International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 181-191   | 1 | 2 |

Section 1.5.5. Examples and discussion

M. I. Aroyoa* and H. Wondratschekb

aDepartamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:  wmpararm@lg.ehu.es

1.5.5. Examples and discussion

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The comparison of the crystallographic classification scheme with the traditional one is illustrated by four examples from the Bilbao Crystallographic Server (1998[link]). The examples are designated by the arithmetic crystal classes.

To each arithmetic crystal class of space groups, cf. Section 1.5.3.2[link], there belongs exactly one reciprocal-space group [({\cal G})^*] which is isomorphic to a type of symmorphic space groups [{\cal G}_0], cf. Sections 1.5.3.2[link] and 1.5.3.4[link].

  • (1) k-vector types of the arithmetic crystal class [m\overline{3}mI] (space groups [Im\overline{3}m] and [Ia\overline{3}d]), reciprocal-space group [({\cal G})^*] isomorphic to [Fm\overline{3}m]. The representation domain [\Phi = \Omega] is equal to the asymmetric unit, see Fig. 1.5.5.1[link] and Table 1.5.5.1[link].

  • (2) k-vector types of the arithmetic crystal class [m\overline{3}I] (space groups [Im\overline{3}] and [Ia\overline{3}]), reciprocal-space group [({\cal G})^*] isomorphic to [Fm\overline{3}]. The representation domain [\Phi > \Omega] is equal to the asymmetric unit; see Fig. 1.5.5.2[link] and Table 1.5.5.2[link].

  • (3) k-vector types of the arithmetic crystal class [4/mmmI] ([I4/mmm], [I4/mcm], [I4_1/amd] and [I4_1/acd]), reciprocal-space group [({\cal G})^*] isomorphic to [I4/mmm]. The representation domains [\Phi = \Omega] are topologically different for different ratios of the lattice parameters a and c whereas the asymmetric units are affinely equivalent; see Figs. 1.5.5.3[link] and 1.5.5.4[link] and Tables 1.5.5.3[link] and 1.5.5.4[link].

  • (4) k-vector types of the arithmetic crystal class [mm2F] ([Fmm2] and [Fdd2]), reciprocal-space group [({\cal G})^*] isomorphic to [Imm2]. The representation domains [\Phi > \Omega] are topologically different for different ratios of the lattice parameters a, b and c whereas the asymmetric units are affinely equivalent; see Figs. 1.5.5.5[link], 1.5.5.6[link] and 1.5.5.7[link], and Tables 1.5.5.5[link], 1.5.5.6[link] and 1.5.5.7[link].

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 k-vector 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 k-vector types in the asymmetric unit are stated. If a k-vector 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 k-vector orbit.

1.5.5.1. Guide to the figures

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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. `[c/a \,\lt\, 1]' for Fig. 1.5.5.3[link] or `[a^{-2} \,\lt\, b^{-2}+c^{-2},\ b^{-2} \,\lt\, c^{-2}+a^{-2}] and [c^{-2} \,\lt\, a^{-2}+b^{-2}]' for Fig. 1.5.5.5[link].

Then the space groups of the arithmetic crystal class are listed with their Hermann–Mauguin symbols, their Schoenflies symbols and their space-group numbers in IT A in parentheses. Following this the type of the reciprocal-space group is denoted, e.g. `[(Imm2)^*], No. 44' for the arithmetic crystal class [mm2F] in Fig. 1.5.5.5[link], 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[link]), pp. 61 f. The coordinate axes are designated [k_x], [k_y] and [k_z]; the [k_z]-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 [\Gamma].

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 upper-case 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 [G_0] is [g_0], of [\Lambda_0] is [\lambda_0] 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[link] and Table 1.5.5.1[link] but a line H in Fig. 1.5.5.5[link] and Table 1.5.5.5[link]. 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: H0, H2, H4; those of lines are always odd: H1, H3.

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 k-vector 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 k-vector types. Non-coloured 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.

  • (a) A representative point of each orbit of symmetry points is designated by a red- or cyan-filled circle with its name also in red or cyan if it belongs to the asymmetric unit or to the representation domain of CDML. If both colours could be used, e.g. if the asymmetric unit coincides with the representation domain, the colour is red.

    Note that a point is coloured red or cyan only if it is really a symmetry point, i.e. its little co-group is a proper supergroup of the little co-groups of all points in its neighbourhood. Such a point has no variable parameters in its coordinates. Points listed by CDML are not coloured if they are part of a symmetry line or symmetry plane only.

  • (b) Coloured lines are always broad. They are solid lines if they are `visible', i.e. if they are not hidden by the Brillouin zone or by the asymmetric unit. A hidden symmetry line or edge of the asymmetric unit is not suppressed but is coloured as a dashed line.

  • (c) The meanings of the different coloured lines and the names used for them in the text are as follows: [Scheme scheme2] Notes:

    • (1) The colour of the line is pink for an edge of the asymmetric unit which is not a symmetry line.

    • (2) The colour is red for a symmetry line of the asymmetric unit, with the name also in red.

    • (3) 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 asymmetric unit.

The representation domain of CDML is displayed in the same figure.

  • (1) The edges of the representation domain are coloured light blue.

  • (2) The symmetry points and lines with their letters are coloured cyan.

  • (3) Edges of the representation domain or common edges of the representation domain and the asymmetric unit are coloured dark blue with the letters in cyan if they are symmetry lines of the representation domain but not of the asymmetric unit.

    Common edges of an asymmetric unit and a representation domain are coloured pink if they are not symmetry lines simultaneously.

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 [\sim], see, e.g. the lines [F\sim F_1=[P\,R]] or the planes [B\sim B_1=[P\,N_2\,H_2]] in Table 1.5.5.1[link].

To enable a uni-arm 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 [F_1] in Fig. 1.5.5.1[link].

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[link].

Within the caption of each figure the following data are listed:

  • (i) a statement of whether the representation domain and the asymmetric unit are identical or not;

  • (ii) the coordinates of auxiliary points if not specified in the corresponding table;

  • (iii) the parameter descriptions of the flagpoles and the wings.

1.5.5.2. Guide to the k-vector tables

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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 co-groups are conjugate under the reciprocal-space group [({\cal G})^*] and they correspond to the same Wyckoff position. Different k vectors with the same CDML label always belong to the same k-vector 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 Wyckoff-position descriptions are preceded by the letters `ex'. Frequently, such k vectors have been transformed (sign `[\sim]' in these tables) to equivalent ones in order to make the k vectors uni-arm, 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 three-dimensional 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 [\leq] or [\lt].

The backslash `\' is used to indicate included parts not belonging to the described region, see e.g. the regions [[\Gamma\, R]\backslash[P]] and [[\Gamma\,N\,N_2\,H_2]\backslash[\Lambda,\,F_3]] in Table 1.5.5.1[link].

Column 2. This column describes the Wyckoff positions (given as the multiplicity, the Wyckoff letter and the site symmetry) of that symmorphic space group [{\cal G}_0] of IT A which is isomorphic to the reciprocal-space group [({\cal G})^*]. Each Wyckoff position of [{\cal G}_0] corresponds to a Wintgen position of [({\cal G})^*], i.e. to a type of k vectors of [({\cal G})^*] 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 space-group tables of IT A. For the nomenclature, see Section 2.2.12[link] of IT A.

Column 3. These are the parameters of that Wyckoff position of [{\cal G}_0] which corresponds to the k-vector 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 k-vector orbit, is listed exactly once.

The following designation is used for the parameter ranges:

  • (1) The statement [0 \,\lt\, x,\,y \,\lt\, \textstyle{{1}\over{2}}] means that x and y may vary independently from 0 to [\textstyle{{1}\over{2}}], 0 and [\textstyle{{1}\over{2}}] both excluded.

  • (2) The statement[\eqalign{GP\quad\alpha,\beta,\gamma\quad 48 \ h\ 1 \quad x,y,z&\!: 0\leq z \,\lt\, x \,\lt\, y \,\lt\,\textstyle{{1}\over{2}}\ \cup\cr \cup\ x,\textstyle{{1}\over{2}},z&\!: 0 \,\lt\, z \,\lt\, x \,\lt\, \textstyle{{1}\over{2}}}]means that the description of the asymmetric unit is split into two adjacent regions, a body and a plane. The boundary plane [z=0] of the body is included, all other boundaries are excluded. Together the regions contain exactly one representative for each k-vector orbit of the general position GP of the reciprocal-space group.

  • (3) The statement [x,\textstyle{{1}\over{2}},z]: [-x \,\lt\, z\leq x,\ z\neq0] means that z may assume any value between −x and +x, z = x included but z = −x and z = 0 excluded.

