# Accounting for Interelement Interferences in Atomic Emission Spectroscopy: A Nonlinear Theory
## 2 Materials and Methods
By expanding \(\Phi_{i}(Y_{ik},I_{k})\) at a constant value \(Y_{ik}\), we have:  
\[\Phi_{i}(Y_{ik},I_{k})=\sum_{l=0}^{\infty}\frac{1}{l!}\frac{\partial^{l}\Phi_ {i}(Y_{ik},I_{k})}{\partial I_{k}{}^{I}}_{I_{k}=0}I_{k}{}^{I}. \tag{5}\]  
Now let us consider the function \(\psi_{ik}=\frac{\partial^{l}\Phi_{i}(Y_{ik}-I_{k})}{\partial I_{k}{}^{I}}_{I_{k }=0}\) and expand it in a Taylor series at \(Y_{ik}=I_{i}\). Then we get:  
\[\psi_{lik}=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\partial^{n}\psi_{lik}(x)}{ \partial x^{n}}_{x=I_{i}}(Y_{ik}-I_{i})^{n}. \tag{6}\]  
Taking (6) into account, after changing the order of summation, (5) takes the following form:  
\[\Phi_{i}(Y_{ik},I_{k})=\sum_{n=0}^{\infty}\sum_{l=0}^{\infty}\frac{1}{n!} \frac{1}{l!}\frac{\partial^{n+l}\Phi_{i}(I_{i},I_{k})}{\partial I_{i}{}^{n} \partial I_{k}{}^{I}}_{I_{k}=0}(Y_{ik}-I_{i})^{n}I_{k}{}^{I}. \tag{7}\]  
Let us consider an arbitrary term \(\psi_{n}(Y_{ik},I_{k})\) in the outer sum (7):  
\[\psi_{n}(Y_{ik},I_{k})=\frac{\left(Y_{ik}-I_{i}\right)^{n}}{n!}\frac{\partial ^{n}}{\partial I_{i}{}^{n}}\left\{F_{i}(I_{i})+\sum_{i=1}^{\infty}\frac{1}{l!} \frac{\partial^{l}\Phi_{i}(I_{i},I_{k})}{\partial I_{k}{}^{I}}_{I_{k}=0}I_{k} {}^{I}\right\}. \tag{8}\]  
Let \(F_{i}(I_{i})\) be a polynomial of degree \(N_{0}\). Suppose that for any concentration of influencing elements, the dependence of \(\Phi_{i}(Y_{ik},I_{k})\) on \(Y_{ik}\) is also described by a polynomial of degree \(N_{0}\). This means that the following holds:  
\[\Phi_{i}(I_{i},I_{k})=\sum_{r=0}^{N_{0}}a_{r}{}^{(ik)}(I_{k})I_{i}{}^{r}, \tag{9}\]  
where \(a_{r}{}^{(ik)}(I_{k})\) are functions of the intensity of \(I_{k}\), and \(a_{r}{}^{(ik)}(0)=a_{I}\). Note that the real form of the functions \(a_{r}{}^{(ik)}(I_{k})\) are shaped by the dependence between the rate constants of the processes occurring when the atoms of the analyte are emitted from the sample into the plasma, as well as by the conditions in the plasma that determine how line emission is generated and transferred. Suppose these functions can be expanded in Taylor series at \(I_{k}=0\). As we are discussing the concentrations of the analyte and the influencing elements, which are relatively small, we will expand it into \(I_{k}\) and limit by \(M_{0}\), that is:  
\[a_{r}{}^{(ik)}(I_{k})=\sum_{q=0}^{M_{0}}\frac{1}{q!}A_{q}{}^{(ikr)}I_{k}{}^{q}, \tag{10}\]  
where \(A_{r}{}^{(ikl)}\) are constants; \(M_{0}\) is a yet undefined integer. Using (9) and (10), and taking into account \(Y_{ik}-I_{i}=d_{ik}I_{k}\), we can rewrite (8) in the following form:  
\[C_{i}=\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1}\right) \tag{2}\]0  
and (7) can be rewritten as follows:  
\[C_{i}=\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1}\right) \tag{2}\]1  
Remember that (7) is valid for an arbitrary dependence of \(\Phi_{i}(Y_{ik},I_{k})\) on the arguments \(Y_{ik},I_{k}\), and (12) is valid when (9) and (10) are satisfied. In what follows, all formulas assume that \(F_{i}(I_{i})\) is a polynomial of \(N_{0}\) degree and relations (9) and (10) are satisfied. Then the analog of formula (A4) (the linear algorithm for taking into account the effects caused by third elements) for the influencing element under the number \(k=m+s\) and the analyte under the number \(i\) has the form:  
\[C_{i}=\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1}\right) \tag{2}\]2  
Thus, in the situation being discussed, the analogue of (A4) contains \(M_{0}\cdot(N_{0}+1)+1\) constants:  
\[C_{i}=\Phi_{i}\left(I_{i}+\sum_{k=m}^{m+V-1}d_{ik}I_{k},I_{m},\ \ldots,I_{m+V-1}\right) \tag{2}\]3  
To find them, it is necessary, as in the linear case, to minimize a functional similar to that defined by (A5). Let us first introduce some corrections to (A5). When applying these results to specific methods for determining the elemental composition of substances (for example, to AES methods), the concentrations of the analyte can be of the third or fourth order of magnitude.