## 2 Theory
### OES Line Ratio Method  
As described above, the major plasma parameters for the plasma heat flux value are compared using OES. The OES sensor monitors the intensity of a particular wavelength produced by an excited electron of an atom as it travels back to a lower state in plasma. Although OES measures optics, the values of electron density and electron temperature can also be obtained using the line-ratio method. The line-ratio method is categorized into a corona model and a collisional-radiative model (CRM). The corona model has low electron density and high electron temperature and is used under very low-pressure (<1 Pa) and low ionization ratio (<10\({}^{-5}\)) conditions. In comparison, the CRM model has wider applications.  
Figure 1: Schematic diagram of an ion behavior in the plasma sheath region.  
The CRM model has high electron density and low electron temperature and is used under relatively high-pressure (1-10 Pa) conditions and high ionization ratio (\(>\)10\({}^{-5}\)). The CRM was used in this study because of its suitable pressure and argon discharge conditions required for the calculation of the plasma heat flux.  
In the case of excited argon atoms, electron impact excitation and spontaneous radiation are the dominant processes at low-pressure levels. The concentration of atoms in excited states is calculated by the ratio of electron impact excitation (\(e+R_{\mathrm{g}}\to e+R_{\mathrm{p}}\)) rate coefficient to the sum of electron impact de-excitation (\(e+R_{\mathrm{p}}\to e+R_{\mathrm{g}}\)) rate coefficient and spontaneous radiation (\(R_{\mathrm{p}}\to R_{\mathrm{r}}+hv\)) rate coefficient. Therefore, it is a function of electron density and electron temperature; in addition, the intensity ratio of the two emission lines is a function of two parameters [32]. By using the ratio coefficient and the ratio balance equation of the electron impact process of the corona model, \(Q\) is expressed as follows.  
\[Q=\int_{0}^{\infty}\sigma(E_{\mathrm{e}})\sqrt{\frac{2E_{\mathrm{e}}}{m_{ \mathrm{e}}}}g_{\mathrm{e}}(E_{\mathrm{e}})dE_{\mathrm{e}} \tag{13}\]  
Here, \(E_{\mathrm{e}}\) denotes electron kinetic energy, \(m_{\mathrm{e}}\) denotes electron mass, \(g_{\mathrm{e}}\) EEDF, and \(\sigma\) denotes the cross-section of the relevant electron impact process. CRM is considered because of the high-lying \(ns\) or \(nd\) level, unlike the process in the corona model. Other than these reactions, the electron impact transition between excited levels (\(e+R_{\mathrm{p}}\to e+R_{\mathrm{r}}\)) and the radiation trapping process (\(R_{\mathrm{g}}+hv\to R_{\mathrm{hr}}\)) are complicated and difficult to consider. Therefore, to simplify CRM, the excitation from metastable in the rate balance equation should be ignored by selecting a specific np level, a zero total angle momentum quantum number (\(J=0\)) of argon gas. Note that data for argon \(n\) p levels with \(J=0\) were quoted from the previous research work [33]. Thus, when \(J=0\), the total excitation rate coefficient is summarized as follows [33].  
\[Q_{\infty,p}=K_{0}\delta exp\left(-\frac{E_{0}}{T_{\mathrm{e}}}\right) \tag{14}\]  
where \(Q_{\infty,p}\) denotes the sum of \(Q_{g-p}\) (ground state to np levels) and \(Q_{g-hr}\) (ground state to high-lying \(ns\) and \(nd\) levels). The constants \(K_{0}\), \(E_{0}\) and \(C_{0}\) can be found in [33], and the rate equation is expressed as follows.  
\[n_{\mathrm{e}}n_{\mathrm{g}}Q_{\infty,p}=A_{\mathrm{p}}n_{\mathrm{p}}\left(1+ \frac{n_{\mathrm{e}}}{n_{\mathrm{e}\mathrm{C},p}}\right) \tag{15}\]  
where \(n_{\mathrm{e}}\) denotes the electron density, \(n_{\mathrm{g}}\) denotes gas density, \(A_{\mathrm{p}}\) denotes total Einstein coefficient, \(n