Trigonal and hexagonal systems

Bertaut, E. F.

doi:10.1107/97809553602060000509

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 4.3, pp. 73-75

Section 4.3.5. Trigonal and hexagonal systems

E. F. Bertaut^a ^‡

^a Laboratoire de Cristallographie, CNRS, Grenoble, France

4.3.5. Trigonal and hexagonal systems

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The trigonal and hexagonal crystal systems are considered together, because they form the hexagonal `crystal family', as explained in Chapter 2.1 . Hexagonal lattices occur in both systems, whereas rhombohedral lattices occur only in the trigonal system.

4.3.5.1. Historical note

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The 1935 edition of International Tables contains the symbols C and H for the hexagonal lattice and R for the rhombohedral lattice. C recalls that the hexagonal lattice can be described by a double rectangular C-centred cell (orthohexagonal axes); H was used for a hexagonal triple cell (see below); R designates the rhombohedral lattice and is used for both the rhombohedral description (primitive cell) and the hexagonal description (triple cell).

In the 1952 edition the following changes took place (cf. pages x, 51 and 544 of IT 1952): The lattice symbol C was replaced by P for reasons of consistency; the H description was dropped. The symbol R was kept for both descriptions, rhombohedral and hexagonal. The tertiary symmetry element in the short Hermann–Mauguin symbols of class 622, which was omitted in IT (1935), was re-established.

In the present volume, the use of P and R is the same as in IT (1952). The H cell, however, reappears in the sub- and supergroup data of Part 7 and in Table 4.3.2.1 of this section, where short symbols for the H description of trigonal and hexagonal space groups are given. The C cell reappears in the subgroup data for all trigonal and hexagonal space groups having symmetry elements orthogonal to the main axis.

4.3.5.2. Primitive cells

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The primitive cells of the hexagonal and the rhombohedral lattice, hP and hR, are defined in Table 2.1.2.1 In Part 7 , the `rhombohedral' description of the hR lattice is designated by `rhombohedral axes'; cf. Chapter 1.2 .

4.3.5.3. Multiple cells

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Multiple cells are frequently used to describe both the hexagonal and the rhombohedral lattice.

(i) The triple hexagonal R cell; cf. Chapters 1.2and 2.1

When the lattice is rhombohedral hR (primitive cell a, b, c), the triple R cell $[{\bf a}',{\bf b}',{\bf c}']$ corresponds to the `hexagonal description' of the rhombohedral lattice. There are three right-handed obverse R cells: $[\matrix{R_{1}: \quad {\bf a}' = {\bf a} - {\bf b}; &{\bf b}' = {\bf b} - {\bf c}; &{\bf c}' = {\bf a} + {\bf b} + {\bf c};\cr R_{2}: \quad {\bf a}' = {\bf b} - {\bf c}; &{\bf b}' = {\bf c} - {\bf a}; &{\bf c}' = {\bf a} + {\bf b} + {\bf c};\cr R_{3}: \quad {\bf a}' = {\bf c} - {\bf a}; &{\bf b}' = {\bf a} - {\bf b}; &{\bf c}' = {\bf a} + {\bf b} + {\bf c}.\cr}]$ Three further right-handed R cells are obtained by changing $[{\bf a}']$ and $[{\bf b}']$ to $[- {\bf a}']$ and $[- {\bf b}']$ , i.e. by a $[180^{\circ}]$ rotation around $[{\bf c}']$ . These cells are reverse. The transformations between the triple R cells and the primitive rhombohedral cell are given in Table 5.1.3.1 and Fig. 5.1.3.6 .

