Determination of atomic parameters from powder diffraction intensity ratios

The basis for a systematic use of intensity ratios is given in order to determine positional and occupational parameters. Given a series of N powder diffraction intensities, the matrix of intensity ratios is defined. Expressing them as vectors, it is shown that they form a subset of $\text{1}\text{2}\text{(N}{}^2\text{? N)}$ elements in a vectorial space with dimension N ? 1. Since any intensity ratio may be obtained from the product or division of other ratios, their vectorial representation is used to define a criterion for independence for a set of ratios, transforming the product in a sum of the corresponding vectors. Any set of N ? 1 independent ratios may be used in the determination of atomic parameters. In order to find the most suitable set, it is proposed to assign an index to each intensity ratio, which depends upon the difference between observed and calculated values and their derivatives with respect to the parameters to be determined. Then, an independent set of intensity ratios is chosen among those with higher indices, for each point of a convenient grid in the space of the parameters, finding the best set at each point. Finally, the best absolute set is defined as that with the minimum value of an agreement indicator between observed and calculated values. An illustrative example of the procedure indicated above is shown, and the results are compared with those obtained by other powder integrated intensity methods. In this work, comparison with the Rietveld profile refinement method is not considered.