Estimation of statistical uncertainties in quantitative phase analysis using the Rietveld method and the whole-powder-pattern decomposition method

Formulae for estimating statistical uncertainties in quantitative phase analysis using the Rietveld method and the whole-powder-pattern decomposition method have been derived. The relative magnitude of statistical uncertainty for a derived weight fraction of a component in a mixture is given by σ(W_m)/W_m = (1/W_m − 1)^1/2F(DY_oi)^−1/2, where W_m is the weight fraction of the mth component, F is the goodness-of-fit index, D (≤1) is a factor depending on the degree of peak overlap, and ∑Y_oi is the total sum of profile intensities in the 2θ range used for whole-powder-pattern fitting. If the step width Δ2θ in step scanning is halved, ∑Y_oi is almost doubled; on the other hand, ∑Y_oi is proportional to the fixed counting time T. Therefore, σ(W_m)/W_m ∝ (Δ2θ/T)^1/2. Extension of the 2θ range for whole-powder-pattern fitting towards the high-angle region is not effective for improving the precision of the derived weight fractions if the profile intensities in that region are weak. The formulae provide guidelines for optimizing experimental parameters in order to obtain a required precision.