Tables of Bravais lattices

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 915-916

Section 9.8.3.1. Tables of Bravais lattices

T. Janssen,^a A. Janner,^a A. Looijenga-Vos^b and P. M. de Wolff^c ^‡

^a Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,^bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and ^cMeermanstraat 126, 2614 AM, Delft, The Netherlands

9.8.3.1. Tables of Bravais lattices

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The (3 + 1)-dimensional lattice $[\Sigma^*]$ is determined by the three-dimensional vectors a*, b*, c* and the modulation vector q. The former three vectors give by duality a, b, and c, the external components of lattice basis vectors, and the products $[-{\bf q}\cdot{\bf a}=-\alpha]$ , $[-{\bf q}\cdot{\bf b}=-\beta]$ , and $[-{\bf q}\cdot{\bf c}=-\gamma]$ the corresponding internal components. Therefore, it is sufficient to give the arithmetic crystal class of the group $[\Gamma_E(K)]$ and the components σ_j (σ₁ = α, σ₂ = β, and σ₃ = γ) of the modulation vector q with respect to a conventional basis a*, b*, c*. The arithmetic crystal class is denoted by a modification of the symbol of the three-dimensional symmorphic space group of this class (see Chapter 1.4 ) plus an indication for the row matrix σ (having entries $[\sigma_j]$ ). In this way, one obtains the so-called one-line symbols used in Tables 9.8.3.1 and 9.8.3.2.

Table 9.8.3.1| top | pdf |
(2 + 1)- and (2 + 2)-Dimensional Bravais classes for incommensurate structures

(a) (2 + 1)-Dimensional Bravais classes. The holohedral point group K_s is given in terms of its external and internal parts, K_E, and K_I, respectively. The reflections are given by ha* + kb* + mq where q is the modulation wavevector. If the rational part q^r is not zero, there is a corresponding centring translation in three-dimensional space. The conventional basis ( $[{\bf a}_c^*]$ , $[{\bf b}_c^*]$ , qⁱ) given for the vector module M* is shown such that q^r = 0. The basis vectors are given by components with respect to the conventional basis a*, b* of the lattice Λ* of main reflections.

No.	Symbol	K_E	K_I	q	Conventional basis	Centring
Oblique
1	2p(αβ)	2	$[\bar 1]$	(αβ)	(10), (01), (αβ)
Rectangular
2	mmp(0β)	mm	$[1\bar 1]$	(0β)	(10), (01), (0β)
3	mmp( $[{{1}\over{2}}]$ β)	mm	$[1\bar 1]$	( $[{{1}\over{2}}]$ β)	( $[{{1}\over{2}}]$ 0), (01), (0β)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$
4	mmc(0β)	mm	$[1\bar 1]$	(0β)	(10), (01), (0β)	$[{{1}\over{2}}]$ $[{{1}\over{2}}]$ 0

(b) (2 + 2)-Dimensional Bravais classes. The holohedral point group K_s is given in terms of its external and internal parts, K_E and K_I, respectively. The basis of the vector module M* contains two modulation wavevectors and the reflections are given by ha* + kb* + m₁q₁ + m₂q₂. If $[{\bf q}_1^r]$ or $[{\bf q}_2^r]$ are not zero, there are corresponding centring translations in four-dimensional space. The conventional basis ( $[{\bf a}_c^*]$ , $[{\bf b}_c^*]$ , $[{\bf q}_1^i]$ , $[{\bf q}_2^i]$ ) for the vector module M* is chosen such that $[{\bf q}_1^r = {\bf q}_2^r =0]$ . The basis vectors are indicated by their components with respect to the conventional basis a*, b* of the lattice Λ* of main reflections.

