# Accounting for Interelement Interferences in Atomic Emission Spectroscopy: A Nonlinear Theory
## 3 Discussion
A low-voltage spark in an argon atmosphere was used in the SPAS-5 spectrometer. The exposure was roughly 4 s at a sparking time of 7 s. The CCD signal accumulation time was 16 ms, and the number of frames was 200. Spectrum excitation source parameters: voltage on the discharge interval at discharge pulse start at a sparking and exposure of 400 V and pulse frequency of 400 Hz. Ar purge speed is 7 L/min. Figure 5 shows the relationship \(C_{c}(C_{p})\), where \(C_{c},\ C_{p}\) are the measured and nominal concentrations of copper in UG0k-UG9k, respectively. When interelement interferences were taken into account, the effects produced by Ni, W, Cr, and V were analyzed as high-speed steels. High W and V concentrations and high-alloy steels are rich in Ni and Cr. The figure demonstrates that the proposed universal theory significantly reduces the error in determining Cu content in steel. The root-mean-square deviation of the measured concentration from the nominal one decreases by a factor of 2.41 (from 14.1 to 5.8%). For a set of low-alloy and medium-alloy steels (where no third-element effects are observed), i.e., at \(C_{c}(I_{i})=F_{i}(I_{i})\), the root-mean-square deviation is equal to 5.6%, i.e., the results are almost equal. This shows that in the case under consideration, the proposed theory adequately takes into account the effects caused by third elements. It should be noted that when these effects are described by means of the empirical theory, the root-mean-square deviation of the measured concentration from the nominal one is 11%, which is almost twice as much as the result produced by the universal theory. It is interesting to note that the range of Cu concentrations for which the relationship \(C_{c}(I_{i})=F_{i}(I_{i})\) was plotted (0.0993 to 0.25%) does not exceed one order of magnitude, while the results of analyzing the effects produced by third elements in the determination of  
Figure 4: Function \(F_{i}(I_{i})\) plotted for the analytical line of Cu using the UG0k–UG9k set; 1—\(C_{p}(I_{i})\); 2—\(F_{i}(I_{i})\). Cu content (0.007 to 8%, as can be seen from Figure 4) cover more than three orders of magnitude. The range of the concentrations of influencing elements varies from hundredths to tens of a percent, covering more than two orders of magnitude. Figure 6 shows data similar to those presented in Figure 5 for Al as an impurity in the same sets of steels. By applying the proposed universal theory, it becomes possible to reduce the root-mean-square relative deviation by a factor of 2.9 (from 25.5 to 8.8%). The use of the empirical method results in a twofold decrease in this value to 12.5%. In the absence of third-element effects when using the UGOk-UG9k set, the value is equal to 4.2%. As can be seen, the difference between the results produced by both theories is not as great as in the case of Cu as an impurity. Apparently, this can be explained by the fact that for a number of reasons (uneven distribution over the sample, etc. ), Al content in steels is determined with large random errors.Therefore, while the random error at a Cu concentration of about 0.01% is approximately 5%, the same concentration of Al will generate a random error of roughly 10%. It is for this reason that the deviation of Al concentration from nominal values in the corresponding standard steels is also roughly 10% (for example, in the UG3k sample, the concentration is 0.015% with an error of 0.0015%). Thus, the root-mean-square relative deviation of \(C_{c}\) from \(C_{p}\) for Al obtained when applying the universal theory is due, for the most part, to a random error in the process of concentration measurement rather than to unaccounted effects caused by third elements.