Abstract
In this chapter we report on the coherent X-ray diffraction imaging study of a single colloidal crystal grain composed of silica spheres. The diffraction data contained Bragg peaks and Bragg rods, which are related to the stacking of the hexagonally close packed layers. The profile of the Bragg rod showed distinct intensity modulations that under our specific experimental conditions are directly related to the stacking sequence of the layers. Using a model for the scattered intensity along a Bragg rod of an exact stacking of a finite number of hexagonally close packed layers, we show that a double hexagonal close packed stacking sequence is present in the colloidal crystal grain. This analysis method opens up ways to obtain crucial structural information from finite sized crystalline samples by employing advanced third generation X-ray sources.
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Acknowledgments
This work is the result of a collaboration with the Vartanyants’ group from Hasylab at DESY, Hamburg. I would like to thank Ivan Vartanyants, Anatoly Shabalin and Johannes Gulden for the close collaboration and useful discussions. Alexey Zozulya, Ulf Lorenz, Dmitry Dzighaev, Oleg Gorobstov, Ruslan Kurta, Andrej Singer, Oleksandr Yefanov and Dmitro Byelov are acknowledged for their contributions to the CXDI experiment. The P10 beamline team of PETRAIII, DESY, Hamburg is thanked for providing technical support. DESY is acknowledged for allocating the beamtime. Jan Hilhorst is thanked for useful suggestions. Part of this work is reproduced with permission of the International Union of Crystallography from Ref. [27].
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Appendices
Appendices
1.1 A.1 Phk(L) and S(L) Contributions at Different hk Indices
In our model the scattering intensity depends on the form factor P hk (l) and the structure factor S(l) and these have different effects on the full intensity I hk (l) along a Bragg rod. P hk (l) is dependent on the specific hk indices of the Bragg rod while S(l) is independent of the hk indices and only determined by the stacking sequence, see Eq. (3.2). Figure 3.8a shows the normalized P hk (l) profiles for the different indices hk(10), hk(20) and hk(21). It is clear that the maxima and minima are located at different l values. The S(l) profile is shown in Fig. 3.8b for a DHCP stacking sequence and shows a specific modulation of the intensity, with a periodicity of l = 1. In Fig. 3.8c the final intensity I hk (l) = P hk (l)S(l) (Eq. 3.4) is plotted for the specific hk indices. This shows the strong influence the specific P hk (l) on the final peak profile of S(l) and explains the large difference of the Bragg rod profiles for the different families shown in Fig. 3.3.
a Normalized profiles of P(l) for the different hk indices of the Bragg rods, the graphs are offset by 1 for clarity. b Normalized profile of S(l) for a DHCP stacking sequence, which is independent of the hk indices. c Final normalized intensity profiles I(l) for different hk indices. Clearly, the effect of P hk (l) and hence the hk indices of the rod is of large influence
1.2 A.2 Sensitivity of Bragg Rod Model for Stacking Sequence
To illustrate the sensitivity of the intensity profile along the Bragg rods to the exact stacking sequence, and thus the strength of this analysis route, we show two DHCP structure with two other small changes in the DHCP sequences for the 21l rod in Fig. 3.9. The two profiles sequences are ABCBABABABCB and ABCBABCBABCA, in which the differences with the DHCP stacking are the middle and last layer, respectively. These both change the number of FCC and HCP layers in the structure and clearly have a dramatic effect on the final model I hk (l), where peak position, shape and intensity are clearly mismatched with the experimental data.
Normalized experimental 21l rod profile with modelled profiles for two different DHCP stacking sequences, with a different middle or end layer than a perfect DHCP structure, showing the sensitivity of the model to a sequence change
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Meijer, JM. (2015). Double Hexagonal Close Packed Structure Revealed in a Single Colloidal Crystal Grain by Bragg Rod Analysis. In: Colloidal Crystals of Spheres and Cubes in Real and Reciprocal Space. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-14809-0_3
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