  • (4) Occasionally the parameter description becomes too clumsy. Then the data listed are abbreviated by replacing the parametrical data by the designation of the corresponding region.

Example. In Table 1.5.5.3[link] one finds for the arithmetic crystal class [4/mmmI] of space groups:[B \quad \alpha,\beta,-\alpha \quad 16 \ m \ ..m \quad x,x,z\!\!:\,[\Gamma\,M\,Z_2\,Z_0]]The parameter description would be:[x,x,z\!\!:\,0 \,\lt\, x \,\lt\, \textstyle{{1}\over{2}},\ 0 \,\lt\, z\leq z_0–2x(2z_0-\textstyle{{1}\over{2}})]

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[link]

The k-vector labels `[\Gamma\quad 0,0,0]' and `[Z\quad \textstyle{{1}\over{2}},\textstyle{{1}\over{2}},0]' of CDML have no free parameter. However, they correspond to the Wyckoff position `[2\ a\ mm2\quad 0,0,z]', which has one free parameter. Therefore, [\Gamma] and [Z] are listed together with `[\Lambda\quad \alpha,\alpha,0]' and `[LE\quad {-\alpha},-\alpha,0]' in the block for the symmetry lines, i.e. for the k vectors with one free parameter: in [(Imm2)^*] there is no parameter-free Wintgen position at all.

The k-vector labels `[\Sigma\quad 0,\alpha,\alpha]' and `[A\quad \textstyle{{1}\over{2}},\textstyle{{1}\over{2}}+\alpha,\alpha]' of CDML have one free parameter each. However, they correspond together with other k-vector labels to the Wyckoff position `[4\ c\ .m.\quad x,0,z]'. Therefore, [\Sigma] and A are listed together with `[J\quad \alpha,\alpha+\beta,\beta]' and `[JA\quad {-\alpha},-\alpha+\beta,\beta]' 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 [4/mmmI,] [c/a \,\lt\, 1], see Table 1.5.5.3[link], 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[link]. 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 [m\overline{3}mI] in Table 1.5.5.1[link]: `Parameter relations: [x=\textstyle{{1}\over{2}}\beta + \textstyle{{1}\over{2}}\gamma,\ y=\textstyle{{1}\over{2}}\alpha+\textstyle{{1}\over{2}}\gamma,\ z=\textstyle{{1}\over{2}}\alpha+\textstyle{{1}\over{2}}\beta]'. These relations may be modified to more convenient parameters without notice, as for the plane B of [m\overline{3}mI] in Table 1.5.5.1[link]:[B\quad \alpha+\beta,-\alpha+\beta,\textstyle{{1}\over{2}}-\beta\quad {ex}\,\,96\ k \ ..m \quad x,\textstyle{{1}\over{2}}-x,z\!\!: 0 \,\lt\, z \,\lt\, x \,\lt\, \textstyle{{1}\over{4}}]instead of [\ldots\textstyle{{1}\over{4}}-\textstyle{{1}\over{2}}\alpha,\textstyle{{1}\over{4}}+\textstyle{{1}\over{2}}\alpha,\beta\!\!:\ 0 \,\lt\, \alpha \,\lt\, \textstyle{{1}\over{2}}-2\beta \,\lt\, \textstyle{{1}\over{2}}.]

1.5.5.3. Figures and tables

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Arithmetic crystal classes [m\overline{3}mI] and [m\overline{3}I]: The reciprocal lattice of a cubic lattice [I] 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 [\pm{{1}\over{2}},0,0] and the eight coordinate triplets of [\pm{{1}\over{4}},\pm{{1}\over{4}},\pm{{1}\over{4}}]. Eleven of these 14 points are visible in the applied projection.

The figure for arithmetic crystal class [m\overline{3}mI] is shown in Fig. 1.5.5.1[link] and the corresponding table is Table 1.5.5.1[link]. The figure for arithmetic crystal class [m\overline{3}I] is shown in Fig. 1.5.5.2[link] and the corresponding table is Table 1.5.5.2[link].

Table 1.5.5.1| top | pdf |
List of k-vector types for arithmetic crystal class [m\overline{3}mI]

See Fig. 1.5.5.1[link]. Parameter relations: [x=\textstyle{1 \over 2}\beta + \textstyle{1 \over 2}\gamma], [y=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\gamma], [z=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0]   [4 \quad a \quad m\overline3m] 0, 0, 0
[H \quad \textstyle{1 \over 2},-\textstyle{1 \over 2},\textstyle{1 \over 2}]   [4 \quad b \quad m\overline3m] [0,\textstyle{1 \over 2},0]
[P \quad \textstyle{1 \over 4}, \textstyle{1 \over 4}, \textstyle{1 \over 4}]   [8 \quad c \quad\overline43m ] [\textstyle{1 \over 4}, \textstyle{1 \over 4}, \textstyle{1 \over 4} ]
[N\quad 0,0,\textstyle{1 \over 2}]   [24 \quad d \quad m.mm] [\textstyle{1 \over 4},\textstyle{1 \over 4},0 ]
[\Delta \quad \alpha,-\alpha,\alpha]   [24 \quad e \quad 4m.m] [0,y,0]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2}]
[\Lambda \quad \alpha,\alpha,\alpha] ex [32 \quad f \quad .3m] [x,x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[F \quad \textstyle{1 \over 2}-\alpha,-\textstyle{1 \over 2}+ 3\alpha,\textstyle{1 \over 2}-\alpha] ex [32 \quad f \quad .3m] [x,\textstyle{1 \over 2}-x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[F\sim F_1=[P\,R]]     [x,x,x]: [\textstyle{1 \over 4} \,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[F\sim F_3=[P\,H_2]]     [x,x,\textstyle{1 \over 2}-x]: [0 \,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[\Lambda\cup F_1=[\Gamma R]\backslash [P] ]   [32 \quad f \quad .3m] [x,x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}, x\neq\textstyle{1 \over 4}]
[D\quad \alpha,\alpha,\textstyle{1 \over 2}-\alpha]   [48 \quad g \quad 2.mm] [\textstyle{1 \over 4},\textstyle{1 \over 4},z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}]
[\Sigma \quad 0,0,\alpha]   [48 \quad h \quad m.m2] [x,x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[G\quad \textstyle{1 \over 2}-\alpha,-\textstyle{1 \over 2}+\alpha,\textstyle{1 \over 2}]   [48 \quad i \quad m.m2] [x,\textstyle{1 \over 2}-x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[A \quad \alpha, -\alpha, \beta]   [96 \quad j \quad m..] [x,y,0]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]
[B\quad \alpha+\beta,-\alpha+\beta,\textstyle{1 \over 2}-\beta] ex [96 \quad k \quad ..m] [x,\textstyle{1 \over 2}-x,z]: [0\,\lt \, z\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[B\sim B_1=[P\,N_2\,H_2]]     [x,x,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}-x\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[C \quad \alpha,\alpha,\beta] ex [96 \quad k \quad ..m] [x,x,z]: [0\,\lt \, z\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[J \quad \alpha, \beta, \alpha] ex [96\quad k \quad ..m] [x,y,x]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]
[J\sim J_1=[\Gamma\,P\,H_2]]     [x,x,z]: [0\,\lt \, x\,\lt \, z\,\lt \, \textstyle{1 \over 2}-x]
[C\cup B_1\cup J_1=[\Gamma\,N\,N_2\,H_2]\backslash[\Lambda,\,F_3]]   [96 \quad k \quad ..m] [x,x,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}, 0\,\lt \, z\,\lt \, \textstyle{1 \over 2}] with [z\neq x, z\neq\textstyle{1 \over 2}-x]
[GP \quad \alpha,\beta,\gamma]   [192 \quad l \quad 1] [x,y,z]: [0\,\lt \, z\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]

Table 1.5.5.2| top | pdf |
List of k-vector types for arithmetic crystal class [m\overline{3}I]

