Bragg intensities

Water on Smectites. Compared to vermiculites, smectites present a more difficult experimental system because of the lack of stacking order of the layers. For these materials, the traditional technique of X-ray diffraction, either using the Bragg or non-Bragg intensities, is of little use. Spectroscopic techniques, especially nuclear magnetic resonance and infrared, as well as neutron and X-ray scattering have provided detailed information about the position of the water molecules, the dynamics of the water molecule motions, and the coordination about the interlayer cations. 

The reduction in intensity due to thermal motion is accompanied by an increase in the incoherent elastic scattering, ensuring conservation of energy. In this respect, thermal motion is much like disorder, with the Bragg intensities... 

Early studies, which did not include many high-order reflections, revealed systematic differences between spherical-atom X-ray- and neutron-temperature factors (Coppens 1968). Though the spherical-atom approximation of the X-ray treatment is an important contributor to such discrepancies, differences in data-collection temperature (for studies at nonambient temperatures) and systematic errors due to other effects cannot be ignored. For instance, thermal diffuse scattering (TDS) is different for neutrons and X-rays. As the effect of TDS on the Bragg intensities can be mimicked by adjustment of the thermal parameters, systematic differences may occur. Furthermore, since neutron samples must be... 

Lucas BW (1969) On contribution of thermal diffuse X-ray scattering to integrated Bragg intensities of single crystals. Acta Crystallogr A 25 627-631... 

Thermal diffuse scattering (TDS) is ascribed to low-frequency lattice vibrations. The atoms in a perfect crystal are not fixed to their sites and oscillate about their positions. The Bragg intensities are reduced by the Debye-Waller factor, proportional to... 

The cell parameters are determined as a= 18.2872(8) A, fe=l 1.2120(4) A, c=l 1.1748(4) A,and /J=107.91(l)° (monoclinic P21),by the Rietveld analysis. The reliability factor (R-factor) based on the Bragg intensities, R(, and weighted profile R-factor, Rwp, were 6.3 and 2.6%, respectively. The MEM charge density based on Rietveld analysis was derived with 502 observed structure factors by dividing the unit cell into 128x72x72 pixels. The reliability factor of the final MEM charge density was 3.8%. 

In normal, routine structure determination experiments, the atomic scattering factors (fj) used are derived from spherically averaged ground state electronic configurations of neutral atoms. The positions of these scattering centers convoluted with thermal motion are then used to calculate structure factors (FH), which are compared with observed structure factors derived from the observed Bragg intensities [3]. 

A correction for TDS, that is, the inelastic phonon scattering, is a difficult problem not yet satisfactorily solved. A proper correction can only be made if the elastic constants of the crystal are known, although an empirical method has been suggested. TDS is inversely correlated with the bulk modulus, thus for soft molecular crystals at room temperature it can be as high as 20-30% of the Bragg intensity, while a much smaller percentage for hard inorganic solids. The effect of no correction is to underestimate the ADP. For neutrons, the problem is even more complicated since the TDS correction will depend on the neutron velocity relative to the velocity of sound in the crystal. ... 

The ratio of the Bragg intensities for positive and negative polarizations (the flipping ratio, R) is then... 

Seiler, P. Measurement of accurate Bragg intensities. In Accurate Molecular Structures. Their Determination and Importance. (Eds., Domenicano, A. and Hargittai, I.) Ch. 7, 170-198. Oxford University Press Oxford (1992). Matsubara, E., and Georgopoulos, P. Diffuse scattering measurements with synchrotron radiation instrumentation and techniques. J. AppL Cryst. 18, 377-383 (1985). 

Juretschke, H. J. Invariant-phase information of X-ray structure factors in the two-beam Bragg intensity near a three-beam point. Phys. Rev. Lett. 48, 1487-1489 (1982). 

The Bragg residual, Rb (this figure of merit is quite important in Rietveld refinement but has little to no use during full pattern decomposition because only observed Bragg intensities are meaningful in both Pawley and Le Bail methods) ... 

A complete description of the texture (or preferred orientation) is formulated as a probability for finding a particular crystallite orientation within the sample this is the orientation distribution function (ODF). For an ideally random powder the ODF is the same everywhere (ODF=l) while for a textured sample the ODF will have positive values both less and greater than unity. This ODF can be used to formulate a correction to the Bragg intensities via a fourdimensional surface general axis equation) that depends on both the direction in reciprocal space and the direction in sample coordinates ... 

The magnitudes of the Bragg intensities used to evaluate mineral abundances are affected by inherent compositional and/or structural variations characteristic of a number of the common minerals found in coal and by non-random crystallite orientation within the sample mount. The subsequent variations in Bragg intensities cause the Bragg intensity-mineral abundance relationship to depart from linearity and result in errors of determination of mineral abundances of 10 percent or more. The procedure must therefore be considered only semiquantitative. 

Table 3.1.1 Expected increase in observed Bragg intensities based on reduction of overall B values from 4.0 A2 (300 K) to 1.5 A2 (100 l<) to 0.75 A2 (20 K). (Adapted from Ref. [10].)...

Figure 9.14 Comparison of the integrated Bragg intensity of different samples. Sample 1 (open triangle) had no capping layer, whereas samples 2-6 had an aluminium oxide capping layer. Variations of the Al/O stoichiometry, the capping layer thickness Laiox heating rate R (in comparison with sample 2 the symbols -, O, and + in...

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