Enlarged unit cell, index 3 or 4

Wondratschek, H.; Aroyo, M. I.

doi:10.1107/97809553602060000542

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 2.1, pp. 50-51 | 1 | 2 |

Section 2.1.4.3.2. Enlarged unit cell, index 3 or 4

Hans Wondratschek^a ^* and Mois I. Aroyo^b

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and ^bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

2.1.4.3.2. Enlarged unit cell, index 3 or 4

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With a few exceptions for trigonal, hexagonal and cubic space groups, k-subgroups with enlarged unit cells and index 3 or 4 are isomorphic. To each of the listed sublattices belong either one or several conjugacy classes with three or four conjugate subgroups or one or several normal subgroups. Only the sublattices with the numbers (5)(a)(v), (5)(b)(i), (5)(c)(ii), (6)(iii) and (7)(i) have index 4, all others have index 3. The different cell enlargements are listed in the following sequence:

(1) Triclinic space groups:

(i) $[{\bf a}'=3{\bf a}]$ ,
(ii) $[{\bf a}'=3{\bf a}]$ , $[{\bf b}'={\bf a}+{\bf b}]$ ,
(iii) $[{\bf a}'=3{\bf a}]$ , $[{\bf b}'=2{\bf a}+{\bf b}]$ ,
(iv) $[{\bf a}'=3{\bf a}]$ , $[{\bf c}'={\bf a}+{\bf c}]$ ,
(v) $[{\bf a}'=3{\bf a}]$ , $[{\bf c}'=2{\bf a}+{\bf c}]$ ,
(vi) $[{\bf a}'=3{\bf a}]$ , $[{\bf b}'={\bf a}+{\bf b}]$ , $[{\bf c}'={\bf a}+{\bf c}]$ ,
(vii) $[{\bf a}'=3{\bf a}]$ , $[{\bf b}'=2{\bf a}+{\bf b}]$ , $[{\bf c}'={\bf a}+{\bf c}]$ ,
(viii) $[{\bf a}'=3{\bf a}]$ , $[{\bf b}'={\bf a}+{\bf b}]$ , $[{\bf c}'=2{\bf a}+{\bf c}]$ ,
(ix) $[{\bf a}'=3{\bf a}]$ , $[{\bf b}'=2{\bf a}+{\bf b}]$ , $[{\bf c}'=2{\bf a}+{\bf c}]$ ,
(x) $[{\bf b}'=3{\bf b}]$ ,
(xi) $[{\bf b}'=3{\bf b}]$ , $[{\bf c}'={\bf b}+{\bf c}]$ ,
(xii) $[{\bf b}'=3{\bf b}]$ , $[{\bf c}'=2{\bf b}+{\bf c}]$ ,
(xiii) $[{\bf c}'=3{\bf c}]$ .

(2) Monoclinic space groups:

(a) Space groups [P121] , [P12_11] , [P1m1] , [P12/m1] , [P12_1/m1] (unique axis b):

(i) $[{\bf b}'=3{\bf b}]$ ,
(ii) $[{\bf c}'=3{\bf c}]$ ,
(iii) $[{\bf a}'={\bf a}-{\bf c}]$ , $[{\bf c}'=3{\bf c}]$ ,
(iv) $[{\bf a}'={\bf a}-2{\bf c}]$ , $[{\bf c}'=3{\bf c}]$ ,
(v) $[{\bf a}'=3{\bf a}]$ .

(b) Space groups [P112] , [P112_1] , [P11m] , [P112/m] , [P112_1/m] (unique axis c):

(i) $[{\bf c}'=3{\bf c}]$ ,
(ii) $[{\bf a}'=3{\bf a}]$ ,
(iii) $[{\bf a}'=3{\bf a},\ {\bf b}'=-{\bf a}+{\bf b}]$ ,
(iv) $[{\bf a}'=3{\bf a},\ {\bf b}'=-2{\bf a}+{\bf b}]$ ,
(v) $[{\bf b}'=3{\bf b}]$ .

