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# Optical clumped isotope thermometry of carbon dioxide

## Abstract

Simultaneous analysis of carbon dioxide isotopologues involved in the isotope exchange between the doubly substituted 13C16O18O molecule and 12C16O2 has become an exciting new tool for geochemical, atmospheric and paleoclimatic research with applications ranging from stratospheric chemistry to carbonate-based geothermometry studies. Full exploitation of this isotope proxy and thermometer is limited due to time consuming and costly analysis using mass spectrometric instrumentation. Here, we present an all optical clumped CO2 isotopologue thermometer with capability for rapid analysis and simplified sample preparation. The current development also provides the option for analysis of additional multiply-substituted isotopologues, such as 12C18O2. Since the instrument unambiguously measures all isotopologues of the 12C16O2 + 13C16O18O $$\rightleftharpoons$$13C16O2 + 12C16O18O exchange, its equilibrium constant and the corresponding temperature are measured directly. Being essentially independent of the isotope composition of the calibration gas, an uncalibrated working reference is sufficient and usage of international calibration standards is obsolete. Other isotopologues and molecules can be accessed using the methodology, opening up new avenues in isotope research. Here we demonstrate the high-precision performance of the instrument with first gas temperature measurements of carbon dioxide samples from geothermal sources.

## Introduction

Mass spectrometry of multiply substituted isotopologues or clumped isotopes has become an extremely powerful tool in the natural sciences. Demonstrated applications which investigated carbon dioxide, methane, nitrous oxide, molecular hydrogen and oxygen range from tectonic history and evolution, geobiology and atmospheric chemistry over the investigation of non-equilibrium processes with correction procedures, diagenesis studies, the investigation of mineral formation conditions, the assessment of hydrothermal flow systems and paleo-evolution to paleothermometry; and there are many more potential applications1,2. The most prominent uses are linked to the oxygen isotope exchange reaction between the main isotopologue and the 13C-18O containing species of CO2

$${}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}+{}^{13}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}\rightleftharpoons {}^{13}{\rm{C}}{}^{16}{\rm{O}}_{2}+{}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}.$$
(R1)

Reaction (R1) involves only a single chemical compound, but could not be exploited scientifically until recently. This is because their very low natural abundance hampers the study of multi-substituted isotopic molecules, such as 13C16O18O, containing two or more rare isotopes (e.g. 13C and 18O) simultaneously. When compared to the main isotopologue 12C16O2, these are well below 10−4 (see Table 1). At the same time, the measurement techniques needed to attain extremely high accuracy levels of a few 0.01‰ (~tens of ppm) in order to trace the natural variability of the corresponding isotopologue and a dynamic range on the order of about 109 or better is therefore required. So far, only mass spectrometer instruments are capable of fulfilling these criteria and clumped CO2 has not yet been measured by optical methods1,2. Doubly substituted 13CH3D methane, which shows higher fractionation values, however, has been investigated using laser-based instruments. The first laser spectrometer setup3 for clumped methane isotopologues based on difference frequency generation (DFG) still suffered from uncertainties in the 20‰ range which exceeds natural 13CH3D variability of about 8‰4. Nevertheless, a more recent diode laser study on doubly substituted methane5 has successfully demonstrated that optical systems can well approach the necessary requirements. The achieved precision level of 200 ppm, however, remains still well above the commonly accepted threshold of 100 ppm (or 0.1‰) required for the study of clumped isotope fractionation in non-hydrogenated molecules.

While mass spectrometers excel in the achieved precision of about 10 to 20 ppm6,7, the instruments have to cope with inherent drawbacks. Not only are they relatively costly and heavy, thus not permitting in-field operation; they also require time consuming measurements and careful sample preparation in order to avoid contamination of the measurement signal. With current mass spectrometric procedures, preparation and analysis of a carbonate sample take about 3 to 6 h8. Importantly, only the largest and most sophisticated instruments can reach the mass resolution required to resolve isobaric interferences in CO29. Typical operation conditions are around ΔM/M ~ 40000 or lower, which is insufficient to separate 13C16O2 from 12C16O17O at m/z = 45 or 13C16O18O from 12C17O18O at m/z = 47, for example. In order to resolve these masses, a resolving power of above 52000 is needed – so far only accessible for large radius instruments in ‘non-normal operation’ mode9. This makes multiply substituted isotopologue analysis by mass spectrometry a very exclusive technology that will remain limited to only a handful of highly specialised laboratories worldwide, likely also constraining industrial or commercial use. Note that only two (12C16O2 and 13C18O2) out of twelve stable CO2 isotopologues can be detected entirely free from isobaric interference using a mass spectrometer (see Table 1). The four isotopologues 13C16O2, 12C16O18O, 13C16O18O, and 12C18O2, strongly dominate (>90%) a cardinal mass signal. They can thus well be assessed by the same technology except 12C18O2, whose quantification suffers from distorting background signals. For minor contributions to a cardinal mass, such as 12C16O17O however, more advanced sample preparation or conversion technologies10,11,12 must be employed at the cost of prolonged measurement time and reduced precision.

