International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 181191
Section 1.5.5. Examples and discussion^{a}Departamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and ^{b}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany 
The comparison of the crystallographic classification scheme with the traditional one is illustrated by four examples from the Bilbao Crystallographic Server (1998). The examples are designated by the arithmetic crystal classes.
To each arithmetic crystal class of space groups, cf. Section 1.5.3.2, there belongs exactly one reciprocalspace group which is isomorphic to a type of symmorphic space groups , cf. Sections 1.5.3.2 and 1.5.3.4.
These examples consist essentially of figures and tables. The Brillouin zones with the representation domains of CDML together with the asymmetric units are displayed in the figures. In the synoptic tables the correlation between the kvector tables of CDML and the tables of (Wyckoff) positions in IT A is presented. One can thus compare the different descriptions and recognize the relations between them. In addition, the parameter ranges of the kvector types in the asymmetric unit are stated. If a kvector type is listed in the table more than once, then the equivalence relations between the k vectors are added such that exactly one representative may be selected for each kvector orbit.
Each figure caption gives the name of the arithmetic crystal class of space groups to which the Brillouin zone belongs. If there is more than one figure for this arithmetic crystal class, then these figures refer to different geometric conditions for the lattice. Therefore, for each of the figures the arithmetic crystal class is followed by the specific conditions for the lattice parameters of this figure, e.g. `' for Fig. 1.5.5.3 or ` and ' for Fig. 1.5.5.5.
Then the space groups of the arithmetic crystal class are listed with their Hermann–Mauguin symbols, their Schoenflies symbols and their spacegroup numbers in IT A in parentheses. Following this the type of the reciprocalspace group is denoted, e.g. `, No. 44' for the arithmetic crystal class in Fig. 1.5.5.5, together with the conditions for the lattice parameters of the reciprocal lattice, if any, and the number of the corresponding table.
The Brillouin zones are objects in reciprocal space. They are displayed in the figures. The reciprocal space is a vector space and its elements are the k vectors. Thus the Brillouin zone is a construction in vector space. Because the Brillouin zones are visualized by drawings consisting of vertices, lines and planes, one usually speaks of points, lines and planes in or on the border of the Brillouin zone, not of vectors. Here we follow this tradition.
The Brillouin zones are projected onto the drawing plane by a clinographic projection which may be found e.g. in Smith (1982), pp. 61 f. The coordinate axes are designated , and ; the coordinate axis points upwards in the projection plane. The diagrams of the Brillouin zones follow those of CDML in order to facilitate the comparison of the data. The origin O with coordinates 0, 0, 0 always forms the centre of the Brillouin zone and is called .
A minimal domain is the smallest fraction of the Brillouin zone which contains exactly one wavevector k from each orbit. In these examples, the representation domain of CDML is compared with the minimal domain, called `asymmetric unit', of the Bilbao Crystallographic Server. This asymmetric unit is a simple body and is often chosen in analogy to that of IT A. It may coincide with the representation domain of Table 3.10 in CDML, but is mostly rather different. Other than the representation domains of CDML, the asymmetric unit is often not fully contained in the Brillouin zone but protrudes from it, in particular by flagpoles and wings, cf. the end of this section.
In the figures the edges of the chosen asymmetric unit are drawn into the frame of the Brillouin zone. The names of points, lines and planes of CDML are retained in this listing. New names have been given to points and lines which are not listed in CDML.
The shape of the Brillouin zone depends on the lattice relations. Therefore, there may be vertices of the Brillouin zone with a variable coordinate. If such a point is displayed and designated in a figure by an uppercase letter, then the label of its variable coordinate in the corresponding table is the same letter but lower case. Thus, the variable coordinate of the point is , of is etc.
In CDML, the same letter may designate items of different quality in different figures and tables. For example, there is a point H in Fig. 1.5.5.1 and Table 1.5.5.1 but a line H in Fig. 1.5.5.5 and Table 1.5.5.5. In the figures and tables of these examples not only lines and points but also their equivalent objects are listed and the parameter ranges of the lines are described. Therefore, the endpoints of the line H, the points equivalent to a point H as well as the lines equivalent to a line H may be also designated by the letter H but distinguished by indices. In order to recognize points and lines easily, the indices of points are always even: H_{0}, H_{2}, H_{4}; those of lines are always odd: H_{1}, H_{3}.
A point is marked in a figure by its name and by a black circle filled with white if it is listed in the corresponding kvector table but is not a point of special symmetry. The same designation is used for the auxiliary points that have been added in order to facilitate the comparison between the two descriptions of the kvector types. Noncoloured parts of the coordinate axes, of the edges of the Brillouin zone or auxiliary lines are displayed by thin solid black lines. Such lines are dashed or omitted if they are not visible, i.e. are hidden by the body of the Brillouin zone or of the asymmetric unit.
The representatives for the orbits of symmetry points or of symmetry lines, as well as the edges of the representation domain of CDML and of the chosen asymmetric unit are shown in colour.
The representation domain of CDML is displayed in the same figure.

