Quantitative XRF analysis using the fundamental algorithm
Authors: Rousseau R M 
Published: November 2013

In X-ray Fluorescence (XRF) analysis, one of the major problems to solve is the correction for matrix effects (absorption and enhancement). During the last three decades, one of the popular solutions proposed was the use of empirical influence coefficients
(Lachance and Traill, 1966) which are numerical coefficients correcting for the effects of each matrix element on the analyte. These coefficients, that depend on the matrix composition, are calculated by a regression analysis of measured intensities and standard compositions. It will be shown that
these empirical coefficients are not very accurate, depend on the composition of standards used, have a limited range of application, are sensitive to any experimental errors, and can be applied only to unknowns similar to the standards used. An easier and much more practical solution would be to
calculate these influence coefficients from theory for any combination of elements and any experimental condition.&amp;lt;%0&amp;gt; Sherman (1955) proposed a set of equations to calculate net X-ray intensities emitted by each element in a specimen of known composition when it is
irradiated by a polychromatic X-ray beam. As proposed, these equations are not very useful to XRF analysts because intensities can be measured. There is therefore no need to calculate them. Furthermore, to calculate intensities the specimen's composition must be known whereas from the start this is
precisely what is being sought. Criss and Birks (1968) proposed the fundamental-parameters method, a new way to use Sherman's equations: the specimen composition is found by an iteration procedure that makes successively closer estimates of concentrations until the calculated intensities agree
satisfactorily with the measured intensities. Rousseau proposed a combination of the two worlds, fundamental parameters and influence coefficients, into a new approach which allows you to calculate theoretical influence coefficients inside of a new algorithm called the fundamental algorithm
(Rousseau, 1984a, b). To calculate the composition of a specimen, a first estimate of composition must be calculated by iteration using the Claisse-Quintin algorithm. Then, from this estimated composition, the theoretical influence coefficients are calculated once only. Finally, with these
coefficients used as constants, the final composition is obtained by applying an iteration process to the fundamental algorithm. Theoretical influence coefficients are powerful tools to correct for matrix effects as long as a link exists for adapting the theory to the experimental reality, knowing
full well that no theory accounts for all experimental variables. It will be shown that an appropriate calibration process can correct for all the imperfections of the theory and has as well, a few other major qualities that facilitate the work of the analyst. The determination of sample
compositions using the fundamental algorithm requires very complex calculations which must be done on a computer. A new computer program has then been written and it runs on any IBM-PC/XT/AT, PS/2 or compatible computer. It will be shown that it is a complete "off-line" program enclosing within
itself all the steps necessary, after the collection of intensities by the XRF spectrometer, to calculate analytical composition: calculation of theoretical influence coefficients, calibration, reading of measured intensities, calculation of net intensities and concentrations of unknowns, print-out
of analytical reports, and data transmission to or from an external computer.