Simple counting statistics prevent using current mass spectrometer technology for the analysis of CO2 isotopologues below the relative abundance level of 10−5, even if these provide the main contribution to a cardinal mass. Assuming the measurement uncertainty being limited by Poisson statistics, the 10 ppm precision is reached after about 3 or 4 h of measurement on m/z = 47 (13C16O18O)13. In order to obtain the same precision for the 12C18O2 isotopologue on m/z = 48, a (n(16 13C16O18O)/n(12C18O2))2 ~ 100 times longer analysis time would be required, thus about two weeks. Even measurement times of two days are impractical and contrary to common practice. Demonstrations of ppm level instrument stabilities over such long time periods are lacking too. The m/z = 48 and 49 signals can therefore only be used as an indicator for sample contamination (hydrocarbons, halocarbons, sulphur monoxide)1,14,15 and must remain useless in exploiting 12C18O2 or 13C18O2 as isotopic tracers with mass spectrometry.

Despite the pioneering achievements of mass spectrometry in rare multi-isotope research, it is evident that alternative technologies are needed to overcome several of the aforementioned limitations. In this paper, we will present the first optical multi-isotopologue analyser for CO2 that is not influenced by most of these limitations, most notably the isobaric interference problem. The instrument achieves a measurement accuracy well below the 100 ppm level in the measurement of 13C16O18O and the technique has the capacity of assessing new tracers such as 12C18O2 that likely provide new and complementary information. In principle the developed method is calibration free and has strong potential of becoming a breakthrough technology, because it provides great isotopologue selectivity at reduced time, size and cost factors, which makes it well suited for widespread scientific, laboratory and commercial applications. The paper focusses on the measurement of carbon dioxide and the isotope exchange in the gas phase and we present the analysis of CO2 from thermal sources in the Upper Rhine Valley. The results will be compared to duplicate mass spectrometer measurements of CO2 from the same source. Apart from direct studies of gaseous carbon dioxide, current applications are concerned with multiply-substituted isotopologues in carbonates. Because carbonate isotopologues are obtained from measurements of gaseous CO2 released during the acid digestion of the carbonates, they can be investigated using the same analysis systems. The direct application of carbon dioxide isotopologue analysis to carbonates is further facilitated by the fact that the carbonate clumped isotope scale has been directly tied into the equilibrium CO2 gas scale6.

## Operating Principles and Instrumental Approach

### Isotopologue absorption spectroscopy

The optical measurement of CO2 isotopologues is based on absorption of isotopologue dependent ro-vibrational transitions in the ν3 fundamental band around 4 µm (Fig. 1), where infra-red absorption of carbon dioxide is strongest. With respect to the 12C containing species, the vibrational bands of 13C containing isotopologues are shifted towards lower energies. Using simultaneously two tuneable inter-band cascade lasers (ICLs) at 4.3 and 4.4 µm and absorption-balanced light paths of 9.2 cm and 10 m length, respectively, absorption signals of several ten percent of the five most abundant isotopologues can be obtained when a few mbar of pure CO2 are analysed in a cylindrical 0.8 L thermostated stainless steel absorption cell. A Herriott cell configuration made out of two concave mirrors at a distance of about 17.6 cm has been realised in order to achieve the long light path with 58 reflections. The short path traverses the cell just once in the perpendicular direction. More information on the absorption lines and a more detailed description of the set up displayed in Fig. 2 are given elsewhere16.

Spectra of pure CO2 at 297 K and 3 to 4 mbar are recorded at a rate of 1.56 kHz by driving laser currents at the same pace. A non-linear current ramp has been chosen for minimising the non-linearity of the laser frequency response over time17. Individual spectra were then averaged over 1 s time interval and analysed using a home-built fitting routine employing Rautian line profiles18. Required line parameters have been taken either from the HITRAN database (position) or were determined in separate experiments (self broadening coefficient and frequency of velocity changing collisions)16. The particle number density n of an isotopologue is obtained from applying the Beer-Lambert law to one of its transitions (see insets in Fig. 1), located at the centre wavenumber ν = νc. For a spectrally narrow laser the CO2 gas number density (n(CO2) = [CO2]) is linked to the optical measurement via19

$$[{{\rm{CO}}}_{2}]=-\frac{\mathrm{ln}(Tr(\nu ))}{S\cdot g(\nu -{\nu }_{c})\cdot L}=\frac{\alpha }{\sigma \cdot L},$$
(1)

