# Accounting for Interelement Interferences in Atomic Emission Spectroscopy: A Nonlinear Theory
## 2 Materials and Methods
In order for the maximum relative error to not exceed 5%, it is sufficient that the number of parameters \(L_{0}\) and the number of samples \(M\) are connected by an empirical relationship:  
\[\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1} \right)_{I_{k}=0,k=m\ldots m+V-1}\equiv F_{i}(I_{i}). \tag{3}\]2where \(\left|\frac{\Delta I_{i,k}}{I_{k,k}}\right|\) is the maximum relative error in measuring the intensities \(I_{i},~{}I_{k}\). Here, the following is satisfied:  
\[\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1} \right)_{I_{k}=0,k=m\ldots m+V-1}\equiv F_{i}(I_{i}). \tag{3}\]3  
It is easy to show that the average values of the parameters found by means of (17) are obtained by using the average intensity values from the results of parallel measurements. Let us discuss in more detail some of the differences between the universal approach and the traditional (empirical) one, which is presented in the first section of the article. When using the empirical approach, the function \(\phi_{i}^{s}(I_{i},I_{k})\) is linear in intensity \(I_{k}\) and product \(I_{i}\cdot I_{k}\). Using the universal approach, the maximum degrees of \(I_{k}\) and \(I_{i}\), on which this function depends, depend on \(N_{0}\) and \(M_{0}\) discussed above. The maximum degree of intensity \(I_{k}\) is \(N_{0}+M_{0}\), therefore, cannot be less than 2, and \(I_{i}\) is equal to \(N_{0}\). Therefore, for \(N_{0}=3\) and \(M_{0}=2\) we get the degree of \(I_{k}\) equal to 5 and the degree of \(I_{i}\) equal to 3, respectively. This is a fundamental difference between the two approaches. In addition, in the empirical approach, there are two constants (for any degree of the polynomial \(F_{i}(I_{i})\)), which need to be found from the condition for minimizing the functional (A5). In the universal approach, the number of constants to be determined from system (17) depends on the degree of the polynomial \(F_{i}(I_{i})\) and is equal to \(M_{0}\cdot(N_{0}+1)+1\). Thus, the minimum number of constants in this case is attained at \(N_{0}=M_{0}=1\) and is equal to three. In practice, when analyzing the elemental composition of substances by means of atomic emission spectroscopy, it is usually (with the exception of analysis using a discharge with inductively coupled plasma in argon) the case that \(N_{0}\geq 2\), consequently, at \(M_{0}=1\) we have at least four constants. In this case, one should expect better results of factoring in interelement effects with an increase in \(M_{0}\) if there is a corresponding increase in the number of reference standards. Finally, let us pay attention to the difference between the functionals being minimized in the empirical and universal approaches (formulas (A1) and (15), respectively). As noted, the difference lies in the fact that the former is the sum of the absolute squares overall reference standards, and the latter is the sum of relative deviations of the calculated concentrations of the analyte from nominal ones that take into account the statistical weights of the points \(W_{ij}\) (see (A1)). Thus, with a decrease in the concentration being analyzed in the standards, their contribution to the functional (A1) decreases (which enhances the effect of the growing statistical weight with a decrease in concentration due to an increase in the random relative measurement error). Consequently, the points with the minimum concentrations of the analyte have little effect on the values of the constants in (A4). On the contrary, the contribution to the functional (17) of points with different concentrations of the analyte does not depend on the value of this concentration, which means that all points with the same relative error have the same effect on the constants being determined (see (14)).