Rietveld method basics

During refinement using the Rietveld method, the following system of equations is solved by means of a non-linear least squares minimization  

Here 1) is the observed and is the calculated intensity of a point i of the powder diffraction pattern, k is the pattern scale factor, which is usually set tk= because scattered intensity is measured on a relative scale and k is absorbed by the phase scale factor (e.g. see Eqs. 7.3 and 7.4, below), and n is the total number of the measured data points. Hence, a powder diffraction pattern in a digital format, in which scattered intensity at every 

Considering Eq. 2.48 and taking into account the one-dimensionality of powder diffraction data, which introduces multiple Bragg reflection overlaps, the expanded form of Eq. 7.2 in the simplest case, i.e. when the powder is a single phase crystalline material, becomes 

A simple analysis of these two equations indicates that the experimental minimization of the background, which generally contains little or no useful 

In the absence of a background and assuming that the measured intensity is only affected by statistical errors (see Eqs. 3.8 and 3.9), the weight can be given as 

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Details of the structure of fibers grown from the melt have been derived from an X-ray fiber analysis for the example of polyfethylene terephthalate) [23-25]. The basic X-ray scattering technique is summarized in Appendix 16. The fuU-pattem fiber analysis used in this analysis, the Rietveld method, is given in [26]. The triclinic crystal structure of polyfethylene terephthalate) is illustrated in Fig. 5.68. The unit cell... 

Quantitative analysis of phase in the multi-phase system has always been a problem. Although the XRD is the most studied technique in the past as quantitative analysis of phase, but because multicrystalline XRD has problems of bulk effect of different X-ray absorption factors, the peak-overlap and changes in the intensity of diffraction hkl is caused by difference of fine structure which cannot be solved by traditional method. Thus, the results are always unsatisfied and considered to be unreliable. Rietveld analysis has completely solved the problem. The basic equation is ... 

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