[B._Beckhoff,_et_al.]_Handbook_of_Practical_X-Ray_(b-ok.org).pdf

# 78 the combination of 576 577 and 578 after some

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(5.78) The combination of (5.76), (5.77), and (5.78), after some algebraic manipula- tions, leads to R i = C i 1 + j ε ij C j 1 + j α ij C j , (5.79a) where ε ij = k W i ( λ k ) δ ij ( λ k ) k W i ( λ k ) (5.79b) α ij = k W i ( λ k ) β ij ( λ k ) k W i ( λ k ) (5.79c)

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356 R.M. Rousseau and where the new weighting factor , W i ( λ k ), present in the two previous equations is defined by W i ( λ k ) = µ i ( λ k ) I 0 ( λ k ) ∆ λ k µ i 1 + j C j β ij ( λ k ) . (5.79d) Equation (5.79a) is still the same Sherman equation (5.70), except that it cal- culates a count-rate ratio, R . Indeed, the intensity R i is still proportional to the concentration C i but also to a ratio on the right-hand side. The numerator contains all the enhancement coeﬃcients δ ij (or ε ij ) of each element j of the matrix, and the denominator contains all the absorption coeﬃcients β ij (or α ij ) of each element j . Thus, here again the count-rate ratio R i will increase with the enhancement effects and decrease with the absorption effects (if β ij is positive). Furthermore, all these matrix effects are weighted by the factor W i , which takes into account the polychromaticity of the incident spectrum and the matrix composition of the sample. Regarding the count-rate ratio, R , the same applies as to the conventional FP methods of Sect. 5.2. When the pure specimen is not available, the cali- bration procedure [49] enables to determine the intensity of the pure analyte from multielement standards (5.20). This intensity is simply equal to the slope of the calibration line. As analysts are interested to calculate concentrations rather than intensi- ties, which are measured, (5.79a) must be reversed: C i = R i 1 + j α ij C j 1 + j ε ij C j . (5.80) If we accept the fundamental nature of the Sherman equation, and since (5.79a) is the only equivalent equation that respects the Sherman equation in every respect (algebraically, mathematically, and physically), the above re- versed expression (5.80) of (5.79a) can be called the fundamental algorithm . Consequently, the α ij and ε ij coeﬃcients defined by the explicit (5.79c) and (5.79b) are the fundamental inﬂuence coeﬃcients correcting for absorption and enhancement effects, respectively. We will see the reason in the next section. Physical Interpretation At first approximation, (5.80) reveals that the concentration of the analyte i is proportional to its measured relative intensity, i.e., C i R i , which is multiplied by a ratio correcting for all matrix effects. In fact, the coeﬃcient α ij , calculated from the β ij and W ij coeﬃcients, includes all mass absorp- tion coeﬃcients µ S (when there is no enhancement) of the Sherman equation.
5 Quantitative Analysis 357 Therefore, we can aﬃrm that the α ij coeﬃcient corrects for all absorption effects caused by element j on analyte i and the numerator of the ratio cor- rects thus for all absorption effects of the matrix, each element j bringing its contribution to the total correction in a proportion C j . If the numerator is greater than unity (it could be lower if the matrix is less absorbent than the analyte), the intensity R i

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