InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 606-607
## Section 4.6.3.3.1.5. Relationships between structure factors at symmetry-related points of the Fourier image |

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

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

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):

### References

Janner, A. (1992).*Decagrammal symmetry of decagonal Al*.

_{78}Mn_{22}quasicrystal*Acta Cryst.*A

**48**, 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.