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

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. 501-502 | 1 | 2 |

Section 4.6.3.3.1.5. Relationships between structure factors at symmetry-related points of the Fourier image

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

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

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The two possible point-symmetry groups in the 1D quasiperiodic case, $[K^{\rm 1D} = 1]$ and $[K^{\rm 1D} = \bar{1}]$ , relate the structure factors to $[\eqalign{1:\qquad \quad F ({\bf H}) &= -F (\bar{\bf H}),\cr \bar{1}:\qquad \quad F ({\bf H}) &= F (\bar{\bf H}).}]$ A 3D structure with 1D quasiperiodicity results from the stacking of atomic layers with distances following a quasiperiodic sequence. The point groups $[K^{\rm 3D}]$ describing the symmetry of such structures result from the direct product $[K^{\rm 3D} = K^{\rm 2D} \otimes K^{\rm 1D}.]$ $[K^{\rm 2D}]$ corresponds to one of the ten crystallographic 2D point groups, $[K^{\rm 1D}]$ can be $[\{1\}]$ or $[\{1, m\}]$ . Consequently, 18 3D point groups are possible.

Since 1D quasiperiodic sequences can be described generically as incommensurately modulated structures, their possible point and space groups are equivalent to a subset of the $[(3 + 1)\hbox{D}]$ superspace groups for IMSs with satellite vectors of the type $[(00\gamma)]$ , i.e. $[{\bf q} = \gamma {\bf c}^{*}]$ , for the quasiperiodic direction [001] (Janssen et al., 2004).

From the scaling properties of the Fibonacci sequence, some relationships between structure factors can be derived. Scaling the physical-space structure by a factor $[\tau^{n}]$ , $[n \in {\bb Z}]$ , corresponds to a scaling of the perpendicular space by the inverse factor $[(-\tau)^{-n}]$ . For the scaling of the corresponding reciprocal subspaces, the inverse factors compared to the direct spaces have to be applied.

The set of vectors r, defining the vertices of a Fibonacci sequence $[s({\bf r})]$ , multiplied by a factor τ coincides with a subset of the vectors defining the vertices of the original sequence (Fig. 4.6.3.10). The residual vertices correspond to a particular decoration of the scaled sequence, i.e. the sequence $[\tau^{2}s({\bf r})]$ . The Fourier transform of the sequence $[s({\bf r})]$ then can be written as the sum of the Fourier transforms of the sequences $[\tau s({\bf r})]$ and $[\tau^{2}s({\bf r})]$ ; $[{\textstyle\sum\limits_{k}} \exp (2\pi i{\bf H}\cdot {\bf r}_{k}) = {\textstyle\sum\limits_{k}} \exp (2\pi i {\bf H}\tau {\bf r}_{k}) + {\textstyle\sum\limits_{k}} \exp [2\pi i{\bf H} (\tau^{2}{\bf r}_{k} + \tau)].]$ In terms of structure factors, this can be reformulated as $[F({\bf H}) = F(\tau {\bf H}) + \exp (2\pi i\tau {\bf H}) F(\tau^{2}{\bf H}).]$

Figure 4.6.3.10| top | pdf |

Part … LSLLSLSL … of a Fibonacci sequence $[s({\bf r})]$ before and after scaling by the factor τ. L is mapped onto $[\tau \hbox{L}]$ , S onto $[\tau \hbox{S} = \hbox{L}]$ . The vertices of the new sequence are a subset of those of the original sequence (the correspondence is indicated by dashed lines). The residual vertices $[\tau^{2}s({\bf r})]$ , which give when decorating $[\tau s({\bf r})]$ the Fibonacci sequence $[s({\bf r})]$ , form a Fibonacci sequence scaled by a factor $[\tau^{2}]$ .

Hence, phases of structure factors that are related by scaling symmetry can be determined from each other.

Further scaling relationships in reciprocal space exist: scaling a diffraction vector $[{\bf H} = h_{1}{\bf d}_{1}^{*} + h_{2}{\bf d}_{2}^{*} = h_{1}a^{*} \pmatrix{1\cr -\tau\cr}_{V} +\ h_{2}a^{*} \pmatrix{\tau\cr 1\cr}_{V}]$ with the matrix $[S = \pmatrix{0 &1\cr 1 &1\cr}_{D},]$ $[\eqalign{\pmatrix{0 &1\cr 1 &1\cr}_{D} \pmatrix{h_{1}\cr h_{2}\cr}_{D} &= \pmatrix{F_{n} &F_{n + 1}\cr F_{n + 1} &F_{n + 2}\cr}_{D} \pmatrix{h_{1}\cr h_{2}\cr}_{D} \cr&= \pmatrix{F_{n}h_{1} + F_{n + 1} h_{2}\cr F_{n + 1}h_{1} + F_{n + 2}h_{2}\cr}_{D},}]$ increases the magnitudes of structure factors assigned to this particular diffraction vector H, $[\big|F(S^{n}{\bf H})\big| \gt \big|F(S^{n - 1}{\bf H}) \big| \gt \ldots \gt \big| F(S{\bf H})\big| \gt \big| F({\bf H})\big|.]$

