Table 9.8.3.1

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

Table 9.8.3.1

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

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	q ₁	q ₂	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), ( $[\bar \alpha2\alpha]$ )