# Isotopologue dependence of the CO2-air broadening and shifting coefficients: Experimental evidence and comparison with theory for 13CO2 and 12CO2

Didier Mondelain, Alain Campargue, Robert Gamache, Jean-Michel Hartmann, Fabien Gibert, Georg Wagner, Manfred Birk, Christian Röske

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Didier Mondelain, Alain Campargue, Robert Gamache, Jean-Michel Hartmann, Fabien Gibert, et al. Isotopologue dependence of the CO2-air broadening and shifting coefficients: Experimental evidence and comparison with theory for 13CO2 and 12CO2. Journal of Quantitative Spectroscopy and Radiative Transfer, 2024, 333, pp.109271. 10.1016/j.jqsrt.2024.109271. hal-05364594

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Isotopologue dependence of the CO₂-air broadening and shifting coefficients: Experimental evidence and comparison with theory for ¹³CO₂ and ¹²CO₂

Didier Mondelain,ª* Alain Campargue,ª Robert R. Gamache,ᵇ

Jean-Michel Hartmann,ᶜ Fabien Gibert,ᶜ Georg Wagner,ᵈ Manfred Birk,ᵈ Christian Röske,ᵈ

ª CNRS, LIPhy, Université Grenoble Alpes, Grenoble, France

ᵇ Department of Environmental, Earth, and Atmospheric Sciences, University of Massachusetts Lowell, USA

ᶜ Laboratoire de Météorologie Dynamique/IPSL, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, Sorbonne Université, Ecole Normale Supérieure, Université PSL, F-91120 Palaiseau, France

ᵈ Remote Sensing Technology Institute, German Aerospace Center (DLR), Germany

*Corresponding author: didier.mondelain@univ-grenoble-alpes.fr; LIPhy, Bat. E, 140 rue de la Physique, 38400 Saint-Martin d'Hères (France).

Key words

Isotopologue dependence; air-induced collisional widths and shifts; 13CO2; 12CO2; Cavity Ring Down Spectroscopy; Fourier transform spectroscopy; Complex Robert Bonamy Ma calculations
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**Abstract**

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Fifteen transitions of the 20012-00001 band of $^{13}$CO$_2$ in air have been recorded at 296 K using a cavity ring-down spectrometer linked to an optical frequency comb referenced to a GPS-disciplined Rb oscillator. In parallel, measurements of $^{12}$CO$_2$ in air were made with a Fourier transform spectrometer for transitions belonging to the 20012-00001 band, enabling comparison of the air-induced half-width and line shift coefficients of both isotopologues. For that a multi-spectrum fit procedure is adopted with a Hartmann-Tran (HT) line profile. A decrease of about 0.4% is demonstrated for the $^{13}$CO$_2$ air-broadening coefficients compared to the corresponding value of the main isotopologue. To the best of our knowledge, this observation is the first experimental evidence of an isotopic effect on the air-broadening coefficients in CO$_2$ (which are generally assumed to be identical in spectroscopic databases).

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On the theoretical side, a refinement to the simple model described in [Lamouroux et al. JQSRT 111;2010:2321] is proposed which enables the determination of $\gamma(^{13}CO_2)$ from $\gamma(^{12}CO_2)$. In addition, complex Robert-Bonamy-Ma (CRBM) calculations have been performed for $^{12}CO_2$ and $^{13}CO_2$ in collision with N$_2$ or O$_2$ for the 20012-00001 band with J'' values from 0 to 85. From these CRBM data, the $^{13}CO_2$-air and $^{12}CO_2$-air half-width and line shift coefficients were computed and the obtained isotopologue effect on these parameters was compared to the experimental data. A good average agreement is achieved for the CRBM calculations for the air-broadening coefficients while the simple model leads to slightly smaller isotopic effect. In contrast, for the air-pressure shift coefficients, a quite large disagreement is observed between the calculations and the experimental data.

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# 1. Introduction

Line-shape parameters are generally assumed to be independent of the isotopologue (as it is the case in the latest version of the HITRAN database [1,2] for CO₂) but a small dependence may exist.

The aim of the present work is to investigate its significance in the case of the ¹³CO₂ and ¹²CO₂ species. This is of importance for example to accurately determine the isotopic ratios from infrared

absorption spectrometry [3] for geosciences and combustion diagnostic applications. Hence, as

discussed in [4,5], pressure broadening effects could introduce biases in the δ¹³C isotopic ratio measurements<fnref n="1" /> when using gas matrices different from the composition of ambient air. Moreover,

absorption spectroscopy is a potentially absolute method to measure the isotopic ratios (avoiding the need for calibration gases) in gas samples if the line intensities and the line shape of the studied isotopologue transitions are accurately known [6]. Accurate line-shape parameters for both isotopologues are also required for measuring the ¹³CO₂ and ¹²CO₂ concentrations (and then δ¹³C) in the Earth's atmosphere using Dl differential Absorption Lidar (DIAL) [7,8]. Given the typical variability of δ¹³C in the atmosphere, the expected relative uncertainty on ¹³CO₂ and ¹²CO₂ absorptions should be lower than 0.2% [9]. In addition, the investigation of the isotopic effect on the line-shape parameters is likely to provide important clues on the intermolecular interactions [6].

