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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, p. 604   | 1 | 2 |

Section 4.6.3.3.1.2. 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

4.6.3.3.1.2. Diffraction symmetry

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The possible Laue symmetry group [K^{\rm 3D}] of the Fourier module [M^{*} = \{{\bf H}^{\parallel} = {\textstyle\sum_{i = 1}^{4}} h_{i}{\bf a}_{i}^{*} | h_{i} \in {\bb Z}\}] is any one of the direct product [K^{\rm 3D} = K^{\rm 2D} \otimes K^{\rm 1D} \otimes \bar{1}]. [K^{\rm 2D}] corresponds to one of the ten crystallographic 2D point groups, [K^{\rm 1D} = \{1\}] in the general case of a quasiperiodic stacking of periodic layers. Consequently, the nine Laue groups [\bar{1}], [2/m], mmm, 4/m, 4/mmm, [\bar{3}], [\bar{3}m], [6/m] and [6/mmm] are possible. These are all 3D crystallographic Laue groups except for the two cubic ones.

The (unweighted) Fourier module shows only 2D lattice symmetry. In the third dimension, the submodule [M_{1}^{*}] remains invariant under the scaling symmetry operation [S^{n} M_{1}^{*} = \tau^{n} M_{1}^{*}] with [n \in {\bb Z}]. The scaling symmetry operators [S^{n}] form an infinite group [s = \{\ldots, S^{-1}, S^{0}, S^{1}, \ldots \}] of reciprocal-basis transformations [S^{n}] in superspace, [\displaylines{S^{n} = \left(\matrix{0 &1 &0 &0\cr 1 &1 &0 &0\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}\right)_{D}^{n}, \ S^{-1} = \left(\matrix{\bar{1} &1 &0 &0\cr 1 &0 &0 &0\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}\right)_{D}, \cr S^{0} = \left(\matrix{1 &0 &0 &0\cr 0 &1 &0 &0\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}\right)_{D},}]and act on the reciprocal basis [{\bf d}_{i}^{*}] in superspace.








































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