Bragg peak analysis

T. Egami, S. Billinge (eds.) Underneath the Bragg Peaks Structural Analysis of Complex Materials, Elsevier Science, Amsterdam, 2003. 

Vol. 7 Underneath the Bragg Peaks Structural Analysis of Complex Materials... 

Egami T, Billinge SJL (2003) Underneath the Bragg peaks Structm-al analysis of complex materials. Pergamon, Oxford... 

Corresponding analysis may be performed for other types of screw axes in different orientations. As an exercise try to derive the relationships between Miller indices of the systematically absent Bragg peaks for a 4i screw axis parallel to the Z direction. Again, the result remains identical whether or not the axis intersects the origin of coordinates. The relationships... 

As shown in section 2.12.3, the presence of translational symmetry causes extinctions of certain types of reflections. This property of infinite symmetry elements finds use in the determination of possible space group(s) symmetry from diffraction data by analyzing Miller indices of the observed Bragg peaks. It is worth noting that only infinite symmetry elements cause systematic absences, and therefore, may be detected from this analysis. Finite symmetry elements, such as simple rotation and inversion axes, mirror plane and center of inversion, produce no systematic absences and therefore, are not distinguishable using this approach. 

As is often the case in diffraction analysis, when the space group symmetry is known, it is quite easy to predict which types of reflections can and which cannot be observed. For example, when the space group symmetry is Pnma, only the following types of Bragg peaks may have nonzero intensity (as established by analyzing Table 2.8 to Table 2.10) ... 

Nonetheless, analysis of the systematic absences (the complete list is found in Table 2.12 to Table 2.17) usually allows one to narrow the choice of space group symmetry to just a few possibilities, and the actual symmetry of the material is usually established in the process of the complete determination of its crystal structure. Especially when powder diffraction data are used, it only makes sense to analyze low angle Bragg peaks to minimize potential influence of the nearly completely overlapped reflections with indices not related by symmetry. An example of the space group determination is shown in Table 2.11. 

Table 2.11. Analysis of the observed combinations of indices in the monoclinic base-centered unit cell (E - even, O - odd). The entries corresponding to Bragg peaks that are theoretically possible but are not observed in the powder diffraction pattern are shaded. The observed Bragg reflections hOl, in which I = 2n are highlighted in bold (zero is considered an even...

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