where L is the path length, Tr(ν) the transmittance, S the line strength and g(ν − νc) the molecular line shape function of the particular transition. Isotopologue concentrations can thus be obtained from the extinction coefficient α = −ln(Tr(νc)) and the absorption cross section σ = Sg(0) at peak centre, which remain constant under fixed experimental conditions. The path length cancels in the measurement of an isotopologue ratio when both isotopologues are detected using the same path, e.g.20. [CO2]1/[CO2]2 = α1/α2(σ1/σ2)−1, making the method immune against eventual slight path length changes. Note that in spite of using the simplifying notation on the right hand side of Eq. (1) spectra are fitted over a whole frequency window and therefore include information from entire absorption lines and not just from peak absorption values.

Duplicate isotope ratio mass spectrometer (IRMS) measurements have been performed using the ThermoFischer MAT253Plus instrument at IUP Heidelberg, that has been equipped with an additional m/z = 47.5 cup and 1013 Ω resistors on m/z = 47−49 mass cups16. The m/z = 47.5 is used for continuous baseline monitoring. The mass spectrometric analysis follows accepted procedures6,14, as does the sample processing and cleaning8.

### Equilibrium constant, thermodynamics and Δ47

The equilibrium constant K1 of the isotope exchange reaction (R1) is strictly proportional to the product of absorption signals $$A={\alpha }_{{}^{13}{\rm{C}}{}^{16}{\rm{O}}_{2}}\,{\alpha }_{{}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}}\,{\alpha }_{{}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}}^{-1}\,{\alpha }_{{}^{13}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}}^{-1}$$, which therefore allows for the optical measurement via

$${K}_{1}=\frac{[{}^{13}{\rm{C}}{}^{16}{{\rm{O}}}_{2}][{}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}]}{[{}^{12}{\rm{C}}{}^{16}{{\rm{O}}}_{2}][{}^{13}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}]}={\rm{\Sigma }}\times A,$$
(2)

where the scaling factor $${\rm{\Sigma }}={\sigma }_{{}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}}\,{\sigma }_{{}^{13}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}}\,{\sigma }_{{}^{13}{\rm{C}}{}^{16}{\rm{O}}_{2}}^{-1}\,{\sigma }_{{}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}}^{-1}$$ depends on the involved molecular line strengths. Lacking the required accuracy, current database values cannot be used for determining Σ, which best is determined experimentally by exploiting the temperature dependence of K1 or its logarithm (see Fig. 3). The latter is widely used, because deviations from the statistical value $$({K}_{1}^{\ast }=1$$, where the *-symbol indicates the high temperature limit) are small and it can be expressed by three individual logarithmic terms, which can be identified with the isotopologue specific enrichment or fractionation values (CO2 denoting any particular isotopologue in the following equation)

$${\rm{\Delta }}({{\rm{CO}}}_{2})=\frac{[{{\rm{CO}}}_{2}]}{{[{{\rm{CO}}}_{2}]}^{\ast }}/\frac{[{}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}]}{{[{}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}]}^{\ast }}-1\simeq \,\mathrm{ln}\,\mathrm{(1}+{\rm{\Delta }}({{\rm{CO}}}_{2})),$$
(3)

commonly used for the quantification of isotopomers21 or multiply substituted isotopologues22,23:

$$-\,\mathrm{ln}(\frac{{K}_{1}}{{K}_{1}^{\ast }})=\,\mathrm{ln}(\frac{[{}^{13}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}]}{[{}^{13}{\rm{C}}{}^{16}{\rm{O}}_{2}]}\frac{[{}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}]}{[{}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}]})=\,\mathrm{ln}(\frac{[{}^{13}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}]}{[{}^{13}{\rm{C}}{}^{16}{\rm{O}}_{2}]})-\,\mathrm{ln}(\frac{[{}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}]}{[{}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}]})\mathrm{.}$$
(4)

The right hand side expression is applicable to the optical measurement, which provides the two particular isotopologue ratios as independent observables. As evident from Eq. (4), the temperature information contained in the equilibrium constant does only depend on two concentration ratios, that may be regrouped differently. ln (K1) is completely independent of the bulk isotope composition. Consequently, the 17O isotopic composition does not at all affect the determination of the equilibrium constant. Bulk isotopic compositions are only introduced when isotopologue concentrations are replaced by Δ values as defined in Eq. (3). Using $${K}_{1}^{\ast }=1={[{}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}]}^{\ast }{[{}^{13}{\rm{C}}{}^{16}{\rm{O}}_{2}]}^{\ast }/$$$$({[{}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}]}^{\ast }{[{}^{13}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}]}^{\ast })$$ and keeping only the leading terms in the Taylor series expansion on both sides, one obtains