This is due to the shrinking of the perpendicular-space component of the diffraction vector by powers of $[(-\tau)^{-n}]$ while expanding the parallel-space component by $[\tau^{n}]$ according to the eigenvalues τ and $[-\tau^{-1}]$ of S acting in the two eigenspaces $[{\bf V}^{\parallel}]$ and $[{\bf V}^{\perp}]$ : $[\displaylines{\eqalign{\pi^{\parallel}(S{\bf H}) &= \left(h_{2} + \tau\left(h_{1} + h_{2}\right)\right) a^{*} = \left(\tau h_{1} + h_{2}\left(\tau + 1 \right)\right) a^{*}\cr&= \tau \left(h_{1} + \tau h_{2}\right) a^{*},\cr \pi^{\perp}(S{\bf H}) &= \left(- \tau h_{2} + h_{1} + h_{2}\right) a^{*} = \left(h_{1} - h_{2}\left(\tau - 1\right)\right) a^{*} \cr&= - (1/\tau) \left(-\tau h_{1} + h_{2}\right) a^{*},\cr}\cr \big| F(\tau^{n}{\bf H}^{\parallel}) \big| \gt \big| F(\tau^{n - 1}{\bf H}^{\parallel}) \big| \gt \ldots \gt \big| F(\tau{\bf H}^{\parallel}) \big| \gt \big| F({\bf H}^{\parallel}) \big|.\cr}]$ Thus, for scaling n times we obtain $[\eqalign{\pi^{\perp}(S^{n}{\bf H}) &= \left( - \tau \left(F_{n}h_{1} + F_{n + 1}h_{2}\right) + \left(F_{n + 1}h_{1} + F_{n + 2}h_{2}\right)\right) a^{*}\cr &= \left(h_{1} \left(- \tau F_{n} + F_{n + 1}\right) + h_{2} \left(- \tau F_{n + 1} + F_{n + 2}\right)\right) a^{*}}]$ with $[\lim\limits_{n \rightarrow \infty} \left(- \tau F_{n} + F_{n + 1}\right) = 0 \hbox{ and } \lim\limits_{n \rightarrow \infty} \left(- \tau F_{n + 1} + F_{n + 2}\right) = 0,]$ yielding eventually $[\lim\limits_{n \rightarrow \infty} \left(\pi^{\perp}(S^{n}{\bf H})\right) = 0 \hbox{ and } \lim\limits_{n \rightarrow \infty} \left(F(S^{n}{\bf H})\right) = F({\bf 0}).]$ The scaling of the diffraction vectors H by $[S^{n}]$ corresponds to a hyperbolic rotation (Janner, 1992) with angle $[n\varphi]$ , where $[\sinh \varphi = 1/2]$ (Fig. 4.6.3.11): $[\eqalign{\left(\matrix{0 &1\cr 1 &1\cr}\right)^{2n} &= \left(\matrix{\cosh 2n\varphi &\sinh 2n\varphi\cr \sinh 2n\varphi &\cosh 2n\varphi\cr}\right),\cr \left(\matrix{0 &1\cr 1 &1\cr}\right)^{2n + 1} &= \left(\matrix{\sinh \left[\left(2n + 1\right)\varphi\right] &\cosh \left[\left(2n + 1\right)\varphi\right]\cr \cosh \left[\left(2n + 1\right)\varphi\right] &\sinh \left[\left(2n + 1\right)\varphi\right]\cr}\right).}]$

Figure 4.6.3.11| top | pdf |

Scaling operations of the Fibonacci sequence. The scaling operation S acts six times on the diffraction vector $[{\bf H} = (4\bar{2})]$ yielding the sequence $[(4\bar{2}) \rightarrow (\bar{2}2) \rightarrow (20) \rightarrow (02) \rightarrow (22) \rightarrow (24) \rightarrow (46)]$ .

References

Janner, A. (1992). Decagrammal symmetry of decagonal Al₇₈Mn₂₂ quasicrystal. Acta Cryst. A48, 884–901.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. 501-502