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

Fluorescent i photons fluorescent i photons

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Fluorescent i -photons Fluorescent i -photons Fluorescent i -photons Incident photons Incident photons i -Excitation i -Excitation i -Excitation j -Excitation j -Excitation j -Excitation Fig. 5.14. Multilayer structure: intralayer and interlayer secondary fluorescence
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372 P.N. Brouwer Intralayer Secondary Fluorescence The secondary fluorescence from element i due to excitation by element j in the same layer n due an incoming photon with wavelength λ is given by N n ijλ = 1 2 A 1 ,n 1 i,ψ A 1 ,n 1 λ,ψ ε ε ij × 4 π sin ψ C n i C n j µ n λ + µ n i τ τ ij µ n j L (5.101) L is a function of the mass absorption coefficients of the layer n for the in- coming radiation, the radiation of analyte i and of element j , the thickness of layer n and the angles of the incoming and outgoing radiation, ψ and ψ , respectively. The factor L can be written as the sum of a downward (element j below element i ) and upward (element j above element i ) contributions: L ( µ n i , µ n j , µ n λ , T n , ψ , ψ ) = L 0 µ n i sin ψ , µ n λ sin ψ , µ n j , T n + L 0 µ n λ sin ψ , µ n i sin ψ , µ n j , T n , (5.102) where L 0 is defined as L 0 ( µ 1 , µ 2 , µ n , T ) = µ n µ 2 µ 1 + µ 2 µ 1 exp ( µ 1 T ) E 1 ( µ n T ) + exp ( ( µ 1 + µ 2 ) T ) E 1 (( µ n µ 2 ) T ) + exp ( ( µ 1 + µ 2 ) T ) ln 1 µ 2 µ 2 + µ 2 µ 1 ln 1 + µ 1 µ n + µ 2 µ 1 E 1 (( µ 1 + µ n ) T ) E 1 is the exponential integral : E 1 ( x ) = x e t t d t. The secondary fluorescence of element i due to element j in the same layer n due to the continuum and characteristic lines of a tube is again obtained by integration over the continuum and summing over the lines as in (5.99). In order to obtain the total intralayer secondary fluorescence of element i originating from a sample, the secondary fluorescence has to be summed over all layers and all elements j in each layer: N i, intra - layer = n j λ abs , i λ 0 N n ijλ N 0 ( λ ) d λ + #tubelines t =1 N n ijλ t N 0 ( λ t ) . (5.103)
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5 Quantitative Analysis 373 For infinitely thick layers L and L 0 reduce to L ( µ n i , µ n j , µ n λ , ψ , ψ ) = µ n j sin ψ µ n λ ln 1 + µ n λ µ n j sin ψ + µ n j sin ψ µ n i ln 1 + µ n i µ n j sin ψ (5.104a) L 0 ( µ 1 , µ 2 , µ n , T → ∞ ) = µ n µ 1 ln 1 + µ 1 µ n . (5.104b) For a bulk sample N ij reduces to N ij = 1 2 ε ij ε 4 π sin ψ L ( µ i , µ j , µ λ , ψ , ψ ) C i C j µ λ + µ i τ τ ij µ sj . (5.105) Interlayer Secondary Fluorescence For interlayer secondary fluorescence, two cases are distinguished: enhance- ment by an element in a layer below the layer of the analyte or by an element in a layer above the layer of the analyte. The first is denoted by N and the latter by N : N kn ij = 1 2 ε ε ij 4 π sin ψ A 1 ,n 1 λ i C n i C k j τ τ ij A 1 ,k 1 λ,ψ X µ k λ sin ψ , µ n i sin ψ , µ n j , d n , µ k j , T k , k 1 b = n +1 µ b j T b N kn ij = 1 2 ε ε ij 4 π sin ψ A 1 ,n 1 λ i C n i C k j τ τ ij A 1 ,k 1 λ,ψ X µ n i sin ψ , µ k λ sin ψ , µ k j , T k , µ n j , T n , n 1 b = k +1 µ b j T b (5.106a) with X ( p, q, µ 1 , T 1 , µ 2 , T 2 , M ) = 1 q Y p, µ 2 , µ 1 T 1 +
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