} + b_{H}.
The k vectors of (No. 78) listed by LW (Fig. 6) are distributed in four kvector types: (i) the Wyckoffposition block 1a formed by the GM point, (ii) the block 2b formed by the K point, (iii) the kvector point M and the kvector lines SM and SN correspond to the block 3c, and (iv) the Wyckoffposition block 6d formed by the kvector lines LD and T and the kvector planes B and BB. The parameter description of a kvector type is given in the last column of the table. Consider for example the line SM which, according to the ITA description, forms part of the kvector type that is assigned to the 3c with a sitesymmetry group `..m'. Its parameter description x, 0: 0 < x < 1/2 indicates that the independent segment of the line x, 0 in the is limited by the special kvector points Γ (x = 0) and M (x = 1/2) with x varying between 0 and 1/2. The parameter descriptions of the uniarm regions of the kvector types are shown in the last row of the corresponding Wyckoffposition block. For example, in the block for position 3c, the kvector line SN is equivalent (by a threefold rotation) to which in turn is equivalent (by a translation) to the line , denoted by SN_{1}. This gives the uniarm description M+SM+SN_{1} for the 3c in Fig. 6.
As the kvector points and lines are brought out in red only if they are special kvector points and lines. For example, the lines T and LD (indicated as Λ on Fig. 8) are not coloured as special lines since they belong to the kvector type of the Wyckoffposition block 6d, and their symmetry coincides with that of the neighbouring points of the symmetry plane B = [GM K M]. The points GM (indicated as Γ on Fig. 8) and K, however, are represented by red circles as they are special kvector points (cf. Fig. 6). Likewise, the line SM (indicated as Σ on Fig. 8) is coloured in brown because it is an edge of the and at the same time is a special kvector symmetry line. The kvector line SN is coloured in dark blue as it is a special symmetry line along the edge of the representation domain. As already indicated, the kvector line SN together with SM and the point M belong to the special kvector type of the Wyckoffposition block 3c, i.e. all these different wavevectors belong to the same kvector type. Although M is explicitly listed by LW as a special kvector point, it is represented by an open circle in Fig. 8: in fact, it joins the symmetry lines SM and SN_{1} to a continuous line as its little cogroup type coincides with those of the points on the lines.
and the representation domain do not coincide, their edges are coloured in pink and light blue, respectively. It has already been pointed out that5.2. kVector table and for the layer group cm2m (No. 35)
The layer group cm2m (No. 35) is an example of orthorhombic layer groups with a ccentred lattice. It belongs to the m2mc which also includes the layer group cm2e (No. 36). The kvector tables and Brillouinzone figures for the layer groups belonging to the m2mc are given in Fig. 9, and Figs. 10 and 11, respectively.
The kvector tables show all special wavevectors with their coefficients and layer little cogroups as specified in Tables 24 and 25 of LW. The wavevector coefficients with respect to the conventional reciprocal basis, i.e. dual to the conventional centred basis, are listed in the column under the heading `Conventional' of the kvector tables. For example, a kvector point of the DT line (Fig. 9) with primitive coefficients is described as (0,1/2) with respect to a basis dual to the conventional basis of cm2m.
The comparison of the wavevector list of LW and the reciprocal planegroup description indicates clearly the redundancy of most of the k vectors given by LW. In fact, for the derivation of a complete set of irreps it is necessary to consider just two kvector types: a general one corresponding to the general 4b, and the kvector symmetry line related to the special 2a. The large number of additional k vectors given in the tables of LW are due to two main reasons:
(a) The more symmetry a layer group has lost compared with its holosymmetric layer group (layer groups whose point groups are holohedral), the more k vectors are introduced in LW. In the case of cm2m, the holosymmetric layer group is cmmm (No. 47), and the lines DA and FA are examples of such additional k vectors. In most cases these additional k vectors can be avoided by extending the parameter range in the kvector space.
(b) In the transition from a holosymmetric to a nonholosymmetric layer group , the order of the little cogroup of a special k vector in may be reduced in and, as a result, the special k vector in may lose its `special nature' in . Such k vectors become part of a more general kvector type (i.e. assigned to a of lower site symmetry) and can be described by an extension of the corresponding parameter range. Consider, for example, the kvector points Γ and Y in the tables of cm2m. They are special kvector points lying on the reflection plane of m_{010} in the holosymmetric layer group cmmm, but in (the nonholosymmetric group) cm2m which does not contain m_{010} the two points form part of the special kvector line with the uniarm description: (cf. the last row of the 2a Wyckoffposition block, Fig. 9).
