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

International Tables for Crystallography (2015). Vol. A, ch. 1.5, pp. 87-88

Section 1.5.3.1. Space groups with more than one description in this volume

G. Chapuis,d H. Wondratscheka and M. I. Aroyob

1.5.3.1. Space groups with more than one description in this volume

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In the description of the space-group symbols presented in Section 1.4.1[link] , 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 space-group tables of Chapter 2.3[link] provide a selection of possible alternative settings. For example, in space group [P2_1/c], two possible orientations of the unit-cell 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 unique-axis position (dummy indices `1' indicate `empty' symmetry directions), and by the corresponding change in the direction of the glide plane: [P12_{1}/c1] or [P112_{1}/a] (cf. Section 1.4.1[link] 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[link] illustrates three cell choices for the monoclinic axis b setting of [P2_1/c].

[Figure 1.5.3.1]

Figure 1.5.3.1 | top | pdf |

Three possible cell choices for the monoclinic space group [P2_{1}/c] (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 [{\cal S}], a second diagram is displayed with the origin at a point with site symmetry [{\cal S}] (`origin choice 1'). Fig. 1.5.3.2[link] 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.

[Figure 1.5.3.2]

Figure 1.5.3.2 | top | pdf |

Two possible origin choices for the orthorhombic space group Pban (50). Origin choice 1 is on 222, whereas origin choice 2 is on [\overline{1}].

Finally, the seven rhombohedral space-group types (i.e. space groups with a rhombohedral lattice) also have alternative descriptions included in the space-group tables of this volume. The rhombohedral lattice is first presented with an R-centred hexagonal cell ([|{\bf a}_{\rm hex}|=|{\bf b}_{\rm hex}|]; [{\bf c}_{\rm hex}\perp {\bf a}_{\rm hex}], [{\bf b}_{\rm hex}]; 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 [a_{\rm rh}=b_{\rm rh}=c_{\rm rh}] and [\alpha_{\rm rh} = \beta_{\rm rh} =\gamma_{\rm rh}]. The relation between the two types of cell is illustrated in Fig. 1.5.3.3[link] 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 [x, x, x]. If we refer to the transformations of the primitive rhombohedral cell cited in Table 1.5.1.1[link], we observe two different centrings with three possible orientations R1, R2 and R3 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[link]. 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; [\textstyle{{2}\over{3}},\textstyle{{1}\over{3}},\textstyle{{1}\over{3}}]; [\textstyle{{1}\over{3}},\textstyle{{2}\over{3}},\textstyle{{2}\over{3}}], whereas the reverse R cells have lattice points at 0, 0, 0; [\textstyle{{1}\over{3}},\textstyle{{2}\over{3}},\textstyle{{1}\over{3}}]; [\textstyle{{2}\over{3}},\textstyle{{1}\over{3}},\textstyle{{2}\over{3}}]. The triple hexagonal cell R1 of the obverse setting (i.e. [{\bf a}_{\rm hex}={\bf a}_{\rm rh}-{\bf b}_{\rm rh}], [{\bf b}_{\rm hex}={\bf b}_{\rm rh}-{\bf c}_{\rm rh}], [{\bf c}_{\rm hex}{}=] [{\bf a}_{\rm rh}+{\bf b}_{\rm rh}+{\bf b}_{\rm rh})] has been used in the description of the rhombohedral space groups in this volume (cf. Table 1.5.1.1[link] and Fig. 1.5.3.3[link]).

[Figure 1.5.3.3]

Figure 1.5.3.3 | top | pdf |

General-position diagram of the space group R3m (160) showing the relation between the hexagonal and rhombohedral axes in the obverse setting: [{\bf a}_{\rm rh}] = [\textstyle{{1}\over{3}}(2{\bf a}_{\rm hex}+{\bf b}_{\rm hex}+{\bf c}_{\rm hex})], [{\bf b}_{\rm rh}] = [\textstyle{{1}\over{3}}(-{\bf a}_{\rm hex}+{\bf b}_{\rm hex}+{\bf c}_{\rm hex})], [{\bf c}_{\rm rh}] = [\textstyle{{1}\over{3}}(-{\bf a}_{\rm hex}] [-2{\bf b}_{\rm hex}+{\bf c}_{\rm hex})].

The hexagonal lattice can be referred to a centred rhombohedral cell, called the D cell (cf. Table 1.5.1.1[link]). The centring points of this cell are [0,0,0], [\textstyle{{1}\over{3}}, \textstyle{{1}\over{3}}, \textstyle{{1}\over{3}}] and [\textstyle{{2}\over{3}},\textstyle{{2}\over{3}},\textstyle{{2}\over{3}}]. However, the D cell is rarely used in crystallography.








































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