Non‐linear least squares refinement with constraints. Evaluation through curve fitting on emulated XPS‐like spectra and application to the analysis of carbon fibres

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Abstract
A non‐linear least squares iterative refinement has been implemented which shows high performance on a multiple‐peak spectrum including baseline or background. Constraints as well as links within a range are introduced to drive the mathematical optimization: each peak parameter (i.e. height, position, Gaussian/Lorentzian mixing ratio and HWHM on both left and right sides) has assigned to it an allowed range of variation and can be strained to be correlated with other parameters belonging either to the same peak (symmetrical peaks) or to other peaks (doublets, triplets, etc.). Peak shapes typical of XP spectra are used and applications in the field of XPS are discussed. Through emulated curves with Poisson distributed noise, the accuracy and precision of back‐calculated (refined) parameters have been estimated. Moreover, a confidence level calculated from X2 and degrees of freedom has been suggested to check the overall fitting of experimental curves where the signal‐to‐noise ratio is a priori unknown. An application to real C ls XP spectra is described as an example and a list of suggestions is given to match operator requirements. Finally, features of NLLSRC are discussed with respect to other approaches.

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