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. 91-92

Section 1.5.4.1.1. Determining the type of a symmetry operation

B. Souvignier,c G. Chapuisd and H. Wondratscheka

1.5.4.1.1. Determining the type of a symmetry operation

| top | pdf |

In this section, a procedure for determining the types of symmetry operations and the corresponding symmetry elements is explained. It is a development of the method of geometrical interpretation discussed in Section 1.2.2.4[link] . The procedure is based on the origin-shift transformations discussed in Sections 1.5.1[link] and 1.5.2[link], and provides an efficient way of analysing the additional symmetry operations and symmetry elements. The key to the procedure is the decomposition of the translation part [{\bi 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 [{\bi 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 space fixed by [{\bi W}] and the space perpendicular to this fixed space are complementary, i.e. their dimensions add up to 3, therefore this decomposition is always possible.

As described in Section 1.2.2.4[link] , the determination of the intrinsic translation part of a symmetry operation [\ispecialfonts{\sfi W} = ({\bi W}, {\bi w})] with linear part [{\bi W}] of order k is based on the fact that the kth power of [\ispecialfonts{\sfi W}] must be a pure translation, i.e. [\ispecialfonts{\sfi W}^k = ({\bi I}, {\bi t})] for some lattice translation [\ispecialfonts{\sfi t}]. The intrinsic translation part of [\ispecialfonts{\sfi W}] is then defined as [{\bi w}_g = ({1}/{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 [{\bi w}]. This terminology is justified by the following observation: As explained in detail in Sections 1.5.1.3[link] and 1.5.2.3[link], under an origin shift by [{\bi p}], a column [{\bi x}] of point coordinates is transformed to[{\bi x}' = ({\bi I}, -{\bi p}) {\bi x} = ({\bi I}, {\bi p})^{-1} {\bi x},]making in particular [{\bi p}] the new origin, and a matrix–column pair [({\bi W}, {\bi w})] is transformed to[({\bi W}', {\bi w}') = ({\bi I}, {\bi p})^{-1} ({\bi W}, {\bi w}) ({\bi I}, {\bi p}).]Applied to the symmetry operation [({\bi W}, {\bi w}_l)], known as the reduced symmetry operation in which the full translation part is replaced by the location part (thereby neglecting the intrinsic translation part), an origin shift by [{\bi p}] results in[\eqalign{({\bi I}, {\bi p})^{-1} ({\bi W}, {\bi w}_l) ({\bi I}, {\bi p}) &= ({\bi I}, -{\bi p}) ({\bi W}, {\bi w}_l) ({\bi I}, {\bi p}) \cr &= ({\bi W}, {\bi W} {\bi p} -{\bi p}+{\bi w}_l) \cr&= ({\bi W}, ({\bi W}-{\bi I}) {\bi p}+{\bi w}_l).}]This means that if it is possible to find an origin shift p such that [({\bi I}-{\bi W}) {\bi p} = {\bi w}_l], then with respect to the new origin the reduced symmetry operation [({\bi W}, {\bi w}_l)] is transformed to [({\bi W}, {\bi o})]. But since the subspace perpendicular to the fixed space of [{\bi W}] clearly does not contain any vector fixed by [{\bi W}], the restriction of [{\bi I} - {\bi W}] to this subspace is an invertible linear transformation, and therefore for every location part [{\bi w}_l] there is indeed a suitable [{\bi p}] perpendicular to the fixed space of [{\bi W}] such that [({\bi I} - {\bi W}) {\bi p} = {\bi w}_l].

The fact that an origin shift by [{\bi p}] transforms the translation part of the reduced symmetry operation [({\bi W}, {\bi w}_l)] to [{\bi o}] is equivalent to [{\bi p}] being a fixed point of [({\bi W}, {\bi w}_l)], which can also be seen directly because[({\bi W}, {\bi w}_l) {\bi p} = {\bi W} {\bi p} + {\bi w}_l = {\bi W} {\bi p} + ({\bi I} - {\bi W}) {\bi p} = {\bi p}.]Note that for one fixed point [{\bi p}] of the reduced symmetry operation [({\bi W}, {\bi w}_l)], the full set of fixed points, as defined in Section 1.2.4[link] , is obtained by adding [{\bi p}] to the fixed vectors of [{\bi W}], because for an arbitrary fixed point [{\bi p}_F] of [({\bi W}, {\bi w}_l)] one has [{\bi W} {\bi p}_F + {\bi w}_l = {\bi p}_F] and since also [{\bi W} {\bi p} + {\bi w}_l = {\bi p}] one finds [{\bi W} ({\bi p}_F - {\bi p}) = {\bi p}_F - {\bi p}], i.e. the difference between two fixed points is a vector that is fixed by [{\bi W}]. In other words, the geometric element of [({\bi W}, {\bi w}_l)] is the space fixed by [{\bi W}], translated such that it runs through [{\bi p}].

Finally, in order to determine the symmetry element of the symmetry operation correctly, it may be necessary to reduce the intrinsic translation part [{\bi w}_g] by a lattice translation in the fixed space of [{\bi W}].

Summarizing, the types of symmetry operations [\ispecialfonts{\sfi W} = ({\bi W}, {\bi w})] and their symmetry elements can be identified as follows:

  • (i) Decompose the translation part [{\bi w}] as [{\bi w} = {\bi w}_g + {\bi w}_l], where [{\bi w}_g] and [{\bi w}_l] are mutually perpendicular and the intrinsic translation part [{\bi w}_g] is fixed by the linear part [{\bi W}] of [\ispecialfonts{\sfi W}].

  • (ii) Determine a shift of origin [{\bi p}] such that [({\bi I} - {\bi W}) {\bi p} = {\bi w}_l], i.e. such that [{\bi p}] is a fixed point of the reduced operation [({\bi W}, {\bi w}_l)].

  • (iii) For the correct determination of the defining operation of the symmetry element it may be necessary to reduce the intrinsic translation part [{\bi w}_g] by a lattice translation in the fixed space of [{\bi W}], thus yielding a coplanar or coaxial equivalent symmetry operation.

This analysis allows one to read off the types of the symmetry operations and of the corresponding symmetry elements that occur for the coset [\ispecialfonts{\cal T}{\sfi W}] of [\ispecialfonts{\sfi W}]. The following two sections provide examples illustrating that in some cases the coset does not contain symmetry operations belonging to symmetry elements of different type, while in others it does.








































to end of page
to top of page