The obverse triple R cell has `centring points' at $[0,0,0; \quad {\textstyle{2 \over 3}, {1 \over 3}, {1 \over 3}; \quad {1 \over 3}, {2 \over 3}, {2 \over 3}},]$ whereas the reverse R cell has `centring points' at $[0,0,0; \quad {\textstyle{1 \over 3}, {2 \over 3}, {1 \over 3}; \quad {2 \over 3}, {1 \over 3}, {2 \over 3}}.]$ In the space-group tables of Part 7 , the obverse $[R_{1}]$ cell is used, as illustrated in Fig. 2.2.6.9 . This `hexagonal description' is designated by `hexagonal axes'.
(ii) The triple rhombohedral D cell

Parallel to the `hexagonal description of the rhombohedral lattice' there exists a `rhombohedral description of the hexagonal lattice'. Six right-handed rhombohedral cells (here denoted by D) with cell vectors $[{\bf a}',{\bf b}',{\bf c}']$ of equal lengths are obtained from the hexagonal P cell a, b, c by the following transformations and by cyclic permutations of $[{\bf a}',{\bf b}',{\bf c}']$ : $[\matrix{D_{1}: &{\bf a}' = \phantom{-}{\bf a} + {\bf c};\hfill &{\bf b}' = \phantom{-}{\bf b} + {\bf c};\hfill &{\bf c}' = - ({\bf a} + {\bf b}) + {\bf c}\hfill\cr D_{2}: &{\bf a}' = - {\bf a} + {\bf c}; &{\bf b}' = - {\bf b} + {\bf c}; &{\bf c}' = \phantom{-(}{\bf a} + {\bf b}\phantom{)} + {\bf c}.\hfill\cr}]$ The transformation matrices are listed in Table 5.1.3.1 . $[D_{2}]$ follows from $[D_{1}]$ by a $[180^{\circ}]$ rotation around [111]. The D cells are triple rhombohedral cells with `centring' points at $[0,0,0; \quad {\textstyle{1 \over 3}, {1 \over 3}, {1 \over 3}; \quad {2 \over 3}, {2 \over 3}, {2 \over 3}}.]$ The D cell, not used in practice and not considered explicitly in the present volume, is useful for a deeper understanding of the relations between hexagonal and rhombohedral lattices.
(iii) The triple hexagonal H cell; cf. Chapter 1.2

Generally, a hexagonal lattice hP is described by means of the smallest hexagonal P cell. An alternative description employs a larger hexagonal H-centred cell of three times the volume of the P cell; this cell was extensively used in IT (1935), see Historical note above.

There are three right-handed orientations of the H cell (basis vectors $[{\bf a}',{\bf b}',{\bf c}']$ ) with respect to the basis vectors a, b, c of the P cell: $[\matrix{H_{1}: &{\bf a}' = \phantom{2}{\bf a} - \phantom{2}{\bf b}{\hbox{;}} &{\bf b}' = \phantom{-2}{\bf a} + 2{\bf b}{\hbox{;}} &{\bf c}' = {\bf c}\hfill\cr H_{2}: \hfill&{\bf a}' = 2 {\bf a} + \phantom{2}{\bf b}{\hbox{;}} &{\bf b}' = - \phantom{2}{\bf a} + {\bf b}{\hbox{;}}\hfill &{\bf c}' = {\bf c}\hfill\cr H_{3}: &{\bf a}' = \phantom{2}{\bf a} + 2{\bf b}{\hbox{;}} &{\bf b}' = - 2 {\bf a} - {\bf b}{\hbox{;}}\hfill &{\bf c}' = {\bf c}.\hfill\cr}]$ The transformations are given in Table 5.1.3.1 and Fig. 5.1.3.8 . The new vectors $[{\bf a}']$ and $[{\bf b}']$ are rotated in the ab plane by $[- 30^{\circ} (H_{1})]$ , $[+ 30^{\circ} (H_{2})]$ , $[+ 90^{\circ} (H_{3})]$ with respect to the old vectors a and b. Three further right-handed H cells are obtained by changing $[{\bf a}']$ and $[{\bf b}']$ to $[- {\bf a}']$ and $[- {\bf b}']$ , i.e. by a rotation of $[180^{\circ}]$ around $[{\bf c}']$ .

