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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 621-623   | 1 | 2 |

Section 4.6.4.1.  Data-collection strategies

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

4.6.4.1. Data-collection strategies

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Theoretically, aperiodic crystals show an infinite number of reflections within a given diffraction angle, contrary to periodic crystals. The number of reflections to be included in a structure analysis of a periodic crystal may be very high (one million for virus crystals, for instance) but there is no ambiguity in the selection of reflections to be collected: all Bragg reflections within a limiting sphere in reciprocal space, usually given by [0 \leq \sin \theta/\lambda \leq 0.7\;\hbox{\AA}^{-1}], are used. All reflections, observed and unobserved, are included to fit a reliable structure model.

However, for aperiodic crystals it is not possible to collect the infinite number of dense Bragg reflections within [0 \leq \sin \theta/\lambda \leq 0.7\;\hbox{\AA}^{-1}]. The number of observable reflections within this limiting sphere depends only on the spatial and intensity resolution.

What happens if not all reflections are included in a structure analysis? How important is the contribution of reflections with large perpendicular-space components of the diffraction vector which are weak but densely distributed? These problems are illustrated using the example of the Fibonacci sequence. An infinite model structure consisting of Al atoms with isotropic thermal parameter B = 1 Å2, and distances S = 2.5 Å and [{\rm L} = \tau{\rm S}], was used for the calculations (Table 4.6.4.1)[link].

Table 4.6.4.1| top | pdf |
Intensity statistics of the Fibonacci chain for a total of 161 322 reflections with [-1000 \leq h_{i} \leq 1000] and [0 \leq \sin \theta/\lambda \leq 2\;\hbox{\AA}^{-1}]

In the upper line, the number of reflections in the respective interval is given; in the lower line the partial sums [ {\textstyle\sum} {I}({\bf H})] of the intensities [I({\bf H})] are given as a percentage of the total diffracted intensity. The F(00) reflection is not included in the sums.

 [F({\bf H})/F({\bf H})_{\max} \geq 0.1][0.1\, \gt\, F({\bf H})/F({\bf H})_{\max} \geq 0.01][0.01 \,\gt\, F({\bf H})/F({\bf H})_{\max} \geq 0.001][F({\bf H})/F({\bf H})_{\max} \,\lt\, 0.001]
[0 \leq \sin \theta/\lambda \leq 0.2\;{\AA}^{-1}] 17 148 1505 14 511
[ {\textstyle\sum} {I}({\bf H})] 52.53% 2.56% 0.27% 0.03%
[0.2 \leq \sin \theta/\lambda \leq 0.4\;\hbox{\AA}^{-1}] 11 107 1066 14 998
[ {\textstyle\sum} {I}({\bf H})] 27.03% 2.03% 0.19% 0.02%
[0.4 \leq \sin \theta/\lambda \leq 0.6\;\hbox{\AA}^{-1}] 9 64 654 15 456
[ {\textstyle\sum} {I}({\bf H})] 9.84% 0.96% 0.12% 0.01%
[0.6 \leq \sin \theta/\lambda \leq 0.8\;\hbox{\AA}^{-1}] 6 27 326 15 823
[ {\textstyle\sum} {I}({\bf H})] 2.94% 0.34% 0.07% 0.01%
[0.8 \leq \sin \theta/\lambda \leq 2\;\hbox{\AA}^{-1}] 1 35 338 96 720
[ {\textstyle\sum} {I}({\bf H})] 0.23% 0.79% 0.06% 0.01%
Total sum 44 381 3389 157 508
92.57% 6.67% 0.70% 0.06%

It turns out that 92.6% of the total diffracted intensity of 161 322 reflections is included in the 44 strongest reflections and 99.2% in the strongest 425 reflections. It is remarkable, however, that in all the experimental data for icosahedral and decagonal quasicrystals collected so far, rarely more than 20 to 50 reflections along reciprocal-lattice lines corresponding to net planes with Fibonacci-sequence-like distances could be observed. The consequences for structure determinations with such truncated data sets are primarily a lower resolution in perpendicular space than in physical space. This corresponds to a smearing of the hyperatoms in the perpendicular space. For the derivation of the local structure-building elements (clusters) of aperiodic crystals this is only a minor problem: the smeared hyperatoms give rise to split atoms and a biased electron-density distribution. The information on the global aperiodic structure, however, which is contained in the detailed shape of the atomic surfaces, is severely reduced when using a low-resolution diffraction data set. A combination of high-resolution electron microscopy, lattice imaging and diffraction techniques allows a good characterization of the local and global order even in these cases. For a more detailed analysis of these problems see Steurer (1995)[link].

References

Steurer, W. (1995). Experimental aspects of the structure analysis of aperiodic materials. In Beyond Quasicrystals, edited by F. Axel & D. Gratias, pp. 203–228. Les Ulis: Les Editions de Physique and Berlin: Springer-Verlag.








































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