See Fig. 1.5.5.2[link]. Parameter relations: [x=\textstyle{1 \over 2}\beta + \textstyle{1 \over 2}\gamma], [y=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\gamma], [z=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma\quad 0,0,0]   [4 \quad a \quad m\overline3.] [0,0,0]
[H \quad \textstyle{1 \over 2},-\textstyle{1 \over 2},\textstyle{1 \over 2}]   [4 \quad b \quad m\overline3.] [0,\textstyle{1 \over 2},0]
[P \quad \textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]   [8 \quad c \quad 23.] [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]
[N \quad 0,0,\textstyle{1 \over 2}]   [24 \quad d \quad 2/m..] [\textstyle{1 \over 4},\textstyle{1 \over 4},0]
[\Delta\quad \alpha,-\alpha,\alpha]   [24 \quad e \quad mm2..] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[\Lambda \quad \alpha,\alpha,\alpha] ex [32 \quad f \quad .3.] [x,x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[F \quad \textstyle{1 \over 2}-\alpha, -\textstyle{1 \over 2}+3\alpha, \textstyle{1 \over 2}-\alpha] ex [32 \quad f \quad .3.] [x,\textstyle{1 \over 2}-x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[F\sim F_1 =[P\,R]]     [x,x,x]: [\textstyle{1 \over 4}\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[\Lambda \cup F_1\sim[\Gamma\, R]\backslash[P]]   [32 \quad f\quad .3.] [x,x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}, x\neq\textstyle{1 \over 4}]
[D\quad \alpha,\alpha,\textstyle{1 \over 2}-\alpha]   [48 \quad g \quad 2..] [\textstyle{1 \over 4},\textstyle{1 \over 4},z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}]
[\Sigma\quad 0,0,\alpha] ex [48 \quad h \quad m..] [x,x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[G \quad\textstyle{1 \over 2}-\alpha,-\textstyle{1 \over 2}+\alpha,\textstyle{1 \over 2}] ex [48 \quad h \quad m..] [x,\textstyle{1 \over 2}-x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[A = [\Gamma\, N\, H] \quad \alpha,-\alpha,\beta] ex [48\quad h \quad m..] [x,y,0]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]
[AA = [\Gamma\,H_2\,N] \quad {-\alpha},\alpha,\beta] ex [48 \quad h \quad m..] [x,y,0]: [0\,\lt \, y\,\lt \, x\,\lt \, \textstyle{1 \over 2}-y]
[\Sigma\cup G\cup A\cup AA]   [48 \quad h \quad m..] [x,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x\,\lt \, \textstyle{1 \over 2}\ \cup] [\cup \ x,\textstyle{1 \over 2}-x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4} ]
[GP \quad \alpha,\beta,\gamma]   [96 \quad i \quad 1] [x,y,z]: [0\,\lt \, z\leq x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x \ \cup] [\cup \ x,y,z]: [0\,\lt \, z\,\lt \, y\,\lt \, x\leq\textstyle{1 \over 2}-y\ \cup] [\cup \ x,x,z]: [0\,\lt \, z\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[Figure 1.5.5.1]

Figure 1.5.5.1 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [m\overline{3}mI]. Space groups: [Im\overline{3}m-O_h^9] (229), [Ia\overline{3}d-O_h^{10}] (230). Reciprocal-space group ([Fm\overline{3}m])*, No. 225 (see Table 1.5.5.1[link]). The representation domain of CDML is identical with the asymmetric unit. Auxiliary points: R: [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 2}]; N2: [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 2}]; H2: [0,0,\textstyle{1 \over 2}]. Flagpole: [F_1=[P\,R]\quad x,x,x]: [\textstyle{1 \over 4} \,\lt \, x \,\lt \, \textstyle{1 \over 2}]. Wing: [B_1 \cup J_1=[\Gamma\,P\,N_2\,H_2]\backslash[F_3]] [ x,x,z]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}]; [x \,\lt \, z \,\lt \,\textstyle{1 \over 2}] with [z \neq \textstyle{1 \over 2} - x].

[Figure 1.5.5.2]

Figure 1.5.5.2 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [m\overline{3}I]. Space groups [Im\overline{3}-T_h^5] (204), [Ia\overline{3}-T_h^7] (206). Reciprocal-space group ([Fm\overline{3}])*, No. 202 (see Table 1.5.5.2[link]). The representation domain of CDML is identical with the asymmetric unit. Auxiliary points: R: [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 2}]; H2: [\textstyle{1 \over 2},0,0]. Flagpole: [F_1=[P\,R]] [ x,x,x]: [\textstyle{1 \over 4} \,\lt \, x \,\lt \, \textstyle{1 \over 2}].

Arithmetic crystal class [4/mmmI]: There are two different types of Brillouin zones for the tetragonal I lattice, one for [c \,\lt\, a] (Fig. 1.5.5.3[link], Table 1.5.5.3[link]) and one for [c\,\gt\, a] (Fig. 1.5.5.4[link], Table 1.5.5.4[link]). The first type of Brillouin zone, Fig. 1.5.5.3[link], 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[link] is a tetragonally deformed cuboctahedron with 14 faces. There are 24 apices; 18 of them are visible.

Table 1.5.5.3| top | pdf |
List of k-vector types for arithmetic crystal class [4/mmmI]: [c/a \,\lt\, 1]

See Fig. 1.5.5.3[link]. Wyckoff positions e and f exchanged. Parameter relations: [x=-\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta], [y=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta+\gamma], [z=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0]   [2 \quad a \quad 4/mmm] [0,0,0]
[M \quad {-\textstyle{1 \over 2}},\textstyle{1 \over 2},\textstyle{1 \over 2}]   [2 \quad b \quad 4/mmm] [\textstyle{1 \over 2},\textstyle{1 \over 2},0]
[M\sim M_2]     [0,0,\textstyle{1 \over 2}]
[X \quad 0,0,\textstyle{1 \over 2}]   [4 \quad c \quad mmm.] [0,\textstyle{1 \over 2},0]
[P \quad \textstyle{1 \over 4}, \textstyle{1 \over 4}, \textstyle{1 \over 4}]   [4 \quad d \quad \overline 4m2] [0,\textstyle{1 \over 2},\textstyle{1 \over 4}]
[N \quad 0, \textstyle{1 \over 2}, 0]   [8 \quad f\quad ..2/m] [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]
[\Lambda\quad \alpha, \alpha, -\alpha] ex [4 \quad e \quad 4mm] [0,0,z]: [0\,\lt \, z\leq z_0]
[V \quad {-\textstyle{1 \over 2}} + \alpha,\textstyle{1 \over 2} + \alpha,\textstyle{1 \over 2} -\alpha] ex [4 \quad e \quad 4mm] [\textstyle{1 \over 2},\textstyle{1 \over 2},z]: [0\,\lt \, z\,\lt \, z_2=\textstyle{1 \over 2}-z_0]
[V \sim \Lambda_1=[Z_0\,M_2]]     [0,0,z]: [\,z_0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[\Lambda \cup \Lambda_1 = [\Gamma\,M_2]]   [4 \quad e \quad 4mm] [0,0,z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[W \quad \alpha, \alpha, \textstyle{1 \over 2}-\alpha]   [8\quad g \quad 2mm.] [0,\textstyle{1 \over 2},z]: [\, 0\,\lt \, z\,\lt \, \textstyle{1 \over 4}]
[\Sigma \quad{-\alpha}, \alpha, \alpha]   [8 \quad h \quad m.2m] [x,x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[\Delta \quad 0,0,\alpha]   [8 \quad i \quad m2m.] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[Y \quad {-\alpha}, \alpha, \textstyle{1 \over 2}]   [8 \quad j \quad m2m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[Q \quad\textstyle{1 \over 4}-\alpha,\textstyle{1 \over 4}+\alpha,\textstyle{1 \over 4}-\alpha]   [16 \quad k \quad ..2] [x,\textstyle{1 \over 2}-x,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[C \quad {-\alpha},\alpha,\beta]   [16 \quad l \quad m..] [x,y,0]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[B \quad \alpha,\beta,-\alpha]   [16 \quad m \quad ..m] [x,x,z]: [[\Gamma\,M\,Z_2\,Z_0]]
[B=B_1\cup B_2] = [[\Gamma\,M\,Z_2\,N\,T]\cup [T\,N\,Z_0]]      
[B_2 \sim B_3]     [x,x,z]: [[N\,Z_2\,T_2]]
[B_1\cup B_3=[\Gamma\,M\,T_2\,T]]   [16 \quad m \quad ..m] [x,x,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2},\,0\,\lt \, z\,\lt \, \textstyle{1 \over 4}\ \cup] [\cup \ x,x,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[A \quad \alpha,\alpha,\beta] ex [16 \quad n \quad .m.] [0,y,z]: [[\Gamma\,X\,P\,Z_0]]
[E\quad \alpha - \beta, \alpha+\beta, \textstyle{1 \over 2}-\alpha] ex [16 \quad n \quad .m.] [x,\textstyle{1 \over 2},z]: [[M\,X\,P\,Z_2]]
[E \sim A_1]     [0,y,z]: [[P\,X_2\,M_2\,Z_0]]
[A \cup A_1=[\Gamma\,X\,X_2\,M_2]]   [16 \quad n \quad .m.] [0,y,z]: [0\,\lt \, y,z\,\lt \, \textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [32 \quad o \quad 1] [x,y,z]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}]; [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}\ \cup] [\cup \ x,y,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]

Table 1.5.5.4| top | pdf |
List of k-vector types for arithmetic crystal class [4/mmmI]: [c/a \,\gt\, 1]