The air-broadening coefficients, γ<sub>air</sub> (in cm⁻<fnref n="1" />/atm) can be approximately calculated for the minor isotopologues from γ<sub>air</sub> of the main isotopologue (¹²CO₂) using Eq. (A4) of the Appendix of [10] which takes into account the isotopologue mass dependence of the line broadenings. Effects at a few per mil level are expected by applying this law (see Table 7 of [11]) but are extremely difficult to measure. In 1998, Devi et al. [12] made measurements of air and nitrogen pressure broadening and pressure shifts for transitions in the 2ν₂ and ν₁ bands of ¹⁶O¹³C¹⁸O and compared their measurements to similar measurements for the laser bands I and II of ¹²C¹⁶O₂ (i.e. 00011-10001 and 00011-10002 bands in HITRAN notation centered at 960.959 and 1063.735 cm⁻<fnref n="1" />, respectively). Both the half-widths and line shifts showed small but no conclusive differences between the two isotopologues. In a later work, Devi et al. [13] measured by FTS air- and N₂-broadening and pressure shift coefficients for transitions belonging to the 9.4 and 10.4 µm bands for ¹²C¹⁶O₂ and ¹³C¹⁶O₂. One of their conclusion was: “Although the differences between corresponding broadening coefficients for ¹²C¹⁶O₂ and ¹³C¹⁶O₂ are small, there is marginal evidence that N₂-broadening parameters for ¹²C¹⁶O₂ may be 1–2% larger than those for ¹³C¹⁶O₂.” In a later work, Le Barbu et al. [14] made diode laser measurements of self-broadening coefficients near 2.04 µm for ¹³C¹⁶O₂ and ¹⁸O¹²C¹⁶O. While a

<footnote n="1">¹ δ¹³C = [((¹³C/¹²C)<sub>meas</sub>/((¹³C/¹²C)<sub>VPDB</sub>) - 1] × 1000 with ¹³C and ¹²C are the number densities of ¹³CO₂ and ¹²CO₂ and VPDB is the Vienna Pee Dee Belemnite reference standard.</footnote>
comparison was made, there were no transitions in common for the two isotopologues, so no
conclusion of the isotopologue dependence could be made.

In this work we use complex Robert-Bonamy-Ma (CRBM) calculations and a refinement to the
simple model proposed in the Appendix of [10], named hereafter ML model (for Modified Lamouroux
model) to obtain the isotopologue dependence of the air-broadening and shifting coefficients for the
$^{12}\text{CO}_2$ and $^{13}\text{CO}_2$ species. Results are then compared to accurate measurements of the air-broadening
(and pressure-shift) coefficients for the same $^{12}\text{CO}_2$ and $^{13}\text{CO}_2$ transitions of the 20012-00001 band
retrieved from spectra recorded by Fourier transform spectroscopy (FTS) at DLR for the main
isotopologue and by cavity ring down spectroscopy (CRDS) at LIPhy for the minor one. The next
section is dedicated to the ML model and the CRBM calculations. Then, in Sec. 3, the experiment
details and the description of the FTS and CRDS datasets are given. Finally, experimental and
theoretical results are compared to each other and discussed (Sec. 4) before some concluding
remarks (part 5).

## 2. ML model and CRBM calculations

### a. The ML model

Let us consider absorbing molecules, denoted $a$, infinitely diluted in a binary mixture with perturbers denoted $p$. We assume binary collisions (valid at the considered pressures) and that the collisional width of a given i→f transition at temperature $T$, $\Gamma_{\text{if}}^{a-b}(T)$, is given by:

$$ \Gamma_{\text{if}}^{a-p}(T) = n_p A_{\text{if}}^{a-p} [\bar{v}_r^{a-p}(T)]^{\alpha_{\text{if}}^{a-p}} \quad (1) $$