$$\frac{{K}_{1}}{{K}_{1}^{\ast }}-1\simeq {\rm{\Delta }}({}^{13}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}})-{\rm{\Delta }}({}^{13}{\rm{C}}{}^{16}{\rm{O}}_{2})-{\rm{\Delta }}({}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}),$$
(5)

where the $$\simeq$$ sign indicates the approximative character of the relation. This equation, which may also be derived directly from Eq. (24) of Wang et al.22, is very similar to the commonly used definition of Δ47 in clumped isotope mass spectrometry of CO214,24,

$${{\rm{\Delta }}}_{47}={\rm{\Delta }}({}^{47}{\rm{C}}{{\rm{O}}}_{2})-{\rm{\Delta }}({}^{46}{\rm{C}}{{\rm{O}}}_{2})-{\rm{\Delta }}({}^{45}{\rm{C}}{{\rm{O}}}_{2}),$$
(6)

where isotopologues are replaced by m/z signals, because they cannot be measured individually. Comparison of Eqs (5) and (6) leads to the identification of $${{\rm{\Delta }}}_{47}\simeq {K}_{1}/{K}_{1}^{\ast }-1$$ with the relative deviation of the equilibrium constant K1 from its statistical value. It has been argued that the last two terms on the right hand side are zero24, but this premise is not completely consistent with the definition of Δ in Eq. (3) and thermodynamic calculations22 that respectively yield −4 and −11 ppm for Δ(45CO2) and Δ(46CO2) for CO2 equilibrated at 300 K. The reason for the conflicting results is that in the one case approximate but precisely measurable and in the other case exact but only approximatively accessible atomic isotope ratios are used in the calculations of the statistical abundances. Nonetheless, the so defined Δ47 is overwhelmingly influenced by the first of the three terms, which in turn is to a large extent (97%, see Table 1) dominated by the 13C16O18O isotopologue.

Unlike the direct measurement of ln (K1) according to Eq. (4), mass spectrometer determinations of Δ47 not only require measurement of heavy isotopologue abundances. The ‘absolute’ or bulk isotope composition must also be known in order to determine the statistical abundance of the m/z = 47 signal. This implies determining atomic 13C/12C, 18O/16O or 17O/16O ratios (traditionally quantified in terms of δ13C, δ18O and δ17O values), necessitating that international standard substances are used and that assumptions on the 17O isotope content are made. In this way systematic biases of up to 40 ppm are introduced24. Equally important, mass spectrometers can only approximately access the clumped 13C16O18O isotopologue (also due to an ion-source dependent scrambling effect) using the m/z = 47 signal and a corresponding scaling factor must be applied1,14.

Using the equilibrium constant of an isotope exchange (or isomerisation) reaction with a particular working gas as a thermometer, temperature is directly measured as a thermodynamic variable. The equilibrium constant of an isotope exchange reaction is linked to the reaction free enthalpy ΔF of that reaction K = exp(−ΔF/(kT)) 25,26,27, where k is the Boltzmann constant and where we have adopted a per molecule rather than a per mole definition of energies. The free energy of the gas is linked to the gas’ molecular partition function, which sums over all energy states εi taking degeneracies di into account and counting internal energy states from the lowest or zero-point energy (ZPE)  ε0 state of the molecule

$$Q=\sum _{i}^{levels}\,{d}_{i}{e}^{-\frac{{\varepsilon }_{i}}{kT}}={Q}_{trans}{Q}_{int}{e}^{-\frac{{\varepsilon }_{0}}{kT}}\mathrm{.}$$
(7)

In the Eq. (7) we have made the usual separation of the centre of mass motion (trans) from the molecular internal degrees of freedom (int). Since the equilibrium constant is given as a product of partition functions of reactant (react) and product (prod) molecules26,27

$$K=\frac{\prod \,{Q}_{prod}}{\prod \,{Q}_{react}}={(\frac{\prod {M}_{prod}}{\prod {M}_{react}})}^{\mathrm{3/2}}\frac{\prod \,{Q}_{int,prod}}{\prod \,{Q}_{int,react}}\exp (-\frac{\sum \,{\varepsilon }_{\mathrm{0,}prod}-\sum \,{\varepsilon }_{\mathrm{0,}react}}{kT}),$$
(8)

it is amenable to quantum statistical mechanics and computational chemistry methods. Here we have followed the usual simplification to evaluate the ratio of translational partition functions to ratios of molecular masses M26. The only non-trivial factors are the total internal partition functions that need to be evaluated separately. If one is mainly interested in isotope fractionation effects, it is convenient to normalise the equilibrium constant by dividing through its classical high temperature limiting value K*, which is given as the product of the classical symmetry numbers of product and reactant molecules:

$$\frac{K}{{K}^{\ast }}=K\frac{\prod \,{\sigma }_{react}}{\prod \,{\sigma }_{prod}}.$$