No.	Symbol	K_E	K_I	q₁	q₂	Conventional basis	Centring
Oblique
1	2p(αβ, λμ)	2	2	(αβ)	(λμ)	(10), (01), (αβ), (λμ)
Rectangular
2	mmp(0β, 0μ)	mm	12	(0β)	(0μ)	(10), (01), (0β), (0μ)
3	mmp( $[{{1}\over{2}}]$ β, 0μ)	mm	12	( $[{{1}\over{2}}]$ β)	(0μ)	( $[{{1}\over{2}}]$ 0), (01), (0β), (0μ)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0
4	mmp(α0, 0μ)	mm	mm	(α0)	(0μ)	(10), (01), (α0), (0μ)
5	mmp(α $[{{1}\over{2}}]$ , 0μ)	mm	mm	(α $[{{1}\over{2}}]$ )	(0μ)	(10), (0 $[{{1}\over{2}}]$ ), (α0), (0μ)	0 $[{{1}\over{2}}{{1}\over{2}}]$ 0
6	mmp(α $[{{1}\over{2}}]$ , $[{{1}\over{2}}]$ μ)	mm	mm	(α $[{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ μ)	( $[{{1}\over{2}}]$ 0), (0 $[{{1}\over{2}}]$ ), (α0), (0μ)	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$ , 0 $[{{1}\over{2}}{{1}\over{2}}]$ 0
7	mmp(αβ)	mm	mm	(αβ)	$[(\alpha{\bar\beta})]$	(10), (01), (α0), (0β)	00 $[{{1}\over{2}}{{1}\over{2}}]$
8	mmc(0β, 0μ)	mm	12	(0β)	(0μ)	(10), (01), (0β), (0μ)	$[{{1}\over{2}}{{1}\over{2}}]$ 00
9	mmc(α0, 0μ)	mm	mm	(α0)	(0μ)	(10), (01), (α0), (0μ)	$[{{1}\over{2}}{{1}\over{2}}]$ 00
10	mmc(αβ)	mm	mm	(αβ)	$[(\alpha{\bar\beta})]$	(10), (01), (α0), (0β)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, 00 $[{{1}\over{2}}{{1}\over{2}}]$
Square
11	4p(αβ)	4	4	(αβ)	$[(\bar \beta\alpha)]$	(10), (01), (αβ), $[(\bar\beta\alpha)]$
12	4mp(α0)	4m	4m	(α0)	(0α)	(10), (01), (α0), (0α)
13	4mp(α $[{{1}\over{2}}]$ )	4m	4m	(α $[{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ α)	( $[{{1}\over{2}}{{1}\over{2}}]$ ), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ ), (γγ), ( $[\delta\bar \delta]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 00, 00 $[{{1}\over{2}}{{1}\over{2}}]$
13	4mp(α $[{{1}\over{2}}]$ )	4m	4m	(α $[{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ α)	$[\gamma]$ = (2α + 1)/4, δ = (2α − 1)/4
14	4mp(αα)	4 $[\dot m]$	4 $[\ddot m]$	(αα)	( $[\bar \alpha\alpha]$ )	(10), (01), (αα), ( $[\bar \alpha\alpha]$ )
Hexagonal
15	6p(αβ)	6	6	(αβ)	( $[\bar \beta\alpha + \beta]$ )	(10), (01), (αβ), ( $[\bar \beta\alpha + \beta]$ )
16	6mp(α0)	6m	6m	(α0)	(0α)	(10), (01), (α0), (0α)
17	6mp(αα)	6 $[\dot m]$	6 $[\ddot m]$	(αα)	( $[\bar \alpha2\alpha]$ )	(10), (01), (αα), ( $[\bar \alpha2\alpha]$ )

Table 9.8.3.2| top | pdf |
(3 + 1)-Dimensional Bravais classes for incommensurate and commensurate structures

(a) (3 + 1)-Dimensional Bravais classes for incommensurate structures. The holohedral point group K_s is given in terms of its external and internal parts, K_E and K_I, respectively. The reflections are given by ha* + kb* + lc* + mq, where q is the modulation wavevector. If the rational part q^r is not zero, there is a corresponding centring translation in four-dimensional space. A conventional basis ( $[{\bf a}_c^*]$ , $[{\bf b}_c^*]$ , $[{\bf c}_c^*]$ , qⁱ) for the vector module M* is then chosen such that q^r = 0. The basis vectors are indicated by their components with respect to the conventional basis a*, b*, c* of the lattice Λ* of main reflections. The Bravais classes can also be found in Janssen (1969) and Brown et al. (1978). The notation of the Bravais classes there is here given in the columns Ref. a and Ref. b, respectively.

