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 [K^{\rm L} = \{R\}] of the Fourier module [M^{*}], [M^{*} = \left\{{\bf H} = {\textstyle\sum\limits_{i = 1}^{3}} h_{i} {\bf a}_{i}^{*} + {\textstyle\sum\limits_{j = 1}^{d}} m_{j} {\bf q}_{j} = {\textstyle\sum\limits_{i = 1}^{3 + d}} h_{i} {\bf a}_{i}^{*}\right\}, \Lambda^{*} = \left\{{\bf H} = {\textstyle\sum\limits_{i = 1}^{3}} h_{i} {\bf a}_{i}^{*}\right\},]is isomorphous to or a subgroup of one of the 11 3D crystallographic Laue groups leaving [\Lambda^{*}] invariant. The action of the point-group symmetry operators R on the reciprocal basis [{\bf a}_{i}^{*}, i = 1, \ldots, 3 + d], can be written as [R{\bf a}_{i}^{*} = {\textstyle\sum\limits_{j = 1}^{3 + d}} \Gamma_{ij}^{T} (R) {\bf a}_{j}^{*}, i = 1, \ldots, 3 + d.]

The [(3 + d) \times (3 + d)] matrices [\Gamma^{T}(R)] 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 [(R^{\parallel}, R^{\perp})] 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. [\Gamma^{T}(R)] is the transpose of [\Gamma (R)] which acts on vector components in direct space.

For the [(3 + d)\hbox{D}] direct-space (superspace) symmetry operator [(R_{\rm s}, {\bf t}_{\rm s})] and its matrix representation [\Gamma (R_{\rm s}, {\bf t}_{\rm s})] on Σ, the following decomposition can be performed: [\Gamma (R_{\rm s}) = \pmatrix{\Gamma^{\parallel}(R) &0\cr \Gamma^{\rm M}(R) &\Gamma^{\perp}(R)\cr} \hbox{ and } {\bf t}_{\rm s} = ({\bf t}_{3}, {\bf t}_{d}).][\Gamma^{\parallel}(R)] is a [3 \times 3] matrix, [\Gamma^{\perp}(R)] is a [d \times d] matrix and [\Gamma^{\rm M}(R)] is a [d \times 3] matrix. The translation operator [{\bf t}_{\rm s}] consists of a 3D vector [{\bf t}_{3}] and a dD vector [{\bf t}_{d}]. According to Janner & Janssen (1979)[link], [\Gamma^{\rm M}(R)] can be derived from [\Gamma^{\rm M}(R) = \sigma \Gamma^{\parallel}(R) - \Gamma^{\perp}(R) \sigma]. [\Gamma^{\rm M}(R)] has integer elements only as it contains components of primitive-lattice vectors of [\Lambda^{*}], whereas σ in general consists of a rational and an irrational part: [\sigma = \sigma^{\rm i} + \sigma^{\rm r}]. Thus, only the rational part gives rise to nonzero entries in [\Gamma^{\rm M}(R)]. With the order of the Laue group denoted by N, one obtains [\sigma^{\rm i} \equiv (1/N) {\textstyle\sum_{R}} \Gamma^{\perp}(R) \sigma \Gamma^{\parallel}(R)^{-1}], where [\Gamma^{\perp}(R) \sigma^{\rm i} \Gamma^{\parallel}(R)^{-1} = \sigma^{\rm i}], implying that [\Gamma^{\rm M}(R) =\sigma^{\rm r} \Gamma^{\parallel}(R)] [ - \ \Gamma^{\perp}(R) \sigma^{\rm r}] and [0 = \sigma^{\rm i} \Gamma^{\parallel}(R) - \Gamma^{\perp}(R) \sigma^{\rm i}].

Example

In the case of a 3D IMS with 1D modulation [(d = 1)] the [3 \times d] matrix [\sigma = \pmatrix{\alpha_{1}\cr \alpha_{2}\cr \alpha_{3}\cr}]has the components of the wavevector [{\bf q} = {\textstyle\sum_{i = 1}^{3}} \alpha_{i} {\bf a}_{i}^{*} = {\bf q}^{\rm i} + {\bf q}^{\rm r}]. [\Gamma^{\perp}(R) = \varepsilon = \pm 1] because for [d = 1], q can only be transformed into [\pm {\bf q}]. Corresponding to [{\bf q}^{\rm i} \equiv (1/N) {\textstyle\sum_{R}} \varepsilon R {\bf q}], one obtains [R^{T} {\bf q}^{\rm i} \equiv \varepsilon {\bf q}^{\rm i}] (modulo [\Lambda^{*}]). The [3 \times 1] row matrix [\Gamma^{\rm M}(R)] is equivalent to the difference vector between [R^{T} {\bf q}] and [\varepsilon {\bf q}] (Janssen et al., 2004[link]).

For a monoclinic modulated structure with point group [2/m] for [M^{*}] (unique axis [{\bf a}_{3}]) and satellite vector [{\bf q} = (1/2) {{\bf a}_{1}}^{*} + \alpha_{3} {{\bf a}_{3}}^{*}], with [\alpha_{3}] an irrational number, one obtains [\eqalign{{\bf q}^{\rm i} &\equiv (1/N) {\textstyle\sum\limits_{R}} \varepsilon R {\bf q}\cr &= {\textstyle{1 \over 4}} \left(+1 \cdot \pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr} \pmatrix{1/2\cr 0\cr \alpha_{3}\cr}\right.\cr &\quad+ 1 \cdot \pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr} \pmatrix{1/2\cr 0\cr \alpha_{3}\cr}- 1 \cdot \pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr} \pmatrix{1/2\cr 0\cr \alpha_{3}\cr}\cr &\quad \left.- 1 \cdot \pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr} \pmatrix{1/2\cr 0\cr \alpha_{3}\cr}\right)\cr &= \pmatrix{0\cr 0\cr \alpha_{3}\cr}.}]From the relations [R^{T} {\bf q}^{\rm i} \equiv \varepsilon {\bf q}^{\rm i}\; (\hbox{modulo } \Lambda^{*})], it can be shown that the symmetry operations 1 and 2 are associated with the perpendicular-space transformations [\varepsilon = 1], and m and [\bar{1}] with [\varepsilon = -1]. The matrix [\Gamma^{\rm M}(R)] is given by [\eqalign{\Gamma^{\rm M}(2) &= \sigma^{\rm r} \Gamma^{\parallel}(2) - \Gamma^{\perp}(2) \sigma^{\rm r}\cr &= \pmatrix{1/2\cr 0\cr 0\cr} \pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr} - (+1) \pmatrix{1/2\cr 0\cr 0\cr} = \pmatrix{\bar{1}\cr 0\cr 0\cr}}]for the operation 2, for instance.

The matrix representations [\Gamma^{T}(R_{\rm s})] of the symmetry operators R in reciprocal [(3 + d)\hbox{D}] superspace decompose according to [\Gamma^{T}(R_{\rm s}) = \pmatrix{\Gamma^{\|T}(R) &\Gamma^{{\rm M}T}(R)\cr 0 &\Gamma^{\perp T}(R)\cr}.]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 [(3 + 1)\hbox{D}] superspace groups (Janssen et al., 2004[link]). 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)[link].

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.








































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