Systematic absences analysis

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. 

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. 

A white crystalline powder, prepared by hydrothermal treatment at 200°C of a mixture of molybdic acid, H2M0O4, and methylammonium ma) chloride, CH3NH3CI, taken in a 1 2 molar ratio and acidified with hydrochloric acid, HCl, to pH = 3.5, resulted in a complex powder diffraction pattern shown in Figure 6.29. It was indexed in the monoclinic crystal system as was discussed in section 5.12.2. The space group C2/c (or its acentric subgroup Cc) was established from the analysis of the systematic absences, and the unit cell dimensions were refined using 120 resolved reflections below 20 = 60° a = 23.0648(6) A, b = 5.5134(2) A, c = 19.5609(5) A, p = 122.931(1)°, and the sample displacement 8 = -0.098(3) mm for a 250 mm goniometer radius. The unit cell volume is 2087.8 A. ... 

Consider the powder diffraction pattern collected from a ground Hf2Ni3Si4 powder, which is shown in Figure 6.46. The pattern has been indexed in the orthorhombic crystal system and the unit cell dimensions are a = 5.18, b = 13.65 and c = 6.85 A. An analysis of the systematic absences indicates that the following groups of reflections have non-zero intensity ... 

The next step in data analysis is the determination of systematic absences to find the space group symmetry. Systematic absences are Bragg peaks that are missing in the diffraction data because of a translational symmetry operation and/or preferential orientation of polycrystalline samples. 

Whenever = 2n- -l, with n integer, exp[7riA ] = —1 and the structure factor vanishes. Thus, the above list of observed structure factors is indeed a direct experimental proof that in the crystal of succinic anhydride any scatterer atx,y,z has an equivalent scatterer at -x, 1/2 -b y, 1/2 - z. The same applies to (hOO) and (00/) reflections because of the other two equivalent positions of the space group. Internal symmetry is revealed by destructive interference of scattered waves from symmetry-related objects. In fact, the analysis of systematic absences is the method normally used for determining the space group from diffraction patterns. 

For exposure of reasons of observable discrepancy of results of the analysis simulated experiment with application synthetic reference samples of aerosols [1]. The models have demonstrated absence of significant systematic errors in results XRF. While results AAA and FMA depend on sort of chemical combination of an elements, method of an ashing of a material and mass of silicic acid remaining after an ashing of samples. The investigations performed have shown that silicic acid adsorbs up to 40 % (rel.) ions of metals. The coefficient of a variation V, describing effect of the indicated factors on results of the analysis, varies %) for Mn and Fe from 5 up to 20, for Cu - from 10 up to 40, for Pb - from 10 up to 70, for Co the ambassador of a dry ashing of samples - exceeds 50. At definition Cr by a method AAA the value V reaches 70 %, if element presences an atmosphere in the form of Cr O. At photometric definition Cr (VI) the value V is equal 40%, when the element is present at aerosols in the form of chromates of heavy metals. 

If several fundamentally different methods of analysis for a given constituent are available, e.g. gravimetric, titrimetric, spectrophotometric, or spectrographic, the agreement between at least two methods of essentially different character can usually be accepted as indicating the absence of an appreciable systematic error in either (a systematic error is one which can be evaluated experimentally or theoretically). 

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