where $\bar{v}_r^{a-p}(T)$ denotes the mean (thermally averaged) relative speed, $n_p$ is the gas density, and $A_{\text{if}}^{a-p}$ and $\alpha_{\text{if}}^{a-p}$ are parameters both supposed independent of the temperature. Assuming the ideal gas law, i.e.: $n_p = P_p/(k_B T)$ with $P_p$ the perturber pressure, and recalling that $\bar{v}_r^{a-p}(T) = (8k_B T/\pi\mu_{\text{ap}})^{1/2}$, with $\mu_{\text{ap}} = M_a M_p/(M_a + M_p)$ the reduced mass of the (a,p) pair, leads to the following expression for the broadening coefficient per unit pressure $\gamma_{\text{if}}^{a-p}(T)$:

$$ \gamma_{\text{if}}^{a-p}(T) = B_{\text{if}}^{a-p} \times T^{\alpha_{\text{if}}^{a-p}/2-1} / \mu_{\text{ap}}^{\alpha_{\text{if}}^{a-p}/2} \quad (2) $$

where $B_{\text{if}}^{a-p}$ is independent of both $\mu_{\text{ap}}$ and $T$. Recalling that the temperature dependence of $\gamma_{\text{if}}^{a-p}(T)$ is generally modeled by a power law [15], i.e.:

$$ \gamma_{\text{if}}^{a-p}(T)/\gamma_{\text{if}}^{a-p}(T_0) = (T_0/T)^{n_{\text{if}}^{a-p}} \quad (3) $$

straightforwardly leads to:

$$ \alpha_{\text{if}}^{a-p} = 2(1 - n_{\text{if}}^{a-p}) \quad (4) $$
so that

$$ \gamma_{\text{if}}^{\text{a-p}}(T) = B_{\text{if}}^{\text{a-p}} \times T^{-n_{\text{if}}^{\text{a-p}}} / \mu_{\text{ap}}^{(1-n_{\text{if}}^{\text{a-p}})} \quad (5) $$

Let us now consider two isotopologues a₁ and a₂ of the “same” molecule which have the same symmetry, such as those, ¹⁶O¹²C¹⁶O and ¹⁶O¹³C¹⁶O, considered in this study. It is reasonable to assume that the (very) small differences between the a₁-p and a₂-p intermolecular potentials and between the rotational structures (i.e. between the energy gaps separating collisionally coupled rotational levels) of a₁ and a₂ have a negligible influence on the broadening, implying that $B_{\text{if}}^{a_1-p} = B_{\text{if}}^{a_2-p}$ and $n_{\text{if}}^{a_1-p} = n_{\text{if}}^{a_2-p}$, which leads to

$$ \gamma_{\text{if}}^{a_2-p}(T) / \gamma_{\text{if}}^{a_1-p}(T) = (\mu_{a_1 p} / \mu_{a_2 p})^{(1-n_{\text{if}}^{a-p})} \quad (6) $$

Note that it is possible to start the derivation “earlier”, by assuming that the dependence on the relative speed $v_r^{a-p}$ of the pressure-broadening coefficient $\gamma_{\text{if}}^{a-p}(v_r^{a-p}, T)$ is proportional to $(v_r^{a-p})\alpha_{\text{if}}^{a-p}$. Note also that this approximation was used in [16], and later in [17] to deduce the dependence of the broadening coefficient on the speed of the absorbing molecule, leading to the frequently used hyper-geometric law [18]. It is easily shown that carrying an average of $\gamma_{\text{if}}^{a-p}(v_r^{a-p}, T)$ over the thermal distribution of $v_r^{a-p}$, here leads to the same result as in Eq. (6).

Equation (6) is more accurate than what was obtained in [10] where a purely resonant model and dominant quadrupole-quadrupole interactions were assumed, leading to a line-independent value $\alpha_{\text{if}}^{a-p}=1/4$ for both O₂ and N₂ as perturbers and thus to air-broadening coefficients for ¹³CO₂ transitions about 0.22% smaller than the corresponding ones for ¹²CO₂.

Assuming that $A_{\text{if}}^{a-p}$ and $\alpha_{\text{if}}^{a-p}$ in Eq. (1) do not depend on temperature, which is valid for atomic perturbers, is an approximation in the case of collisions between the absorber a and a molecule p because of the change of the rotational populations of p with T. This likely explains a large part of the differences between the isotope effect predicted using Eq. (6) for ¹²CO₂ and ¹³CO₂ and direct predictions from semi-classical calculations of $\gamma_{\text{if}}^{12\text{CO}_2\text{-p}}(T)$ and $\gamma_{\text{if}}^{13\text{CO}_2\text{-p}}(T)$ detailed in the next section and observed in Figure 2.

b. The CRBM calculations

Complex Robert-Bonamy-Ma (CRBM) calculations [19,20] were performed for the ¹²CO₂-N₂, ¹²CO₂-O₂, ¹³CO₂-N₂ and ¹³CO₂-O₂ collision systems. The intermolecular potentials used consisted of the electrostatic quadrupole-quadrupole interaction, the London induction and dispersion potentials and an atom-atom potential expanded to 20th order and rank 4. The atom-atom parameters were
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adjusted by fitting the ¹²CO₂-N₂ and ¹²CO₂-O₂ halfwidths and line shifts measured in [21] and [22].