See Fig. 1.5.5.4[link]. Wyckoff positions [e] and [f] exchanged. Parameter relations: [x=-\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta], [y=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta+\gamma], [z=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0]   [2 \quad a \quad 4/mmm] [0,0,0]
[M \quad \textstyle{1 \over 2}, \textstyle{1 \over 2}, -\textstyle{1 \over 2}]   [2 \quad b \quad 4/mmm] [0,0,\textstyle{1 \over 2}]
[M\sim M_2]     [\textstyle{1 \over 2},\textstyle{1 \over 2},0]
[X \quad 0,0,\textstyle{1 \over 2}]   [4 \quad c \quad mmm.] [0,\textstyle{1 \over 2},0]
[P \quad \textstyle{1 \over 4}, \textstyle{1 \over 4}, \textstyle{1 \over 4}]   [4 \quad d \quad \overline 4m2] [0,\textstyle{1 \over 2},\textstyle{1 \over 4}]
[N \quad 0, \textstyle{1 \over 2}, 0]   [8 \quad f \quad ..2/m] [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]
[\Lambda\quad \alpha, \alpha, -\alpha]   [4 \quad e \quad 4mm] [0,0,z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[W \quad \alpha, \alpha, \textstyle{1 \over 2}-\alpha]   [8 \quad g \quad 2mm.] [0,\textstyle{1 \over 2},z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}]
[\Sigma\quad {-\alpha}, \alpha, \alpha] ex [8 \quad h \quad m.2m] [x,x,0]: [0\,\lt \, x\leq s_2]
[F \quad \textstyle{1 \over 2}-\alpha,\textstyle{1 \over 2}+\alpha,-\textstyle{1 \over 2}+\alpha] ex [8\quad h \quad m.2m] [x,x,\textstyle{1 \over 2}]: [0\,\lt \, x\,\lt \, s=\textstyle{1 \over 2}-s_2]
[F\sim\Sigma_1=[S_2\,M_2]]     [x,x,0]: [s_2\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[\Sigma\cup \Sigma_1=[\Gamma M_2]]   [8 \quad h \quad m.2m] [x,x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[\Delta\quad 0,0,\alpha]   [8 \quad i \quad m2m.] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[Y\quad {-\alpha}, \alpha, \textstyle{1 \over 2}] ex [8 \quad j \quad m2m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\leq r]
[U \quad \textstyle{1 \over 2},\textstyle{1 \over 2},-\textstyle{1 \over 2}+\alpha] ex [8 \quad j \quad m2m.] [0,y, \textstyle{1 \over 2}]: [0\,\lt \, y\,\lt \, g=\textstyle{1 \over 2}-r]
[U\sim Y_1=[R\,M_2]]     [x,\textstyle{1 \over 2},0]: [r\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[Y\cup Y_1=[X\,M_2]]   [8 \quad j \quad m2m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[Q\quad\textstyle{1 \over 4}-\alpha,\textstyle{1 \over 4}+\alpha,\textstyle{1 \over 4}-\alpha]   [16 \quad k \quad ..2] [x,\textstyle{1 \over 2}-x,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[C\quad {-\alpha},\alpha,\beta] ex [16 \quad l \quad m..] [x,y,0]: [[\Gamma\,S_2\,R\,X]]
[D \quad\textstyle{1 \over 2}-\alpha,\textstyle{1 \over 2}+\alpha,-\textstyle{1 \over 2}+\beta] ex [16 \quad l \quad m..] [x,y,\textstyle{1 \over 2}]: [[M\,S\,G]]
[D\sim C_1]     [x,y,0]: [[M_2\,R\,S_2]]
[C\cup C_1=[\Gamma\,M_2\,X]]   [16 \quad l \quad m..] [x,y,0]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[B\quad\alpha,\beta,-\alpha]   [16 \quad m \quad ..m] [x,x,z]: [[\Gamma\,S_2\,S\,M]]
[B=B_1\cup B_2] = [[\Gamma\,S_2\,N\,T]\cup[T\,N\,S\,M]]      
[B_2\sim B_3]     [x,x,z]: [[T_2\,N\,S_2\,M_2]]
[B_1\cup B_3=[\Gamma\,M_2\,T_2\,T]]   [16 \quad m \quad ..m] [x,x,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2},0\,\lt \, z\,\lt \, \textstyle{1 \over 4}\ \cup] [\cup \ x,x,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[A\quad\alpha,\alpha,\beta] ex [16 \quad n \quad .m.] [0,y,z]: [[\Gamma\, X\,P\,G\,M]]
[E \quad\alpha - \beta, \alpha+\beta, \textstyle{1 \over 2}-\alpha] ex [16 \quad n \quad .m.] [x,\textstyle{1 \over 2},z]: [[X\,P\,R]]
[E\sim A_1]     [0,y,z]: [[X_2\,G\,P]]
[A\cup A_1=[\Gamma\,X\,X_2\,M]]   [16 \quad n \quad .m.] [0,y,z]: [0\,\lt \, y,z\,\lt \, \textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [32 \quad o \quad 1] [x,y,z]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}]; [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}\ \cup] [\cup \ x,y,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]
[Figure 1.5.5.3]

Figure 1.5.5.3 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [4/mmmI]: [c/a \,\lt \, 1]. Space groups [I4/mmm-D_{4h}^{17}] (139) to [I4_1/acd-D_{4h}^{20}] (142). Reciprocal-space group ([I4/mmm])*, No. 139: [c^{*}/a^{*}\,\gt \, 1] (see Table 1.5.5.3[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T: [0,0,\textstyle{1 \over 4}]; T2: [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 4}]; X2: [0,\textstyle{1 \over 2},\textstyle{1 \over 2}]. Flagpole: [[T\,M_2]\quad 0,0,z]: [\textstyle{1 \over 4} \,\lt \, z \,\lt \, \textstyle{1 \over 2}]. Wing: [[T\,P\,X_2\,M_2]\quad 0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2}], [\textstyle{1 \over 4} \,\lt \, z \,\lt \, \textstyle{1 \over 2}].

[Figure 1.5.5.4]

Figure 1.5.5.4 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [4/mmmI]: [c/a\,\gt \, 1]. Space groups [I4/mmm-D_{4h}^{17}] (139) to [I4_1/acd-D_{4h}^{20}] (142). Reciprocal-space group ([I4/mmm])*, No. 139: [c^{*}/a^{*} \,\lt \, 1] (see Table 1.5.5.4[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: X2: [0,\textstyle{1 \over 2},\textstyle{1 \over 2}]; T: [0,0,\textstyle{1 \over 4}]; T2: [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 4}]. Flagpole: [[T\,M]\quad 0,0,z]: [\textstyle{1 \over 4} \,\lt \, z \,\lt \, \textstyle{1 \over 2}]. Wing: [[T\,P\,X_2\,M]\quad 0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2},\textstyle{1 \over 4} \,\lt \, z \,\lt \, \textstyle{1 \over 2}].

Arithmetic crystal class [mm2F]: 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 [b^{-2}\,\gt \, c^{-2}+a^{-2}] of Fig. 3.6(c) of CDML has been omitted in these examples; it is obtained from [a^{-2}\,\gt \, c^{-2}+b^{-2}] 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[link] (see also Table 1.5.5.5[link]), Fig. 1.5.5.6[link] (see also Table 1.5.5.6[link]) and Fig. 1.5.5.7[link] (see also Table 1.5.5.7[link]). Fig. 1.5.5.5[link] 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[link] and 1.5.5.7[link] are distorted elongated rhombdodecahedra. There are 12 faces, 28 edges and 18 apices; 14 of them are visible.

