A practical approach to the direct-derivation method for QPA: use of observed powder patterns of individual components without background subtraction in whole-powder-pattern fitting

Abstract

The direct‐derivation (DD) method for quantitative phase analysis (QPA) can be used to derive weight fractions of individual phases in a mixture from the sums of observed intensities along with the chemical composition data [Toraya (2016). J. Appl. Cryst. 49, 1508–1516]. The whole‐powder‐pattern fitting (WPPF) technique can be used as one of the tools for deriving the observed intensities of individual phases. In WPPF, the observed powder pattern of a single‐phase sample after background (BG) subtraction can be used as the fitting function in combination with the fitting functions widely used in Pawley and Rietveld refinements. The direct fitting of the observed pattern is a very useful technique when the target component is a low‐crystallinity or amorphous material [Toraya (2018). J. Appl. Cryst. 51, 446–455]. Technical problems in utilizing the BG‐subtracted pattern are the uncertainty associated with the determination of BG height and the parameter interaction between the BG function (BGF) and the BG‐subtracted pattern in the least‐squares fit. In this study, a practical approach in which single‐phase observed patterns are used for the direct fitting without subtracting their BG intensities is proposed. In QPA, the contribution of BG intensities can be neutralized by converting the sum of BG‐included intensities into the sum of BG‐subtracted intensities by multiplying by a conversion factor. When the magnitudes of the conversion factors are almost identical for all components, they can be canceled out under the normalization condition in deriving weight fractions, and they are not required in QPA. The magnitude of the conversion factor for each component can be determined by one of two experimental techniques: using a single‐phase powder of the target component or a mixture containing the target component in a known weight ratio. The theoretical basis of the present procedure is given, and the procedure is experimentally verified. In this procedure, the interaction between the BGF and the BG‐included observed pattern is negligibly small. Least‐squares fitting with a few adjustable parameters is very fast and stable. Accurate QPA could be conducted, as indicated by the average deviation of 0.05% from weighed values in QPA of α‐Al2O3 + γ‐Al2O3 mixtures with five different weight ratios and 0.4% in QPA of an α‐SiO2 + SiO2 glass mixture

Accurate and time-saving quantification of a component present in a very small amount in a mixture by the direct derivation method

“…a À1 k is given by a À1 k ¼ ð1=M k Þ P i n 2 ki , where M k is the chemical formula weight and n ki is the number of electrons belonging to the ith atom in the chemical formula unit (Toraya, 2016;Toraya & Omote, 2019). In deriving the observed quantities fS k g for individual components by using WPPF techniques, we can currently use four types of fitting functions, designated as types A, B, C and C 2 , which are summarized in Table 1 of Toraya (2019). Since the equation for calculating a À1 has been derived by integrating all scattered intensities from an assemblage of atoms or, in other words, summing jFðhklÞj 2 to infinity, the corresponding observed quantity S k should also be free from the termination effect in summing/integrating the observed intensities.…”

“…The substitution of this equation into equation (6) can also derive equation 17. In the case that the relation (Toraya, 2019), equation 17can be rewritten as…”

“…It is based on the simple principle that the weight proportion of a component material in a mixture can be calculated by dividing the total sum of scattered intensities from the component material by the scattered intensity per unit weight. The scattered intensity per unit weight for each component can be calculated only with the chemical composition data, whereas total sums of scattered intensities for individual component materials can be obtained as observed quantities by decomposing the diffraction pattern of the mixture into individual component patterns (Toraya, 2016;Toraya & Omote, 2019;Toraya, 2019). For decomposing powder XRD patterns, various techniques such as individual profile fitting techniques (Taupin, 1973;Parrish et al, 1976) or whole-powder-pattern decomposition (WPPD) methods (Pawley, 1981;Toraya, 1986;Le Bail et al, 1988) can be used.…”

“…The ready-made profile models are not flexible, and this approach requires that the individual patterns be measured under the same experimental conditions as those for target mixtures to reduce model bias. These models are, however, very stable in the least-squares fitting and considered to be suitable for QPA, in particular, of very small amounts in mixtures by the DD method (Toraya, 2018(Toraya, , 2019.…”

Review of the direct derivation method: quantitative phase analysis with observed intensities and chemical composition data

“…As will be described later, the DD method itself is based on a simple principle. There are, however, various ways to obtain the total sums of diffracted/scattered intensities from the target mixture pattern, and they have been reported in several papers (Toraya, 2016(Toraya, , 2017(Toraya, , 2018(Toraya, , 2019(Toraya, , 2020. In this report, it is tried to review comprehensively the theoretical background of the DD method and the techniques for the pattern decomposition.…”