Combined Rietveld refinement

In this section, we are concerned with a powder diffraction experiment, which consists of a single pattern (profile). The Rietveld technique may also be used to conduct refinement of the crystal structure employing multiple patterns collected from the same material. For example, powder diffraction data collected using conventional x-ray sources with different wavelengths, conventional and synchrotron x-rays, conventional or synchrotron x-rays and neutron source may be used simultaneously in a combined Rietveld refinement. The fundamentals of the combined Rietveld refinement are briefly considered in section 7.3.8. 

Table 7.7. The progress of the combined Rietveld refinement of the crystal structure of LaNi4,85Sno 15 employing two sets of experimental data obtained using Cu Ka and Mo Ka radiations.

When multiple sets of data are used in a combined Rietveld refinement, the first set has a fixed scale, k= 1, but all other sets have their own scales in addition to a phase scale. 

When more than one set of experimental diffraction data is employed in the combined Rietveld refinement, the minimized function (in the simplest case of Eq. 7.3) becomes... 

In Eq. 7.10, /t is the number of different sets of powder diffraction data, is the number of data points collected in the 5" set, and is the scale factor for the x diffraction pattern, which appears because scattered intensity is measured on a relative scale. Other notations are identical to Eq. 7.3. Different scale factors, and K, are simple multipliers. Hence, they strongly correlate, and usually are not refined simultaneously. Constraining one of the scale factors (usually k, for the first diffraction data set) at 1 enables successful refinement of the phase scale K) and scale factors of all remaining sets of diffraction data ki, k, . .., k/,). Equations 7.4, 7.6 and 7.7 are modified in the same way as Eq. 7.3 for a combined Rietveld refinement. Furthermore, it is often the case that x-ray and neutron, or conventional x-ray and synchrotron data are used in combined refinements, therefore, the... 

As expected, adding the contribution from the impurity phase (again as Le Bail s approximation) results in further reduction of the profile residuals, see row six in Table 7.7. The final model of this crystal structure, as determined using the combined Rietveld refinement in the two-phase approximation, is found in the data file Ch7Ex01k.inp on the CD. Structural parameters are listed in Table 7.8. 

Figure 7.10. The observed and calculated powder diffraction patterns of LaNi4 gsSno.is (Cu Ka radiation) after the completion of the combined Rietveld refinement.

This crystal structure was solved earlier (see sections 6.10 and 6.11), first using x-ray and then using neutron powder diffraction data. The x-ray data (Mo Ka radiation) were collected at room temperature, while the neutron scattering experiment (K = 1.494 A) was conducted at 200 K. Hence, combined Rietveld refinement is not feasible because of the differences in the lattice and structural parameters of the alloy due to thermal expansion, and we will use the two sets of data independently. 

Table 7.18. Atomic parameters and interatomic distances (in A) after the completion of the combined Rietveld refinement based on both the x-ray and neutron powder diffraction data collected from NiMn02(0H) powder. The refined chemical composition is NiMnOs (OH)j where 8 = 0.62(5). The unit cell parameters are a = 2.86112(4), b = 14.6516(1), c =...

In this paper the relation between cation mobility and catalytic activity in lean SCR NOx by CH4 on (Ag,Co)- and (Co,Ag)-FER have been studied by combining XRD Rietveld refinement in fresh and used catalysts with SCR catalytic testing performed in dry and wet conditions. UV-Vis DRS measurements were also performed. 

The major difficulty in applying an absorption correction (Eq. 2.74) arises from usually unknown peff. Obviously, the linear absorption coefficient, p, can be easily calculated when the dimensions of the unit cell and its content are known (see Eq. 2.10) but it is applicable only for a fully dense sample. When a pulverized sample is used (and typically it is), Peff cannot be determined easily without measuring sample density. Often the combined parameter ( aeff/) can be refined or estimated and accounted in intensity calculations during Rietveld refinement (Chapter 7). 

Given the observed small variations in the Rietveld refinement results, it is much better to employ all available data while performing the combined least squares fit of the model. This can be done using the majority of available Rietveld refinement programs and is illustrated in the next section. 

R.B. Von Dreele, Combined Rietveld and stereochemical restraint refinement of a protein crystal structure, J. Appl. Cryst. 32, 1084 (1999). 

A Quench D5niamics protocol (a combination of high temperature Molecular Dynamics and Energy Minimization techniques) is used for predicting the location of triethylmethylammonium (TEMA) cations in zeolite MFI. Rietveld refinement of the high-resolution synchrotron X-ray diffraction data confirms these predictions. The TEMA cations are located at the channels intersections in two different conformations with two ethyl groups located in the linear channel. 

Advanced structure refinement. The Rietveld refinement structural method has been taken further in a number of directions. One of the most appealing is to combine it with the maximum entropy method (MEM), in which... 

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