Accurate Estimation of Uncertainties in Quantitative Electron Energy-Loss Spectrometry

Abstract

To determine the background under an ionization edge, the pre-edge background intensities are extrapolated using a function which describes the intensity decrease as a function of the energy loss. When intensities and energy losses are logarithmically transformed, a linear relationship is found. The uncertainty of the background, which determines to a large extent the minimal detectable signal and the uncertainty of the calculated concentration, depend on the uncertainties of the slope and intercept of this linear relationship. The uncertainties in slope and intercept must be estimated by a weighted linear regression procedure if the variances of the intensities are non-uniform. If the variances of the normal distributed intensities are determined with few degrees of freedom, they are not suited as weight factors in the least squares minimalisation, however they can be replaced by smoothed values. These smoothed values are calculated with variance functions which describe the relation between the variance and the intensity. The results of the weighted least squares estimation with the smoothed variances did not differ much from the results of an un weighted regression. The weighted regression with the experimental variances led to erroneous results.