Solving for $${\rm{\Delta }}{\varepsilon }_{0}=\sum \,{\varepsilon }_{\mathrm{0,}prod}-\sum \,{\varepsilon }_{\mathrm{0,}react}$$, one obtains the ZPE change of the reaction in terms of measurable quantities:

$${\rm{\Delta }}{\varepsilon }_{0}=-\,kT[\mathrm{ln}\,K-\frac{3}{2}\,\mathrm{ln}(\frac{\prod \,{M}_{prod}}{\prod \,{M}_{react}})-\,\mathrm{ln}(\frac{\prod \,{Q}_{int,prod}}{\prod \,{Q}_{int,react}})].$$
(9)

The different terms in Eq. (9) can be identified with the reaction enthalpy or energy ΔH = ΔU = Δε0, the reaction free enthalpy ($${\rm{\Delta }}F=-\,kT\,\mathrm{ln}\,K$$), and the free enthalpy change associated to the reaction entropy (TΔS), which is given by the two remaining terms on the right hand side. Eq. (9) takes into account the energy change associated with the translational and internal molecular motion. In the following we adopt spectroscopic conventions and use term energies in wavenumber units Δν = Δε/hc, where h and c are the Planck constant and the speed of light, respectively. Different methods have been proposed to calculate the internal partition functions in Eq. (7). The traditional method based on work of Urey, Bigeleisen and Goeppert-Mayer (BMU)25,26,27 is to consider molecules as rigid rotor – harmonic oscillators and use corresponding spectroscopic constants. Combination of this approach with the Teller-Redlich rule28, usually attributed to Urey29, leads to a very simple description. For better accuracy, anharmonic corrections to vibrational energies and rotation-vibration interactions can be taken into account25,30,31, but for reasons of convenience or lack of parameters mostly only the anharmonic corrections to the ZPE are applied22,29. This can lead to significant uncertainties and more elaborate methods have been proposed, such as calculating the direct sum as a path integral using Monte-Carlo (PIMC) methods32,33. If highly accurate potential surfaces with spectroscopic quality are available or global effective Hamiltonians have been determined, such as for CO234, the total internal partition function in Eq. (7) might also be calculated very accurately by summing directly over all terms. In any case, the standard BMU approach must fail at low temperatures and masses (e.g. H2) due to neglecting the quantisation of rotational states, which is only taken into account approximately. In such a case the sum in Eq. (7) must be evaluated directly25. Conversely, the computational cost associated with numerical methods, such as PIMC and direct summation will increase when the temperature augments, because the number of thermally accessible states increases strongly. In addition the potential energy surface properties far from the equilibrium configuration become important when temperatures raise, leading to numerical convergence problems and artefacts22,35. In this limiting case, where isotope fractionation must vanish and a high precision is required, the BMU approach in combination with the Teller-Redlich rule might serve as a particularly useful guide, because its convergence towards the statistical limit is always assured.

### Implementation of the spectroscopic measurement

Unlike mass spectrometry, a laser absorption instrument can unambiguously measure all four (or three) required isotopologues of a homogenous CO2 isotope exchange reaction. The measurement is conceptually straight forward and does not depend on additional determinations and hypotheses on the bulk isotopic composition, as it directly determines ln K or $$\mathrm{ln}\,(K/{K}^{\ast })$$ and its temperature dependence, disregarding the ln Σ term (see Eqs (2) and (4)), which needs to be determined experimentally using a reference measurement:

$${\rm{l}}{\rm{n}}\,{K}_{1}(T)=\,{\rm{l}}{\rm{n}}\,A(T)+\,{\rm{l}}{\rm{n}}\,{\rm{\Sigma }}=\,{\rm{l}}{\rm{n}}\,{K}_{1}({T}_{ref})+\,{\rm{l}}{\rm{n}}(A(T)/{A}_{ref}).$$
(10)

