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

# They also introduced a third element term to

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approximating a curve by linear fitting. They also introduced a “third element term” to compensate for the fact that the algorithm cannot be simply “a sum of binary correction terms.” The COLA model sought to replace the linear curve fitting with a hyper- bolic curve fitting, involving a correction algorithm of the type: 1 + { α ij, hyp + α ijk C k } C j where the subscript “hyp” is used to denote that the binary inﬂuence coeﬃ- cients are approximated by an hyperbola (compare Figs. 5.6 and 5.7). In 1981 Lachance [32] proposed the following expression for quantifying the correction term, namely: C i = R i 1 + j α 1 + α 2 C M 1 + α 3 (1 C M ) + α ijk C k C j (5.50) and the better quality of the hyperbolic fit to the binary inﬂuence coeﬃcient values is shown in Fig. 5.7. The COLA model also takes into consideration the observation by Tertian that the inﬂuence coeﬃcient α ij should be calculated at the C i concentration level. For the Claisse–Quintin model, this translates into the introduction of C M , where C M is equal to the sum of the matrix elements, i.e., C M = C j + C k + · · · + C n . 1 2.4 2.3 2.2 2.1 2 0 0.5 a FeCr a FeCr a 1 a 3 a 2 Fig. 5.7. Explanation of the three inﬂuence coeﬃcients in the COLA algorithm

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344 J.P. Willis and G.R. Lachance This approach meant that, in the absorption context, { . . . } ijk values were for all practical purposes equal to 0, and in the cases where enhancement was present, the values as computed by C–Q were reduced by as much as a factor of 2 to 3. If the C–Q model is retained, this is equivalent to using the formulation [1 + { α ij + α ijj C M } C j ] instead of [1 + { α ij + α ijj C j } C j ] in the case of a linear approximation. Tao et al. [55] published “NBSGSC – A FORTRAN . . . ” program for perform- ing quantitative analysis of bulk specimens by X-ray ﬂuorescence spectrome- try. This program corrects for absorption/enhancement phenomena using the comprehensive alpha coeﬃcient algorithm proposed by Lachance (COLA). NBSGSC is a revision of the program ALPHA and CARECAL originally developed by R.M. Rousseau of the Geological Survey of Canada. The process can be described in the following steps: The value of the correction term, i.e., [ . . . ], is calculated using the classical Criss and Birks approach; The value of [ . . . ] is calculated using the six binary inﬂuence coeﬃcients α 1 , α 2 , and α 3 in Table 5.4, where α 1 are the values when C i approaches the limit 1.0 and are calculated for C i = 0 . 999: 2.3456 and 0 . 1954 for α FeCr and α FeNi , respectively; α 2 is the difference between those values calculated for C i = 0 . 001 and the above values, i.e., 2 . 0837 2 . 3456 = 0 . 2619 and 0 . 4861 ( 0 . 1954) = 0 . 2906, respectively; α 3 is the value that defines the “degree of curvature” of the hyperbola (deviation from the straight line at C i = 0 . 5) and causes α ij,hyp to match the theoretical value at the mid point. α 3 is defined as α 3 = α 2 m ij, bin ,C i =0 . 5 α 1 2 .
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