The H cell has `centring' points at $[0,0,0; \quad {\textstyle{2 \over 3}, {1 \over 3}, 0; \quad {1 \over 3}, {2 \over 3}}, 0.]$ Secondary and tertiary symmetry elements of the P cell are interchanged in the H cell, and the general position in the H cell is easily obtained, as illustrated by the following example.

Example

The space-group symbol P3m1 in the P cell a, b, c becomes H31m in the H cell $[{\bf a}',{\bf b}',{\bf c}']$ . To obtain the general position of H31m, consider the coordinate triplets of P31m and add the centring translations $[0,0,0{\hbox{;}}\ {2 \over 3},{1 \over 3},0{\hbox{;}}\ {1 \over 3},{2 \over 3},0]$ .
(iv) The double orthohexagonal C cell

The C-centred cell which is defined by the so-called `orthohexagonal' vectors $[{\bf a}']$ , $[{\bf b}']$ , $[{\bf c}']$ has twice the volume of the P cell. There are six right-handed orientations of the C cell, which are $[C_{1}]$ , $[C_{2}]$ and $[C_{3}]$ plus three further ones obtained by changing $[{\bf a}']$ and $[{\bf b}']$ to $[-{\bf a}']$ and $[-{\bf b}']$ : $[\matrix{C_{1}: {\bf a}' = {\bf a}\hfill{\hbox{;}} &{\bf b}' = \phantom{-2}{\bf a} + 2\hfill{\bf b}{\hbox{;}}\hfill &{\bf c}' = {\bf c}\cr C_{2}: {\bf a}' = {\bf a} + {\bf b}{\hbox{;}} &{\bf b}' = -\phantom{2}{\bf a} + \phantom{2}{\bf b}{\hbox{;}}\hfill &{\bf c}' = {\bf c}\cr C_{3}: {\bf a}' = \hfill{\bf b}{\hbox{;}} &{\bf b}' = -2{\bf a} - \phantom{2}\hfill{\bf b}{\hbox{;}}\hfill &{\hbox to 2pt{}}{\bf c}' = {\bf c}.\cr}]$ Transformation matrices are given in Table 5.1.3.1 and illustrations in Fig. 5.1.3.7 . Here $[{\bf b}']$ is the long axis.

4.3.5.4. Relations between symmetry elements

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In the hexagonal crystal classes 62(2), 6m(m) and $[\bar{6}2]$ (m) or $[\bar{6}m]$ (2), where the tertiary symmetry element is between parentheses, the following products hold: $[6 \times 2 = (2) = \bar{6} \times m;\quad 6 \times m = (m) = \bar{6} \times 2]$ or $[6 \times 2 \times (2) = 6 \times m \times (m) = \bar{6} \times 2 \times (m) = \bar{6} \times m \times (2) = 1.]$ The same relations hold for the corresponding Hermann–Mauguin space-group symbols.

4.3.5.5. Additional symmetry elements

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Parallel axes 2 and $[2_{1}]$ occur perpendicular to the principal symmetry axis. Examples are space groups R32 (155), P321 (150) and P312 (149), where the screw components are $[{1 \over 2},{1 \over 2},0]$ (rhombohedral axes) or $[{1 \over 2},0,0]$ (hexagonal axes) for R32; $[{1 \over 2},0,0]$ for P321; and $[{1 \over 2},1,0]$ for P312. Hexagonal examples are P622 (177) and $[P\bar{6}2c]$ (190).

Likewise, mirror planes m parallel to the main symmetry axis alternate with glide planes, the glide components being perpendicular to the principal axis. Examples are P3m1 (156), P31m (157), R3m (160) and P6mm (183).

