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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 55-56

Section 1.4.2.4. Additional symmetry operations and symmetry elements

M. I. Aroyo,a G. Chapuis,b B. Souvignierd and A. M. Glazerc

1.4.2.4. Additional symmetry operations and symmetry elements

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The symmetry operations of a space group are conveniently partitioned into the cosets with respect to the translation subgroup. All operations which belong to the same coset have the same linear part and, if a single operation from a coset is given, all other operations in this coset are obtained by composition with a translation. However, not all symmetry operations in a coset with respect to the translation subgroup are operations of the same type and, furthermore, they may belong to element sets of different symmetry elements. In general, one can distinguish the following cases:

  • (i) The composition [\ispecialfonts{\sfi W}' = {\sfi t} {\sfi W}] of a symmetry operation [\ispecialfonts{\sfi W}] with a translation [\ispecialfonts{\sfi t}] is an operation of the same type as [\ispecialfonts{\sfi W}], with the same or a different type of symmetry element.

  • (ii) The composition [\ispecialfonts{\sfi W}' = {\sfi t} {\sfi W}] is an operation of a different type to [\ispecialfonts{\sfi W}] with the same or a different type of symmetry element.

In order to distinguish the different cases, a closer analysis of the type of a symmetry operation and its symmetry element is required. These types, however, might be obscured by two obstacles:

  • (1) The origin in the chosen coordinate system might not lie on the geometric element of the symmetry operation. For example, the symmetry operation represented by the coordinate triplet [\bar{x}+1,\bar{y}+1,\bar{z}] (cf. Section 1.4.2.3[link] ) is in fact an inversion through the point [1/2, 1/2, 0] and thus of the same type as the inversion [\{\bar{1}| 0\}] through the origin.

  • (2) The screw or glide part might not be reduced to a vector within the unit cell. For example, the symmetry operation [\bar{x},\bar{y},z+1], which is a twofold screw rotation [2\ (0,0,1) \ 0,0,z] along the c axis, is the composition of the twofold rotation [\bar{x},\bar{y},z] with the lattice translation [t(0,0,1)] along the screw axis. Although the two operations [\bar{x},\bar{y},z] and [\bar{x},\bar{y},z+1] are of different types, they are coaxial equivalents and belong to the element set of the same symmetry element (cf. Section 1.2.3[link] ).

These issues can be overcome by decomposing the translation part w of a symmetry operation [\ispecialfonts{\sfi W} = ({\bi W}, {\bi w})] into an intrinsic translation part [{\bi w}_g] which is fixed by the linear part W of [\ispecialfonts{\sfi W}] and thus parallel to the geometric element of [\ispecialfonts{\sfi W}], and a location part [{\bi w}_l], which is perpendicular to the intrinsic translation part. Note that the subspace of vectors fixed by W and the subspace perpendicular to this space of fixed vectors are complementary subspaces, i.e. their dimensions add up to 3, therefore this decomposition is always possible.

The procedure for determining the intrinsic translation part of a symmetry operation is described in Section 1.2.2.4[link] , and is based on the fact that the kth power of a symmetry operation [\ispecialfonts{\sfi W} = ({\bi W}, {\bi w})] with linear part W of order k must be a pure translation, i.e. [\ispecialfonts{\sfi W}^k = ({\bi I}, {\bi t})] for some lattice translation [{\bi t}]. The intrinsic translation part of [\ispecialfonts{\sfi W}] is then defined as [{\bi w}_g = \textstyle{{1}\over{k}} {\bi t}].

The difference [{\bi w}_l = {\bi w} - {\bi w}_g] is perpendicular to [{\bi w}_g] and it is called the location part of w. This terminology is justified by the fact that the location part can be reduced to o by an origin shift, i.e. the location part indicates whether the origin of the chosen coordinate system lies on the geometric element of [\ispecialfonts{\sfi W}].