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Calculations used the same intermolecular potentials for both CO₂ isotopologues but different

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resonance functions and energy levels. The velocity integral was computed using 35 velocities. For
both isotopologues, CRBM calculations of the half-widths and line shifts for CO₂ in collision with N₂ or
O₂ for the 20012-00001 band with J'' values from 0 to 85 were made at 13 temperatures between

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200 and 3000 K including the temperature at which the experiments were carried out. The air-
broadened half-width and air-induced line shift were determined from the N₂ and O₂ data by the
standard routine.

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3. The experimental data sets

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a. The ¹²CO₂ in air data set by FTS

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Self- and air-broadened CO₂ measurements in the 2 µm region have been performed at DLR
within the ESA-funded project ISOGG (Improved spectroscopy for satellite measurements of
greenhouse gases), covering the 4750-5175 cm⁻¹ wavenumber range of the 20011-00001, 20012-
00001 and 20013-00001 bands. Measurement and line parameter databases for self- and air-
broadening are available on ZENODO [23,24]. Measurements and analysis will be detailed in a
dedicated publication. Here, only a brief description will be given.

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Fourier-transform measurements with a Bruker IFS 125HR and a multireflection cell have been
done for pure and air-broadened CO₂. Five pure CO₂ spectra have been recorded with pressures
ranging from 0.3 to 40 mbar and 17 CO₂/air spectra with total pressures ranging from 100 to 1000
mbar and a volume mixing ratio varying between 0.01 and 0.05 requiring the self-broadening to be
taken into account. The temperature range was 200-294 K, the absorption path 14.6 m, and the
resolution 0.005 cm⁻¹ (1/(2 maximum optical path difference)). Gas mixtures were prepared in a 800 l
stainless steel mixing chamber. Line parameters were deduced with multispectrum fitting applying
the Hartmann-Tran profile (with the correlation parameter η equal to zero) with Rosenkranz line
mixing and a dedicated instrumental line shape function. First, pure CO₂ spectra were analyzed, then
ambient temperature air-broadened CO₂ spectra, and finally low temperature air-broadened spectra.
In each step, parameters from the previous steps were kept fixed. Spectra were split into
microwindows. In order to determine the air pressure-induced line shift a common calibration factor
was fitted for all spectra with the same volume mixing ratio. There were four measurements with
total pressures between 100 and 1000 mbar for each volume mixing ratio. For all lines investigated in
the present paper γ₀air, γ₂air, and δ₀air were fitted. In case of the Dicke narrowing parameter, β, this
one could not be determined for the opaque lines P(14) and P(16) and was thus fixed to 0.

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The statistical uncertainties of the parameters given in the line parameter database are obtained from the multispectrum fitting ignoring the off-diagonal elements of the covariance/variance matrix. Systematic uncertainty for γ₀air is 0.03% and for δ₀air 3x10⁻⁵ cm⁻¹atm⁻¹ (ca. 0.6%). In **Table 1** the root square of the sum (RSS) of statistical and systematic uncertainties are given. Especially pressure shifts are sensitive to instrumental line shape (ILS) errors. The ILS influences the fitted line position and hence the pressure shift. Unfortunately, the multireflection cell has to be realigned when the pressure changes and the spectrum used for the ILS analysis is not the same as the actual sample spectra. The ILS changes slightly between the individual spectra, causing a systematic error in the line position. Line positions were analyzed for single spectra and compared to line positions calculated from unperturbed line positions and pressure shifts. The differences of measured and calculated line positions were averaged for each spectrum and used for an uncertainty estimate.

b. *The <sup>13</sup>CO<sub>2</sub> in air data set by CRDS*

Fifteen transitions belonging to the 20012-00001 band near 2.05 µm have been recorded at LIPhy-Grenoble for a mixture of <sup>13</sup>CO<sub>2</sub> in air at room temperature using a cavity ring-down spectrometer linked to an optical frequency comb referenced to a GPS-disciplined Rb oscillator. The comb-coherence transfer (CCT) technique was applied to obtain a RF tunable narrow-line laser comb-disciplined source using an external cavity diode laser and a simple electro-optic modulator. The set-up is described in detail in [25,26]. The only difference is the replacement of the single side-band modulator, limited to 6 GHz, by one working between 2 and 18 GHz allowing scanning a 16 GHz-broad spectral window centered on the studied transition.