No.	Symbol	K_E	K_I	q	Conventional basis	Centring translation(s)	Ref. a	Ref. b
Triclinic
1	$[\bar 1]$ P(αβγ)	$[\bar 1]$	$[\bar 1]$	(αβγ)	(100), (010), (001), (αβγ)		I P	I/I
Monoclinic
2	2/mP(αβ0)	2/m	$[\bar 11]$	(αβ0)	(100), (010), (001), (αβ0)		II P	II/I
3	2/mP(αβ $[{{1}\over{2}}]$ )	2/m	$[\bar 11]$	(αβ $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ ), (αβ0)	00 $[{{1}\over{2}}{{1}\over{2}}]$	II I	II/II
4	2/mB(αβ0)	2/m	$[\bar 11]$	(αβ0)	(100), (010), (001), (αβ0)	$[{{1}\over{2}}0{{1}\over{2}}0]$	II I	II/II
5	2/mP(00γ)	2/m	$[1\bar 1]$	(00γ)	(100), (010), (001), (00γ)		III P	III/I
6	2/mP( $[{{1}\over{2}}]$ 0γ)	2/m	$[1\bar 1]$	( $[{{1}\over{2}}]$ 0γ)	( $[{{1}\over{2}}]$ 00), (010), (001), (00γ)	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$	III I	III/II
7	2/mB(00γ)	2/m	$[1\bar 1]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	III I	III/II
8	2/mB(0 $[{{1}\over{2}}]$ γ)	2/m	$[1\bar 1]$	(0 $[{{1}\over{2}}]$ γ)	(100), (0 $[{{1}\over{2}}]$ 0), (001), (00γ)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0, 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	III G	III/III
Orthorhombic
9	mmmP(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)		IV P	IV/I
10	mmmP(0 $[{{1}\over{2}}]$ γ)	mmm	$[11\bar 1]$	(0 $[{{1}\over{2}}]$ γ)	(100), (0 $[{{1}\over{2}}]$ 0), (001), (00γ)	0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV B	IV/III
11	mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	mmm	$[11\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (001), (00γ)	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$ , 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV F	IV/VI
12	mmmI(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	IV I	IV/IV
13	mmmC(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 00	IV C	IV/II
14	mmmC(10γ)	mmm	$[11\bar 1]$	(10γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV I	IV/IV
15	mmmA(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)	0 $[{{1}\over{2}}{{1}\over{2}}]$ 0	IV B	IV/III
16	mmmA( $[{{1}\over{2}}]$ 0γ)	mmm	$[11\bar 1]$	( $[{{1}\over{2}}]$ 0γ)	( $[{{1}\over{2}}]$ 00), (010), (001), (00γ)	0 $[{{1}\over{2}}{{1}\over{2}}]$ 0, $[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$	IV G	IV/V
17	mmmF(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	IV F	IV/VI
18	mmmF(10γ)	mmm	$[11\bar 1]$	(10γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}{{1}\over{2}}]$	IV G	IV/V
Tetragonal
19	4/mmmP(00γ)	4/mmm	$[1\bar 111]$	(00γ)	(100), (010), (001), (00γ)		VII P	VI/I
20	4/mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	4/mmm	$[1\bar 111]$	( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	VII I	VI/II
21	4/mmmI(00γ)	4/mmm	$[1\bar 111]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	VII I	VI/II
Trigonal
22	$[\bar 3]$ mR(00γ)	$[\bar 3m]$	$[\bar 11]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{3}}{{2}\over{3}}{{2}\over{3}}]$ 0	VI P	VII/I
23	$[\bar 3]$ 1mP( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	$[\bar 31m]$	$[\bar 111]$	( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	( $[{{1}\over{3}}{{1}\over{3}}]$ 0), ( $[{{\bar 1}\over{3}}{{2}\over{3}}]$ 0), (001), (00γ)	$[{{1}\over{3}}{{2}\over{3}}]$ 0 $[{{2}\over{3}}]$	VI P	VII/I
Hexagonal
24	6/mmmP(00γ)	6/mmm	$[1\bar 111]$	(00γ)	(100), (010), (001), (00γ)		V P	VII/II

(b) (3 + 1)-Dimensional Bravais classes for commensurate structures. The holohedral point group K_s is given in terms of its external and internal parts, K_E and K_I, respectively. The reflections are given by ha* + kb* + lc* + mq, where q is the modulation wavevector. Here q is a commensurate vector having rational components with respect to a*, b*, c*. The rank of the vector module M* is three. Therefore, there are three basis vectors for M*. They are given by their components with respect to the conventional basis a*, b*, c* of the lattice of main reflections. If they do not coincide with the primitive basis vectors of the lattice Λ* of main reflections, there is a centring in four-dimensional space. The notation of the Bravais classes in Janssen (1969) is here given in the column Ref. a. Notice that for a commensurate one-dimensional modulation cubic symmetry is also possible.

