International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 599-600   | 1 | 2 |

## Section 4.6.3.1.2. Diffraction symmetry

W. Steurera* and T. Haibacha

aLaboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland
Correspondence e-mail:  walter.steurer@mat.ethz.ch

#### 4.6.3.1.2. Diffraction symmetry

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The Laue symmetry group of the Fourier module , is isomorphous to or a subgroup of one of the 11 3D crystallographic Laue groups leaving invariant. The action of the point-group symmetry operators R on the reciprocal basis , can be written as

The matrices form a finite group of integral matrices which are reducible, since R is already an orthogonal transformation in 3D physical space. Consequently, R can be expressed as pair of orthogonal transformations in 3D physical and dD perpendicular space, respectively. Owing to their mutual orthogonality, no symmetry relationship exists between the set of main reflections and the set of satellite reflections. is the transpose of which acts on vector components in direct space.

For the direct-space (superspace) symmetry operator and its matrix representation on Σ, the following decomposition can be performed: is a matrix, is a matrix and is a matrix. The translation operator consists of a 3D vector and a dD vector . According to Janner & Janssen (1979), can be derived from . has integer elements only as it contains components of primitive-lattice vectors of , whereas σ in general consists of a rational and an irrational part: . Thus, only the rational part gives rise to nonzero entries in . With the order of the Laue group denoted by N, one obtains , where , implying that and .

#### Example

In the case of a 3D IMS with 1D modulation the matrix has the components of the wavevector . because for , q can only be transformed into . Corresponding to , one obtains (modulo ). The row matrix is equivalent to the difference vector between and (Janssen et al., 2004).

For a monoclinic modulated structure with point group for (unique axis ) and satellite vector , with an irrational number, one obtains From the relations , it can be shown that the symmetry operations 1 and 2 are associated with the perpendicular-space transformations , and m and with . The matrix is given by for the operation 2, for instance.

The matrix representations of the symmetry operators R in reciprocal superspace decompose according to Phase relationships between modulation functions of symmetry-equivalent atoms can give rise to systematic extinctions of different classes of satellite reflections. The extinction rules may include indices of both main and satellite reflections. A full list of systematic absences is given in the table of superspace groups (Janssen et al., 2004). Thus, once point symmetry and systematic absences are found, the superspace group can be obtained from the tables in a way analogous to that used for regular 3D crystals. A different approach for the symmetry description of IMSs from the 3D Fourier-space perspective has been given by Dräger & Mermin (1996).

### References

Dräger, J. & Mermin, N. D. (1996). Superspace groups without the embedding: the link between superspace and Fourier-space crystallography. Phys. Rev. Lett. 76, 1489–1492.
Janner, A. & Janssen, T. (1979). Superspace groups. Physica A, 99, 47–76.
Janssen, T., Janner, A., Looijenga-Vos, A. & de Wolff, P. M. (2004). Incommensurate and commensurate modulated crystal structures. In International Tables for Crystallography, Vol. C, edited by E. Prince, ch. 9.8. Dordrecht: Kluwer Academic Publishers.