The m2mc is a nonregular hexagon (cf. Figs. 10 and 11). Depending on the relation between the lattice parameters a and b, two topologically different are to be distinguished: (i) the acute case with and (ii) the obtuse case with . Because of the reflection (with normal k_{x}) of the reciprocal plane group (c1m1)^{*}, the representation domain is only one half of the hexagon: for example, in the acute case (Fig. 10) it is the trapezium with vertices (light blue boundary). The is different from the representation domain: it is the rectangle with vertices J_{2}, J_{4}, V_{4}, V_{2} (pink boundary), i.e. the points . While the representation domains of the acute and obtuse unit cells have the more complicated form of a trapezium, the asymmetric units in both cases have the topologically identical and relatively simple shape of a rectangle.
of the layer groups of theAs already indicated, the points Γ, Y_{2} and S (acute case), and Γ, Y and S (obtuse case) are not special kvector points but form part of special lines and planes and in the diagrams they are represented by open circles. The line SM is not a symmetry line and is represented by a thin black line because it is located inside the The lines DT and DA are coloured in brown because they are symmetry lines and at the same time are edges of the Parts of DT and DA are also coloured in red because they correspond to flagpoles. The kvector lines F and FA (acute case) are coloured in dark blue as they are symmetry lines along the edges of the representation domain.
Because of the special shape of the kvector line corresponding to the Wyckoffposition block 2a splits into several segments: the lines DT and DA, located inside the and the lines F and FA (coloured dark blue) at the border of the (cf. Figs. 9 and 10). For the description of the end points of the segments, it is necessary to introduce additional parameters as dt_{0} and f_{0} whose values depend on the specific relations between the lattice parameters. [In fact, the vertices and F_{0} of the representation domain have the coordinates 0, 1 / 4+1 / 4(a^{*} / b^{*}) and F_{0}: .] The use of flagpoles enables the uniarm description: the flagpole [J_{2} Y_{2}] is equivalent to the segment [V_{4} Y] and the flagpole [V_{2} Y_{4}] 0,y: is equivalent to the segment [Y J_{4}] . The uniarm description of the kvector type of the 2a is shown in the last row of the Wyckoffposition block and it is formed by the union of the points GM and Y_{2}, the lines DT, DA, DT_{1}(∼FA) and DA_{1} (∼F). Its parameter description (0, y) with y varying in the range (−1/2, 1/2) coincides with that of the acute case. The parameter description of the flagpole and its parameter range with respect to the basis of the reciprocal group are given below the kvector table.
and the representation domain for the acute case (), the special6. Conclusions
In this paper we have presented the layergroup crystallographic and wavevector databases of the BCS, together with the programs which give access to these data. Like the rest of the programs on the server, these tools are freely available and can be accessed via userfriendly web interfaces. The programs GENPOS, WYCKPOS and MAXSUB provide access to generators/general positions, Wyckoff positions and maximal information, respectively.
The wavevector database which contains the Brillouinzone figures and wavevector tables for all 80 layer groups was recently implemented on the BCS. One can access the database through the program LKVEC. In this compilation, the representation domains and lists of special k vectors in the tables on layergroup representations by Litvin & Wike (1991) are compared with the figures and wavevector data derived applying the reciprocalspacegroup approach. This new database provides a solution to the completeness problem of layergroup representations by specifying the independent parameter ranges of general and special k vectors within the representation domains.
Acknowledgements
Open access funding enabled and organized by Projekt DEAL.
Funding information
This work has been supported by the Spanish Ministry of Science and Innovation (project MINECOG19/P21) and the Government of the Basque Country (project IT130119).
