# Accounting for Interelement Interferences in Atomic Emission Spectroscopy: A Nonlinear Theory
## 3 Discussion  
The proposed theory can be verified using a simulation problem, testing its applicability to analyzing steels, and comparing the results with those produced by the empirical theory. Let us consider how the proposed universal approach can be applied to AES experiments in which the results of determining the concentration of a certain element are affected by the presence of other elements. We verified the proposed method by using a simulation problem, software, and test results produced by commercially available optical emission spectrometers. We will start by discussing the results of solving the simulation problem. Appendix B gives a detailed description of the algorithm. It was assumed that there are three elements affecting the results of measuring the concentration of the analyte. It was also assumed that the relationship between the effects produced by these elements and the intensities of their analytical lines could be described as a Gaussian function (see Appendix B). The problem consisted in finding whether it is possible to compensate for these effects with the help of the proposed algorithm if there are significant systematic errors (with maximum values approaching 100%) caused by these effects. Figure 1 shows the results of simulating the process of measuring the relative concentration of the \(i\)-th element in a set of 37 reference standards depending on the intensity \(I_{i}^{j}\), as well as the model function \(F_{i}(I_{i})=F_{im}(I_{i})\), which is defined by (A12). The simulation parameters are given in the figure caption. It can be seen that in this simulation, the effects caused by the three elements are manifested at different intensities, that is, when different standards are used. The effects are significant and reach a value of 100% in relation to the concentration. Next, we tested the algorithm described above for \(N_{0}=3;M_{0}=1\) and tried to describe the complex effects introduced in the simulation by means of (12). The results are presented in Figures 2 and 3. Figure 2 show the data on the relationship between the estimated concentration \(C_{ic}\) and the observed one \(C_{ie}\). Figure 3 demonstrate their values \(D_{i}\)\(uFd_{i}\) and the relative differences \(C_{ic}\) and \(C_{ie}\) when factoring in the effects and without doing it, respectively. It can be seen that \(C_{ic}\) and \(C_{ie}\) are so close that \(D_{i}\) turns out to be of the order of the variance of \(C_{ie}\), which is 5% in this example. At the same time, \(Fd_{i}\) reaches 200% (at low concentrations). Next, we tested the proposed algorithm using analytical techniques realized in a commercially available optic emission spectrometer. As an example, we determined Cu and Al concentrations in steels. We used SPAS-02 and SPAS-05 optical emission spectrometers and sets of state standard reference samples of steels: UGOk-UG9k (low-alloy and medium-alloy steels); RG10-RG18 (high-speed steels); LG57, LG32-LG36, LG60-LG65 (high-alloy steels). The function \(F_{i}(I_{i})\) for the analytical line of Cu was found as a third-degree polynomial using the UGOk-UG9k set (see Figure 4). Figure 3: Relationship between concentration values \(D_{i}\), \(Fd_{i}\), and the concentration \(C_{ie}\); 1—\(D_{i}\), 2—\(Fd_{i}\). Figure 2: Relationships between the concentration calculated using the proposed algorithm, the concentration calculated without taking into account the effects, and the observed concentration: 1—\(C_{ic}(C_{ie})\); 2—\(F_{im}(C_{ie})\); 3—straight line \(C_{ic}=C_{ie}\). Cu I 219.958 nm and Al I 394.401 nm analytical lines were used with Fe I 219.604 nm and Fe I 389.566 nm comparison lines, respectively. To determine the arrays of intensities and the corresponding mean square root errors necessary to implement the above-described algorithm accounting for the effect of the third elements, the spectra of each of the standards were taken three times. After the measurements were completed, the specified arrays were uploaded from the service libraries of the spectrometer software.