# Accounting for Interelement Interferences in Atomic Emission Spectroscopy: A Nonlinear Theory
## Appendix A  
As before, we will assume that we have \(N\) analytes and \(M\) measurements of their concentrations in state standard reference samples or quality control samples, and the relationship between the concentration of the \(i\)-th element and the intensity of its analytical line in the absence of third-element effects can be described by (3). As detailed above, programs for processing data from emission spectrometers usually represent the function \(F_{i}(I_{i})\) as a polynomial of some degree, which is determined by an analytical method. If \(C_{i}^{(j)}\) is the nominal value of the concentration of the \(i\)-th element in the \(j\)-th reference standard, then the coefficients in the polynomial (1) are found by the least squares method (LSM) from minimizing the functional:  
\[S_{i}=\sum_{j=1}^{M}\Bigl{[}C_{i}^{(j)}-F_{i}\Bigl{(}I_{i}^{(j)}\Bigr{)}\Bigr{]} ^{2}\cdot\frac{1}{W_{ij}};W_{ij}=\sqrt{\left(\frac{\Delta I_{i}^{(j)}}{I_{i} ^{(j)}}\right)^{2}+\left(\frac{\Delta C_{i}^{(j)}}{C_{i}^{(j)}}\right)^{2}}, \tag{1}\]  
where \(W_{ij}\) is the statistical weight when measuring the analytical line of the \(i\)-th element according to the results of analyzing the \(j\)-th reference standard; \(\Delta I_{i}^{(j)}\), \(\Delta C_{i}^{(j)}\) are absolute errors in measuring the intensity \(I_{i}^{(j)}\) and the reference value of the concentration \(C_{i}^{(j)}\). If (3) is a polynomial in the \(K\)-order variable \(I_{i}^{(j)}\) with coefficients \(a_{1},\dots,a_{K}\), the following is fulfilled:  
\[\frac{\partial S_{i}}{\partial a_{k}}=0;\frac{\partial^{2}S_{i}}{\partial a_{ k}{}^{2}}>0;k=1,\dots,K;\ F_{i}(I_{i})\equiv F_{i}(I_{i},a_{1},\dots,a_{K}). \tag{2}\]  
Thus, to find the function \(F_{i}(I_{i})\), it is necessary to use a set of reference standards in which there are no third-element effects, for example, standards of one type of alloys with small (to a certain extent) concentrations of impurity elements. The description of the relationship between the concentration of the \(i\)-th element and the intensities of the analytical lines of other elements has the form:  
\[\varphi_{i}=C_{i}-F_{i}(I_{i})\equiv\varphi_{i}(I_{i},I_{m},\dots,I_{m+V-1}). \tag{3}\]  
\(m,V,s\) were defined earlier. We will approximate the dependence of the function \(\varphi_{i}\) on the intensity \(I_{m+s}\) by a linear dependence in the form:  
\[\varphi_{i}^{s}(I_{m+s},I_{i})=b_{m+s}I_{m+s}+d_{m+s}I_{i}I_{m+s}, \tag{4}\]where the first term takes into account the so-called additive effect, while the second one represents the multiplicative effect. In this case, we will assume that the effects produced by different elements are summed additively. As a rule, in programs for factoring in third-element effects on the results of AES experiments, the multiplicative term includes the concentration \(C_{i. }\) rather than the intensity \(I_{i}\). This makes it necessary to solve systems of equations for the relative concentrations of the elements in the sample by successive approximation methods. When using (A4), it can be avoided, and the relative concentration of each element can be found independently of the others. We will solve the problem sequentially.