EMAS-2002, Szczyrk (Poland) Lábár: Quantitative 1 / 10 Quantitative electron microprobe analysis of homogeneous bulk samples J.L. Lábár 1 Research Institute for Technical Physics and Materials Science, H-1121, Budapest, Konkoly-Thege u. 29-33, Hungary Introduction As we mentioned in the introductory paper, the measured input for quantitative analysis consists the intensities of the analytical X-ray lines (one line per element). Based on physical reasoning, we can give a formula that relates the X-ray intensity emitted by the selected analytical line as a function of both materials constants and experimental parameters. That formula is given by () () ( )(){}Ω⋅⋅⋅⋅⋅−⋅⋅⋅⋅⋅⋅=∫επωρρχρϕρ4exp0*azdzzNActNEQIiwiii (1) where Q*i is the cross section per atom for ionization of the selected subshell of the ith element, taking into account both direct ionization by the electrons of energy E and the effect of indirect ionizations at that subshell caused by the Coster-Kronig transitions. N0, ρ, t and N are Avogadro's number, the density and thickness of a very thin tracer layer made of the ith element and the number of primary electrons, respectively. The mass fraction, atomic weight, fluorescence yield and weight of line for the ith element is designated by cwi, Ai, ω and a, respectively, while Ω and ε are the solid angle and the detection efficiency of the detector. The integral in curly bracket in the center describes the depth distribution of ionization in the bulk sample with respect to the tracer layer and the self-absorption of the X-rays. We shall deal with these quantities in more details later. If this formula is accurate enough, all the data like atomic constants and geometric parameters are known and the measurement is done on the absolute basis (giving the number of photons per primary electron) the composition of the sample can be given directly and no additional measurements are needed. The above idealised situation means a true standardless analysis. The traditional approach is different, however. The ingenious in Castaing's approach was that he realized that by comparing the measured intensity to another measurement on a known material (the so-called standard), we can get rid of many of the unknown constants and parameters. For a standard, he originally selected a (known) sample made of a single chemical element only. He showed (for a selected X-ray line of the ith element in the unknown sample) that the ratio of the intensity, measured in the unknown (unkiI) to that, measured in the elementary standard (stdiI ) represents a good approximation of the mass fraction of the ith element in the sample. The correction factor needed to obtain a better approxiation of the mass fraction is of the order of unity and depends on all the n elements present in the sample (in contrast to depending only on the selected element for which the ratio is formed). 1 e-mail: [email protected], Szczyrk (Poland) Lábár: Quantitative 2 / 10 ()stdiunkin10ZAFiwiIIc,..,c,,ECorrectionc =ψ⋅ (2) The additional dependence on the primary beam energy (E0) and on the geometry of the measurement (ψ) is also indicated in (2). Variants of quantitative analysis are nothing else than different approaches to the calculation of the mentioned correction factor. Present paper explains the relation between equations (1) and (2) and gives a short overview of the most frequently used approximations in the calculation of the correction factors. The possibility of and the limitations in calculating the standard intensities (in contrast to measuring them) is also outlined. Although, before we start, let's take a diversion. There is a regularly returning confusion, why the generated X-ray intensity is proportional to the mass fraction of the elements present, in contrast to the atomic fraction. This fact is contradictory to our first physical anticipation, since the ionization cross section is defined for one atom, so it is the number of atoms that should be important (irrespective of their masses). It is a correct assumption and we recall it below, how the mass fraction results from the starting number of atoms. Avogadro's number, N0 (=6.02*1023) gives the number of molecules in that many gramms of a chemical compound as the value of its molecular weight. If, for the sake of simplicity, we assume that the chemical formula of the molecule is XnYm than its molecular weight is nAX+mAY and there are N0n atoms of the X element and N0m atoms of the Y element in the nAX+mAY gramms of that compound. The importan feature of the molecule here is the constant proportion of its constituents (and not the type of bond between them). Let's take the excited volume in a thin layer first, for which we want to calculate the generated intensity during our EPMA analyisis. Extension to a bulk sample will be examined in a next step. The excited volume is assumed to be homogeneous, so it can be characterized by a constant composition (that is what we want to determine with EPMA). Let the atomic fractions be ci (i=1,n for the the n elements present in the layer). Than, the material can be thought of being similar to a chemical compound (irrespective of the chemical bounds) made up of fixed "compositional blocks" with "block weight" ΣciAi. If we take ΣciAi gramms of that material, it will contain N0 "compositional blocks " [and (within that) ciN0 atoms of the ith element]. There will be N0/ΣciAi "compositional blocks" in 1 gramms of this material and ρN0/ΣciAi "compositional blocks" in 1 cm3. If we take a layer of thickness t, the number of "compositional blocks" is tρN0/ΣciAi and within it, the number of the ith atoms is iiiiAcctN∑⋅⋅⋅ρ0 (3a) per unit surface area. You can see that linear dependence on the number of atoms of an element in a given piece of material does not mean linear dependence on the atomic fraction of that element. Since the mass fraction can be calculated from the atomic fractions as cwi=ciAi/Σ ciAi, the number of the ith atoms in unit surface area of this
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