For each transition, spectra were recorded at five total pressures (50, 100, 250, 500, 750 Torr) measured by a heated absolute capacitance manometer (Model AA02A Baratron from MKS; 1000 mbar full scale; accuracies are 0.04%, 0.05%, 0.05%, 0.07% and 0.10% for 750, 500, 250, 100 and 50 Torr, respectively, according to the calibration certificate). The <sup>13</sup>CO<sub>2</sub> in air mixture was obtained by first filling the empty cell with pure <sup>13</sup>CO<sub>2</sub> gas (99.0 atom % <sup>13</sup>C; <3 atom % <sup>18</sup>O from Aldrich) whose pressure was measured with a 10 mbar absolute capacitance manometer (Model AA01A Baratron from MKS; accuracy better than 0.02% of the reading according to the calibration certificate). Pure <sup>13</sup>CO<sub>2</sub> was then complemented with dry air (20.75% of O<sub>2</sub>, 1.00% of Ar and 78.25% of N<sub>2</sub> from Air Liquide) up to a total pressure of 750 Torr to achieve a mixture of ~10 ppm of <sup>13</sup>CO<sub>2</sub> in air. The measurements were carried out starting from 750 Torr down to 50 Torr by pumping from one pressure to the next one.

To improve the sampling of the profile for a given absorption line, four spectra were recorded at each pressure from a starting frequency, f<sub>start</sub>= 2000 MHz, for the first one, and from f<sub>start</sub> shifted by ¼, ½ and ¾ of the FSR value for the others (i.e. 2083, 2166 and 2249 MHz). The acquisition time of one
spectrum was typically 5 minutes. The high finesse cavity described in [27] (and called V2 in that paper) was temperature regulated at 296.0 K leading to the knowledge of the gas sample temperature with an uncertainty of ~0.05 K.

For each studied transition, the recorded spectra were analyzed with a multi-spectrum fit procedure using the *Multi-spectrum Analysis Tool for Spectroscopy (MATS)* fitting program developed at NIST [28]. In this procedure, a quadratic speed dependent Nelkin-Ghatak profile (qSDNGP) corresponding to the Hartmann Tran profile [29] with the correlation parameter $\eta$ set to zero, with line-mixing (first-order approximation) was adopted. The wavenumber of the spectral line at zero pressure, $v_0$, the line-broadening and pressure-shift coefficients $\gamma_0$ and $\delta_0$, their respective speed dependence components $\gamma_2$ and $\delta_2$, the Dicke narrowing parameter $\beta$ and the 1st order line-mixing coefficient $\zeta$ were simultaneously adjusted to all spectra. The $^{13}$CO$_2$ mole fraction and a linear base line were fitted for each spectrum with line intensities fixed to their HITRAN2020 values. The relative abundance of the weak interfering $^{12}$CO$_2$ lines (see **Figure 1**) were also fitted during the fit procedure.

An example of the spectra recorded for the P(14) transition in the 20012-00001 band of $^{13}$CO$_2$ together with those obtained from the multi-spectrum fit procedure are shown in **Figure 1**. The obtained (Exp.-Fit) residuals are at the noise level, leading to quality factors (defined as the ratio of the absorption at the peak to the RMS of the residuals) between 4494 and 7924 in this case.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Upper panel: CRDS spectra of the P(14) transition of the 20012-00001 band recorded at 50, 100, 250, 500 and 750 Torr with the mixture of $^{13}$CO$_2$ in air. Lower panel: Corresponding (exp.-fit) residuals obtained after the multi-spectrum fit</figcaption></figure>
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procedure. The two columns on the right correspond respectively to the quality factors (QF) and to the standard deviation of the residuals (in cm⁻¹). Note that the two weak lines visible on each side of the studied line correspond to ¹²CO₂ lines.

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The retrieved air-broadening, γ<sub>air</sub>, and pressure shift, δ<sub>air</sub>, parameters are reported in **Table 1** for both data sets. The error budget for these CRDS spectroscopic parameters is estimated in the same way as described in [30]. As shown in this reference and in [31] for the same noise level conditions, the type-A uncertainties correspond to the fit errors. Type-B uncertainties are estimated by assuming a bias of 0.1% for the total pressure, corresponding mostly to the uncertainty of the pressure gauge, and a bias of 0.05 K for the gas temperature [27]. As the ¹³CO₂ mole fraction is fitted for each spectrum, it is not considered here as a source of error. The final uncertainty is obtained as the RSS of the type-A and type-B errors is reported in **Table 1** for the ¹³CO₂ isotopologue. Final uncertainties at the 0.8‰ and 2×10⁻⁵ cm⁻¹atm⁻¹ levels are obtained for γ<sub>air</sub> and δ<sub>air</sub>, respectively.