Table 1.5.5.5| top | pdf |
List of k-vector types for arithmetic crystal class [mm2F]: [a^{-2} \,\lt\, b^{-2}+c^{-2}], [b^{-2} \,\lt\, c^{-2}+a^{-2}] and [c^{-2} \,\lt\, a^{-2}+b^{-2}]

See Fig. 1.5.5.5[link]. Parameter relations: [x=-\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [y=\textstyle{1 \over 2} \alpha-\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [z=\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta-\textstyle{1 \over 2}\gamma].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0] ex [2 \quad a \quad mm2] [0,0,0]
[Z \quad \textstyle{1 \over 2},\textstyle{1 \over 2} ,0] ex [2 \quad a \quad mm2] [0,0,\textstyle{1 \over 2}]
[\Lambda \quad \alpha, \alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[LE \quad {-\alpha}, -\alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [-\textstyle{1 \over 2}\,\lt \, z\,\lt \, 0]
[\Gamma \cup \Lambda \cup Z \cup LE]   [2 \quad a \quad mm2] [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \leq \textstyle{1 \over 2}]
[T\quad 0,\textstyle{1 \over 2},\textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,0]
[T \sim T_2]     [0,\textstyle{1 \over 2} ,\textstyle{1 \over 2}]
[Y \quad \textstyle{1 \over 2}, 0, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},0]
[G \quad \alpha, \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,z]: [0\,\lt \, z\leq g_0]
[G \sim H_3=[H_2\,T_4]]     [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \leq -\textstyle{1 \over 2}+g_0=h_2]
[GA \quad {-\alpha}, \textstyle{1 \over 2}-\alpha, \textstyle{1 \over 2}] ex [2 \quad b\quad mm2] [\textstyle{1 \over 2},0,z]: [g_2=-g_0\,\lt \, z\,\lt \, 0]
[GA \sim H_1=[H_0\,T_2]]     [0,\textstyle{1 \over 2},z]: [\textstyle{1 \over 2}-g_0=h_0 \,\lt \, z\,\lt \,\textstyle{1 \over 2}]
[H \quad \textstyle{1 \over 2}+\alpha, \alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [0\,\lt \, z \leq h_0]
[HA \quad \textstyle{1 \over 2}-\alpha,-\alpha,\textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [h_2=-h_0\,\lt \, z\,\lt \, 0]
[T_2 \cup H_1 \cup H \cup Y \cup HA \cup H_3]   [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \leq \textstyle{1 \over 2}]
[\Sigma \quad 0,\alpha,\alpha] ex [4 \quad c \quad .m.] [x,0,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[A \quad \textstyle{1 \over 2},\textstyle{1 \over 2}+\alpha,\alpha] ex [4 \quad c \quad .m.] [x,0,\textstyle{1 \over 2}]: [0\,\lt \, x\leq a_0]
[C \quad \textstyle{1 \over 2},\alpha,\textstyle{1 \over 2}+\alpha] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\,\lt \, c_0=\textstyle{1 \over 2} -a_0]
[C \sim A_1]     [x,0,\textstyle{1 \over 2}]: [\textstyle{1 \over 2}-a_0=c_0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[J \quad \alpha, \alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\, Z\,A_0\,G_0\,T]]
[JA \quad {-\alpha}, -\alpha+\beta, \beta] ex [4\quad c \quad .m.] [x,0,z]: [[\Gamma\,T\,G_2\,A_2\,Z_2]]
[K \quad \textstyle{1 \over 2}+\alpha, \alpha+\beta, \textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,H_0\,C_0]]
[K \sim J_1]     [x,0,z]: [[Y_4\,G_2\,A_2]]
[KA \quad \textstyle{1 \over 2}-\alpha, -\alpha+\beta, \textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,C_0\,H_2]]
[KA \sim J_3]     [x,0,z]: [[Y_2\,G_0\,A_0]]
[A \cup A_1 \cup J\cup J_3 \cup\Sigma\cup JA \cup J_1]   [4 \quad c \quad .m.] [x,0,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]; [0\,\lt \, z\leq\textstyle{1 \over 2}]
[\Delta\quad \alpha, 0, \alpha] ex [4\quad d \quad m..] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[B \quad \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}, \alpha] ex [4 \quad d \quad m..] [0,y,\textstyle{1 \over 2}]: [0\,\lt \, y\,\lt \, b_0]
[D \quad \alpha, \textstyle{1 \over 2}, \textstyle{1 \over 2}+\alpha] ex [4\quad d\quad m..] [\textstyle{1 \over 2},y,0]: [0\,\lt \, y\leq d_0]
[D \sim B_1]     [0,y,\textstyle{1 \over 2}]: [\textstyle{1 \over 2}-d_0=b_0\leq y\,\lt \, \textstyle{1 \over 2}]
[E \quad \alpha+\beta, \alpha, \beta] ex [4\quad d \quad m..] [0,y,z]: [[\Gamma\,Y\,H_0\,B_0\,Z]]
[EA \quad {-\alpha}+\beta, -\alpha, \beta] ex [4 \quad d \quad m..] [0,y,z]: [[\Gamma\,Z_2\,B_2\,H_2\,Y]]
[F\quad \alpha+\beta, \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}+\beta] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,z]: [[T\,D_0\,G_0]]
[F \sim E_3]     [0,y,z]: [[B_2\,T_4\,H_2]]
[FA \quad {-\alpha}+\beta, \textstyle{1 \over 2}-\alpha, \textstyle{1 \over 2}+\beta] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,z]: [[T\,G_2\,D_0]]
[FA \sim E_1]     [0,y,z]: [[T_2\,B_0\,H_0]]
[\Delta \cup B \cup B_1 \cup E \cup E_1 \cup EA \cup E_3]   [4\quad d \quad m..] [0,y,z]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]; [-\textstyle{1 \over 2}\,\lt \, z\leq\textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [8 \quad e \quad 1] [x,y,z]: [0\,\lt \, x,y\,\lt \, \textstyle{1 \over 2}]; [0 \,\lt \, z \leq \textstyle{1 \over 2}]

Table 1.5.5.6| top | pdf |
List of k-vector types for arithmetic crystal class [mm2F]: [c^{-2}>a^{-2}+b^{-2}]

See Fig. 1.5.5.6[link]. Parameter relations: [x=-\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [ y=\textstyle{1 \over 2} \alpha-\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [z=] [\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta-\textstyle{1 \over 2}\gamma].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0] ex [2 \quad a \quad mm2] [0,0,0]
[Z \quad \textstyle{1 \over 2},\textstyle{1 \over 2} ,1] ex [2 \quad a \quad mm2] [\textstyle{1 \over 2},\textstyle{1 \over 2},0]
[Z \sim Z_2]     [0,0,\textstyle{1 \over 2}]
[\Lambda \quad \alpha, \alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [0\,\lt \, z\leq\lambda_0]
[LE \quad {-\alpha}, -\alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [\lambda_2=-\lambda_0 \,\lt \, z\,\lt \, 0]
[Q \quad \textstyle{1 \over 2}+\alpha,\textstyle{1 \over 2}+ \alpha, 1] ex [2 \quad a \quad mm2] [\textstyle{1 \over 2},\textstyle{1 \over 2},z]: [0\,\lt \, z\leq q_0]
[Q \sim \Lambda_3=[\Lambda_2\,Z_4]]     [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \leq -\textstyle{1 \over 2} + q_0 = -\lambda_0]
[QA \quad \textstyle{1 \over 2}-\alpha,\textstyle{1 \over 2}- \alpha, 1] ex [2 \quad a \quad mm2] [\textstyle{1 \over 2},\textstyle{1 \over 2},z]: [q_2= -q_0\,\lt \, z\,\lt \, 0]
[QA\sim \Lambda_1=[\Lambda_0\,Z_2]]     [0,0,z]: [\textstyle{1 \over 2}-q_0=\lambda_0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[Z_2 \cup \Lambda_1 \cup \Lambda \cup \Gamma \cup LE \cup \Lambda_3]   [2 \quad a \quad mm2] [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \leq \textstyle{1 \over 2}]
[T\quad 0,\textstyle{1 \over 2},\textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,0]
[T\sim T_2]     [0,\textstyle{1 \over 2},\textstyle{1 \over 2}]
[Y \quad \textstyle{1 \over 2}, 0, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},0]
[G \quad \alpha, \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,z]: [0\,\lt \, z\leq g_0]
[G \sim H_3=[H_2\,T_4]]     [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \leq -\textstyle{1 \over 2}+g_0]
[GA \quad {-\alpha}, \textstyle{1 \over 2}-\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,z]: [g_2=-g_0\,\lt \, z\,\lt \, 0]
[GA \sim H_1=[H_0\,T_2]]     [0,\textstyle{1 \over 2},z]: [\textstyle{1 \over 2}-g_0=h_0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[H \quad \textstyle{1 \over 2}+\alpha, \alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [0\,\lt \, z\leq h_0]
[HA \quad \textstyle{1 \over 2}-\alpha, -\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [h_2=-h_0\,\lt \, z\,\lt \, 0]
[T_2 \cup H_1 \cup H\cup Y \cup HA \cup H_3]   [2\quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2}\,\lt \, z \leq \textstyle{1 \over 2}]
[\Sigma \quad 0, \alpha, \alpha] ex [4 \quad c \quad .m.] [x,0,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[C \quad \textstyle{1 \over 2}, \alpha, \textstyle{1 \over 2}+\alpha] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[C \sim A=[Z_2\,Y_2]]     [x,0,\textstyle{1 \over 2}]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[J \quad \alpha, \alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\, \Lambda_0\,G_0\,T]]
[JA \quad {-\alpha}, -\alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\,T\,G_2\,\Lambda_2]]
[K \quad \textstyle{1 \over 2}+\alpha, \alpha+\beta, \textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,H_0\,Q_0\,Z]]
[K \sim J_3]     [x, 0 ,z]: [[Y_4\,G_2\,\Lambda_2\,Z_4]]
[KA \quad \textstyle{1 \over 2}-\alpha, -\alpha+\beta, \textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Z\,Q_2\,H_2\,Y]]
[KA \sim J_1]     [x,0,z]: [[Z_2\,Y_2\,G_0\,\Lambda_0]]
[A \cup J_1 \cup J \cup \Sigma \cup JA \cup J_3]   [4 \quad c \quad .m.] [x,0,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]; [-\textstyle{1 \over 2} \,\lt \, z \leq \textstyle{1 \over 2}]
[\Delta\quad\alpha, 0, \alpha] ex [4 \quad d\quad m..] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[D\quad \alpha, \textstyle{1 \over 2}, \textstyle{1 \over 2}+\alpha] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[D \sim B]     [0,y, \textstyle{1 \over 2}]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[E \quad \alpha+\beta, \alpha, \beta] ex [4 \quad d \quad m..] [0,y,z]: [[\Gamma\,Y\,H_0\,\Lambda_0]]
[EA \quad {-\alpha}+\beta, -\alpha, \beta] ex [4 \quad d \quad m..] [0,y,z]: [[\Gamma\,\Lambda_2\,H_2\,Y]]
[F \quad \alpha+\beta, \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}+\beta] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,z]: [[T\,Z\,Q_0\,G_0]]
[F \sim E_3]     [0,y,z]: [[Z_4\,\Lambda_2\,H_2\,T_4]]
[FA \quad {-\alpha}+\beta, \textstyle{1 \over 2}-\alpha, \textstyle{1 \over 2}+\beta] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,z]: [[T\,G_2\,Q_2\,Z]]
[FA \sim E_1]     [0,y,z]: [[Z_2\,\Lambda_0\,H_0\,T_2]]
[B \cup E_1 \cup E \cup \Delta \cup EA \cup E_3]   [4 \quad d \quad m..] [0,y,z]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]; [-\textstyle{1 \over 2}\,\lt \, z\leq\textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [8 \quad e \quad 1] [x,y,z]: [0\,\lt \, x,y\,\lt \, \textstyle{1 \over 2}]; [0\,\lt \, z \leq \textstyle{1 \over 2}]