Exactly one element of each point orbit, line orbit or orbit of planes is contained in the asymmetric unit. Exceptionally, different elements of the same orbit have been coloured because of their special meaning. In these cases the different elements are connected in the corresponding table by the equivalence sign , see, e.g. the lines or the planes in Table 1.5.5.1.
To enable a uniarm description, symmetry lines outside the asymmetric unit may be selected as orbit representatives. Such a piece of a line is called a flagpole. Flagpoles are always coloured red, see, e.g., the line in Fig. 1.5.5.1.
Symmetry planes are not distinguished in the figures. However, in analogy to the flagpoles, symmetry planes outside the asymmetric unit may be selected as orbit representatives. Such a piece of a plane is called a wing. Wings are always coloured pink, see, e.g., Fig. 1.5.5.1.
Within the caption of each figure the following data are listed:
Each figure is followed by a table with the same number. As for the figures, each table caption gives the name of the arithmetic crystal class of space groups. If there is more than one table for this arithmetic crystal class, then the symbol for the arithmetic crystal class is followed by the specific conditions for the lattice parameters, as for the figures.
Column 1. Label of the k vectors in CDML, Tables 3.9 and 3.11 and parameter description of CDML for the set of k vectors which belong to the label. No ranges for the parameters are listed in CDML.
If two k vectors belong to the same type of k vectors, then their little cogroups are conjugate under the reciprocalspace group and they correspond to the same Wyckoff position. Different k vectors with the same CDML label always belong to the same kvector type. k vectors with different CDML labels may either belong to the same or to different types of k vectors. If such k vectors belong to the same type, the corresponding Wyckoffposition descriptions are preceded by the letters `ex'. Frequently, such k vectors have been transformed (sign `' in these tables) to equivalent ones in order to make the k vectors uniarm, see the tables in this section.
The parameter range of a region may be described by the vertices of that region in brackets […]. One point in brackets, e.g. [P], means the point P. Two points within the brackets, e.g. [A B] means the line from A to B. Three points within the brackets, e.g. [A B C] means the triangular region of a plane with the vertices A, B and C. Four or more points may mean a region of a plane or a threedimensional body, depending on the positions of the points. The meaning can be recognized by studying the corresponding figure. Commas between the points, e.g. [A, B, C] indicate the set {A, B, C} of the three points A, B and C.
A symbol […] does not indicate whether the vertices, boundary lines or boundary planes of the region are themselves included or not. All or part of them may belong to the region, all or part of them may not. In the parameter description of the region in Column 3 the inclusion or exclusion is stated by the symbols or .
The backslash `\' is used to indicate included parts not belonging to the described region, see e.g. the regions and in Table 1.5.5.1.
Column 2. This column describes the Wyckoff positions (given as the multiplicity, the Wyckoff letter and the site symmetry) of that symmorphic space group of IT A which is isomorphic to the reciprocalspace group . Each Wyckoff position of corresponds to a Wintgen position of , i.e. to a type of k vectors of and vice versa.
`Multiplicity' is the number of points in the conventional unit cell of IT A. Here it is the number of arms of the star of the k vector, multiplied by the number of centring vectors of the conventional unit cell in IT A.
Unlike in IT A, each table starts with the Wyckoff letter a for a Wyckoff position of highest site symmetry and proceeds in alphabetical order until the general position GP is reached. The sequence of the CDML labels is not that of CDML but is determined essentially by the alphabetical sequence of the Wyckoff positions.
The symbol for the site symmetry is `oriented', as given in the spacegroup tables of IT A. For the nomenclature, see Section 2.2.12 of IT A.
Column 3. These are the parameters of that Wyckoff position of which corresponds to the kvector label in CDML, see Column 1. The parameter description and the parameter range are listed. This range is chosen such that each orbit of the Wyckoff position of IT A, i.e. also each kvector orbit, is listed exactly once.
The following designation is used for the parameter ranges:

Example. In Table 1.5.5.3 one finds for the arithmetic crystal class of space groups:The parameter description would be:
Horizontal lines. The horizontal lines extending across the tables separate blocks with different numbers of free parameters. Decisive for this subdivision is the number of free parameters of the Wyckoff position to which the Wintgen position is assigned, not the number of free parameters of CDML.
Example. Arithmetic crystal class mm2F, see Table 1.5.5.5
The kvector labels `' and `' of CDML have no free parameter. However, they correspond to the Wyckoff position `', which has one free parameter. Therefore, and are listed together with `' and `' in the block for the symmetry lines, i.e. for the k vectors with one free parameter: in there is no parameterfree Wintgen position at all.
The kvector labels `' and `' of CDML have one free parameter each. However, they correspond together with other kvector labels to the Wyckoff position `'. Therefore, and A are listed together with `' and `' and others in the block for the planes, i.e. for the k vectors with two free parameters.
In general the sequence of the Wyckoff letters in IT A follows the falling number of free parameters. In the few cases where the sequence in IT A is different, the Wyckoff letters are exchanged. The exchange is noted at the top of the table.
Example. In the arithmetic crystal class , see Table 1.5.5.3, Wyckoff position e has one free parameter, whereas Wyckoff position f has constant parameters, i.e. no free parameter. Therefore, f is listed above the horizontal line, e is listed below, see Table 1.5.5.3. The note at the top of the table states `Wyckoff positions e and f exchanged'.
Parameter relations. The relations between the parameters of CDML and the parameters referred to the asymmetric unit are listed at the top of the table, e.g. for in Table 1.5.5.1: `Parameter relations: '. These relations may be modified to more convenient parameters without notice, as for the plane B of in Table 1.5.5.1:instead of
Arithmetic crystal classes and : The reciprocal lattice of a cubic lattice is a cubic lattice F. Its Brillouin zone is a rhombic dodecahedron and has 12 faces, 24 edges and 14 apices, the coordinates of which are the six permutations of and the eight coordinate triplets of . Eleven of these 14 points are visible in the applied projection.
The figure for arithmetic crystal class is shown in Fig. 1.5.5.1 and the corresponding table is Table 1.5.5.1. The figure for arithmetic crystal class is shown in Fig. 1.5.5.2 and the corresponding table is Table 1.5.5.2.



Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class . Space groups: (229), (230). Reciprocalspace group ()*, No. 225 (see Table 1.5.5.1). The representation domain of CDML is identical with the asymmetric unit. Auxiliary points: R: ; N_{2}: ; H_{2}: . Flagpole: : . Wing: : ; with . 

