Quantitative phase analysis of amorphous components in mixtures by using the direct-derivation method

The direct-derivation (DD) method is a new technique for quantitative phase analysis (QPA) [Toraya (2016). J. Appl. Cryst. 49, 1508–1516]. A simple equation, called the intensity–composition (IC) formula, is used to derive weight fractions of individual components (w_k; k = 1–K) in a mixture. Two kinds of parameters are required as input data of the formula. One is the parameter S_k, which is the sum of observed powder diffraction intensities for each component, measured in a wide 2θ range and corrected for the Lorentz–polarization factor. The other is the parameter a_k⁻¹, defined by a_k⁻¹ = M_k⁻¹∑n_ik², where M_k is the chemical formula weight and n_ik is the number of electrons belonging to the ith atom in the chemical formula unit. The parameter a_k⁻¹ was originally derived by using the relationship between the peak height and the integrated value of the peak at the origin of the Patterson function, implicitly assuming the presence of periodic structures like crystals. In this study, the formula has been derived theoretically from a general assemblage of atoms resembling amorphous material, and the same expression as the original formula has been obtained. The physical meaning of a_k⁻¹, which represents `the total scattering power per chemical formula weight', has been reconfirmed in the present formulation. The IC formula has been tested experimentally by using two-, three- and four-component mixtures containing SiO₂ or GeO₂ glass powder. In the whole-powder-pattern fitting (WPPF) procedure, incorporated into the DD method, a background-subtracted halo pattern is directly fitted as one of the components in the mixture, together with profile models for crystalline components. In the WPPF, an interaction was observed between the parameters of the background function (BGF) and the parameter for scaling the halo pattern, and this resulted in systematic deviations of w_k from weighed values. The deviations were ≤0.7% in the case of binary mixtures when the BGF was fixed at the correct background height, supporting the hypothesis that the DD method is applicable to the QPA of amorphous components.