No.	Symbol	K_E	K_I	q	Conventional basis	Centring translation(s)	Ref. a
Triclinic
1	$[\bar 1]$ P(000)	$[\bar 11]$	$[1\bar 1]$	(000)	(100), (010), (001)		II P
2	$[\bar 1]$ P( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	$[\bar 11]$	$[1\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{\bar 1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ ), ( $[{{1}\over{2}}{{\bar 1}\over{2}}{{1}\over{2}}]$ ), $[{{1}\over{2}}{{1}\over{2}}{{\bar 1}\over{2}}]$	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	II I
Monoclinic
3	2/mP(000)	2/m1	$[11\bar 1]$	(000)	(100), (010), (001)		IV P
4	2/mP( $[{{1}\over{2}}{{1}\over{2}}]$ 0)	2/m1	$[11\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}]$ 0)	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV B
5	2/mP(00 $[{{1}\over{2}}]$ )	2/m1	$[11\bar 1]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$	IV C
6	2/mP( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	2/m1	$[11\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	IV F
7	2/mB(000)	2/m1	$[11\bar 1]$	(000)	(100), (010), (001)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	IV B
8	2/mB(100)	2/m1	$[11\bar 1]$	(100)	(100), (010), (001)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}{{1}\over{2}}]$	IV I
9	2/mB(0 $[{{1}\over{2}}]$ 0)	2/m1	$[11\bar 1]$	(0 $[{{1}\over{2}}]$ 0)	(100), (0 $[{{1}\over{2}}]$ 0), (001)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0, 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV G
Orthorhombic
10	mmmP(000)	mmm1	$[111\bar 1]$	(000)	(100), (010), (001)		VIII P
11	mmmP(00 $[{{1}\over{2}}]$ )	mmm1	$[111\bar 1]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$	VIII A
12	mmmP(0 $[{{1}\over{2}}{{1}\over{2}}]$ )	mmm1	$[111\bar 1]$	(0 $[{{1}\over{2}}{{1}\over{2}}]$ )	(100), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	VIII F
13	mmmP( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	mmm1	$[111\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$ , 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	VIII S
14	mmmI(000)	mmm1	$[111\bar 1]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	VIII E
15	mmmI(111)	mmm1	$[111\bar 1]$	(111)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	VIII I
16	mmmI( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	mmm	$[{\bar 1}{\bar 1}{\bar 1}]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{4}}{{1}\over{4}}{{1}\over{4}}{{1}\over{4}}]$ , $[{{\bar 1}\over{4}}{{1}\over{4}}{{1}\over{4}}{{\bar 1}\over{4}}]$ , $[{{1}\over{4}}{{1}\over{4}}{{\bar 1}\over{4}}{{\bar 1}\over{4}}]$	VIII K
17	mmmF(000)	mmm1	$[111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	VIII F
18	mmmF(001)	mmm1	$[111{\bar 1}]$	(001)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}{{1}\over{2}}]$	VIII H
19	mmmC(000)	mmm1	$[111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 00	VIII A
20	mmmC(100)	mmm1	$[111{\bar 1}]$	(100)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	VIII E
21	mmmC(00 $[{{1}\over{2}}]$ )	mmm1	$[111{\bar 1}]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 00, 00 $[{{1}\over{2}}{{1}\over{2}}]$	VIII G
22	mmmC(10 $[{{1}\over{2}}]$ )	mmm1	$[111{\bar 1}]$	(10 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	VIII H
Tetragonal
23	4/mmmP(000)	4/mmm1	$[1111{\bar 1}]$	(000)	(100), (010), (001)		XII P
24	4/mmmP(00 $[{{1}\over{2}}]$ )	4/mmm1	$[1111{\bar 1}]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$	XII A
25	4/mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ 0)	4/mmm1	$[1111{\bar 1}]$	( $[{{1}\over{2}}{{1}\over{2}}]$ 0)	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	XII E
26	4/mmmP( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	4/mmm1	$[1111{\bar 1}]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	XII H
27	4/mmmI(000)	4/mmm1	$[1111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	XII E
28	4/mmmI(111)	4/mmm1	$[1111{\bar 1}]$	(111)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	XII I
29	4/mmmI( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	4/mmm	$[{\bar 1}{\bar 1}{\bar 1}1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	(100), (010), (001)		XII N
Trigonal
30	$[{\bar 3}]$ m1R(000)	$[{\bar 3}m1]$	$[11{\bar 1}]$	(000)	(100), (010), (001)	$[{{2}\over{3}}{{1}\over{3}}{{1}\over{3}}]$ 0	X R
31	$[{\bar 3}]$ m1R(00 $[{{1}\over{2}}]$ )	$[{\bar 3}m1]$	$[11{\bar 1}]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$ , $[{{2}\over{3}}{{1}\over{3}}{{1}\over{6}}{{1}\over{6}}]$	X RI
Hexagonal
32	6/mmmP(000)	6/mmm1	$[1111{\bar 1}]$	(000)	(100), (010), (001)		X P
33	6/mmmP(00 $[{{1}\over{2}}]$ )	6/mmm1	$[1111{\bar 1}]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$	X A
34	6/mmmP( $[{{1}\over{3}}{{1}\over{3}}]$ 0)	6/mmm	$[{\bar 1}11{\bar 1}]$	( $[{{1}\over{3}}{{1}\over{3}}]$ 0)	( $[{{1}\over{3}}{{1}\over{3}}]$ 0), ( $[{{\bar 1}\over{3}}{{2}\over{3}}]$ 0), (001)	$[{{1}\over{3}}{{2}\over{3}}]$ 0 $[{{2}\over{3}}]$	X R
35	6/mmmP( $[{{1}\over{3}}{{1}\over{3}}{{1}\over{2}}]$ )	6/mmm	$[{\bar 1}11{\bar 1}]$	( $[{{1}\over{3}}{{1}\over{3}}{{1}\over{2}}]$ )	( $[{{1}\over{3}}{{1}\over{3}}]$ 0), ( $[{{\bar 1}\over{3}}{{2}\over{3}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{3}}{{2}\over{3}}]$ 0 $[{{2}\over{3}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	X RI
Cubic
36	m3mP(000)	$[m{\bar 3}m1]$	$[111{\bar 1}]$	(000)	(100), (010), (001)		XIV P
37	m3mP( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	$[m{\bar 3}m1]$	$[111{\bar 1}]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$ , 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	XIV S
38	m3mI(000)	$[m{\bar 3}m1]$	$[111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	XIV V
39	m3mI(111)	$[m{\bar 3}m1]$	$[111{\bar 1}]$	(111)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	XIV I
40	m3mI( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	$[m{\bar 3}m]$	$[{\bar 1}{\bar 1}1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{4}}{{1}\over{4}}{{1}\over{4}}{{1}\over{4}}]$ , $[{{\bar 1}\over{4}}{{1}\over{4}}{{1}\over{4}}{{\bar 1}\over{4}}]$ , $[{{1}\over{4}}{{1}\over{4}}{{\bar 1}\over{4}}{{\bar 1}\over{4}}]$	XIV K
41	m3mF(000)	$[m{\bar 3}m1]$	$[111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	XIV F

