Quantitative analysis using Toraya's "Direct Derivation" method
Using the so-called "Direct Derivation" (DD) method (proposed by H. Toraya in 2018 ) you can perform a quantitative phase analysis, just based on the known composition and integrated peak intensities of the phases (details see below).
In order to perform this analysis, you first have to finish the qualitative analysis of the phases in your sample. Afterwards, simply run the command "Direct Derivation (Toraya)" in the “Quantify” menu (which is available as soon as two (or more) phases are selected as “matching”). The amounts in the match list will be replaced by the ones obtained from the Toraya “DD” analysis. A corresponding information is also given in the report.
The "Direct Derivative" (DD, Toraya) method for quantitative analysis derives the phase amounts as follows:
- The amounts are calculated from the chemical composition (given as the formula sum) and the integrated peak intensities.
- The chemical composition (formula sum) is used to calculate the molecular mass as well as the number of electrons in the formula unit. Both values are required to calculate αk in Toraya's formula (8).
- For the calculation of the integrated peak intensities Ijk, the peak heights (calculated from the scaled peak heights of the reference patterns) as well as the FWHM (full width at half maximum) values of the correlated experimental peaks are used.
- The intensity scaling factors of the reference patterns (phases/entries selected as "matching") are obtained from a least-squares fit of the profile calculated from the reference pattern's peak data to the experimental profile. The whole profile is used for the fit (and not just the peak heights or even just the height of the strongest peak).
- If no experimental peak can be assigned to a reference peak, the standard (default) FWHM value is used for calculating the profile as well as the integrated peak intensities.
The "DD method" is advantageous to other quantitative analysis methods for several reasons:
- No I/Ic values are required. These values sometimes cause problems when using the older "Reference Intensity Ratio"-(RIR)-method, either because in some reference databases (like the PDF) quite a lot of entries do not contain this value, and/or because these values tend to vary significantly between different entries for the same phase. As a result, using the RIR-method may result in different amounts being obtained, just by selecting a different entry for a certain phase!
- No crystal structure data are required for the selected phases, which is a big advantage compared to Rietveld refinement.
- All that is required for the "DD"-analysis are the composition (formula sums) of the phases and the integrated peak intensities. This information is typically available for all qualitative analysis results.
- The experimental pattern should cover a 2θ range up to at least 80° (for Cu K α1 radiation). A smaller range may have a negative impact on the accuracy of the method. The Toraya method uses peaks up to a maximum of 2θ=110°.
- The Toraya DD-method results will be overwritten if you select a new phase, run the “Quantify / Reference intensity ratio (RIR)” command, or perform a Rietveld refinement calculation.
 Hideo Toraya, "A new method for quantitative phase analysis: Direct derivation of weight fractions from observed intensities and chemical composition data of individual crystalline phases", Rigaku Journal 34(1), 3-8 (2018).