Indirect method for the determination of x-ray tube primary spectra: Application to x-ray fluorescence analysis

Abstract

The spectral distribution of a fluorescence-exciting x-ray beam is related via a Volterra integral equation of the first kind to the fluorescence excited on a set of thick targets. The inversion of this equation is unstable, but the equation of the first kind can be transformed, by classical methods in integral equations, into a stable Volterra integral equation of the second kind for the cumulative spectral distribution. The x-ray beam spectral distribution is the derivative of the cumulative spectral distribution. As the numerical differentiation of experimental data is also unstable, the intrinsic instability associated with a Volterra integral equation of the first kind appears again. Nevertheless, the cumulative spectral distribution of the x-ray beam is, in itself, a complete functional description of the fluorescence exciting x-ray. Further, it is convenient for use with the Sherman equations of x-ray fluorescence production because these equations rely on the calculation of integrals of the spectral distribution weighted with functions which depend on the concrete composition of the sample used. When integration by parts is performed, the integrand becomes a product of the cumulative spectral distribution times the derivative of the weighting functions. Thus, instead of the unstable spectral distribution, the stable cumulative spectral distribution is used. Copyright © 2002 John Wiley & Sons, Ltd.