Here, A(T) and Aref indicate the measured product of absorbances in Eq. (2) for CO2, once for the sample and once for a reference gas with known equilibrium constant K1 that has been equilibrated at the reference temperature Tref. Consequently, the method makes the quantification of the statistical distribution of isotopes obsolete (as indicated by Eq. (4)). Since absolute abundances of C and O isotopes don’t need to be known, the absorption measurement dispenses in principle the use and measurement of international standard substances. It only requires the comparison with a working gas whose value of ln K is known and remains stable over time. The extremely slow gas phase isotope exchange at ambient temperatures assures that any equilibrated CO2 gas with an isotopic composition close to natural sample gas composition in principle suffices for determining ln Σ (see Eq. (10)). This should make laser-based clumped isotopologue analysis much easier applicable than mass spectrometer investigations. In our setup, however, a slight cross-sensitivity of ln K on the difference in δ13C between the sample and the working reference gas of −4 ppm/‰ has been observed. This entails the determination of δ13C in our samples such that the interference can be corrected empirically16. The correction requires only one extra measurement and evaluation step relative to the sophisticated and error-prone mass spectrometric procedure. On the contrary, a cross correlation between ln K and δ18O has not been observed. The δ13C interference is much stronger than the isotope dependence of the thermodynamic equilibrium composition of about −0.01 ppm/‰. The effect is likely due to an insufficient modelling of the baseline originating from strong nearby absorption features of 13CO2. An improved fitting algorithm and a better choice of the spectral micro-window for the absorption lines should eliminate the effect, but this hypothesis requires further examination.

A natural candidate for the calibration of the optical method using Eq. (10) are measurements of the equilibrium constant at the high temperature limit $$(\mathrm{ln}\,{K}_{1}(T > 2000\,{\rm{K}})\simeq \,\mathrm{ln}\,{K}_{1}^{\ast }=0)$$. As these conditions are difficult to realise experimentally, we use a two step calibration, involving a heated working reference gas at a lower temperature and an ambient temperature working reference gas. The hot CO2 serves as calibration point, that determines the origin of the optical ln K1 measurements. The gas has been equilibrated at 1000 °C for about 5 h. We use the calculated value of $$\mathrm{ln}\,{K}_{1}=-\,26$$ ppm at that temperature (see Table 2) to determine Aref in Eq. (10). The uncertainty of this calibration is very small: different calculations at 1000 K are given in the literature22,33,36 and our calculation based on partition functions evaluated as direct sums of energy levels provided by ab initio calculations that were refined by spectroscopic measurements34, indicate that the error at that temperature is about 5 ppm. This deviates by only 2 ppm from the BMU method using the molecular constants of Wang et al.22. If we assume that the relative uncertainty remains the same, the systematic bias of the calibration should only be about 2 to 3 ppm at 1000 °C. Calculated room temperature (300 K) values of ln (K1) show a larger spread between −951 and −968 ppm, giving an order of ±10 ppm agreement (see Table 2). This uncertainty span is significantly larger than the spread in the high temperature values, rendering high temperature measurement preferable for calibration. As does the temperature gradient, which is 82 times smaller than the room temperature gradient of $$d\,\mathrm{ln}\,{K}_{1}/dT=5.7$$ ppm/K and makes the high temperature calibration less sensitive to instabilities in the temperature than its room temperature counterpart. This first calibration led us assign a value of $${\rm{l}}{\rm{n}}\,{K}_{1,ref}=-(954\pm 20)$$ ppm to our room temperature reference gas (lab grade purity N4.8 from Air Liquide, δ13CVPDB = −(40.0 ± 0.3)‰, $${\delta }^{18}{{\rm{O}}}_{{\rm{VPDB}}-{{\rm{CO}}}_{{\rm{2}}}}=-\,\mathrm{(27.3}\pm \mathrm{0.3)}$$‰). If not noted otherwise, we indicate measurement uncertainties as combined standard uncertainties at the 68% level of confidence.

In the second step, individual samples are measured by alternating acquisition sequences of sample and working gas. Each sequence starts by filling the spectrometer with the working reference gas and acquiring spectra for about 30 s with 1 s integration time. Then fast (~2 min) removal of the working reference occurs and the sample gas is analysed following the same acquisition procedure. Pressures of sample and reference gases are matched to agree within 0.01%. For repeated analysis, the sample is recovered using cryogenic trapping for about 5 minutes. At the end of the evacuation period the optical base line is determined. These repeated sample-reference comparisons, where Eq. (10) is applied each time, allow to take into account slow instrumental drifts that may have an impact on the determination of ln Σ.

## Results and Discussion

### CO2 thermometry with 13C16O18O and case application

The newly developed laser instrument has first been employed to demonstrate its capacity as CO2 isotopologue thermometer using ln K1 as directly observable temperature proxy. Four different samples of equilibrated CO2 have been prepared, with equilibration temperatures at 1 °C, 21 °C, 131 °C, and 1000 °C. For measurements at 1000 °C, pure CO2 gas was filled into quartz vials and kept in a lab oven for about 5 h. For the lower temperatures, droplets of liquid water were added to facilitate isotope exchange between isotopologues of CO2. At 1 °C equilibration times were about one month, and they were about a week for the intermediate temperature at 131 °C. Figure 4 shows the results of the measurements in comparison to the theoretically calculated curve. As a reference we use our evaluations of K1 from partition functions determined as direct sums and from the BMU method with harmonic frequencies and ZPE values given by Wang et al.22 (see Table 2). The maximum deviation of 59 ppm between either of the theoretical calculations in Fig. 4 and the measurements has been observed at 274 K. It is within twice the combined standard uncertainty (61 ppm) of the laser spectroscopic measurements at that temperature. At room temperature or above, the observed agreement is well within one standard uncertainty, which is 25 and 33 ppm, respectively.