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Note that among all the ¹²CO₂ transitions studied by FTS in the ISOGG project near 1.6 µm and 2 µm, nine have also been studied by CRDS in [30] and [31]. For these transitions, the observed average relative deviation between the FTS- and CRDS-measured broadening parameters is 0.1±0.1% and -0.8±1.2% for γₐir and δₐir, respectively (see **Table 2**).
<table><caption>**Table 1.** Fitted air-broadening and pressure shift parameters (in cm⁻¹atm⁻¹) obtained for the fifteen transitions studied in this work for the ¹²CO₂ and ¹³CO₂ isotopologues and belonging to the 20012-00000 band. Fitted parameters are in cm⁻¹atm⁻¹ at 296 K. Numbers in parenthesis correspond to the fit uncertainty in units of the last digit.</caption><thead><tr><th rowspan="2">Transition</th><th colspan="2">¹³CO₂ (CRDS)</th><th colspan="2">¹²CO₂ (FTS)</th></tr><tr><th>γ<sub>air</sub></th><th>δ<sub>air</sub></th><th>γ<sub>air</sub></th><th>δ<sub>air</sub></th></tr></thead><tbody><tr><td>P24e</td><td>0.07080(6)</td><td>-0.00548(2)</td><td>0.07111(2)</td><td>-0.00558(3)</td></tr><tr><td>P22e</td><td>0.07153(6)</td><td>-0.00545(2)</td><td>0.07195(4)</td><td>-0.00562(5)</td></tr><tr><td>P20e</td><td>0.07266(6)</td><td>-0.00542(2)</td><td>0.07296(2)</td><td>-0.00549(3)</td></tr><tr><td>P18e</td><td>0.07360(7)</td><td>-0.00527(2)</td><td>0.07398(2)</td><td>-0.00539(3)</td></tr><tr><td>P16e</td><td>0.07483(6)</td><td>-0.00523(2)</td><td>0.07526(2)</td><td>-0.00530(3)</td></tr><tr><td>P14e</td><td>0.07649(6)</td><td>-0.00510(2)</td><td>0.07680(2)</td><td>-0.00515(3)</td></tr><tr><td>P12e</td><td>0.07814(6)</td><td>-0.00492(2)</td><td>0.07837(2)</td><td>-0.00496(3)</td></tr><tr><td>P10e</td><td>0.07977(6)</td><td>-0.00464(2)</td><td>0.08015(2)</td><td>-0.00474(3)</td></tr><tr><td>P8e</td><td>0.08181(7)</td><td>-0.00439(2)</td><td>0.08233(3)</td><td>-0.00446(4)</td></tr><tr><td>P6e</td><td>0.08382(7)</td><td>-0.00406(2)</td><td>0.08430(3)</td><td>-0.00413(3)</td></tr><tr><td>R6e</td><td>0.08253(7)</td><td>-0.00356(2)</td><td>0.08281(4)</td><td>-0.00365(4)</td></tr><tr><td>R12e</td><td>0.07715(7)</td><td>-0.00390(2)</td><td>0.07717(2)</td><td>-0.00404(3)</td></tr><tr><td>R16e</td><td>0.07406(6)</td><td>-0.00422(1)</td><td>0.07424(3)</td><td>-0.00440(4)</td></tr><tr><td>R20e</td><td>0.07198(6)</td><td>-0.00454(1)</td><td>0.07207(2)</td><td>-0.00468(3)</td></tr><tr><td>R24e</td><td>0.07020(6)</td><td>-0.00475(1)</td><td>0.07048(2)</td><td>-0.00488(3)</td></tr></tbody></table>

