Diffraction symmetry

Steurer, W.; Haibach, T.

doi:10.1107/97809553602060000568

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

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International Tables for Crystallography (2006). Vol. B. ch. 4.6, pp. 495-496 | 1 | 2 |

Section 4.6.3.1.2. Diffraction symmetry

W. Steurer^a ^* and T. Haibach^a

^aLaboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland
Correspondence e-mail: w.steurer@kristall.erdw.ethz.ch

4.6.3.1.2. Diffraction symmetry

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The Laue symmetry group $[K^{L} = \{R\}]$ 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 $[(R_{s}, {\bf t}_{s})]$ and its matrix representation $[\Gamma (R_{s}, {\bf t}_{s})]$ on Σ, the following decomposition can be performed: $[\Gamma (R_{s}) = \pmatrix{\Gamma^{\parallel}(R) &0\cr \Gamma^{M}(R) &\Gamma^{\perp}(R)\cr} \hbox{ and } {\bf t}_{s} = ({\bf t}_{3}, {\bf t}_{d}).]$ $[\Gamma^{\parallel}(R)]$ is a $[3 \times 3]$ matrix, $[\Gamma^{\perp}(R)]$ is a $[d \times d]$ matrix and $[\Gamma^{M}(R)]$ is a $[d \times 3]$ matrix. The translation operator $[{\bf t}_{s}]$ consists of a 3D vector $[{\bf t}_{3}]$ and a dD vector $[{\bf t}_{d}]$ . According to Janner & Janssen (1979), $[\Gamma^{M}(R)]$ can be derived from $[\Gamma^{M}(R) = \sigma \Gamma^{\parallel}(R) - \Gamma^{\perp}(R) \sigma]$ . $[\Gamma^{M}(R)]$ 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: $[\sigma = \sigma^{i} + \sigma^{r}]$ . Thus, only the rational part gives rise to nonzero entries in $[\Gamma^{M}(R)]$ . With the order of the Laue group denoted by N, one obtains $[\sigma^{i} \equiv (1/N) {\textstyle\sum_{R}} \Gamma^{\perp}(R) \sigma \Gamma^{\parallel}(R)^{-1}]$ , where $[\Gamma^{\perp}(R) \sigma^{i} \Gamma^{\parallel}(R)^{-1} = \sigma^{i}]$ , implying that $[\Gamma^{M}(R) = \sigma^{r} \Gamma^{\parallel}(R) - \Gamma^{\perp}(R) \sigma^{r}]$ and $[0 = \sigma^{i} \Gamma^{\parallel}(R) - \Gamma^{\perp}(R) \sigma^{i}]$ .

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 $[{\bf q} = {\textstyle\sum_{i = 1}^{3}} \alpha_{i} {\bf a}_{i}^{*} = {\bf q}^{i} + {\bf q}^{r}]$ . $[\Gamma^{\perp}(R) = \varepsilon = \pm 1]$ because for [d = 1] , q can only be transformed into $[\pm {\bf q}]$ . Corresponding to $[{\bf q}^{i} \equiv (1/N) {\textstyle\sum_{R}} \varepsilon R {\bf q}]$ , one obtains $[R^{T} {\bf q}^{i} \equiv \varepsilon {\bf q}^{i}]$ (modulo $[\Lambda^{*}]$ ). The $[3 \times 1]$ row matrix $[\Gamma^{M}(R)]$ is equivalent to the difference vector between $[R^{T} {\bf q}]$ and $[\varepsilon {\bf q}]$ (Janssen et al., 2004).

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 $[\eqalign{{\bf q}^{i} &\equiv (1/N) {\textstyle\sum\limits_{R}} \varepsilon R {\bf q}\cr &= {1 \over 4} \left(+1 \cdot \pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr} \pmatrix{1/2\cr 0\cr \alpha_{3}\cr}\right.\cr &\quad+ 1 \cdot \pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr} \pmatrix{1/2\cr 0\cr \alpha_{3}\cr}- 1 \cdot \pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr} \pmatrix{1/2\cr 0\cr \alpha_{3}\cr}\cr &\quad \left.- 1 \cdot \pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr} \pmatrix{1/2\cr 0\cr \alpha_{3}\cr}\right)\cr &= \pmatrix{0\cr 0\cr \alpha_{3}\cr}.}]$ From the relations $[R^{T} {\bf q}^{i} \equiv \varepsilon {\bf q}^{i}\; (\hbox{modulo } \Lambda^{*})]$ , 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 $[\Gamma^{M}(R)]$ is given by $[\eqalign{\Gamma^{M}(2) &= \sigma^{r} \Gamma^{\parallel}(2) - \Gamma^{\perp}(2) \sigma^{r}\cr &= \pmatrix{1/2\cr 0\cr 0\cr} \pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr} - (+1) \pmatrix{1/2\cr 0\cr 0\cr} = \pmatrix{\bar{1}\cr 0\cr 0\cr}}]$ for the operation 2, for instance.

The matrix representations $[\Gamma^{T}(R_{s})]$ of the symmetry operators R in reciprocal $[(3 + d)\hbox{D}]$ superspace decompose according to $[\Gamma^{T}(R_{s}) = \pmatrix{\Gamma^{\|T}(R) &\Gamma^{MT}(R)\cr 0 &\Gamma^{\perp T}(R)\cr}.]$ 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). 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.Google Scholar

Janner, A. & Janssen, T. (1979). Superspace groups. Physica A, 99, 47–76.Google Scholar

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.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.6, pp. 495-496