International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.1, p. 712

Table 3.1.3.1 

P. M. de Wolffc

Table 3.1.3.1| top | pdf |
The parameters [D = {\bf b}\cdot {\bf c}], [E = {\bf a}\cdot {\bf c}] and [F = {\bf a}\cdot {\bf b}] of the 44 lattice characters ([A = {\bf a}\cdot {\bf a},\ B = {\bf b}\cdot {\bf b},\ C = {\bf c}\cdot {\bf c}])

The character of a lattice given by its reduced form (3.1.3.1)[link] is the first one that agrees when the 44 entries are compared with that reduced form in the sequence given below (suggested by Gruber). Such a logical order is not always obeyed by the widely used character numbers (first column), which therefore show some reversals, e.g. 4 and 5.

No.TypeDEFLattice symmetryBravais type of latticeTransformation to a conventional basis (cf. footnote [\ddag] to Table 3.1.3.2[link])
[A = B = C]
1 I [A/2] [A/2] [A/2] Cubic cF [1\bar{1}1/11\bar{1}/\bar{1}11]
2 I D D D Rhombohedral hR [1\bar{1}0/\bar{1}01/\bar{1}\bar{1}\bar{1}]
3 II 0 0 0 Cubic cP [100/010/001]
5 II [- A/3] [- A/3] [- A/3] Cubic cI [101/110/011]
4 II D D D Rhombohedral hR [1\bar{1}0/\bar{1}01/\bar{1}\bar{1}\bar{1}]
6 II D D F Tetragonal tI [011/101/110]
7 II D E E Tetragonal tI [101/110/011]
8 II D E F Orthorhombic oI [\bar{1}\bar{1}0/\bar{1}0\bar{1}/0\bar{1}\bar{1}]
[A = B], no conditions on C
9 I [A/2] [A/2] [A/2] Rhombohedral hR [100/\bar{1}10/\bar{1}\bar{1}3]
10 I D D F Monoclinic mC [110/1\bar{1}0/00\bar{1}]
11 II 0 0 0 Tetragonal tP [100/010/001]
12 II 0 0 [- A/2] Hexagonal hP [100/010/001]
13 II 0 0 F Orthorhombic oC [110/\bar{1}10/001]
15 II [- A/2] [- A/2] 0 Tetragonal tI [100/010/112]
16 II D D F Orthorhombic oF [\bar{1}\bar{1}0/1\bar{1}0/112]
14 II D D F Monoclinic mC [110/\bar{1}10/001]
17 II D E F Monoclinic mC [1\bar{1}0/110/\bar{1}0\bar{1}]
[B = C], no conditions on A
18 I [A/4] [A/2] [A/2] Tetragonal tI [0\bar{1}1/1\bar{1}\bar{1}/100]
19 I D A/2 A/2 Orthorhombic oI [\bar{1}00/0\bar{1}1/\bar{1}11]
20 I D E E Monoclinic mC [011/01\bar{1}/\bar{1}00]
21 II 0 0 0 Tetragonal tP [010/001/100]
22 II [- B/2] 0 0 Hexagonal hP [010/001/100]
23 II D 0 0 Orthorhombic oC [011/0\bar{1}1/100]
24 II D [- A/3] [- A/3] Rhombohedral hR [121/0\bar{1}1/100]
25 II D E E Monoclinic mC [011/0\bar{1}1/100]
No conditions on A, B, C
26 I [A/4] [A/2] [A/2] Orthorhombic oF [100/\bar{1}20/\bar{1}02]
27 I D [A/2] [A/2] Monoclinic mC [\bar{1}20/\bar{1}00/0\bar{1}1]
28 I D [A/2] 2D Monoclinic mC [\bar{1}00/\bar{1}02/010]
29 I D 2D [A/2] Monoclinic mC [100/1\bar{2}0/00\bar{1}]
30 I [B/2] E 2E Monoclinic mC [010/01\bar{2}/\bar{1}00]
31 I D E F Triclinic aP [100/010/001]
32 II 0 0 0 Orthorhombic oP [100/010/001]
40 II [- B/2] 0 0 Orthorhombic oC [0\bar{1}0/012/\bar{1}00]
35 II D 0 0 Monoclinic mP [0\bar{1}0/\bar{1}00/00\bar{1}]
36 II 0 [- A/2] 0 Orthorhombic oC [100/\bar{1}0\bar{2}/010]
33 II 0 E 0 Monoclinic mP [100/010/001]
38 II 0 0 [- A/2] Orthorhombic oC [\bar{1}00/120/00\bar{1}]
34 II 0 0 F Monoclinic mP [\bar{1}00/00\bar{1}/0\bar{1}0]
42 II [- B/2] [- A/2] 0 Orthorhombic oI [\bar{1}00/0\bar{1}0/112]
41 II [- B/2] E 0 Monoclinic mC [0\bar{1}\bar{2}/0\bar{1}0/\bar{1}00]
37 II D [- A/2] 0 Monoclinic mC [102/100/010]
39 II D 0 [- A/2] Monoclinic mC [\bar{1}\bar{2}0/\bar{1}00/00\bar{1}]
43 II D§ E F Monoclinic mI [\bar{1}00/\bar{1}\bar{1}\bar{2}/0\bar{1}0]
44 II D E F Triclinic aP [100/010/001]
The symbols for Bravais types of lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985[link]). The capital letter of the symbols in this column indicates the centring type of the cell as obtained by the transformation in the last column. For this reason, the standard symbols mS and oS are not used here.
[2|D + E + F| = A + B].
§[2|D + E + F| = A + B] plus [|2D + F| = B].