<table><caption>**Table 2.** Fitted air-broadening and pressure shift parameters (in cm⁻¹atm⁻¹) obtained for the nine ¹²CO₂ transitions studied by CRDS in [30,31] and by FTS in the frame of the ISOGG project. Fitted parameters are in cm⁻¹atm⁻¹ at 296 K. Numbers in parenthesis correspond to the fit uncertainty in units of the last digit.</caption><thead><tr><th rowspan="2">Transition</th><th colspan="2">¹²CO₂ (CRDS)</th><th colspan="2">¹²CO₂ (FTS)</th></tr><tr><th>γ<sub>air</sub></th><th>δ<sub>air</sub></th><th>γ<sub>air</sub></th><th>δ<sub>air</sub></th></tr></thead><tbody><tr><td><strong>30013-00001</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>R(6)</td><td>0.08295(3)</td><td>-0.00465(3)</td><td>0.082882(36)</td><td>-0.004528 (41)</td></tr><tr><td>R(12)</td><td>0.07726(1)</td><td>-0.00506(2)</td><td>0.077164(28)</td><td>-0.005093(32)</td></tr><tr><td>R(16)</td><td>0.07437(1)</td><td>-0.00539(1)</td><td>0.074193(26)</td><td>-0.005382(25)</td></tr><tr><td>R(18)</td><td>0.07318(3)</td><td>-0.00558(1)</td><td>0.073091(26)</td><td>-0.005571(25)</td></tr><tr><td>R(20)</td><td>0.07218(2)</td><td>-0.00574(2)</td><td>0.072019(26)</td><td>-0.005760(25)</td></tr><tr><td><strong>20013-00001</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>R(24)</td><td>0.07049(2)</td><td>-0.00493(2)</td><td>0.070546(20)</td><td>-0.004841(12)</td></tr><tr><td>R(30)</td><td>0.06890(2)</td><td>-0.00520(1)</td><td>0.068799(35)</td><td>-0.005123(23)</td></tr><tr><td><strong>20012-00001</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>P(16)</td><td>0.07531(2)</td><td>-0.00531(2)</td><td>0.075265(12)</td><td>-0.005298(8)</td></tr><tr><td>P(28)</td><td>0.06986(5)</td><td>-0.00563(2)</td><td>0.069853(14)</td><td>-0.005531(16)</td></tr></tbody></table>

## 4. Results and discussion

An overview comparison of the ¹²CO₂ FTS values of γ<sub>air</sub> and δ<sub>air</sub> in the 20012-00001 band with CRBM calculations is presented in **Figure 2** and **Figure 3**, respectively. Regarding the broadening coefficients, the CRBM values agree with experiment within ±1% for |m|<40 (except for m=1,-2,3) where m equals to -J for the P-branch and J+1 for the R-branch. Note the agreement at the per mil level between the FTS measurements and the CRDS data of the P(16) and P(28) lines reported in [31], confirming the quality of the FTS measurements. In the case of the pressure shift, δ<sub>air</sub>, a clearly
different *m*-dependence is measured in the P- and R-branches. This asymmetric behavior, also observed previously for several CO₂ bands (see [32] and references therein), is not predicted by the CRBM calculations. Nevertheless, the latter reproduce quite well the experimental shifts for the P-branch for *m* values between -26 and -6 but start to diverge for *m*>-26 and *m*>-6. Note again the good agreement between the present FTS data and the CRDS data of Ref. [16].

269

<figure><img src="image_3.png" /></figure>

270

271 **Figure 2.** Upper panel: Air-broadening coefficients for the <sup>12</sup>CO<sub>2</sub> transitions of the 20012-00001 band measured by FTS (This work) and CRDS [31] and obtained by CRBM calculations. Lower panel: Relative differences to the FTS values.

272
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Air-pressure shift coefficients for the <sup>12</sup>CO<sub>2</sub> transitions of the 20012-00001 band measured by FTS in this work and CRDS in [31] and obtained by CRBM calculations.</figcaption></figure>

The isotopologue dependence of the air-broadening coefficient is evaluated by calculating the quantity 100(1-γ<sub>air_13CO2</sub>/γ<sub>air_12CO2</sub>). The experimental values of this quantity are obtained by using the γ<sub>air</sub> values of the same transitions measured by FTS in <sup>12</sup>CO<sub>2</sub> and by CRDS in <sup>13</sup>CO<sub>2</sub>. In **Figure 4**, the experimental results are compared to the CRBM values and to the ML model predictions. The latter were obtained by determining values of O<sub>2</sub> (and N<sub>2</sub>) broadening coefficients of <sup>13</sup>CO<sub>2</sub> lines using Eq. (6) with input values of γ<sub>if</sub> and n<sub>if</sub> for <sup>12</sup>CO<sub>2</sub> predicted by the CRBM model for O<sub>2</sub> (and N<sub>2</sub>). CRBM calculations show an isotopologue effect of about +0.40%, γ<sub>air</sub> being larger for <sup>12</sup>CO<sub>2</sub>, with a slight rotational dependence of 0.14% peak-peak while the ML model predicts a mean value of +0.29% and a smaller rotational dependence of 0.10% peak-peak. The experimental data confirm the amplitude of the isotopologue effect obtained from CRBM calculations with a mean value of +0.40% given with a standard error of the mean of 0.04% (N= 15). Due to its weakness and the dispersion of the experimental data it is not possible to confirm the rotational dependence of the effect.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Isotopologue dependence on γair obtained from FTS (¹²CO₂) and CRDS (¹³CO₂) measurements and from CRBM calculations, ML model and model proposed in the Appendix of Lamouroux2010[10]. The blue solid and dashed lines represent the mean of the experimental values and the standard error of the mean.</figcaption></figure>

The same type of comparison can be made for the air-induced pressure shift as shown in **Figure 5**.
In that case, a quite large disagreement is observed between the CRBM calculations and the
experimental data for which mean values of 0.08% and 2.1±0.2% (N=15) are obtained, respectively.
From the error analysis of the FTS results, the average systematic uncertainty for line shifts was
found to be 0.6%, which is significantly smaller than the observed isotopologue difference of about
2%.

