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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, p. 46

## Section 1.4.1.4.4. Monoclinic space groups

H. Wondratscheke

#### 1.4.1.4.4. Monoclinic space groups

| top | pdf |

Monoclinic space groups have exactly one symmetry direction, often called the monoclinic axis. The b axis is the symmetry direction of the (most frequently used) conventional setting, called the b-axis setting. Another conventional setting has c as its symmetry direction (c-axis setting). In earlier literature, the unique-axis c setting was called the first setting and the unique-axis b setting the second setting (cf. Section 2.1.3.15 ). In addition to the primitive lattice P there is a centred lattice which is taken as C in the b-axis setting, A in the c-axis setting. The (possible) glide reflections are c (or a). In this volume, more settings are described, cf. Sections 1.5.4 and 2.1.3.15 and the space-group tables of Chapter 2.3 .

The full HM symbol consists of the lattice symbol and three possible positions for the symmetry directions. The symmetry in the a direction is described first, followed by the symmetry in the b direction and last in the c direction. The two positions of the HM symbol that are not occupied by the monoclinic symmetry direction are marked by 1. The symbol is thus similar to the orthorhombic HM symbol and the monoclinic axis is clearly visible. P1m1 or P11m may designate the same space group but in different settings. Pm11 is a possible but not conventional setting.

The short HM symbols of the monoclinic space groups are independent of the setting of the space group. They form the monoclinic standard symbols and are not variable: P2, , C2, Pm, Pc, Cm, Cc, P2/m, , C2/m, P2/c, and C2/c. Altogether there are 13 monoclinic space-group types.

There are several reasons for the many conventional settings.

 (1) As only one of the three coordinate axes is fixed by symmetry, there are two conventions related to the possible permutations of the other axes. (2) The sequence of the three coordinate axes may be chosen because of the lengths of the basis vectors, i.e. not because of symmetry. (3) If two different crystal structures have related symmetries, one being a subgroup of the other, then it is often convenient to choose a non-conventional setting for one of the structures to make their structural relations transparent. Such similarity happens in particular in substances that are related by a non-destructive phase transition. Monoclinic space groups are particularly flexible in their settings.