Structure analysis by Rietveld fitting

In multicrystal diffraction, the diffraction of many small crystals in the three-dimensional space are compressed into one-dimensional, and lose the independence of all diffraction hkl, where the overlapping of the symmetry or the unsynunetry of diffraction peak obscures the profile of distribution curve of hkl diffraction intensity. So the abundant structural information contained in the powder diffraction pattern cannot be extracted, so for a long time the powder diffraction could only be used as the identification of phases. 

In 1967, Rietveld proposed least squares fitting structure amending for allpowder diffraction pattern in the powder neutron diffraction analysis of the structure using computer. In 1974, Malmros and Young ° introduced it into X-ray powder diffraction analysis of the structure, and with the widespread use of computers, the method has been developed rapidly and used widely. 

The outline of Rietveld analysis of the structure is to compare the digital intensity y of a powder diffraction pattern obtained by experiment to intensity yc of calculation by a model, and the difference between the two models leads to the excursion amendatory value of structure parameters by the least squares method, and by adding it to the corresponding model structure parameters, the difference between y, and yci achieves the minimum value. 

The use of the least squares method first need to establish a normal equation, including the differential of all the calculation intensity of y to each adjustable parameter, and then solve the corresponding formal matrix inversion. The element of the formal matrix Mjk can be calculated by the following equation  

Rietveld structural analysis of multicrystalline X-ray is based on the intensity value and position of diffraction hkl. Small changes in the structure are reflected in the intensity, so it demands precise experiment and cannot use general conventional method. 

---