We also looked in detail at the differences observed between the FTS and CRDS measurements
for the nine common ¹²CO₂ transitions in the 1.6 and 2 µm regions (see **Table 2**). On average the
difference is 0.8±1.2%. For the 2 µm region only, the difference is 1.3±0.4%. If we assume that this
difference represents a systematic error of the FTS shifts then the average isotopologue difference
between CRBM theoretical calculations and experimental data shrinks from 2% to 0.7%. However,
the validation was carried out using only four transitions, where one has a smaller difference of
1.24x10⁻⁵ cm⁻¹atm⁻¹, while for the three others the differences range from 7.7 to 9.9x10⁻⁵ cm⁻¹atm⁻¹.
For one of these three transitions, the P(28) transition of 20012-00001 band, the δ₂air parameter,
corresponding to the speed dependence component of the pressure shift coefficient, was fitted in
the case of the FTS fit procedure but not in the CRDS fits. Omitting the δ₂air in the FTS fits decreases
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309
310
311

the difference to almost zero. Among the transitions used in the paper there is only the R(6)
transition for which δ₂air was fitted in the fit procedure of the CRDS spectra. Thus, the validation of
FTS data with CRDS ones does not support the idea that the FTS data have a systematic bias on the
order of 2%.

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313
314

Note that the experimental data seems to show a branch dependence of the isotopologue effect for both γ<sub>air</sub> and δ<sub>air</sub>. This behavior which is not predicted by the CRBM calculations and ML model remains to be confirmed.

<figure><img src="image_6.png" /></figure>

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316
317
318

<caption>Figure 5. Isotopologue dependence on δ<sub>air</sub> obtained from FTS (<sup>12</sup>CO<sub>2</sub>) and CRDS (<sup>13</sup>CO<sub>2</sub>) measurements and from CRBM calculations. The blue solid and dashed lines represent the mean of the experimental values and the standard error of the mean.</caption>

319

5. Conclusion

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321
322
323
324
325
326
327
328
329
330
331

Air-broadening and air-pressure shift coefficients have been accurately determined from series of
spectra recorded for (i) a selection of fifteen transitions belonging to the 20012-00001 band of <sup>13</sup>CO<sub>2</sub>
using a comb-referenced CRDS setup and (ii) transitions of the 20012-00001 band of <sup>12</sup>CO<sub>2</sub> with a FT
spectrometer. These experimental data sets are compared to the output values of a refined version
of the simple model described in [10], and to CRBM calculations. These latter agree within ±1% for
|m|<40 (except for m=-2, 1, 3) with the experimental γ<sub>air</sub> values for the <sup>12</sup>CO<sub>2</sub> isotopologue and
reproduce also quite well the experimental shifts for the P-branch for m values between -26 and -6.
Nevertheless, the asymmetric behavior of the latter between the P- and R- branches evidenced from
the FTS data is not reproduced. In agreement with the predictions of CRBM calculations, the
comparison of the <sup>12</sup>CO<sub>2</sub> and <sup>13</sup>CO<sub>2</sub> air-broadening coefficients has revealed a small isotopic effect of
about 0.4%, the <sup>12</sup>CO<sub>2</sub> value being larger than the <sup>13</sup>CO<sub>2</sub> value. To the best of our knowledge, this is
the first experimental evidence of an isotopic effect on the air-broadening coefficients in CO<sub>2</sub>. Note
332

that, in the present version of the HITRAN database, no isotopic dependence is assumed for the air-
broadening coefficients of the various CO₂ isotopologues [1,2].

333

334

Acknowledgements

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338

339

340

341

342

A part of this work (¹²CO₂ FTS measurements) is funded by the European Space Agency (ESA) through the contract No. 4000132228/20/I-NS with Deutsches Zentrum fuer Luft- und Raumfahrt) entitled *Improved Spectroscopy for Carbon Dioxide, Oxygen, and Water Vapour Satellite Measurements*. DM and AC are grateful to Equipex REFIMEVE+ (ANR-11-EQPX-0039) for the ultra-stable reference frequency delivered. F. Gibert from LMD and P. Cacciani from PhLAM are acknowledged for lending the Thulium doped fiber amplifier and the external cavity diode laser, respectively.
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