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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 610-611   | 1 | 2 |

Section 4.6.3.3.2.2. 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

4.6.3.3.2.2. Diffraction symmetry

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The diffraction symmetry of decagonal phases can be described by the Laue groups [10/mmm] or [10/m]. The set of all vectors H forms a Fourier module [M^{*} = \{{\bf H}^{\parallel} = {\textstyle\sum_{i = 1}^{5}} h_{i}{\bf a}_{i}^{*} | h_{i} \in {\bb Z}\}] of rank 5 in physical space which can be decomposed into two submodules [M^{*} = M_{1}^{*} \oplus M_{2}^{*}]. [M_{1}^{*} = \{h_{1}{\bf a}_{1}^{*} + h_{2}{\bf a}_{2}^{*} + h_{3}{\bf a}_{3}^{*} + h_{4}{\bf a}_{4}^{*}\}] corresponds to a [{\bb Z}] module of rank 4 in a 2D subspace, [M_{2}^{*} = \{h_{5}{\bf a}_{5}^{*}\}] corresponds to a [{\bb Z}] module of rank 1 in a 1D subspace. Consequently, the first submodule can be considered as a projection from a 4D reciprocal lattice, [M_{1}^{*} = \pi^{\parallel} (\Sigma^{*})], while the second submodule is of the form of a regular 1D reciprocal lattice, [M_{2}^{*} = \Lambda^{*}]. The diffraction pattern of the Penrose tiling decorated with equal point scatterers on its vertices is shown in Fig. 4.6.3.17[link]. All Bragg reflections within [10^{-2} |F({\bf 0})|^{2} \,\lt\, |F({\bf H})|^{2} \,\lt\, |F({\bf 0})|^{2}] are depicted. Without intensity-truncation limit, the diffraction pattern would be densely filled with discrete Bragg reflections. To illustrate their spatial and intensity distribution, an enlarged section of Fig. 4.6.3.17[link] is shown in Fig. 4.6.3.18[link]. This picture shows all Bragg reflections within [10^{-4} |F({\bf 0})|^{2} \,\lt\, |F({\bf H})|^{2} \,\lt\, |F({\bf 0})|^{2}]. The projected 4D reciprocal-lattice unit cell is drawn and several reflections are indexed. All reflections are arranged along lines in five symmetry-equivalent orientations. The perpendicular-space diffraction patterns (Figs. 4.6.3.19[link] and 4.6.3.20[link]) show a characteristic star-like distribution of the Bragg reflections. This is a consequence of the pentagonal shape of the atomic surfaces: the Fourier transform of a pentagon has a star-like distribution of strong Fourier coefficients.

[Figure 4.6.3.17]

Figure 4.6.3.17 | top | pdf |

Schematic diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs ar = 4.04 Å). All reflections are shown within [10^{-2} |F({\bf 0})|^{2} \lt |F({\bf H})|^{2} \lt |F({\bf 0})|^{2}] and [0 \leq |{\bf H}^{\parallel}| \leq] [2.5\;\hbox{\AA}^{-1}].

[Figure 4.6.3.18]

Figure 4.6.3.18 | top | pdf |

Enlarged section of Fig. 4.6.3.17[link]. All reflections shown are selected within the given limits from a data set within [10^{-4} |F({\bf 0})|^{2} \;\lt\; |F({\bf H})|^{2} \;\lt\; |F({\bf 0})|^{2}] and [0 \leq |{\bf H}^{\parallel}| \leq 2.5\;\hbox{\AA}^{-1}]. The projected 4D reciprocal-lattice unit cell is drawn and several reflections are indexed.

[Figure 4.6.3.19]

Figure 4.6.3.19 | top | pdf |

The perpendicular-space diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs ar = 4.04 Å). All reflections are shown within [10^{-4} |F({\bf 0})|^{2} \;\lt\; |F({\bf H})|^{2} \;\lt\; |F({\bf 0})|^{2}] and [0 \leq |{\bf H}^{\parallel}| \leq 2.5\;\hbox{\AA}^{-1}].

[Figure 4.6.3.20]

Figure 4.6.3.20 | top | pdf |

Enlarged section of Fig. 4.6.3.19[link] showing the projected 4D reciprocal-lattice unit cell.

The 5D decagonal space groups that may be of relevance for the description of decagonal phases are listed in Table 4.6.3.1.[link] These space groups are a subset of all 5D decagonal space groups fulfilling the condition that the 5D point groups they are associated with are isomorphous to the 3D point groups describing the diffraction symmetry. Their structures are comparable to 3D hexagonal groups. Hence, only primitive lattices exist. The orientation of the symmetry elements in the 5D space is defined by the isomorphism of the 3D and 5D point groups. However, the action of the tenfold rotation is different in the subspaces [{\bf V}^{\parallel}] and [{\bf V}^{\perp}]: a rotation of [\pi/5] in [{\bf V}^{\parallel}] is correlated with a rotation of [3\pi/5] in [{\bf V}^{\perp}]. The reflection and inversion operations are equivalent in both subspaces.

Table 4.6.3.1| top | pdf |
3D point groups of order k describing the diffraction symmetry and corresponding 5D decagonal space groups with reflection conditions (see Rabson et al., 1991[link])

3D point groupk5D space groupReflection condition
[\displaystyle{{10 \over m} {2 \over m} {2 \over m}}] 40 [\displaystyle{P{10 \over m} {2 \over m} {2 \over m}}] No condition
[\displaystyle{P{10 \over m} {2 \over c} {2 \over c}}] [h_{1}h_{2}h_{2}h_{1}h_{5}]: [h_{5} = 2n]
[h_{1}h_{2}\overline{h_{2}}\overline{h_{1}}h_{5}]: [h_{5} = 2n]
[\displaystyle{P{10_{5} \over m} {2 \over m} {2 \over c}}] [h_{1}h_{2}\overline{h_{2}}\overline{h_{1}}h_{5}]: [h_{5} = 2n]
[\displaystyle{P{10_{5} \over m} {2 \over c} {2 \over m}}] [h_{1}h_{2}h_{2}h_{1}h_{5}]: [h_{5} = 2n]
[\displaystyle{{10 \over m}}] 20 [\displaystyle{P{10 \over m}}] No condition
[\displaystyle{P{10_{5} \over m}}] [0000h_{5}]: [h_{5} = 2n]
1022 20 P1022 No condition
[P10_{j}22] [0000h_{5}]: [jh_{5} = 10n]
10mm 20 P10mm No condition
P10cc [h_{1}h_{2}h_{2}h_{1}h_{5}]: [h_{5} = 2n]
  [h_{1}h_{2}\overline{h_{2}}\overline{h_{1}}h_{5}]: [h_{5} = 2n]
[P10_{5}mc] [h_{1}h_{2}\overline{h_{2}}\overline{h_{1}}h_{5}]: [h_{5} = 2n]
[P10_{5}cm] [h_{1}h_{2}h_{2}h_{1}h_{5}]: [h_{5} = 2n]
[\overline{10}m2] 20 [P\overline{10}m2] No condition
[P\overline{10}c2] [h_{1}h_{2}h_{2}h_{1}h_{5}]: [h_{5} = 2n]
[P\overline{10}2m] No condition
[P\overline{10}2c] [h_{1}h_{2}\overline{h_{2}}\overline{h_{1}}h_{5}]: [h_{5} = 2n]
[\overline{10}] 10 [P\overline{10}] No condition
10 10 P10 No condition
[P10_{j}] [0000h_{5}]: [jh_{5} = 10n]








































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