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class . Space groups (204), (206). Reciprocalspace group ()*, No. 202 (see Table 1.5.5.2). The representation domain of CDML is identical with the asymmetric unit. Auxiliary points: R: ; H_{2}: . Flagpole: : . 
Arithmetic crystal class : There are two different types of Brillouin zones for the tetragonal I lattice, one for (Fig. 1.5.5.3, Table 1.5.5.3) and one for (Fig. 1.5.5.4, Table 1.5.5.4). The first type of Brillouin zone, Fig. 1.5.5.3, is a tetragonal elongated rhombdodecahedron with 12 faces, four of them being hexagons. There are 18 apices; 14 of them are visible. The Brillouin zone of Fig. 1.5.5.4 is a tetragonally deformed cuboctahedron with 14 faces. There are 24 apices; 18 of them are visible.



Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : . Space groups (139) to (142). Reciprocalspace group ()*, No. 139: (see Table 1.5.5.3). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T: ; T_{2}: ; X_{2}: . Flagpole: [: . Wing: : , . 

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : . Space groups (139) to (142). Reciprocalspace group ()*, No. 139: (see Table 1.5.5.4). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: X_{2}: ; T: ; T_{2}: . Flagpole: : . Wing: : . 
Arithmetic crystal class : Depending on the lattice ratios a:b:c, there are four figures in CDML for the Brillouin zone of an orthorhombic crystal with an F lattice, see Fig. 3.6 on p. 26 in CDML. Only three of them are really necessary. Therefore, the case of Fig. 3.6(c) of CDML has been omitted in these examples; it is obtained from of Figure 3.6(d) by a rotation by 90° about the c* axis. The three remaining Brillouin zones are displayed in Fig. 1.5.5.5 (see also Table 1.5.5.5), Fig. 1.5.5.6 (see also Table 1.5.5.6) and Fig. 1.5.5.7 (see also Table 1.5.5.7). Fig. 1.5.5.5 is a distorted cuboctahedron with 14 faces, 36 edges and 24 apices, 18 of which are visible. The Brillouin zones of Figs. 1.5.5.6 and 1.5.5.7 are distorted elongated rhombdodecahedra. There are 12 faces, 28 edges and 18 apices; 14 of them are visible.




Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : , and . Space groups (42), (43). Reciprocalspace group ()*, No. 44: , and (see Table 1.5.5.5). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T_{4}: ; Y_{2}: ; Y_{4}: ; Z_{2}: . Flagpoles: : ; : . Wings: : ; : . 

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : . Space groups (42), (43). Reciprocalspace group ()*, No. 44: (see Table 1.5.5.6). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T_{4}: ; Y_{4}: ; Z_{4}: . Flagpoles: : ; : . Wings: : , ; : , . 

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class : . Space groups (42), (43). Reciprocalspace group ()*, No. 44: (see Table 1.5.5.7). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T_{4}: ; Y_{4}: ; Z_{4}: . Flagpoles: : ; : . Wings: : ; : . 
When the symmetry of the reciprocal lattice allows, the shape of the asymmetric unit may be chosen to be much simpler than that of the representation domain.
Examples

The Brillouin zone as well as the unit cell are always convex bodies; the same holds for the representation domain of CDML and for the choice of the asymmetric unit. It is thus sometimes unavoidable that the kvector types are split and that the different parts belong to different arms and to different stars of k vectors. Sometimes this splitting of kvector types may be avoided by an appropriate choice of the asymmetric unit; sometimes the introduction of flagpoles and wings is necessary to make the kvector types uniarm.
Examples

The kvector labels of CDML are primarily listed for the holosymmetric space groups. These lists are kept and supplemented for the nonholosymmetric space groups. In this way many superfluous kvector labels are introduced.
Examples

In Section 1.5.4.3 a method for the determination of the parameter ranges was described. A few examples shall display the procedure.

In the way just described the inner part of the parameter range can be fixed. The boundaries of the parameter range must be determined in addition:

References
Bilbao Crystallographic Server (1998). http://www.cryst.ehu.es/ .Smith, J. V. (1982). Geometrical and Structural Crystallography. New York: John Wiley & Sons.