LAUSR

A formula for quantitative phase analysis (QPA), called the intensity–composition (IC) formula, can be used for deriving weight fractions of individual crystalline phases in a mixture from sets of observed integrated intensities, measured in a wide 2θ range, with chemical composition data [Toraya (2016). J. Appl. Cryst. 49, 1508–1516; Toraya (2017). J. Appl. Cryst. 50, 820–829]. In this study, the IC formula has been incorporated into the whole-powder-pattern fitting (WPPF) procedure to conduct QPA. The fitting function for calculating the profile intensity at each step of the scattering angle consists of three sub-functions that represent the individual component diffraction patterns. The first sub-function calculates the diffraction pattern using a set of integrated intensities, the parameter values of which are determined by the least-squares fitting of the whole-powder pattern as is usually done by the whole-powder-pattern decomposition (WPPD) method. The second sub-function uses a set of integrated intensity parameters, which are preliminarily prepared by WPPD or may be calculated from a crystal structure model. These intensity parameters, multiplied by a scale factor, are fixed at their original values while the scale factor is adjusted in WPPF. The third sub-function uses an observed or calculated diffraction pattern multiplied by a scale factor. This diffraction pattern can be fitted directly by adjusting the scale factor. Therefore, one can fit patterns consisting of heavily broadened and degraded diffraction lines, like those of clay minerals, without being concerned with the problem of peak overlap in decomposing the diffraction pattern. The IC formula uses the total sums of the intensities under the diffraction patterns of individual phases as observed data sets; therefore, it can equally treat these intensity data sets irrespective of differences in the profile models used by the three sub-functions. The three sub-functions can arbitrarily be chosen and linearly combined, and then they can simultaneously be fitted to the observed diffraction pattern of a target mixture. The capability of the above method has been demonstrated with QPA of mixtures consisting of α-quartz, albite and kaolinite. Theories of currently used QPA techniques are reviewed from a viewpoint of the present theory and they can be interpreted as being based on the same principle, whereby the total observed intensities of individual phases are divided by the standard reference intensity per unit weight.