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Rietveld refinement
The Rietveld method uses a least squares approach to refine a theoretical line profile until it matches the measured profile. The introduction of this technique was a significant step forward in the diffraction analysis of powder samples as, unlike other techniques at that time, it was able to deal reliably with strongly overlapping reflections. The method was first reported for the diffraction of monochromatic neutrons where the peak position is reported in terms of the Bragg angle 2θ. This terminology will be used here although the technique is equally applicable to alternative scales such as xray energy or neutron timeofflight. The only wavelength and technique independent scale is in reciprocal space units or momentum transfer Q, which is historically rarely used in powder diffraction but very common in all other diffraction and optics techniques. The relation is
Additional recommended knowledge
Peak shapeThe shape of a powder diffraction peak is influenced by the characteristics of the beam, the experimental arrangement, and the sample size and shape. In the case of monochromatic neutron sources the convolution of the various effects has been found to result in a peak almost exactly Gaussian in shape. If this distribution is assumed then the contribution of a given peak to the profile y_{i} at position 2θ_{i} is:
where H_{k} is the full width at half peak height (fullwidth halfmaximum), 2θ_{k} is the centre of the peak, and I_{k} is the calculated intensity of the peak (determined from the structure factor, the Lorentz factor, and multiplicity of the reflection) At very low diffraction angles the peaks may acquire an asymmetry due to the vertical divergence of the beam. Reitveld used a semiempirical correction factor, A_{s} to account for this asymmetry
where P is the asymmetry factor and s is +1,0,1 depending on the difference 2θ_{i}2θ_{k} being positive, zero or negative respectively. At a given position more than one diffraction peak may contribute to the profile. The intensity is simply the sum of all peaks contributing at the point 2θ_{i}. Peak widthThe width of the diffraction peaks are found to broaden at higher Bragg angles. This angular dependency was originally represented by
where U, V and W are the halfwidth parameters and may be refined during the fit. Preferred orientationIn powder samples there is a tendency for plate or rodlike crystallites to align themselves along the axis of a cylindrical sample holder. In solid polycrystalline samples the production of the material may result in greater volume fraction of certain crystal orientations (commonly referred to as texture). In such cases the peak intensities will vary from that predicted for a completely random distribution. Rietveld allowed for moderate cases of the former by introducing a correction factor:
where I_{obs} is the intensity expected for a random sample, G is the preferred orientation parameter and α is the acute angle between the scattering vector and the normal of the crystallites. RefinementThe principle of the Rietveld Method is to minimise a function M which represents the difference between a calculated profile y(calc) and the observed data y(obs). Rietveld defined such an equation as:
where W_{i} is the statistical weight and c is an overall scale factor such that y^{calc} = cy^{obs} References
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Categories: Crystallography  Diffraction  Neutron related techniques 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Rietveld_refinement". A list of authors is available in Wikipedia. 