The transformation of point coordinates and matrix–column pairs under an origin shift is explained in detail in Sections 1.5.1.3[link] and 1.5.2.3[link] , and the complete procedure for determining the additional symmetry operations will be discussed in the context of the synoptic tables in Section 1.5.4 . In this section we will restrict ourselves to a detailed discussion of two examples which illustrate typical phenomena.

Example 1

Consider a space group of type Fmm2 (42). The information on the general position and on the symmetry operations given in the space-group tables are reproduced in Fig. 1.4.2.2[link]. From this information one deduces that coset representatives with respect to the translation subgroup are the identity element [\ispecialfonts{\sfi W}_1 = x,y,z], a rotation [\ispecialfonts{\sfi W}_2 = \bar{x},\bar{y},z] with the c axis as geometric element, a reflection [\ispecialfonts{\sfi W}_3 = x,\bar{y},z] with the plane [x,0,z] as geometric element and a reflection [\ispecialfonts{\sfi W}_4 = \bar{x},y,z] with the plane [0,y,z] as geometric element (with the indices following the numbering in the table).

Composing these coset representatives with the centring translations [t(0,\textstyle{{1 \over 2}},\textstyle{{1 \over 2}})], [t(\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}})] and [t(\textstyle{{1 \over 2}},\textstyle{{1 \over 2}},0)] gives rise to elements in the same cosets, but with different types of symmetry operations and symmetry elements in several cases.

  • (i) [(0,\textstyle{{1 \over 2}},\textstyle{{1 \over 2}})]: The composition of the rotation [\ispecialfonts{\sfi W}_2] with [t(0,\textstyle{{1 \over 2}},\textstyle{{1 \over 2}})] results in the symmetry operation [\bar{x},\bar{y}+\textstyle{{1 \over 2}},z+\textstyle{{1 \over 2}}], which is a twofold screw rotation with screw axis [0,\textstyle{{1 \over 4}},z]. This means that both the type of the symmetry operation and the location of the geometric element are changed. Composing the reflection [\ispecialfonts{\sfi W}_3] with [t(0,\textstyle{{1 \over 2}},\textstyle{{1 \over 2}})] gives the symmetry operation [x,\bar{y}+\textstyle{{1 \over 2}},z+\textstyle{{1 \over 2}}], which is a c glide with the plane [x,\textstyle{{1 \over 4}},z] as geometric element, i.e. shifted by [\textstyle{{1 \over 4}}] along the b axis relative to the geometric element of [\ispecialfonts{\sfi W}_3]. In the composition of [\ispecialfonts{\sfi W}_4] with [t(0,\textstyle{{1 \over 2}},\textstyle{{1 \over 2}})], the translation lies in the plane forming the geometric element of [\ispecialfonts{\sfi W}_4]. The geometric element of the resulting symmetry operation [\bar{x},y+\textstyle{{1 \over 2}},z+\textstyle{{1 \over 2}}] is still the plane [0,y,z], but the symmetry operation is now an n glide, i.e. a glide reflection with diagonal glide vector.

  • (ii) [(\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}})]: Analogous to the first centring translation, the composition of [\ispecialfonts{\sfi W}_2] with [t(\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}})] results in a twofold screw rotation with screw axis [\textstyle{{1 \over 4}},0,z] as geometric element. The roles of the reflections [\ispecialfonts{\sfi W}_3] and [\ispecialfonts{\sfi W}_4] are interchanged, because the translation vector now lies in the plane forming the geometric element of [\ispecialfonts{\sfi W}_3]. Therefore, the composition of [\ispecialfonts{\sfi W}_3] with [t(\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}})] is an n glide with the plane [x,0,z] as geometric element, whereas the composition of [\ispecialfonts{\sfi W}_4] with [t(\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}})] is a c glide with the plane [\textstyle{{1 \over 4}},y,z] as geometric element.

