# Accounting for Interelement Interferences in Atomic Emission Spectroscopy: A Nonlinear Theory
## 2 Materials and Methods  
Derivation of the basic formulas.Appendix A describes the traditional method for factoring in the effects caused by third elements. Some of the formulas within its framework will be used in the Discussion section. We will assume that we have \(M\) reference standards, in each of which \(N\) elements are known. Let us imagine a situation in which the concentration of the \(i\)-th element (\(C_{i}\)), determined by the analytical method, depends not only on the intensity of its analytical line but also on the concentrations (and hence on the intensities of the analytical lines) of other elements. Let us assume that the number of these elements \(V<N\) and number them from \(m\) to \(m+V-1\). It is obvious that:  
\[m+s\
eq i,\text{ where }s=0,\ldots,V-1. \tag{1}\]  
Let us also assume that the relationship between the \(i\)-th element being determined and the intensities of the analytical lines of the analyte \(I_{i}\) and the influencing elements \(I_{m},\ \ldots,I_{m+V-1}\) has the form:  
\[C_{i}=\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1}\right) \tag{2}\]  
where \(d_{ik}\) are yet unknown dimensionless numerical coefficients, and the relationship between \(\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1}\right)\) and the arguments \(I_{m},\ \ldots,I_{m+V-1}\) has not yet been determined. The form of the first argument of the function on the right side of (2) \(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k}\) has a simple physical meaning. It takes into account possible spectral line interferences and the dependence of the background where the analytical line of the analyte is located on the presence of influencing elements in the plasma. For simplicity, we have taken this relationship to be linear depending on the intensity of the influencing elements \(I_{k}\). In modern optical emission spectrometers, when there are interferences of the lines of elements that affect the intensity of the line of the analyte, the real relationship is apparently close to linear. On the contrary, the relationship between the intensity of the plasma background and the intensities of the lines of the influencing elements (which are determined by their concentrations) is more complex. We will assume that this relationship can be expanded in a Taylor series at \(I_{k}=0\); thus, (2) takes into account the first term of this series. Strictly speaking, it is possible that the coefficient of a linear Taylor series in the \(I_{k}\) term is zero, and this series begins with \(I_{k}\)2. If there are no line interferences and the background of the analytical line of analyte does not depend on the concentration of the influencing elements in the plasma, \(d_{ik}\equiv 0\) and \(\Phi_{i}(I_{i},I_{k})\) should be written in (2). It is quite obvious that the identity holds:  
\[\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1} \right)_{I_{k}=0,k=m\ldots m+V-1}\equiv F_{i}(I_{i}). \tag{3}\]  
That is, \(F_{i}(I_{i})\) represents the dependence of the concentration of the \(i\)-th element on the intensity of its analytical line if there are no effects caused by third elements. Let us suppose that these effects are additive and consider first the case of one influencing element with number \(k\) (\(m\leq k\leq m+V-1\)):  
\[C_{i}=\Phi_{i}(I_{i}+d_{ik}I_{k},I_{k}). \tag{4}\]  
Note that, as a rule, the function \(F_{i}(I_{i})\) is represented in AES as a polynomial of degree \(N_{0}\leq 4\). Let us expand the function (4) of two variables \(\Phi_{i}(Y_{ik},I_{k})\), where \(Y_{ik}=I_{i}+d_{ik}I_{k}\), in a double Taylor series at a point \(I_{k}=0;\ Y_{ik}=I_{i}\).