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A method of quantitative phase analysis using reference samples with absent phases

X-ray diffraction quantitative phase analysis is a technique widely used in materials science and engineering research. The method proposed by Zevin [L. S. Zevin, J. Appl. Cryst. 10, 147 (1977)] has proven very useful in practice because standards or pure crystalline phases are not needed, but, Zevin only described the case of n samples, each of which contain different concentrations of the same n phases. An extension of this method, in which the reference samples could contain less phases than the analyzed sample is proposed in this paper. The absence of phases in reference samples is not arbitrary but depends on certain conditions. The conditions required to solve the equations are discussed in detail using the concepts of the set theory, and the results of confirmation experiments agree well with the theory.