  • (iii) [(\textstyle{{1 \over 2}},\textstyle{{1 \over 2}},0)]: Because this translation vector lies in the plane perpendicular to the rotation axis of [\ispecialfonts{\sfi W}_2], the composition of [\ispecialfonts{\sfi W}_2] with [t(\textstyle{{1 \over 2}},\textstyle{{1 \over 2}},0)] is still a twofold rotation, i.e. a symmetry operation of the same type, but the rotation axis is shifted by [\textstyle{{1 \over 4}}, \textstyle{{1 \over 4}}, 0] in the xy plane to become the axis [\textstyle{{1 \over 4}}, \textstyle{{1 \over 4}}, z]. The composition of [\ispecialfonts{\sfi W}_3] with [t(\textstyle{{1 \over 2}},\textstyle{{1 \over 2}},0)] results in the symmetry operation [x+\textstyle{{1 \over 2}},\bar{y}+\textstyle{{1 \over 2}},z], which is an a glide with the plane [x,\textstyle{{1 \over 4}},z] as geometric element, i.e. shifted by [\textstyle{{1 \over 4}}] along the b axis relative to the geometric element of [\ispecialfonts{\sfi W}_3]. Similarly, the composition of [\ispecialfonts{\sfi W}_4] with [t(\textstyle{{1 \over 2}},\textstyle{{1 \over 2}},0)] is a b glide with the plane [\textstyle{{1 \over 4}},y,z] as geometric element.

In this example, all additional symmetry operations are listed in the symmetry-operations block of the space-group tables of Fmm2 because they are due to compositions of the coset representatives with centring translations.

The additional symmetry operations can easily be recognized in the symmetry-element diagrams (cf. Section 1.4.2.5[link]). Fig. 1.4.2.3[link] shows the symmetry-element diagram of Fmm2 for the projection along the c axis. One sees that twofold rotation axes alternate with twofold screw axes and that mirror planes alternate with `double' or e-glide planes, i.e. glide planes with two glide vectors. For example, the dot–dashed lines at [x = \textstyle{{1 \over 4}}] and [x = \textstyle{{3}\over{4}}] in Fig. 1.4.2.3[link] represent the b and c glides with normal vector along the a axis [for a discussion of e-glide notation, see Sections 1.2.3[link] and 2.1.2[link] , and de Wolff et al., 1992[link]].

[Figure 1.4.2.3]

Figure 1.4.2.3 | top | pdf |

Symmetry-element diagram for space group Fmm2 (42) (orthogonal projection along [001]).

Example 2

In a space group of type P4mm (99), representatives of the space group with respect to the translation subgroup are the powers of a fourfold rotation and reflections with normal vectors along the a and the b axis and along the diagonals [110] and [[1\bar{1}0]] (cf. Fig. 1.4.2.4[link]).

[Figure 1.4.2.4]

Figure 1.4.2.4 | top | pdf |

General-position and symmetry-operations blocks as given in the space-group tables for space group P4mm (99).

In this case, additional symmetry operations occur although there are no centring translations. Consider for example the reflection [\ispecialfonts{\sfi W}_8] with the plane [x,x,z] as geometric element. Composing this reflection with the translation [t(1,0,0)] gives rise to the symmetry operation represented by [y+1,x,z]. This operation maps a point with coordinates [x+\textstyle{{1 \over 2}},x,z] to [x+1,x+\textstyle{{1 \over 2}},z] and is thus a glide reflection with the plane [x+\textstyle{{1 \over 2}},x,z] as geometric element and [(\textstyle{{1 \over 2}},\textstyle{{1 \over 2}},0)] as glide vector. In a similar way, composing the other diagonal reflection with translations yields further glide reflections.

These glide reflections are symmetry operations which are not listed in the symmetry-operations block, although they are clearly of a different type to the operations given there. However, in the symmetry-element diagram as shown in Fig. 1.4.2.5[link], the corresponding symmetry elements are displayed as diagonal dashed lines which alternate with the solid diagonal lines representing the diagonal reflections.

[Figure 1.4.2.5]

Figure 1.4.2.5 | top | pdf |

Symmetry-element diagram for space group P4mm (99) (orthogonal projection along [001]).

References

Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Hahn, Th., Senechal, M., Shoemaker, D. P., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1992). Symbols for symmetry elements and symmetry operations. Final report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A48, 727–732.








































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