## 3 Results  
### Key formulas. Calibration-curve recalibration accounting for background precise value. As it is known, due to diverse physical reasons, the continuous operation of an emission spectrometer may cause a change in the spectral-transmission coefficient of the device. This may be, for instance, due to contamination of the slit-light system's optical elements, or the size reduction of the entrance slit itself (also due to contamination). Besides, changes may occur in the parameters of the spectrometer systems that are responsible for impure-atom emission from the analyzed sample and their optical excitation. The influencing factors may include the changes over time in the parameters of the electronic units of the spectrum-excitation system when using spark and arc spectrometers, changes in the shape of the electrodes due to various technological reasons, etc. Finally, the changes may affect sensitive element parameters of the recording system and the system of its signal conversion, for example, as a result of wavelength-selectiveaging of the radiation receivers of a spectrometer recording system, i.e., a CCD sensitivity shift, etc. This necessitates the adjustment of calibration curves, which is called recalibration. This procedure is of great importance to ensure the emission spectrometers operate correctly. The existing methods of emission-spectrometer recalibration are based on the following assumptions:  
* The calibration-curve shape of the dependence of the measured (not emitted by impure atoms in the light source) analytical-line intensity of the impure atoms on their concentration in an analyzed sample under the parameter changes of an in-service spectrometer is constant [17; 18; 19; 20; 21];
* The relationship between the measured analytical-line intensities before and after the spectrometer's parameter changes is linear [17; 18; 19; 20; 21]. As experience with various brands of spectrometers shows, the first assumption is quite valid (we will discuss it in detail further on). At the same time, the second assumption is apparently not always valid. As a rule, the authors who assume a linear dependence between the analytical-line intensities before and after the spectrometer parameters change do not analyze the reasons for this relationship and the area of its application. We will conduct such an analysis below. Firstly, we will determine how these linear-conversion parameters are related to the spectrometer parameters. Secondly, we will determine the conditions under which the relationship between the intensities \(I\) and \(I^{\prime}\) is linear. We consider the recalibration algorithm in the case when the measured intensity \(I\) of the analytical line of a particular wavelength and the impurity concentration in SS \(C\) are related by Equation (2). After the spectral-transmission coefficient has been changed, this analytical-line intensity \(I^{\prime}\) (similarly, all other values related to the spectrometer with a modified transmission coefficient will also be indicated with a stroke) of some impure element at the same impurity concentration is changed. It is related to intensity \(I\) by the correlation:  
\[I=F_{I}(I^{\prime}), \tag{13}\]  
where \(F_{I}(I^{\prime})\) is some yet unknown function. Then, for the spectrometer with a modified transmission coefficient, there are correlations to Equations (5) and (9):  
\[\Delta C=F^{\prime}{}_{a}(I^{\prime}-I^{\prime}{}_{1});C=F^{\prime}{}_{A}(I^{ \prime}-I^{\prime}{}_{F}). \tag{14}\]  
We suppose that assumption (1) is satisfied, i.e., the shape of calibration curves is the same, and the equality is valid:  
\[F^{\prime}{}_{a}(I^{\prime})=F_{a}[F_{I}(I^{\prime})],\]  
where \(z\) is the arbitrary analytical-line intensity. As shown in the Appendix A, this requires that analytical-line-intensity-formation conditions stay unchanged before the spectrometer slit-light entrance system.