Table 1.5.5.7| top | pdf |
List of k-vector types for arithmetic crystal class [mm2F]: [a^{-2}\,\gt \, b^{-2}+c^{-2}]

See Fig. 1.5.5.7[link]. Parameter relations: [x=-\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [y=\textstyle{1 \over 2} \alpha-\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [z=\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta-\textstyle{1 \over 2}\gamma].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0] ex [2 \quad a \quad mm2] [0,0,0]
[Z \quad \textstyle{1 \over 2},\textstyle{1 \over 2} ,0] ex [2 \quad a \quad mm2] [0,0,\textstyle{1 \over 2}]
[\Lambda \quad \alpha, \alpha, 0] ex [2 \quad a\quad mm2] [0,0,z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[LE \quad {-\alpha}, -\alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [-\textstyle{1 \over 2}\,\lt \, z\,\lt \, 0]
[\Gamma\cup Z\cup\Lambda\cup LE]   [2 \quad a\quad mm2] [0,0,z]: [-\textstyle{1 \over 2}\,\lt \, z\leq\textstyle{1 \over 2}]
[T\quad 1,\textstyle{1 \over 2},\textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},\textstyle{1 \over 2}]
[Y \quad \textstyle{1 \over 2}, 0, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},0]
[H \quad \textstyle{1 \over 2}+\alpha, \alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[HA \quad \textstyle{1 \over 2}-\alpha, -\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2}\,\lt \, z\,\lt \, 0]
[T \cup Y \cup H \cup HA]   [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2}\,\lt \, z\leq \textstyle{1 \over 2}]
[\Sigma \quad 0,\alpha,\alpha] ex [4 \quad c \quad .m.] [x,0,0]: [0\,\lt \, x\leq\sigma_0]
[U \quad 1,\textstyle{1 \over 2}+\alpha,\textstyle{1 \over 2}+\alpha] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},\textstyle{1 \over 2}]: [0\,\lt \, x\,\lt \, u_0]
[U \sim \Sigma_1=[\Sigma_0\,T_2]]     [x,0,0]: [\textstyle{1 \over 2}-u_0=\sigma_0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[A \quad \textstyle{1 \over 2}, \textstyle{1 \over 2}+\alpha, \alpha] ex [4\quad c \quad .m.] [x,0,\textstyle{1 \over 2}]: [0\,\lt \, x\,\lt \, a_0]
[C \quad\textstyle{1 \over 2}, \alpha, \textstyle{1 \over 2}+\alpha] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\leq c_0]
[C \sim A_1=[A_0\,Y_2]]     [x,0,\textstyle{1 \over 2}]: [a_0=\textstyle{1 \over 2} -c_0\leq x\,\lt \, \textstyle{1 \over 2}]
[J \quad \alpha, \alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\,Z\,A_0\,\Sigma_0]]
[JA \quad {-\alpha}, -\alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\,\Sigma_0\,A_2\,Z_4]]
[K \quad \textstyle{1 \over 2}+\alpha,\alpha+\beta,\textstyle{1 \over 2}+\beta] ex [4\quad c\quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,T\,U_0\,C_0]]
[K\sim J_3]     [x,0,z]: [[T_2\,\Sigma_0\,A_2\,Y_4]]
[KA \quad \textstyle{1 \over 2}-\alpha,-\alpha+\beta,\textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,C_0\,U_2\,T_4]]
[KA\sim J_1]     [x,0,z]: [[T_2\,\Sigma_0\,A_0\,Y_2]]
[A \cup A_1 \cup J \cup J_1 \cup \Sigma\cup\Sigma_1 \cup JA \cup J_3]   [4 \quad c \quad .m.] [x,0,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}, -\textstyle{1 \over 2} \,\lt \, z\leq \textstyle{1 \over 2}]
[\Delta\quad \alpha, 0, \alpha] ex [4 \quad d \quad m..] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[B \quad \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}, \alpha] ex [4 \quad d \quad m..] [0,y,\textstyle{1 \over 2}]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[E \quad \alpha+\beta, \alpha, \beta] ex [4 \quad d \quad m..] [0,y,z]: [0\,\lt \, y,z\,\lt \, \textstyle{1 \over 2}]
[EA \quad {-\alpha+\beta}, -\alpha, \beta] ex [4\quad d \quad m..] [0,y,z]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}, -\textstyle{1 \over 2}\,\lt \, z\,\lt \, 0]
[\Delta\cup B\cup E\cup EA]   [4 \quad d \quad m..] [0,y,z:] [0\,\lt \, y\,\lt \, \textstyle{1 \over 2},-\textstyle{1 \over 2}\,\lt \, z\leq \textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [8 \quad e \quad 1] [x,y,z]: [0\,\lt \, x,y\,\lt \, \textstyle{1 \over 2}, 0\leq z\,\lt \, \textstyle{1 \over 2}]
[Figure 1.5.5.5]

Figure 1.5.5.5 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [mm2F]: [a^{-2} \,\lt \, b^{-2}+c^{-2}], [b^{-2} \,\lt \, c^{-2}+a^{-2}] and [c^{-2} \,\lt \, a^{-2}+b^{-2}]. Space groups [Fmm2-C_{2v}^{18}] (42), [Fdd2-C_{2v}^{19}] (43). Reciprocal-space group ([Imm2])*, No. 44: [a^{*2} \,\lt \, b^{*2}+c^{*2}], [b^{*2} \,\lt \, c^{*2}+a^{*2}] and [c^{*2} \,\lt \, a^{*2}+b^{*2}] (see Table 1.5.5.5[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T4: [0,\textstyle{1 \over 2},-\textstyle{1 \over 2}]; Y2: [\textstyle{1 \over 2},0,\textstyle{1 \over 2}]; Y4: [\textstyle{1 \over 2},0,-\textstyle{1 \over 2}]; Z2: [0,0, -\textstyle{1 \over 2}]. Flagpoles: [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]. Wings: [x,0,z]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 2}, -\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2}, -\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0].

[Figure 1.5.5.6]

Figure 1.5.5.6 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [mm2F]: [c^{-2}\,\gt \, a^{-2}+b^{-2}]. Space groups [Fmm2-C_{2v}^{18}] (42), [Fdd2-C_{2v}^{19}] (43). Reciprocal-space group ([Imm2])*, No. 44: [c^{*2}\,\gt \, a^{*2}+b^{*2}] (see Table 1.5.5.6[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T4: [0,\textstyle{1 \over 2},-\textstyle{1 \over 2}]; Y4: [\textstyle{1 \over 2},0,-\textstyle{1 \over 2}]; Z4: [0,0,-\textstyle{1 \over 2}]. Flagpoles: [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]. Wings: [x,0,z]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 2}], [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2}], [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0].

