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

Defining the variable β ij ? k µ j µ i 1 574 where

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Defining the variable β ij ( λ k ) = µ j µ i 1 , (5.74) where µ j = µ j ( λ k ) csc φ + µ j ( λ i ) csc φ µ i = µ j ( λ k ) csc φ + µ i ( λ i ) csc φ produces µ s = µ i 1 + j C j β ij ( λ k ) . Note that the variable β ij ( λ k ), as we will see later, is defined as the absorption influence coefficient in the case of a monochromatic incident source of wavelength λ k . This coefficient corrects for the absorption effects of the ma- trix element j on the analyte i and can be positive or negative. If (5.74) is rewritten in the following form: β ij ( λ k ) = µ j µ i µ i , (5.75a) it is easier to determine when the influence coefficient β ij ( λ k ) is positive or negative. The values for β ij ( λ k ) depend on the matrix composition. For example, if Fe is determined in the presence of Mg (a lighter matrix element), then µ i > µ j and β ij ( λ k ) is negative. If Fe is determined in the presence of Ni (a heavier matrix element), then µ i < µ j and β ij ( λ k ) is positive. Furthermore, the coefficient β ij ( λ k ) is the ratio of the difference between the mass absorption coefficients of elements j and i relative to the mass absorption coefficient of element i . In other words, the coefficient β ij ( λ k ) shows, in a relative way, how much greater or smaller the absorption of ele- ment j is compared to that of element i . The relative absorption of element i compared to itself is therefore equal to 0. Indeed, β ii ( λ k ) = µ i µ i µ i = 0 . (5.75b) In the following paragraphs, we continue to modify Sherman’s equation to make it easier to understand and manipulate. Defining a second variable:
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5 Quantitative Analysis 355 W i ( λ k ) = µ i ( λ k ) µ i I o ( λ k ) ∆ λ k and combining with the above equations leads to I i ( λ i ) = g i C i λ = λ Edge i λ k = λ min W i ( λ k ) 1 + j C j δ ij ( λ k ) 1 + j C j β ij ( λ k ) . (5.76) This preliminary modified Sherman equation is already simpler and more revealing. Indeed, the intensity is still proportional to the concentration C i and also to a ratio on the right-hand side. The numerator contains all the enhancement coefficients δ ij of each element j of the matrix, and the denom- inator contains all the absorption coefficients β ij of each element j . Thus, I 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. We will return to this subject later. Because of the difficulty in determining the experimental constant g i and for making the measured intensities independent of the instrument, the second important step is to replace the absolute intensity I i of element i by the relative X-ray intensity , R i , which is defined as follows: R i = I i ( λ i ) I ( i ) ( λ i ) , (5.77) where I ( i ) ( λ i ) is the intensity emitted by the pure element i . For a specimen composed only of the pure analyte i , C i = 1, all C j = 0, and the modified Sherman equation (5.76) becomes I ( i ) = g i k W i ( λ k ) . (5.78)
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