Optimized Method for Correcting Smearing Aberrations: Complex X-Ray Spectra

The problem considered is one of obtaining a true spectrum from an observed spectrum containing the effects of finite resolution and noise. The finite resolution gives rise to a smearing aberration which may be represented by the convolution or fold of the true spectrum with a smearing function. The noise introduces an uncertainty which deserves careful consideration if a meaningful correction for the smearing aberration is to be effected. In this paper an optimized practical solution to this problem is developed for the case of complex x-ray spectra as measured with modern high-precision spectrometers. The solution is based on the conventional Fourier transform solution of the convolution integral and is evaluated by means of a high-speed digital computer. The noise term is handled by a modification of the Wiener smoothing theory in which the stationarity condition is relaxed to permit application to complex x-ray spectra. Introduction of the inherent line shape permits easy evaluation of the required autocorrelation properties of the spectrum. Special consideration is given to the complicating difficulty of long ``tails'' or ``wings'' on the smearing functions encountered in x-ray spectroscopy. The corrected Cl K emission and absorption spectra of crystalline potassium chloride are presented as examples of the correction. Although the present solution was developed for the specific purpose indicated, it may be applied immediately to perform smearing corrections or differentiation of any data adequately represented by the present model and assumption regarding the errors. One possible additional application is the differentiation of electron energy-loss spectra recorded by the retarding field method.