Rietveld refinement multiple phase

Rietveld refinement (Chapter 7, section 7.3.5 and the example in section 7.3.8) of multiple phase samples may be used for relatively accurate quantitative analysis. It requires knowledge of the atomic structure for each phase present in the mixture. Structural data are needed to calculate corresponding intensities. Scale factors for every phase present in the mixture, which are determined quite accurately during Rietveld refinement, are proportional to the fraction of the unit cells present in the irradiated volume of the sample. The latter directly follows from Eqs. 

Considering Eqs. 7.6 and 7.7, it is clear that each additional crystalline phase adds multiple Bragg peaks plus a new seale factor along with a set of eorresponding peak shape and structural parameters into the non-linear least squares. Even though mathematically they are easily aceounted for, the finite accuracy of measurements as well as the limited resolution of even the most advanced powder diffractometer, usually result in lowering the quality and stability of the Rietveld refinement in the case of multiple phase samples. Thus, when the precision of structural parameters is of concern, it is best to work with single-phase materials, where Eqs. 7.3 and 7.4 are applicable. On the other hand, since individual scale factors may be independently... 

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. 

XND XND code from X ray laboratory data to incommensurately modulated phases. Rietveld modelling of complex materials, J. F. Berar and G. Baldinozzi, Int. Union Crystallogr., Commission Powder Diffr. Newsletter, 1998, No. 20, 3 5 Incommensurate and composite structures, refinement of multiple powder data sets... 

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