Clumped isotope thermometry of gas phase CO2 originating from hydrothermal systems might provide a new and unique tracer for hydrothermal reservoir temperatures. The application is particularly relevant for studying the feasibility of the construction of hydrothermal power plants and estimating the associated risks, because the isotope thermometer may provide additional information on the related geological system and involved aquifers. For this case study we compare tuneable laser direct absorption spectroscopy (TLDAS) and IRMS measurements of the 13C16O18O isotopologue in a case study of natural carbon dioxide extracted from an operating hydrothermal power plant (Soultz) and two shallow wells (Landgrafenbrunnen and Stahlbrunnen), all located in the Upper Rhine Valley. The geothermal reservoir in Soultz (Alsace, France) has a temperature of about 200 °C at a depth of 5000 m37. During power plant operation, the water cools down to ~150 °C at the surface. Carbon dioxide for the analysis has been sampled from a separate sampling line, where the water has been rapidly cooled down to 38.5 °C. The mass spectrometric and laser measurements show clumped isotope temperatures between 92 and 108 °C and the two methods agree well within the respective uncertainties of 8 °C for the mass spectrometer determinations and 11 °C for the laser measurements (Fig. 5). Stahlbrunnen and Landgrafenbrunnen are two hydrothermal wells in Bad Homburg, Germany. The CO2 from the first one has been sampled directly from the well in the gas phase, whereas sampling of the carbon dioxide dissolved in water has been performed for the latter. The preparation of the gas samples for laser spectroscopic analysis follows a simplified procedure. It involves cryogenic separation from water and removal of non-condensable species through vacuum pumping. Compared to preparation for IRMS analysis, which requires additional cleaning by passage through a Porapak column38, the total preparation time is reduced by a factor of two. Laser spectroscopy and IRMS analysis of Landgrafenbrunnen CO2 apparent equilibrium temperatures show values of (10 ± 4) °C and (15 ± 6) °C, respectively, which is in good agreement with the temperature of the well’s water, Tw = 13.5 °C. A slight deviation from the expected water-CO2 equilibrium towards higher temperature has been observed for Stahlbrunnen, Tw = 12.4 °C versus TIRMS = (20 ± 5) °C and TTLDAS = (28 ± 7) °C. The interpretation of eventual discrepancies between measured apparent equilibration temperatures and parent water temperatures requires further investigation and is beyond the scope of this paper.

### Future developments

The standard uncertainty of the laser measurements in the 50 ppm range is obtained with samples of about 100 µmol and about 10 sample reference comparisons, which take between 1.5 and 2 h. This is still slightly larger than what can be obtained by mass spectrometry. However, this type of optical measurements is still in its infancy and is expected to improve soon. Already, our 0.8 L Herriott cell can be replaced with a very compact 40 to 140 mL multi-pass cell39,40 that provides a similar absorption length. Such small volumes imply reduced sample sizes on the order of 10 μmol or below and lead to faster evacuation times due to much simpler geometry without dead volumes. This shortens the time lapse between sample and reference measurement, thus limiting the impact of instrument drift and reducing the overall measurement uncertainty. We further anticipate that the introduction of an automated pressure balance system will allow for more reproducible conditions that further improve the measurement uncertainty.

It is also worth noting that optical measurements in the ν3 fundamental region of CO2 around 4.4 µm are not exclusively limited to the detection of the 13C16O18O clumped isotopologue. Figure 1 already shows that 13C16O17O can be measured as well. Inspection of spectral data41,42 further indicates that the P(12)e transition at 2305.365327 cm−1 provides a well isolated absorption line of 12C18O2. Its reported line strength (S = 1.11  10−23 cm molecule−1) is similar to the intensities of 12C16O17O and 13C16O2 used in this work (Fig. 1) and 12C18O2 should thus well be amenable to quantitative analysis. Given a suitable laser source, the isotopologue can be detected simultaneously with 12C16O2, 13C16O2, 12C16O18O and 13C16O18O and its measurement would provide a second and independent thermometer via the homogeneous

$${}^{12}{\rm{C}}{}^{18}{\rm{O}}_{2}+{}^{12}{\rm{C}}{}^{16}{\rm{O}}_{2}\rightleftharpoons 2{}^{12}{\rm{C}}{}^{16}{\rm{O}}{}^{18}{\rm{O}}$$
(R2)

