A Monte Carlo error simulation applied to calibration-free X-ray diffraction phase analysis

Quantitative phase analysis of a system of n phases can be effected without the need for calibration standards provided at least n different mixtures of these phases are available. A series of linear equations relating diffracted X-ray intensities, weight fractions and quantitation factors coupled with mass balance relationships can be solved for the unknown weight fractions and factors. Uncertainties associated with the measured X-ray intensities, owing to counting of random X-ray quanta, are used to estimate the errors in the calculated parameters utilizing a Monte Carlo simulation. The Monte Carlo approach can be generalized and applied to any quantitative X-ray diffraction phase analysis method. Two examples utilizing mixtures of CaCO₃, Fe₂O₃ and CaF₂ with an α-SiO₂ (quartz) internal standard illustrate the quantitative method and corresponding error analysis. One example is well conditioned; the other is poorly conditioned and, therefore, very sensitive to errors in the measured intensities.