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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 606-607   | 1 | 2 |

Section 4.6.3.3.1.5. 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

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

| top | pdf |

The two possible point-symmetry groups in the 1D quasi­periodic case, and , relate the structure factors to A 3D structure with 1D quasiperiodicity results from the stacking of atomic layers with distances following a quasiperiodic sequence. The point groups describing the symmetry of such structures result from the direct product corresponds to one of the ten crystallographic 2D point groups, can be or . 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 superspace groups for IMSs with satellite vectors of the type , i.e. , 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 , , corresponds to a scaling of the perpendicular space by the inverse factor . 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 , 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 . The Fourier transform of the sequence then can be written as the sum of the Fourier transforms of the sequences and ; In terms of structure factors, this can be reformulated as

 Figure 4.6.3.10 | top | pdf |Part … LSLLSLSL … of a Fibonacci sequence before and after scaling by the factor τ. L is mapped onto , S onto . 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 , which give when decorating the Fibonacci sequence , form a Fibonacci sequence scaled by a factor .

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 vectorwith the matrixincreases the magnitudes of structure factors assigned to this particular diffraction vector H,

This is due to the shrinking of the perpendicular-space component of the diffraction vector by powers of while expanding the parallel-space component by according to the eigenvalues τ and of S acting in the two eigenspaces and :Thus, for scaling n times we obtainwithyielding eventually The scaling of the diffraction vectors H by corresponds to a hyperbolic rotation (Janner, 1992) with angle , where (Fig. 4.6.3.11):

 Figure 4.6.3.11 | top | pdf |Scaling operations of the Fibonacci sequence. The scaling operation S acts six times on the diffraction vector yielding the sequence .

References

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