As an example, the symbol $[2/mB(0{1\over2}\gamma)]$ denotes a Bravais class for which the main reflections belong to a B-centred monoclinic lattice (unique axis c) and the satellite positions are generated by the point-group transforms of $[{1\over2}{\bf b}^*+\gamma{\bf c}^*]$ . Then the matrix σ becomes $[\sigma=(0{1\over2}\gamma)]$ . It has as irrational part $[\sigma^i=(00\gamma)]$ and as rational part $[\sigma^r=(0{1\over2}0)]$ . The external part of the (3 + 1)-dimensional point group of the Bravais lattice is 2/m. By use of the relation [cf. (9.8.2.4)] $[R{\bf q}^i=\varepsilon{\bf q}^i,\quad R{\bf q}^r\equiv\varepsilon{\bf q}^r\hbox{ (modulo {\bf b}}{^*}), \eqno (9.8.3.1)]$ we see that the operations 2 and m are associated with the internal space transformations ɛ = 1 and ɛ = −1, respectively. This is denoted by the one-line symbol $[(2/m,1\bar1)]$ for the (3 + 1)-dimensional point group of the Bravais lattice . In direct space, the symmetry operation {R, ɛ(R)} is represented by the matrix Γ(R) which transforms the components $[v_j, j=1,\ldots,4]$ , of a vector [v_s] to: $[v'_j=\textstyle\sum\limits^4_{k=1}\Gamma(R)_{jk}v_k.]$ The operations (2, 1) and $[(m,\bar1)]$ are represented by the matrices: $[\Gamma(2)=\left(\matrix{ -1&\hfill0&0&0 \cr \hfill0&-1&0&0 \cr \hfill0&\hfill0&1&0 \cr \hfill0&-1&0&1}\right)\semi \quad \Gamma(m)=\left(\matrix{ 1&0&\hfill0&\hfill0 \cr 0&1&\hfill0&\hfill0 \cr 0&0&-1&\hfill0 \cr 0&1&\hfill0&-1}\right). \eqno (9.8.3.2)]$ The 3 × 3 part $[\Gamma_E(R)]$ of each matrix is obtained by considering the action of R on the external part v of [v_s] . The 1 × 1 part $[\Gamma_I(R)]$ is the value of the ɛ associated with R and the remaining part $[\Gamma_M(R)]$ follows from the relation $[\Gamma_M(R)=-\Gamma_I(R)\sigma^r+\sigma^r\Gamma_E(R). \eqno (9.8.3.3)]$