[Figure 1.5.5.7]

Figure 1.5.5.7 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [mm2F]: [a^{-2}\,\gt \, b^{-2}+c^{-2}]. Space groups [Fmm2-C_{2v}^{18}] (42), [Fdd2-C_{2v}^{19}] (43). Reciprocal-space group ([Imm2])*, No. 44: [a^{*2}\,\gt \, b^{*2}+c^{*2}] (see Table 1.5.5.7[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T4: [0,\textstyle{1 \over 2},-\textstyle{1 \over 2}]; Y4: [\textstyle{1 \over 2},0,-\textstyle{1 \over 2}]; Z4: [0,0,-\textstyle{1 \over 2}]. Flagpoles: [LE=[Z_4\,\Gamma] \quad 0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [HA=[T_4\,Y] \quad 0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]. Wings: [JA \cup J_3] [=] [[\Gamma\,T_2\,Y_4\,Z_4]] [x,0,z]: [ 0 \,\lt \, x \,\lt \, \textstyle{1 \over 2},-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [EA=[\Gamma\,Z_4\,T_4\,Y] \quad 0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2},-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0].

1.5.5.4. Discussion

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1.5.5.4.1. Representation domains and asymmetric units

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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

  • (1) Arithmetic crystal class [4/mmmI]. The parameter ranges for the special lines and planes of the asymmetric unit and for general k vectors of the reciprocal-space group [(F4/mmm)^*] [setting [(I4/mmm)^*]] are listed in Tables 1.5.5.3[link] and 1.5.5.4[link]. One can describe the corresponding conditions of the representation domain by the boundary plane [x,y,z] = [\{1+( c/ a)^2[1–2(x+ y)]\}/4] which for [c/a \,\lt \, 1] forms the triangle [[Z_0\,Z_2\,P]] in Fig. 1.5.5.3[link] but for [c/a\,\gt \, 1] the pentagon [[S_2\,R\,P\,G\,S]] in Fig. 1.5.5.4[link]. The inner points of this boundary plane are points of the general position GP with the exception of the line [Q = x,\textstyle{1 \over 2}-x,\textstyle{1 \over 4}], which is a twofold rotation axis. The boundary conditions for the representation domain depend on [c/a]; they are much more complicated than those, [x,y,z=\textstyle{1 \over 4}], for the asymmetric unit.

  • (2) Arithmetic crystal class [mm2F], see Figs. 1.5.5.5[link] to 1.5.5.7[link][link]. In the reciprocal-space group [(Imm2)^*] the lines [\Lambda] and LE belong to Wintgen position [2\ a\ mm2], as do the lines [Q,QA,\Lambda_1] and [\Lambda_3] if present. The lines [H] and [HA] belong to the Wintgen position [2\ b\ mm2]; as do the lines [G, GA,H_1] and [H_3] if present. The lines [\Sigma], [\Sigma_1], A, A1, C and U belong to the plane [x,0,z]; the lines [\Delta], B, B1 and D belong to the plane [0,y,z]. The decisive boundary plane of the representation domain is [x a^{*2} + yb^{*2} + zc^{*2} = d^{*2}/4], where [d^{*2} = a^{*2} + b^{*2} + c^{*2}]; it is a hexagon for Fig. 1.5.5.5[link] and a parallelogram for Figs. 1.5.5.6[link] and 1.5.5.7[link]. There is no relation of the lattice parameters for which all the above-mentioned lines are realized on the surface of the representation domain simultaneously; either two or three of them do not appear and the length of the others depends on the boundary plane, see Tables and Figs. 1.5.5.5 to 1.5.5.7.

    The boundary conditions for the asymmetric unit are independent of the lattice parameters and the boundary plane is always represented by the simple equation [x,y,\textstyle{1 \over 2}]: [0 \,\lt \, x,y \,\lt \, \textstyle{1 \over 2}]. By introducing flagpoles and wings, the description may become uni-arm.

1.5.5.4.2. Splitting of k-vector types

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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 k-vector types are split and that the different parts belong to different arms and to different stars of k vectors. Sometimes this splitting of k-vector types may be avoided by an appropriate choice of the asymmetric unit; sometimes the introduction of flagpoles and wings is necessary to make the k-vector types uni-arm.

Examples

  • (1) In the reciprocal-space group [({\cal G})^*=(Fm\overline{3}m)^*], No. 225 of the arithmetic crystal class [m\overline{3}mI] there are the lines of k vectors Λ (α, α, α) and F [(\textstyle{1 \over 2}-\alpha, -\textstyle{1 \over 2}+3\alpha, \textstyle{1 \over 2}-\alpha)] of CDML, p. 41. By Figure 1.5.5.1[link] one sees that the line Λ connects the points Γ and P, the line F connects the points P and H. One takes from the corresponding Table 1.5.5.1[link] the coefficients of [P=\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}] and [H=0,\textstyle{1 \over 2},0]. From these points or from the transformation listed at the top of Table 1.5.5.1[link] as `Parameter relations' the coefficients of the line F are obtained as [F=x,\textstyle{1 \over 2}-x,x\semi 0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}].

    The inspection of the symmetry diagram of [Fm\overline{3}m], No. 225, in IT A shows that a twofold rotation [\sf 2] (represented by the [{\sf{4}_2}] [\textstyle{1 \over 4},y,\textstyle{1 \over 4}] screw-rotation axis) leaves the point P invariant, whereas the point H is mapped onto the point R [\textstyle{1 \over 2}, \textstyle{1 \over 2}, \textstyle{1 \over 2}]. More formally: the rotation is described by [x,\textstyle{1 \over 2}-x,x \ \rightarrow] [\textstyle{1 \over 2}-x,\textstyle{1 \over 2}-x,\textstyle{1 \over 2}-x], where [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}]. The result is the line [F_1=[R\,P]]. It is uni-arm to the line Λ = x, x, x and the union [\Lambda \cup F_1] forms the Wintgen position [32\ f\ 3m]. An analoguous result is obtained for the same lines in the arithmetic crystal class [m\overline{3}I].

  • (2) In the following example the splitting of a Wintgen position happens if a representation domain of the Brillouin zone is chosen. The splitting can be avoided by the choice of the asymmetric unit. We consider the plane [x,y,0] in the arithmetic crystal class [4/mmmI], see Fig. 1.5.5.4[link] and Table 1.5.5.4[link]. In CDML this plane is split into the parts [C=[\Gamma\, S_2\, R\, X]] and [D=[M\, S\, G]\sim [M_2\, S_2\, R]]. By the choice of the asymmetric unit the independent region of the Wintgen position is uni-arm: [[\Gamma M_2 X]=16\ l\ m..] [x,y,0]: [0 \,\lt \, x \,\lt \, y \,\lt \, \textstyle{1 \over 2}].

  • (3) The splitting of a Wintgen position can be avoided if flagpoles and wings are admitted, i.e. if the minimal domain is described by a non-convex body. If one chooses in the first example of the arithmetic crystal classes [m\overline{3}mI] and [m\overline{3}I] the union [\Lambda\cup F_1] for the line [x,x,x], then [F_1=[P\,R]] forms a flagpole, whereas Λ forms an edge of the asymmetric unit, see Figs. 1.5.5.1[link] and 1.5.5.2[link].

    The same holds for the Wintgen position [96\ k\ ..m\quad x,x,z] of [m\overline{3}mI]. In the representation domain which is simultaneously the asymmetric unit, this Wintgen position is split into three parts B, C and J, which form three of the four walls of the (tetrahedral) minimal domain. By proper symmetry operations these three parts can be made uni-arm to the part C, such that their union [C\cup B_1\cup J_1] describes the independent part of that Wintgen position, see Fig. 1.5.5.1[link]. The part C forms a wall of the asymmetric unit; the part [B_1\cup J_1] forms a wing, see Fig. 1.5.5.1[link].

1.5.5.4.3. k-vector types for non-holosymmetric space groups

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The k-vector labels of CDML are primarily listed for the holosymmetric space groups. These lists are kept and supplemented for the non-holosymmetric space groups. In this way many superfluous k-vector labels are introduced.