References
Aroyo, M. I. (2016). Editor. International Tables for Crystallography, Vol. A, Spacegroup Symmetry, 6th ed. Chichester: Wiley. Google Scholar
Aroyo, M. I., Orobengoa, D., de la Flor, G., Tasci, E. S., PerezMato, J. M. & Wondratschek, H. (2014). Acta Cryst. A70, 126–137. Web of Science CrossRef CAS IUCr Journals Google Scholar
Aroyo, M. I., PerezMato, J. M., Capillas, C., Kroumova, E., Ivantchev, S., Madariaga, G., Kirov, A. & Wondratschek, H. (2006). Z. Kristallogr. 221, 15–27. Web of Science CrossRef CAS Google Scholar
Aroyo, M. I., PerezMato, J. M., Orobengoa, D., Tasci, E. S., de la Flor, G. & Kirov, A. (2011). Bulg. Chem. Commun. 43, 183–197. CAS Google Scholar
Aroyo, M. I. & Wondratschek, H. (1995). Z. Kristallogr. 210, 243–254. CrossRef CAS Web of Science Google Scholar
Boyle, L. L. (1986). Proceedings of the 14th International Colloquium on Group Theoretical Methods in Physics, pp. 405–408. Singapore: World Scientific. Google Scholar
Bradley, C. J. & Cracknell, A. P. (1972). The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups. Oxford: Clarendon Press. Google Scholar
Cracknell, A. P., Davies, B. L., Miller, S. C. & Love, W. F. (1979). Kronecker Product Tables, Vol. 1, General Introduction and Tables of Irreducible Representations of Space Groups. New York: IFI/Plenum. Google Scholar
Dresselhaus, M. S., Dresselhaus, G. & Jorio, A. (2008). Group Theory Application to the Physics of Condensed Matter. Berlin: Springer. Google Scholar
Elcoro, L., Bradlyn, B., Wang, Z., Vergniory, M. G., Cano, J., Felser, C., Bernevig, B. A., Orobengoa, D., de la Flor, G. & Aroyo, M. I. (2017). J. Appl. Cryst. 50, 1457–1477. Web of Science CrossRef CAS IUCr Journals Google Scholar
Flor, G. de la, Orobengoa, D., Evarestov, R. A., Kitaev, Y. E., Tasci, E. & Aroyo, M. I. (2019). J. Appl. Cryst. 52, 1214–1221. Web of Science CrossRef IUCr Journals Google Scholar
Gallego, S. V., Etxebarria, J., Elcoro, L., Tasci, E. S. & PerezMato, J. M. (2019). Acta Cryst. A75, 438–447. Web of Science CrossRef IUCr Journals Google Scholar
Hatch, D. M. & Stokes, H. T. (1986). Phase Transit. 7, 87–279. CrossRef Google Scholar
Ipatova, I. P. & Kitaev, Y. E. (1985). Prog. Surf. Sci. 18, 189–246. CrossRef CAS Web of Science Google Scholar
Kopský, V. & Litvin, D. (2010). Editors. International Tables for Crystallography, Vol. E, Subperiodic Groups, 2nd ed. Chichester: Wiley. Google Scholar
Litvin, D. B. & Kopský, V. (2014). Acta Cryst. A70, 677–678. Web of Science CrossRef IUCr Journals Google Scholar
Litvin, D. B. & Wike, T. R. (1991). Character Tables and Compatibility Relations of the Eighty Layer Groups and Seventeen Plane Groups. New York: Plenum Press. Google Scholar
Miller, S. C. & Love, W. F. (1967). Tables of Irreducible Representations of Space Groups and CoRepresentations of Magnetic Space Groups. Boulder: Pruett Press. Google Scholar
Milošević, I., Nikolić, B., Damnjanović, M. & Krčmar, M. (2008). J. Phys. A, 31, 3625–3648. Google Scholar
Stokes, H. T. & Hatch, D. M. (1988). Isotropy Subgroups of the 230 Crystallographic Space Groups. Singapore: World Scientific. Google Scholar
Tasci, E. S., de la Flor, G., Orobengoa, D., Capillas, C., PerezMato, J. M. & Aroyo, M. I. (2012). EJP Web Conf. 22, 00009. Google Scholar
Wintgen, G. (1941). Math. Ann. 118, 195–215. CrossRef CAS Google Scholar
Wondratschek, H. & Müller, U. (2010). Editors. International Tables for Crystallography, Vol. A1, Symmetry Relations between Space Groups, 2nd ed. Chichester: John Wiley & Sons. Google Scholar
This is an openaccess article distributed under the terms of the Creative Commons Attribution (CCBY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.