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

# Note that by this definition all α ii are 0 the

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Note that by this definition all α ii are 0. The specification of i = j in the sum is therefore not necessary, but kept for clarity. The matrix-inﬂuence factor, M , can be expressed by M = λ = λ Edge , i λ = λ min τ i N 0 ( λ ) d λ µ i λ = λ Edge , i λ = λ min τ i N 0 ( λ ) d λ µ i 1 + n j =1 , j = i C j α ij . (5.25) The α ij -terms are of course a function of the sample composition (concen- trations) and of wavelength in the integral. The expression for M simplifies greatly by assuming monochromatic incident radiation: M mono = 1 + n j =1 , j = i C j α ij . (5.26) In the resulting equation for the concentration of the analyte element C i = R i 1 + j = i α ij C j = C i, apparent (1 + correction terms) (5.27) the α ij -terms are constants and can be interpreted as inﬂuence coeﬃcients of the matrix element j upon the analyte element, i . The uncorrected con- centration , C i = R i , is often referred to as apparent concentration . Involved

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5 Quantitative Analysis 331 assumptions are that the incident radiation is monochromatic and that no contributions by indirect excitation occur. For polychromatic incident radiation it is possible to apply the mean value theorem for integrals, which states that there exists a value λ equiv within the integration interval λ min , λ Edge ,i such that λ = λ Edge , i λ = λ min f ( λ ) g ( λ ) d λ = f ( λ equiv ) λ = λ Edge , i λ = λ min g ( λ ) d λ hence 2 M = λ = λ Edge , i λ = λ min τ i N 0 ( λ ) d λ µ i λ = λ Edge ,i λ = λ min τ i N 0 ( λ ) d λ µ i 1 + n j =1 , j = i C j α ij = 1 + n j =1 , j = i C j α ij ( λ equiv ) . (5.28) In many practical cases, the equivalent wavelength , λ equiv varies only slowly with composition and can be treated as constant. The concept of equiv- alent wavelengths (often called effective wavelengths ) was an important step in the development of the inﬂuence coeﬃcient methods [56]. Note that the above equation for M contains no approximations (but the restriction to direct excitation). Despite its simple appearance, an accurate (theoretical) determination of the equivalent wavelength and of α ij ( λ equiv ) is not a simple matter and requires solving the integrals. Indirect excitation . In analogy to direct excitation, the starting point is (5.17) for indirect (secondary) excitation. k encompasses all elements in the specimen and K all lines of element k which are capable of exciting the analyte line of element i : N i, sec = G i C i k K C k g K τ i,K λ = λ Edge K λ = λ min τ k N 0 ( λ ) µ × 1 2 1 µ E ln 1 + µ E µ K + 1 µ i ln 1 + µ i µ K d λ = G i C i k K C k g K τ i,K × λ = λ Edge K λ = λ min τ k N 0 ( λ ) µ i 1 + n j =1 , j = i C j α ij L iK d λ. 2 A mathematical requirement for general validity is that the integrand is a contin- uous function. Since the integrand is (practically) discontinuous at the λ -values of absorption edges and/or tube lines, a mean value ( λ equiv ) may not necessarily exist.
332 M. Mantler For monochromatic radiation, this simplifies similar to direct excitation, and by using the previously derived expression for α -coeﬃcients one obtains: N i, sec , mono N ( i ) = C i k

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