(i) $[{\bf b}'=3{\bf b}]$ ,
(ii) $[{\bf c}'=3{\bf c}]$ ,
(iii) $[{\bf a}'=3{\bf a}]$ ,
(iv) $[{\bf a}'=3{\bf a},\ {\bf c}'=-2{\bf a}+{\bf c}]$ ,
(v) $[{\bf a}'=3{\bf a},\, {\bf c}' = -4{\bf a} + {\bf c}]$ .

(d) Space groups [P11a] , [P112/a] , [P112_1/a] (unique axis c):

(i) $[{\bf c}'=3{\bf c}]$ ,
(ii) $[{\bf a}'=3{\bf a}]$ ,
(iii) $[{\bf b}'=3{\bf b}]$ ,
(iv) $[{\bf a}'={\bf a}-2{\bf b},\, {\bf b}'=3{\bf b}]$ ,
(v) $[{\bf a}'={\bf a}-4{\bf b},\, {\bf b}'=3{\bf b}]$ .

(e) All space groups with C lattice (unique axis b):

(i) $[{\bf b}'=3{\bf b}]$ ,
(ii) $[{\bf c}'=3{\bf c}]$ ,
(iii) $[{\bf a}'={\bf a}-2{\bf c},\ {\bf c}'=3{\bf c}]$ ,
(iv) $[{\bf a}'={\bf a}-4{\bf c},\, {\bf c}'=3{\bf c}]$ ,
(v) $[{\bf a}'=3{\bf a}]$ .

(f) All space groups with A lattice (unique axis c):

(i) $[{\bf c}'=3{\bf c}]$ ,
(ii) $[{\bf a}'=3{\bf a}]$ ,
(iii) $[{\bf a}'=3{\bf a},\, {\bf b}'=-2{\bf a}+{\bf b}]$ ,
(iv) $[{\bf a}'=3{\bf a},\, {\bf b}'=-4{\bf a}+{\bf b}]$ ,
(v) $[{\bf b}'=3{\bf b}]$ .

(3) Orthorhombic space groups:

(i) $[{\bf a}'=3{\bf a}]$ ,
(ii) $[{\bf b}'=3{\bf b}]$ ,
(iii) $[{\bf c}'=3{\bf c}]$ .

(4) Tetragonal space groups:

(i) $[{\bf c}'=3{\bf c}]$ .

(5) Trigonal space groups:

(a) Trigonal space groups with hexagonal P lattice:

(i) $[{\bf c}'=3{\bf c}]$ ,
(ii) $[{\bf a}'=3{\bf a},\, {\bf b}'=3{\bf b},]$ H-centring,
(iii) $[{\bf a}'={\bf a}-{\bf b},\, {\bf b}'={\bf a}+2{\bf b},\, {\bf c}'=3{\bf c}]$ , R lattice,
(iv) $[{\bf a}'=2{\bf a}+{\bf b}, {\bf b}'=-{\bf a}+{\bf b},\, {\bf c}'=3{\bf c}]$ , R lattice,
(v) $[{\bf a}'=2{\bf a},\, {\bf b}'=2{\bf b}]$ .

(b) Trigonal space groups with rhombohedral R lattice and hexagonal axes:

(i) $[{\bf a}'=-2{\bf b},\, {\bf b}'=2{\bf a}+2{\bf b}]$ .

(i) $[{\bf a}'={\bf a}-{\bf b},\, {\bf b}'= {\bf b}-{\bf c},\, {\bf c}'={\bf a}+{\bf b}+{\bf c}]$ ,
(ii) $[{\bf a}'={\bf a}-{\bf b}+{\bf c},\, {\bf b}'={\bf a}+{\bf b}-{\bf c},\, {\bf c}'= -{\bf a}+{\bf b}+{\bf c}]$ .

(6) Hexagonal space groups:

(i) $[{\bf c}'=3{\bf c}]$ ,
(ii) $[{\bf a}'=3{\bf a},\, {\bf b}'=3{\bf b},]$ H-centring,
(iii) $[{\bf a}'=2{\bf a},\, {\bf b}'=2{\bf b}]$ .

(7) Cubic space groups with P lattice:

(i) $[{\bf a}'=2{\bf a},\, {\bf b}'=2{\bf b},\, {\bf c}'=2{\bf c}]$ , I lattice.

References

International Tables for Crystallography (2006). Vol. A1. ch. 2.1, pp. 50-51