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

International Tables for Crystallography (2011). Vol. A1, ch. 2.1, pp. 80-81   | 1 | 2 |

Section 2.1.4.3.1. Enlarged unit cell, index 2

Hans Wondratscheka* and Mois I. Aroyob

aInstitut 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

2.1.4.3.1. Enlarged unit cell, index 2

| top | pdf |

For sublattices with twice the volume of the unit cell, the sequence of the different cell enlargements is as follows:

  • (1) Triclinic space groups:

    • (i) [{\bf a}' = 2{\bf a}],

    • (ii) [{\bf b}' = 2{\bf b}],

    • (iii) [{\bf c}' = 2{\bf c}],

    • (iv) [{\bf b}' = 2{\bf b},\, {\bf c}' = 2{\bf c},] A-centring,

    • (v) [{\bf a}' = 2{\bf a},\, {\bf c}' = 2{\bf c},] B-centring,

    • (vi) [{\bf a}' = 2{\bf a},\, {\bf b}' = 2{\bf b},] C-centring,

    • (vii) [{\bf a}' = 2{\bf a},\, {\bf b}' = 2{\bf b},\, {\bf c}' = 2{\bf c},] F-centring.

  • (2) Monoclinic space groups:

    • (a) with P lattice, unique axis b:

      • (i) [{\bf b}' = 2{\bf b}],

      • (ii) [{\bf c}' = 2{\bf c}],

      • (iii) [{\bf a}' = 2{\bf a}],

      • (iv) [{\bf a}' = 2{\bf a},\, {\bf c}' = 2{\bf c},] B-centring,

      • (v) [{\bf a}' = 2{\bf a},\, {\bf b}' = 2{\bf b},] C-centring,

      • (vi) [{\bf b}' = 2{\bf b},\, {\bf c}' = 2{\bf c},] A-centring,

      • (vii) [{\bf a}' = 2{\bf a},\, {\bf b}' = 2{\bf b},\, {\bf c}' = 2{\bf c},] F-centring.

    • (b) with P lattice, unique axis c:

      • (i) [{\bf c}' = 2{\bf c}],

      • (ii) [{\bf a}'=2{\bf a}],

      • (iii) [{\bf b}' = 2{\bf b}],

      • (iv) [{\bf a}' = 2{\bf a},\, {\bf b}' = 2{\bf b},] C-centring,

      • (v) [{\bf b}' = 2{\bf b},\, {\bf c}' = 2{\bf c},] A-centring,

      • (vi) [{\bf a}' = 2{\bf a},\, {\bf c}' = 2{\bf c},] B-centring,

      • (vii) [{\bf a}' = 2{\bf a},\, {\bf b}' = 2{\bf b},\, {\bf c}' = 2{\bf c},] F-centring.

    • (c) with C lattice, unique axis b: There are three sublattices of index 2 of a monoclinic C lattice. One has lost its centrings such that a P lattice with the same unit cell remains. The subgroups with this sublattice are listed under `Loss of centring translations'. The block with the other two sublattices consists of [{\bf c}' = 2{\bf c}], C-centring and I-centring. The sequence of the subgroups in this block is determined by the space-group number of the subgroup.

    • (d) with A lattice, unique axis c: There are three sublattices of index 2 of a monoclinic A lattice. One has lost its centrings such that a P lattice with the same unit cell remains. The subgroups with this sublattice are listed under `Loss of centring translations'. The block with the other two sublattices consists of [{\bf a}' = 2{\bf a}], A-centring and I-centring. The sequence of the subgroups in this block is determined by the No. of the subgroup.

  • (3) Orthorhombic space groups:

    • (a) Orthorhombic space groups with P lattice: Same seq­uence as for triclinic space groups.

    • (b) Orthorhombic space groups with C (or A) lattice: Same sequence as for monoclinic space groups with C (or A) lattice.

    • (c) Orthorhombic space groups with I and F lattice: There are no subgroups of index 2 with enlarged unit cell.

  • (4) Tetragonal space groups:

    • (a) Tetragonal space groups with P lattice:

      • (i) [{\bf c}' = 2{\bf c}].

      • (ii) [{\bf a}' = 2{\bf a}, \,\, {\bf b}' = 2{\bf b},] C-centring. The conventional set­ting results in a P lattice.

      • (iii) [{\bf a}' = 2{\bf a},\, {\bf b}' = 2{\bf b},\, {\bf c}' = 2{\bf c},] F-centring. The conventional setting results in an I lattice.

    • (b) Tetragonal space groups with I lattice: There are no subgroups of index 2 with enlarged unit cell.

  • (5) For trigonal and hexagonal space groups, [{\bf c}' = 2{\bf c}] holds.

    For rhombohedral space groups referred to hexagonal axes, [{\bf a}' = -{\bf b}], [{\bf b}' = {\bf a} + {\bf b}], [{\bf c}' = 2{\bf c}] or [{\bf a}' = {\bf a} + {\bf b}], [{\bf b}' = -{\bf a}], [{\bf c}' = 2{\bf c}] holds.

    For rhombohedral space groups referred to rhombohedral axes, [{\bf a}' = {\bf a} + {\bf c}], [{\bf b}' = {\bf a} + {\bf b}], [{\bf c}' = {\bf b} + {\bf c}] or [{\bf a}' = 2{\bf a}], [{\bf b}' = 2{\bf b}], [{\bf c}' = 2{\bf c}], F-centring holds.

  • (6) Only cubic space groups with a P lattice have subgroups of index 2 with enlarged unit cell. For their lattices the following always holds: [{\bf a}' = 2{\bf a}], [{\bf b}' = 2{\bf b}], [{\bf c}' = 2{\bf c}], F-centring.








































to end of page
to top of page