## 4 Spectroscopy of \({}^{12}\)C/\({}^{13}\)C Isotopologues
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **\({}^{12}\)CO\({}_{2}\)** & **\({}^{13}\)CO\({}_{2}\)** & **\({}^{12}\)CH\({}_{4}\)** & **\({}^{13}\)CH\({}_{4}\)** \\ \hline \(\
u\), cm\({}^{-1}\) & 1388 & 1370 & 2919 & 2915 \\ \hline \(I_{\rm signal}\), counts & 7630 & 13754 & 37487 & 64654 \\ \(I_{\rm noise}\), counts & 382 & 229 & 523 & 311 \\ (\(r\).m.s.)\({}_{\rm noise}\), counts & 28 & 27 & 30 & 28 \\ \(\Delta I_{\rm signal}\), counts & 7248 & 13525 & 36964 & 64343 \\ \hline LOD (1 atm.), \% & 1.1731 & 0.5952 & 0.2428 & 0.1303 \\ \hline LOD (50 atm.), \% & _0.0227_ & _0.0119_ & _0.0049_ & _0.0026_ \\ LOD (50 atm.), ppm & _227_ & _119_ & _49_ & _26_ \\ \hline LOD (50 atm., 300 s), ppm & 95 & _50_ & _20_ & _11_ \\ \hline \hline \end{tabular}  
* The Raman spectra are shown in Figure 4, gas pressure: 1 atm., exposure time: 60 s. The estimated limit of detection (LOD) values for the pressure of 50 atm. are according to Figure 5. The error in the specified LOD values is 14%. The estimated values are given in _italics_.  
\end{table} 
Table 2: Analysis * of the sensitivity of the Raman gas spectrometer according to the study of chemically pure isotopically enriched carbon dioxide and methane gases. Figure 4: Raman spectra of isotopically enriched (**a**) carbon dioxide (\({}^{12}\)CO\({}_{2}\) and \({}^{13}\)CO\({}_{2}\)) and (**b**) methane (\({}^{12}\)CH\({}_{4}\) and \({}^{13}\)CH\({}_{4}\)) gases. The excitation wavelength \(\lambda_{\rm exc}\) = 532.123 nm, the resolution is \(\sim\)1 cm\({}^{-1}\). Gas pressure: 1 atm. Exposure time: 60 s.  
First, the Raman intensity at the characteristic Raman shift (\(\
u\)) was determined both for the signal (\(I_{\rm signal}\)) and noise (\(I_{\rm noise}\)) yielding the useful signal, \(\Delta I_{\rm signal}=I_{\rm signal}-I_{\rm noise}\). The room mean square value of the noise, (\(r.m.s.\))\({}_{\rm noise}\), was also determined. The noise analysis was performed using the spectra of isotopically enriched gases in the wavenumber range free of Raman lines covering about \(\sim\)1000 cm\({}^{-1}\). The LOD is:  
\[\mathrm{LOD}_{\rm vol\%}=X_{\rm vol\%}\frac{3(r.m.s.)_{\rm noise}}{\Delta I _{\rm signal}}. \tag{1}\]  
here, \(X_{\rm vol\%}\) is the volume fraction of the studied gas, which is close to 100% for chemically pure isotopically enriched gases, cf. Table 1. The results on the LOD are shown in Table 2, being expressed both in vol% and parts per million (ppm), \(\mathrm{LOD}_{\rm ppm}=10^{4}\times\mathrm{LOD}_{\rm vol\%}\). The upper limit for the error in the LOD estimation was determined by analyzing 10 repetitive measurements for the \({}^{12}\)CH\({}_{4}\) gas. The maximum deviation from the average LOD value was \(\sim\)14%. The LOD can be decreased in two ways. One is the increase in the gas pressure. The limit of detection is proportional to inverse of the gas pressure \(p\), LOD \(\sim\) 1/\(p\), see Equation (1). Indeed, the useful Raman signal of a gas is proportional to the gas pressure [37], while the noise level is mostly determined by the detector and thus is independent of the pressure. To confirm this, we have measured the Raman spectra of chemically pure isotopically enriched \({}^{12}\)CO\({}_{2}\) gas at various gas pressures in the range of 1-9 atm., keeping the same exposure time of 60 s. The resulting LOD values (according to the 1388 cm\({}^{-1}\) Raman line) are shown in Figure 5. The data agree well with the LOD \(\sim\) 1/\(p\) law, yielding an estimation of the limit of detection at the maximum available gas pressure (50 atm.) of 227 ppm (for 60 s exposure). Note that particularly for CO\({}_{2}\) at high pressures (\(>\)50 atm. ), it was pointed out in [10] that the exponential law better describes the dependence of the Raman signal on the gas pressure. Further studies are thus required at such elevated pressures. Using the data measured at 1 atm., we interpolated the dependences for other gases, as shown in Figure 5 by dashed lines. The corresponding LOD (50 atm.)