## 4 Spectroscopy of \({}^{12}\)C/\({}^{13}\)C Isotopologues  
The study of greenhouse gas mixtures containing the gases of interest (i.e., carbon dioxide and methane) requires a preliminary study of the reference gases with high chemical and isotopic purity. The Raman spectra of the carbon isotopologues of carbon dioxide and methane are known. Here, we briefly describe and interpret the measured spectra. The CO\({}_{2}\) molecule is linear and centrosymmetric. The Raman spectrum of CO\({}_{2}\), see Figure 4a, contains two intense and sharp Q-branches, which form the diad \(\
u_{1}\) (1388/1370 cm\({}^{-1}\)) and 2\(\
u_{2}\) (1285/1266 cm\({}^{-1}\)) for the \({}^{12}\)C/\({}^{13}\)C containing molecules, respectively [33]. Adjacent to these lines, there are much weaker ones assigned as \(\
u_{1}+\
u_{2}-\
u_{2}\) (1410 cm\({}^{-1}\) for \({}^{12}\)C, for \({}^{13}\)C, it overlaps with the \(\
u_{1}\) line) and 3\(\
u_{2}-\
u_{2}\) (1266/1248 cm\({}^{-1}\) for \({}^{12}\)C/\({}^{13}\)C, respectively). The easiest attribution of the carbon isotopologues of CO\({}_{2}\) is according to the \(\
u_{1}\) and 2\(\
u_{2}\) diad, the wavenumber shift between the corresponding lines for the \({}^{12}\)CO\({}_{2}\) and \({}^{13}\)CO\({}_{2}\) molecules is \(\sim\)18-19 cm\({}^{-1}\). The Raman lines of isotopologues of carbon dioxide are easily detectable even at low gas pressure (1 atm.) using isotopically-enriched gases. The Raman spectra of the \({}^{12}\)CH\({}_{4}\) isotopologue have been systematically studied. Much less information is present about the second most abundant natural isotopologues, \({}^{13}\)CH\({}_{4}\)[34]. Only recently have several studies focused on filling in this gap [35]. CH\({}_{4}\) has tetrahedral symmetry (point group \(T_{\rm d}\)) with four normal modes of vibration. They are labeled by irreducible representations of the \(T_{\rm d}\) point group. The fundamental frequencies exhibit a simple relation, \(\
u_{1}(\rm A_{1})\approx\
u_{3}(\rm F_{2})\approx 2\
u_{2}(\rm E)\approx 2\
u_{4} (\rm F_{2})\). It leads to the so-called polyad structure in the Raman and infrared (IR) spectra [36]. A polyad \(P_{\rm n}\) gathers all vibrational states (\(\
u_{1}\), \(\
u_{2}\), \(\
u_{3}\) and \(\
u_{4}\)) satisfying the condition \(n=2(\
u_{1}+\
u_{3})+\
u_{2}+\
u_{4}\), where \(\
u_{\rm i}\) are the vibrational quantum numbers for \(i\)-th (\(i=1\), \(2\), \(3\), \(4\)) mode. Each set of \(\{v_{\rm i}\}\) defines a vibrational level. The pentad \(P_{2}\) corresponds to five vibrational levels: \(2\
u_{4}\), \(\
u_{2}+\
u_{4}\), \(2\
u_{2}\), \(\
u_{1}\) and \(\
u_{3}\). The lines in the Raman spectra corresponding to the pentad of methane are observed at \(\sim\)3000 cm\({}^{-1}\) and they can be used for detection of isotopologues of CH\({}_{4}\). The Raman spectra of isotopologues of methane around \(\sim\)3000 cm\({}^{-1}\) have a rich structure due to the complex vibration-rotation polyad interaction. The most intense line is observed at \(\sim\)2919 cm\({}^{-1}\) (\({}^{12}\)CH\({}_{4}\)) and 2915 cm\({}^{-1}\) (\({}^{13}\)CH\({}_{4}\)) and is assigned as \(\
u_{1}\). The isotope wavenumber shift is only \(\sim\)4 cm\({}^{-1}\). This line is well detected even at low gas pressure and short exposure, as shown in Figure 4b. Other prominent lines are those at \(\sim\)3024 cm\({}^{-1}\) (\({}^{12}\)CH\({}_{4}\)) and 3011 cm\({}^{-1}\) (\({}^{13}\)CH\({}_{4}\)), showing higher isotope wavenumber shift whilst showing much lower intensity. The spectra from Figure 4 were analyzed to determine the limit of detection (LOD) for each of the gases, as shown in Table 2.