Estimation of errors in the measurement of unit-cell parameters. I. Statistical uncertainties of peak positions of powder diffraction lines determined by individual profile fitting

A formula for estimating the statistical uncertainty of peak positions, determined by individual profile fitting, has been derived. The magnitude of the statistical uncertainty is given by (sum of peak profile intensities)^-1/2 × (full width at half-maximum) × F × G × Q × (goodness-of-fit index). The factor F depends on the shape of the diffraction profile; it becomes smaller by 40% when the profile shape changes from Lorentzian to Gaussian. The factor G represents the parameter correlation between the peak position and the parameter for profile asymmetry; it becomes unity when the profile shape is symmetric or the parameter for profile asymmetry is fixed during the least-squares fitting. The factor Q depends on the peak-to-background ratio. The formula was experimentally verified by using diffraction data sets for CeO₂ and Si powders. The statistical uncertainty is essentially based on the counting statistics and profile width. Therefore, the peak position of a strong and sharp peak in the low-angle region can be determined more precisely than that of a weak and broadened peak in the high-angle region. The formula provides a guideline for optimizing experimental parameters for required precision.