Bravais classes can be denoted in an alternative way by two-line symbols . In the two-line symbol, the Bravais class is given by specifying the arithmetic crystal class of the external symmetry by the symbol of its symmorphic space group, the associated elements $[\Gamma_I(R)=\varepsilon]$ by putting their symbol under the corresponding symbols of $[\Gamma_E(R)]$ , and by the rational part $[\sigma^r]$ indicated by a prefix. In the following table, this prefix is given for the components of $[{\bf q}^r]$ that play a role in the classification. $[\let\normalbaselines\relax\openup4pt\matrix{ P \quad(000)\hfill& R\quad(\,{1\over3},{1\over3},0)\hfill \cr A\quad(\,{1\over2},0,0)\hfill& B\quad(0,{1\over2},0)\hfill& C\quad(0,0,{1\over2}\,)\hfill \cr L\quad(1,0,0)\hfill& M\quad(0,1,0)\hfill& N\quad(0,0,1)\hfill \cr U\quad (0,{1\over2},{1\over2}\,)\hfill& V\quad(\,{1\over2},0,{1\over2}\,)\hfill& W\quad(\,{1\over2},{1\over2},0).}]$ Note that the integers appearing here are not equivalent to zero because they express components with respect to a conventional lattice basis (and not a primitive one). For the Bravais class mentioned above, the two-line symbol is $[B^{2/mB}_{1\;\;\bar1}]$ . This symbol has the advantage that the internal transformation (the value of ɛ) is explicitly given for the corresponding generators. It has, however, certain typographical drawbacks. It is rare for the printer to put the symbol together in the correct manner: $[B^{2/mB}_{1\;\;\bar1}]$ .

In Tables 9.8.3.1 and 9.8.3.2 the symbols for the (2 + d)- and (3 + 1)-dimensional Bravais classes are given in the one-line form. It is, however, easy to derive from each one-line symbol the corresponding two-line symbol because the bottom line for the two-line symbol appears in the tables as the internal part of the point-group symbol.

The number of symbols in the bottom line of the two-line symbol should be equal to that of the generators given in the top line. A symbol `1' is used in the bottom line if the corresponding [R_I] is the unit transformation. If necessary, a mirror perpendicular to a crystal axis is indicated by $[\dot m]$ and one that is not by $[\ddot m]$ . This situation only occurs for $[d\ge2]$ . So the (2 + 2)-dimensional class $[P^{4mp}_{4m}]$ is actually $[P^{4 {\dot m} p}_{4\dot m}]$ and is different from the class $[P^{4{\dot m}p}_{4{\ddot m}}]$ . In a one-line symbol, their difference is apparent, the first being 4mp(α0), whereas the second is 4mp(αα).

References

Brown, H., Bülow, H., Neubüser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of four-dimensional space. New York: John Wiley.Google Scholar

Janssen, T. (1969). Crystallographic groups in space and time. III. Four-dimensional Euclidean crystal classes. Physica (Utrecht), 42, 71–92.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 915-916