exchange reaction, whose equilibrium constant has a temperature coefficient ($$d\,\mathrm{ln}\,({K}_{2}/\mathrm{4)/}dT=3.5$$ ppm/K) of the same magnitude than K1 at 300 K (see Fig. 3). The advantage of using this clumped CO2 thermometer along with 13C16O18O thermometry is its independence from 13C. The presence of kinetic fractionation effects that possibly compromise equilibrium thermometer readings would thus likely be different in the 13C containing and in the 13C free clumped isotope systems43,44,45,46. These effects could thus potentially be identified and corrected for. Therefore, optical measurements could provide an entirely new level of temperature information in the future. As an aside, we mention that clumped isotopes are often discussed in terms of bond ordering4,47, i.e. whether two rare isotopes form a common bond, such as 13C-18O in 13C16O18O. Reaction R2 is an example of indirect isotope clumping, where the two rare isotopes do not share the same bond1. In larger molecules, such as propane, ethane etc. this will be the predominant clumping mechanism. By definition, statistical combination of two different isotopic reservoirs leads to position-independent (anti-)clumping48,49. The similar magnitude of isotope fractionation in both reactions R2 and R1 (Fig. 3) demonstrates that thermodynamic clumping effects should also be considered as concentrating two (or more) rare isotopes in the same molecule, leading to a molecular configuration which is thermodynamically more stable than when these isotopes are redistributed over two (or more) different molecules – irrespective whether these isotopes share the same chemical bond or not.

Finally, the direct measurement of the equilibrium constant of an homogeneous exchange reaction at ppm accuracy may provide an interesting benchmark for molecular quantum calculations and potential energy surfaces. At low temperatures, different models are particularly sensitive to ZPE differences (Δν0 and the energies of the lowest states (see Eqs (8) or (9)). At 300 K the ZPE difference factor $$\exp (\,-\,{c}_{2}{\rm{\Delta }}{\nu }_{0}/T)$$ deviates from unity by 5 parts in 106 if Δν0 = 0.001 cm−1. This implies that an uncertainty of a few ppm – a range that will be amenable to measurements in the near future – is sufficient to determine ZPE differences at the 0.001 cm−1 uncertainty level, irrespective whether the ZPE differences are large, as in the case of the H2 + D2$$\rightleftharpoons$$ 2 HD reaction where Δν0 = 54.867 cm−150,51, or small – as in reaction R1, where calculated values22,33,34 range from 0.433 to 0.435 cm−1.

## Summary and Conclusion

We provide the first optical measurement of multiply-substituted isotopologues of CO2 at the accuracy level of better than 100 ppm. New advances in laser absorption spectroscopy, such as evidenced by the recent measurement of 12C16O17O at the precision level of 10 ppm within a time frame of 10 min52, indicate that laser instruments will favourably compete with mass spectrometer technology very soon. The comparatively high selectivity of laser-based instruments and their large potential of assessing new tracers, such as 12C18O2 for the homogeneous isotope exchange with 12C16O2, will open up new horizons in clumped isotope science and thermometry. The most important advantage of the technology is that the temperature can be obtained easily and directly via an unambiguous measurement of the equilibrium constant of the isotope exchange reaction. The optical CO2 isotopologue thermometer is a strong example showcasing the full potential for this and other molecules in the future. We have shown how the technology can be used for the thermometry of gaseous CO2 and that new areas related to molecular chemistry and physics are opened up for clumped isotope research. The relatively low cost and size factors of the technology will be additional parameters for developing and easing the spread of this exciting frontier science technology to more laboratories and technological applications.

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## Acknowledgements

I.P. and T.K. acknowledge funding from the Heidelberg Graduate School of Fundamental Physics (HGSFP). We acknowledge the technical help of the ‘physics of environmental archives’ team to maintain the IRMS instrument that was funded through the grant DFG-INST 35/1270-1. We thank Thomas Neumann, Elisabeth Eiche and Michael Kraml for support in selecting and sampling of hydrothermal wells. We are grateful for assistance by Maximilian Kalb and Andreas Weise in sampling CO2 at Soultz. We also thank Nicolas Cuenot for enabling access to the hydrothermal plant at Soultz and for technical support. C.J. would like to acknowledge Norbert Frank for invitation to a visiting professorship at IUP Heidelberg during this work.

## Author information

I.P. and C.J. planned the work, I.P. set up the instrument, made the laser measurements and performed the analysis. C.J. contributed to the instrumental set up. I.P. and T.K. collected geothermal CO2 samples. T.K. performed the IRMS analysis. All authors wrote and reviewed the manuscript.

### Competing Interests

The authors declare no competing interests.

Correspondence to Christof Janssen.