Examples

  • (1) Arithmetic crystal class [m\overline{3}I]. In its reciprocal-space group [(Fm\overline{3})^*], the introduction of the plane [AA = [\Gamma \,H_2\,N]] is unnecessary because the plane [A = [\Gamma \,N\,H]] of Wintgen position 96 [j\ m..] of [(Fm\overline{3}m)^*] can be extended to [A \cup AA = [\Gamma \,H_2\,H]] in the reciprocal-space group [(Fm\overline{3})^*], cf. Fig. 1.5.5.2[link] and Table 1.5.5.2[link]. In [(Fm\overline{3})^*], both planes, A and AA, belong to Wintgen position [48\ h\ m..]. The parameter description is extended from [x,y,0]: [0 \,\lt \, x \,\lt \, y \,\lt \, \textstyle{1 \over 2} - x( \lt \,\textstyle{1 \over 4})] to [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2} - x \,\lt \, \textstyle{1 \over 2}].

  • (2) In the previous example, during the transition from the group [(Fm\overline{3}m)^*] to the subgroup [(Fm\overline{3})^*] the order of the little co-group of the special k vectors of [(Fm\overline{3}m)^*] was not changed. In other cases, the little co-group may be reduced to a subgroup. Such k vectors may then be incorporated into a more general Wintgen position and described by an extension of the parameter range.

    Arithmetic crystal class [m\overline{3}mI], plane [[\Gamma \,H \,N] = x, y,0]. In [(Fm\overline{3}m)^*], see Fig. 1.5.5.1[link], all points (Γ, H, N) and lines (Δ, Σ, G) of the boundary of the asymmetric unit are special. In [(Fm\overline{3})^*], see Fig. 1.5.5.2[link], the lines Δ and [[\Gamma\,H_2]\sim\Delta] are special but Σ, G and [[N\,H_2]\sim G] belong to the plane ([A \cup AA]). The free parameter range on the line G is [0\,\lt\,y\,\lt\,\textstyle{1 \over 4}]. Therefore, the parameter ranges of ([A \cup AA\cup G \cup\Sigma]) in [x,y,0] can be taken as: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 2} - y \,\lt \, \textstyle{1 \over 2}] for [A \cup AA \cup \Sigma] and [0 \,\lt \, y = \textstyle{1 \over 2} - x \,\lt \, \textstyle{1 \over 4}] for G.

1.5.5.4.4. Ranges of independent parameters

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

  • (1) Arithmetic crystal class [m\overline{3}mI], line [\Lambda \cup F_1]: In the reciprocal-space group [(Fm\overline{3}m)^*] of the arithmetic crystal class [m\overline{3}mI], the line [x,x,x] has stabilizer [\overline{3}m] and little co-group [\overline{{\cal G}}^{\bf k} = 3m]. Therefore, the divisor is 12:6 = 2 and [x] is running from 0 to [\textstyle{1 \over 2}].

    The same result holds for the line [\Lambda \cup F_1] in the reciprocal-space group [(Fm\overline{3})^*] of the arithmetic crystal class [m\overline{3}I]: the stabilizer generated by [\overline{\sf 3}] is of order 6, [|\overline{{\cal G}}\,^{\bf k}|= |\{{\sf 3}\}| = 3], the quotient is again [\textstyle{1 \over 2}], the parameter range is the same as for [(Fm\overline{3}m)^*].

  • (2) Arithmetic crystal class [m\overline{3}mI], plane [B_1\cup C \cup J_1]: In [(Fm\overline{3}m)^*], the stabilizer of [x,x,z] is generated by [m.mm] and the centring translation [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)] mod (integer translations). They generate a group of order 16; [\overline{{\cal G}}\,^{\bf k}] is [..m] of order 2. The fraction of the plane is [\textstyle{2 \over 16} = \textstyle{1 \over 8}] of the area [\sqrt{2}a^{*2}] in the (centred) unit cell, as expressed by the parameter ranges [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}, 0 \,\lt \, z \,\lt \, \textstyle{1 \over 2}]. There are six arms of the star of [x,x,z]: [x,x,z]; [\overline{x},x,z]; [x,y,x]; [x,y,\overline{x}]; [x,y,y]; [x,\overline{y},y]. Three of them ([x,x,z], [\overline{x},x,z] and [x,y,x]) are represented in the boundaries of the representation domain: C = [Γ N P], B = [H N P] and J = [Γ H P], see Fig. 1.5.5.1[link]. The areas of their parameter ranges are [\textstyle{1\over 32}, \textstyle{1 \over 32}] and [\textstyle{1\over 16}], respectively; the sum is [\textstyle{1 \over 8}].

    Arithmetic crystal class [m\overline{3}I], the same result holds in the reciprocal-space group [(Fm\overline{3})^*]. The stabilizer generated by [2/m..] and by the centring translation [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)] mod (integer translations) forms a group of order 8; the order of the little co-group [|\overline{{\cal G}}\,^{\bf k}|= |\{{\sf 1}\}| = 1]. The quotient is again [\textstyle{1\over 8}], the parameter range is the same as for [(Fm\overline{3}m)^*] but the plane belongs to the general position GP because the little co-group is trivial.

  • (3) Arithmetic crystal class [m\overline{3}mI], reciprocal-space group [(Fm\overline{3}m)^*], plane [x,y,0]: the stabilizer of the plane A is generated by [4/mmm] and [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)], order 32, [\overline{{\cal G}}\,^{\bf k}] (site-symmetry group) [m..], order 2. Consequently, [Γ H N] is [\textstyle{1\over 16}] of the unit square [a^{*2}]: [0 \,\lt \, x \,\lt \, y \,\lt \, \textstyle{1 \over 2}-x]. In [(Fm\overline{3})^*], the stabilizer of [x,y,0], here [A \cup AA], is [mmm.] and [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)], order 16, with the same group [\overline{{\cal G}}\,^{\bf k}=m..] of order 2. Therefore, [Γ H2 H] is [\textstyle{1 \over 8}] of the unit square [a^{*2}] in [(Fm\overline{3})^*]; [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2} - x \,\lt \, \textstyle{1 \over 2}].

  • (4) Arithmetic crystal class [m\overline{3}mI], line [x,x,0]: In [(Fm\overline{3}m)^*] the stabilizer is generated by [m.mm] and [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)] mod (integer translations), order 16, [\overline{{\cal G}}\,^{\bf k}] is [m.2m] of order 4. The divisor is 4 and thus [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}]. In [(Fm \overline{3})^*] the stabilizer is generated by [2/m..] and [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)] mod (integer translations), order 8, and [\overline{{\cal G}}\,^{\bf k} = m.. ], order 2; the divisor is 4 again and [0 \,\lt \, x \,\lt\, \textstyle{1 \over 4}] is restricted to the same range.

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:

  • (1) Arithmetic crystal classes [m\overline{3}mI] and [m\overline{3}I], i.e. [(Fm \overline{3} m)^*] and [(Fm \overline{3})^*], line [x,x,x]: The points [0,0,0]; [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 2}] (and [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]) are special points; the parameter ranges are open: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}], [\textstyle{1 \over 4} \,\lt \, x \,\lt \, \textstyle{1 \over 2}].

  • (2) Arithmetic crystal class [m\overline{3}mI], plane [x,x,z]: In [(Fm\overline{3}m)^*] all corners Γ, N, N2, H2 and all edges are special points or lines. Therefore, the parameter ranges are open: [x,x,z]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}, 0 \,\lt \, z \,\lt \, \textstyle{1 \over 2}], where the lines [x,x,x]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] and [x, x, \textstyle{1 \over 2} -x]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] are special lines and thus excepted.

  • (3) Arithmetic crystal classes [m\overline{3}mI] and [m\overline{3}I], plane [x,y,0]: In both reciprocal-space groups, [(Fm\overline{3}m)^*] and [(Fm\overline{3})^*], [0 \,\lt \, x] and [0 \,\lt \, y] holds. The line [0,y,0 = \Delta] is a special line, its k vectors have little co-groups of higher order than that of the planes [x,y,0] and the boundaries of both planes are open. The same holds for the boundary [x,0,0 \sim 0,y,0] for [(Fm\overline{3})^*]. The k vectors of the lines [x,x,0] and [x,\textstyle{1 \over 2}-x,0], Σ and G, also have little co-groups of higher order and belong to other Wintgen positions in the representation domain (or asymmetric unit) of [(Fm\overline{3}m)^*]. Therefore, for the arithmetic crystal class [m\overline{3}mI], the plane [A = x,y,0] is open at its boundaries [x,x,0] and [x,\textstyle{1 \over 2}-x,0] in the range [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}]. In the asymmetric unit of [(Fm\overline{3})^*] the lines [x,x,0]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] and [x,\textstyle{1 \over 2}-x,0]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] belong to the plane, and the boundary of the plane A is here closed. The boundary line [x,\textstyle{1 \over 2}-x,0]: [\textstyle{1 \over 4} \,\lt \, x \,\lt \, \textstyle{1 \over 2}] of the plane AA is equivalent to the range [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] of the part A and thus does not belong to the asymmetric unit; here the boundary of the plane [A\cup AA] is open.

References

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








































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