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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 619-621   | 1 | 2 |

Section 4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image

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

4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image

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The weighted 3D reciprocal space [M^{*} = \{{\bf H}^{\parallel} =] [{\textstyle\sum_{i = 1}^{6}} h_{i} {\bf a}_{i}^{*} | h_{i} \in {\bb Z}\}] exhibits the icosahedral point symmetry [K = m \bar{3} \bar{5}]. It is invariant under the action of the scaling matrix [S^{3}]: [\displaylines{S = {\textstyle{1 \over 2}} \pmatrix{1\hfill &1\hfill &1\hfill &1\hfill &1\hfill &1\hfill\cr 1\hfill &1\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill\cr 1\hfill &1\hfill &1\hfill &1\hfill &-1\hfill &-1\hfill\cr 1\hfill &-1\hfill &1\hfill &1\hfill &1\hfill &-1\hfill\cr 1\hfill &-1\hfill &-1\hfill &1\hfill &1\hfill &1\hfill\cr 1\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill &1\hfill\cr}_{D}, S^{3} = \pmatrix{2\hfill &1\hfill &1\hfill &1\hfill &1\hfill &1\hfill\cr 1\hfill &2\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill\cr 1\hfill &1\hfill &2\hfill &1\hfill &-1\hfill &-1\hfill\cr 1\hfill &-1\hfill &1\hfill &2\hfill &1\hfill &-1\hfill\cr 1\hfill &-1\hfill &-1\hfill &1\hfill &2\hfill &1\hfill\cr 1\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill &2\hfill\cr}_{D},\cr \pmatrix{2\hfill &1\hfill &1\hfill &1\hfill &1\hfill &1\hfill\cr 1\hfill &2\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill\cr 1\hfill &1\hfill &2\hfill &1\hfill &-1\hfill &-1\hfill\cr 1\hfill &-1\hfill &1\hfill &2\hfill &1\hfill &-1\hfill\cr 1\hfill &-1\hfill &-1\hfill &1\hfill &2\hfill &1\hfill\cr 1\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill &2\hfill\cr}_{D} \pmatrix{{\bf a}_{1}^{*}\cr {\bf a}_{2}^{*}\cr {\bf a}_{3}^{*}\cr {\bf a}_{4}^{*}\cr {\bf a}_{5}^{*}\cr {\bf a}_{6}^{*}\cr} = \tau^{3} \pmatrix{{\bf a}_{1}^{*}\cr {\bf a}_{2}^{*}\cr {\bf a}_{3}^{*}\cr {\bf a}_{4}^{*}\cr {\bf a}_{5}^{*}\cr {\bf a}_{6}^{*}\cr}.}]The scaling transformation [(S^{-3})^{T}] leaves a primitive 6D reciprocal lattice invariant as can easily be seen from its application on the indices: [\openup2pt{\pmatrix{h^{'}_{1}\cr h^{'}_{2}\cr h^{'}_{3}\cr h^{'}_{4}\cr h^{'}_{5}\cr h^{'}_{6}\cr}} = \pmatrix{-2\hfill &1\hfill &1\hfill &1\hfill &1\hfill &1\hfill\cr 1\hfill &-2\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill\cr 1\hfill &1\hfill &-2\hfill &1\hfill &-1\hfill &-1\hfill\cr 1\hfill &-1\hfill &1\hfill &-2\hfill &1\hfill &-1\hfill\cr 1\hfill &-1\hfill &-1\hfill &1\hfill &-2\hfill &1\hfill\cr 1\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill &-2\hfill\cr}_{D} \pmatrix{h_{1}\cr h_{2}\cr h_{3}\cr h_{4}\cr h_{5}\cr h_{6}\cr}.]The matrix [(S^{-1})^{T}] leaves [M^{*} = \{{\bf H}^{\parallel} = {\textstyle\sum_{i = 1}^{6}} h_{i} {\bf a}_{i}^{*} | h_{i} \in {\bb Z}\}] invariant, [\openup2pt\pmatrix{h^{'}_{1}\cr h^{'}_{2}\cr h^{'}_{3}\cr h^{'}_{4}\cr h^{'}_{5}\cr h^{'}_{6}\cr} = {\textstyle{1 \over 