Glide planes c parallel to the main axis are interleaved by glide planes n. Examples are P3c1 (158), P31c (159), R3c (161, hexagonal axes), $[P\bar{6}c2]$ (188). In R3c and $[R\bar{3}c]$ , the glide component $[0,0,{1 \over 2}]$ for hexagonal axes becomes $[{1 \over 2},{1 \over 2},{1 \over 2}]$ for rhombohedral axes, i.e. the c glide changes to an n glide. Thus, if the space group is referred to rhombohedral axes, diagonal n planes alternate with diagonal a, b or c planes (cf. Section 1.4.4 ).

In R space groups, all additional symmetry elements with glide and screw components have their origin in the action of an integral lattice translation. This is also true for the axes $[3_{1}]$ and $[3_{2}]$ which appear in all R space groups (cf. Table 4.1.2.2 ). For this reason, the `rhombohedral centring' R is not included in Table 4.1.2.3 , which contains only the centrings A, B, C, I, F.

4.3.5.6. Group–subgroup relations

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4.3.5.6.1. Maximal k subgroups

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Maximal k subgroups of index [3] are obtained by `decentring' the triple cells R (hexagonal description), D and H in the trigonal system, H in the hexagonal system. Any one of the three centring points may be taken as origin of the subgroup.

(i) Trigonal system

Examples

(1) P3m1 (156) (cell a, b, c) is equivalent to H31m (a′, b′, c). Decentring of the H cell yields maximal non-isomorphic k subgroups of type P31m. Similarly, P31m (157) has maximal subgroups of type P3m1; thus, one can construct infinite chains of subgroup relations of index [3], tripling the cell at each step: $[P3m1 \rightarrow P31m \rightarrow P3m1 \ldots]$
(2) R3 (146), by decentring the triple hexagonal R cell $[{\bf a}',{\bf b}',{\bf c}']$ , yields the subgroups P3, $[P3_{1}]$ and $[P3_{2}]$ of index [3].
(3) Likewise, decentring of the triple rhombohedral cells $[D_{1}]$ and $[D_{2}]$ gives rise, for each cell, to the rhombohedral subgroups of a trigonal P group, again of index [3].

Combining (2) and (3), one may construct infinite chains of subgroup relations, tripling the cell at each step: $[P3 \rightarrow R3 \rightarrow P3 \rightarrow R3 \ldots]$ These chains illustrate best the connections between rhombohedral and hexagonal lattices.
(4) Special care must be applied when secondary or tertiary symmetry elements are present. Combining (1), (2) and (3), one has for instance: $[P31c \rightarrow R3c \rightarrow P3c1 \rightarrow P31c \rightarrow R3c \ldots]$
(5) Rhombohedral subgroups, found by decentring the triple cells $[D_{1}]$ and $[D_{2}]$ , are given under block IIb and are referred there to hexagonal axes, $[{\bf a}',{\bf b}',{\bf c}]$ as listed below. Examples are space groups P3 (143) and $[P\bar{3}1c]$ (163) $[\matrix{{\bf a}' = \phantom{2}{\bf a} - {\bf b}, &{\bf b}' = \phantom{-}{\bf a} + 2{\bf b}, &{\bf c}' = 3{\bf c};\cr {\bf a}' = 2{\bf a} + {\bf b}, &{\bf b}' = -{\bf a} + \phantom{2}{\bf b}, &{\bf c}' = 3{\bf c}.\cr}]$

(ii) Hexagonal system

Examples

(1) $[P\bar{6}2c]$ (190) is described as $[H\bar{6}c2]$ in the triple cell $[{\bf a}',{\bf b}',{\bf c}']$ ; decentring yields the non-isomorphic subgroup $[P\bar{6}c2]$ .
(2) (192) (cell a, b, c) keeps the same symbol in the H cell and, consequently, gives rise to the maximal isomorphic subgroup with cell $[{\bf a}',{\bf b}',{\bf c}']$ . An analogous result applies whenever secondary and tertiary symmetry elements in the Hermann–Mauguin symbol are the same and also to space groups of classes 6, $[\bar{6}]$ and .

