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

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 917-918

Table 9.8.3.2 

T. Janssen,a A. Janner,a A. Looijenga-Vosb and P. M. de Wolffc

aInstitute 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

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 Ks is given in terms of its external and internal parts, KE and KI, respectively. The reflections are given by ha* + kb* + lc* + mq, where q is the modulation wavevector. If the rational part qr 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 qr = 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[link]) and Brown et al. (1978[link]). The notation of the Bravais classes there is here given in the columns Ref. a and Ref. b, respectively.

No.SymbolKEKIqConventional basisCentring translation(s)Ref. aRef. 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 Ks is given in terms of its external and internal parts, KE and KI, 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[link]) is here given in the column Ref. a. Notice that for a commensurate one-dimensional modulation cubic symmetry is also possible.

No.SymbolKEKIqConventional basisCentring 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