2}} \pmatrix{-1\hfill &1\hfill &1\hfill &1\hfill &1\hfill &1\hfill\cr 1\hfill &-1\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill\cr 1\hfill &1\hfill &-1\hfill &1\hfill &-1\hfill &-1\hfill\cr 1\hfill &-1\hfill &1\hfill &-1\hfill &1\hfill &-1\hfill\cr 1\hfill &-1\hfill &-1\hfill &1\hfill &-1\hfill &1\hfill\cr 1\hfill &1\hfill &-1\hfill &-1\hfill &1\hfill &-1\hfill\cr}_{D} \pmatrix{h_{1}\cr h_{2}\cr h_{3}\cr h_{4}\cr h_{5}\cr h_{6}\cr},]for any [{\bf H} = {\textstyle\sum_{i = 1}^{6}} h_{i} {\bf d}_{i}^{*}] with [h_{i}] all even or all odd, corresponding to a 6D face-centred hypercubic lattice. In a second case the sum [{\textstyle\sum_{i = 1}^{6}} h_{i}] is even, corresponding to a 6D body-centred hypercubic lattice. Block-diagonalization of the matrix S decomposes it into two irreducible representations. With [WSW^{-1} = S_{V} = S_{V}^{\parallel} \oplus S_{V}^{\perp}] we obtain [S_{V} = \pmatrix{\tau &0 &0{\,\vrule height 8pt depth 8pt} &0 &0 &0\cr\noalign{\vskip -5pt} 0 &\tau &0{\,\vrule height 8pt depth 8pt} &0 &0 &0\cr\noalign{\vskip -5pt} 0 &0 &\tau{\hskip 1.7pt}{\vrule height 8pt depth 8pt} &0 &0 &0\cr\noalign{\vskip -4pt}\noalign{\hrule}\noalign{\vskip 2pt} 0 &0 &0{\,\vrule height 8pt depth 8pt} &-1/\tau &0 &0\cr\noalign{\vskip -5pt} 0 &0 &0{\,\vrule height 8pt depth 8pt} &0 &-1/\tau &0\cr\noalign{\vskip -5pt} 0 &0 &0{\,\vrule height 8pt depth 2pt} &0 &0 &-1/\tau\cr}_{V} = \pmatrix{S^{\parallel}{\hskip 0.7pt}{\vrule height 8pt depth 6pt} &{\hskip -6pt}0\cr\noalign{\vskip -2pt}\noalign{\hrule}\noalign{\vskip 1pt}0{\hskip 5.0pt}{\vrule height 10pt depth 2pt} &S^{\perp}\cr}_{V},]the scaling properties in the two 3D subspaces: scaling by a factor τ in parallel space corresponds to a scaling by a factor [(-\tau)^{-1}] in perpendicular space. For the intensities of the scaled reflections analogous relationships are valid, as discussed for decagonal phases (Figs. 4.6.3.36[link] and 4.6.3.37[link], Section 4.6.3.3.2.5[link]).

[Figure 4.6.3.36]

Figure 4.6.3.36 | top | pdf |

Parallel-space distribution of (a) positive and (b) negative structure factors of the 3D Penrose tiling of the 6D P lattice type decorated with point atoms (edge lengths of the Penrose unit rhombohedra ar = 5.0 Å). The magnitudes of the structure factors are indicated by the diameters of the filled circles. All reflections are shown within [10^{-4} |F({\bf 0})|^{2} \;\lt\; |F({\bf H})|^{2} \;\lt\; |F({\bf 0})|^{2}] and [-6 \leq h_{i} \leq 6, i = 1,\ldots, 6].

[Figure 4.6.3.37]

Figure 4.6.3.37 | top | pdf |

Perpendicular-space distribution of (a) positive and (b) negative structure factors of the 3D Penrose tiling of the 6D P lattice type decorated with point atoms (edge lengths of the Penrose unit rhombohedra ar = 5.0 Å). The magnitudes of the structure factors are indicated by the diameters of the filled circles. All reflections are shown within [10^{-4} |F({\bf 0})|^{2} \;\lt\; |F({\bf H})|^{2} \;\lt\; |F({\bf 0})|^{2}] and [-6 \leq h_{i} \leq 6, i = 1,\ldots, 6].








































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