4.3.5.6.2. Maximal t subgroups

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Maximal t subgroups of index [2] are read directly from the full symbol of the space groups of classes 32, 3m, $[\bar{3}m]$ , 622, 6mm, $[\bar{6}2m]$ , [6/mmm] .

Maximal t subgroups of index [3] follow from the third power of the main-axis operation. Here the C-cell description is valuable.

(i) Trigonal system

(a) Trigonal subgroups

Examples

(1) $[R\bar{3}2/c]$ (167) has R3c, R32 and $[R\bar{3}]$ as maximal t subgroups of index [2].
(2) $[P\bar{3}c1]$ (165) has P3c1, P321 and $[P\bar{3}]$ as maximal t subgroups of index [2].

(b) Orthorhombic subgroups

No orthorhombic subgroups of trigonal space groups exist, in spite of the existence of an orthohexagonal C cell.
(c) Monoclinic subgroups

All trigonal space groups with secondary or tertiary symmetry elements have monoclinic C-centred maximal t subgroups of index [3].

Example

$[P\bar{3}1c]$ (163), $[P\bar{3}c1]$ (165) and $[R\bar{3}c]$ (167) have subgroups of type .
(d) Triclinic subgroups

All trigonal space groups without secondary or tertiary symmetry elements have triclinic maximal t subgroups of index [3].

Example

$[P\bar{3}]$ (147) and $[R\bar{3}]$ (148) have subgroups $[P\bar{1}]$ .

(ii) Hexagonal system

(a) Hexagonal subgroups

Example

$[P6_{3}/m\;2/c\;2/m]$ (193) has maximal t subgroups $[P6_{3}/m]$ , $[P6_{3}22]$ , $[P6_{3}cm]$ , $[P\bar{6}2m]$ and $[P\bar{6}c2]$ of index [2].

(b) Trigonal subgroups

The second and fourth powers of sixfold operations are threefold operations; thus, all hexagonal space groups have maximal trigonal t subgroups of index [2]. In space groups of classes 622, 6mm, $[\bar{6}2m]$ , [6/mmm] with secondary and tertiary symmetry elements, trigonal t subgroups always occur in pairs.

Examples

(1) $[P6_{1}]$ (169) contains $[P3_{1}]$ of index [2].
(2) $[P\bar{6}2c]$ (190) has maximal t subgroups P321 and P31c; $[P6_{1} 22]$ (178) has subgroups $[P3_{1}21]$ and $[P3_{1}12]$ , all of index [2].
(3) $[P6_{3}/mcm]$ (193) contains the operation $[\bar{3}\;[= (6_{3})^{2} \times \bar{1}]]$ and thus has maximal t subgroups $[P\bar{3}c1]$ and $[P\bar{3}1m]$ of index [2].

The third power of the sixfold operation is a twofold operation: accordingly, maximal orthorhombic t subgroups of index [3] are derived from the C-cell description of space groups of classes 622, 6mm, $[\bar{6}2m]$ and [6/mmm] . Monoclinic P subgroups of index [3] occur in crystal classes 6, $[\bar{6}]$ and [6/m] .

Examples

(1) $[P\bar{6}2c]$ (190) becomes $[C\bar{6}2c]$ in the C cell; with $[(\bar{6})^{3} = m]$ , one obtains C2cm (sequence a, b, c) as a maximal t subgroup of index [3]. The standard symbol is Ama2.
(2) $[P6_{3}/mcm]$ (193) has maximal orthorhombic t subgroups of type Cmcm of index [3]. With the examples under (a) and (b), this exhausts all maximal t subgroups of $[P6_{3}/mcm]$ .
(3) $[P6_{1}]$ (169) has a maximal t subgroup $[P2_{1}]$ ; $[P6_{3}/m]$ (176) has $[P2_{1}/m]$ as a maximal t subgroup.

References

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]Google Scholar

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 4.3, pp. 73-75