International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.5, pp. 8790

In the description of the spacegroup symbols presented in Section 1.4.1 , we have already seen that in the conventional, unique axis b description of monoclinic space groups, the unique symmetry direction is chosen as b; it is normal to c and a, which form the angle β. However, it is often the case that this standard direction is not the most appropriate choice and that another choice would be more convenient. An example of this would be when following a phase transition from an orthorhombic parent phase to a monoclinic phase. Here, it would often be preferable to keep the same orientation of the axes even if the resulting monoclinic setting is not standard.
In some of the space groups, and especially in the monoclinic ones, the spacegroup tables of Chapter 2.3 provide a selection of possible alternative settings. For example, in space group , two possible orientations of the unitcell axes are provided, namely with unique axis b and c. This is reflected in the corresponding full Hermann–Mauguin symbols by the explicit specification of the uniqueaxis position (dummy indices `1' indicate `empty' symmetry directions), and by the corresponding change in the direction of the glide plane: or (cf. Section 1.4.1 for a detailed treatment of Hermann–Mauguin symbols of space groups).
It is not just the unique monoclinic axis that can be varied: the choice of the other axes can vary as well. There are cases where the selection of the conventional setting leads to an inconvenient monoclinic angle that deviates greatly from 90°. If another cell choice minimizes the deviation from 90°, it is preferred. Fig. 1.5.3.1 illustrates three cell choices for the monoclinic axis b setting of .

Three possible cell choices for the monoclinic space group (14) with unique axis b. Note the corresponding changes in the full Hermann–Mauguin symbols. The glide vector is indicated by an arrow. 
In centrosymmetric space groups the origin of the unit cell is located at an inversion centre (`origin choice 2'). If, however, another point has higher site symmetry , a second diagram is displayed with the origin at a point with site symmetry (`origin choice 1'). Fig. 1.5.3.2 illustrates the space group Pban with two possible origins. The origin of the first choice is located on a point with site symmetry 222, whereas the origin for the second choice is located on an inversion centre. Among the 230 space groups, this volume lists 24 centrosymmetric space groups with an additional alternative origin.

Two possible origin choices for the orthorhombic space group Pban (50). Origin choice 1 is on 222, whereas origin choice 2 is on . 
Finally, the seven rhombohedral spacegroup types (i.e. space groups with a rhombohedral lattice) also have alternative descriptions included in the spacegroup tables of this volume. The rhombohedral lattice is first presented with an Rcentred hexagonal cell (; , ; γ = 120°) with a volume three times larger than that of the primitive rhombohedral cell. The second presentation is given with a primitive rhombohedral cell with and . The relation between the two types of cell is illustrated in Fig. 1.5.3.3 for the space group R3m (160). In the hexagonal cell, the coordinates of the special position with site symmetry 3m are 0, 0, z, whereas in the rhombohedral cell the same special position has coordinates . If we refer to the transformations of the primitive rhombohedral cell cited in Table 1.5.1.1, we observe two different centrings with three possible orientations R_{1}, R_{2} and R_{3} which are related by ±120° to each other. The two kinds of centrings, called obverse and reverse, are illustrated in Fig. 1.5.1.6. A rotation of 180° around the rhombohedral axis relates the obverse and reverse descriptions of the rhombohedral lattice. The obverse triple R cells have lattice points at 0, 0, 0; ; , whereas the reverse R cells have lattice points at 0, 0, 0; ; . The triple hexagonal cell R_{1} of the obverse setting (i.e. , , has been used in the description of the rhombohedral space groups in this volume (cf. Table 1.5.1.1 and Fig. 1.5.3.3).

Generalposition diagram of the space group R3m (160) showing the relation between the hexagonal and rhombohedral axes in the obverse setting: = , = , = . 
The hexagonal lattice can be referred to a centred rhombohedral cell, called the D cell (cf. Table 1.5.1.1). The centring points of this cell are , and . However, the D cell is rarely used in crystallography.
In the spacegroup tables of this volume, the monoclinic space group (14) is described in six different settings: for each of the `unique axis b' and `unique axis c' settings there are three descriptions specified by different cell choices (cf. Section 2.1.3.15 ). The different settings are identified by the appropriate full Hermann–Mauguin symbols. The basis transformations between the different settings are completely specified by the linear part of the transformation, the 3 × 3 matrix P [cf. equation (1.5.1.4)], as all settings of refer to the same origin, i.e. . The transformation matrices P necessary for switching between the different descriptions of can either be read off directly or constructed from the transformationmatrix data listed in Table 1.5.1.1.
(A) Transformation from (unique axis b, cell choice 1) to (unique axis c, cell choice 1). The change of the direction of the screw axis indicates that the unique direction b transforms to the unique direction c, while the glide vector along c transforms to a glide vector along a. These changes are reflected in the transformation matrix P between the basis of and of , which can be read directly from Table 1.5.1.1:

(B) Transformation from (unique axis c, cell choice 3) to (unique axis b, cell choice 1): = . A transformation matrix from directly to is not found in Table 1.5.1.1, but it can be constructed in two steps from transformation matrices that are listed there. For example:
Step 1. Unique axis c fixed: transformation from `cell choice 3' to `cell choice 1':
The zircon example of Section 1.5.1.1 illustrates how the atomic coordinates change under an originchoice transformation. Here, the case of the two originchoice descriptions of the same space group I4_{1}/amd (141) will be used to demonstrate how the rest of the crystallographic quantities are affected by an origin shift.
The two descriptions of I4_{1}/amd in the spacegroup tables of this volume are distinguished by the origin choices of the reference coordinate systems: the origin statement of the origin choice 1 setting indicates that its origin is taken at a point of symmetry, which is located at with respect to the origin of origin choice 2, taken at a centre (2/m). Conversely, the origin is taken at a centre (2/m) at from the origin . These origin descriptions in fact specify explicitly the originshift vector p necessary for the transformation between the two settings. For example, the shift vector listed for origin choice 2 expresses the origin with respect to , i.e. the corresponding transformation matrixtransforms the crystallographic data from the origin choice 1 setting to the origin choice 2 setting.