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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 604-605   | 1 | 2 |

## Section 4.6.3.3.1.3. Structure factor

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

#### 4.6.3.3.1.3. Structure factor

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The structure factor of a periodic structure is defined as the Fourier transform of the density distribution of its unit cell (UC): The same is valid in the case of the nD description of a quasiperiodic structure. The parallel- and perpendicular-space components are orthogonal to each other and can be separated. In the case of the 1D Fibonacci sequence, the Fourier transform of the parallel-space component of the electron-density distribution of a single atom gives the usual atomic scattering factor . Parallel to , adopts values only within the interval and one obtains The factor results from the normalization of the structure factors to . With and the integrand can be rewritten as yielding Using gives Thus, the structure factor has the form of the function with x a perpendicular reciprocal-space coordinate. The upper and lower limiting curves of this function are given by the hyperbolae (Fig. 4.6.3.6). The continuous shape of as a function of allows the estimation of an overall temperature factor and atomic scattering factor for reflection-data normalization (compare Figs. 4.6.3.6 and 4.6.3.7).

 Figure 4.6.3.7 | top | pdf |The structure factors of the Fibonacci chain decorated with aluminium atoms (Uoverall = 0.005 Å2) as a function of the parallel (above) and the perpendicular (below) component of the diffraction vector. The short distance is , all structure factors within have been calculated and normalized to .

In the case of a 3D crystal structure which is quasiperiodic in one direction, the structure factor can be written in the form The sum runs over all n averaged hyperatoms in the 4D unit cell of the structure. The geometric form factor corresponds to the Fourier transform of the kth atomic surface, normalized to , the area of the 2D unit cell projected upon , and , the area of the kth atomic surface.

The atomic temperature factor can also have perpendicular-space components. Assuming only harmonic (static or dynamic) displacements in parallel and perpendicular space one obtains, in analogy to the usual expression (Willis & Pryor, 1975), with The elements of the type represent the average values of the atomic displacements along the ith axis times the displacement along the jth axis on the V basis.

### References

Willis, B. T. M. & Pryor, A. W. (1975). Thermal Vibrations in Crystallography. Cambridge University Press.