Calibration in a closed system: the case of X-ray diffraction analysis

Knudsen [X-ray Spectrom. (1981), 10, 54-561] proposed and demonstrated a least-squares approach to estimating the unknown parameters of a system of equations required for calibration in X-ray diffraction analysis. The approach is an ordinary least-squares approach which does not incorporate information on the errors of the measured intensities for a set of samples used as standards. The purpose of the present paper is to show that a functional relationship model can be applied to the problem to account for all the variation due to sampling and measurement error in the peak intensities. It is also shown that Knudsen's calibration estimator can be regarded as an approximation to a more general and potentially more efficient weighted least-squares estimator derived from the functional relationship model. The closeness of the approximation depends on the nature of the covariance structure of the intensity measurements.