Absolute ¹³C/¹²C Isotope Amount Ratio for Vienna Pee Dee Belemnite from Infrared Absorption Spectroscopy

Abstract

Measurements of isotope ratios are predominantly made with reference to standard specimens that have been characterized in the past. In the 1950s, the carbon isotope ratio was referenced to a belemnite sample collected by Heinz Lowenstam and Harold Urey¹ in South Carolina’s Pee Dee region. Due to the exhaustion of the sample since then, reference materials that are traceable to the original artefact are used to define the Vienna Pee Dee Belemnite (VPDB) scale for stable carbon isotope analysis². However, these reference materials have also become exhausted or proven to exhibit unstable composition over time³, mirroring issues with the international prototype of the kilogram that led to a revised International System of Units⁴. A campaign to elucidate the stable carbon isotope ratio of VPDB is underway⁵, but independent measurement techniques are required to support it. Here we report an accurate value for the stable carbon isotope ratio inferred from infrared absorption spectroscopy, fulfilling the promise of this fundamentally accurate approach⁶. Our results agree with a value recently derived from mass spectrometry⁵, and therefore advance the prospects of SI-traceable isotope analysis. Further, our calibration-free method could improve mass balance calculations and enhance isotopic tracer studies in CO₂ source apportionment.

Small variations in the isotopic abundances of common nuclei such as carbon, oxygen and hydrogen occur naturally. The isotope-delta notation is a useful expression for precision stable isotope analysis and is generally defined as δ ⁱE = R(ⁱE/^jE)_sample/R(ⁱE/^jE)_ref − 1, where i is the heavier isotope mass number of element E (e.g., ¹³C), j is the lighter isotope mass number of E (e.g., ¹²C), and R(ⁱE/^jE)_sample and R(ⁱE/^jE)_ref are the absolute isotope ratios of an unknown sample and a reference material, respectively (e.g., R(¹³C/¹²C)_sample and R(¹³C/¹²C)_ref)⁷. The δ notation—expressed per mille using the symbol ‰, where 1 ‰ ≡ 1 × 10⁻³—allows for the convenient expression of exceedingly small differences in isotope composition, and thus an accessible discussion of subjects as diverse as variability in atmospheric composition, climatology, geochemistry, ecology, and dietary evolution and networking.

To measure small differences on a δ scale, ultraprecise analytical techniques such as isotope ratio mass spectrometry (IRMS)⁸ and Fourier transform infrared spectroscopy⁹, based on the identical treatment of sample and reference, are commonly used. Although δ values can be determined with a precision of ≤0.01 ‰, absolute isotope ratio measurements of R(ⁱE/^jE)_sample with commensurate accuracy have not been possible because of either systematic uncertainties in the values of R(ⁱE/^jE)_ref or the achievable uncertainty in primary methods such as calibration using synthetic isotope mixtures. For example, the most recent evaluation of the VPDB reference ratio that underpins the δ ¹³C scale, R(¹³C/¹²C)_VPDB, reported uncertainties that were more than 100-fold greater than the achievable δ-value precision⁵. The lack of independent experimental techniques for stable carbon isotope ratio measurements has generally impeded the assessment of uncertainty in R(¹³C/¹²C)_ref values and confounded efforts to correct for their respective drifts over time³.

The δ notation has additional limits. For example, δ values are approximately a linear function of mole fraction only for very small differences between R(ⁱE/^jE)_sample and R(ⁱE/^jE)_ref, and multiple reference materials are required to establish linearity². For samples with large differences between R(ⁱE/^jE)_sample and R(ⁱE/^jE)_ref (e.g., samples enriched or depleted in ¹³C, or extraterrestrial samples), δ values are a highly nonlinear function of mole fraction⁷, and precision is generally degraded when measuring sample values far from the scale definition (δ ⁱE = 0). For several applications, direct measurements of R(ⁱE/^jE)_sample or atom (mole) fraction would be advantageous. These include tracer studies for enriched compound- or position-specific isotope analysis¹⁰ and the calculation of mixing ratios in complex systems using a mass balance equation (e.g., systems with two or more sources and/or sinks of variable isotopic composition⁷). From a standards and metrology perspective, direct measurements of R(ⁱE/^jE)_sample would enable detailed studies of drifts in primary reference materials³ and SI traceable isotope ratio determinations⁵. Accurate measurements of R(ⁱE/^jE)_sample that are traceable to intrinsic physical invariants would also maintain compatibility with historical δ notation scales by underpinning the isotope ratios of the primary and secondary reference materials. This is highly desirable, given that the ratio-of-ratios approach conveniently expressed by the δ notation is inherently susceptible to temporal instability and physical inhomogeneity in the required traceability chain and two-point calibration schemes. Finally, reliable scales that eliminate artefacts altogether are desirable when measuring isotope ratios in extreme environments like those encountered during solar system exploration. There, carrying bulky and consumable specimens can be cost-prohibitive and pre-launch calibration routines are necessary to validate measurements like the isotopic composition of the modern Martian atmosphere¹¹.

Optical methods to measure R(ⁱE/^jE)_sample directly—circumventing the δ notation—have been proposed for decades⁶. Several notable efforts were focused on improving measurement precision and studied infrared transitions over a narrow wavelength range^12–15. That approach is problematic even when the absorption cross-sections—intrinsic molecular quantities which scale the isotope amounts—are well known¹⁶. Transitions involving different isotopic substitutions of the same molecule that occur at wavelengths near to one another often originate from very different rotational states, and consequently substantial temperature biases are introduced. Those biases can be mitigated through a high degree of temperature control^17,18, or by studying pairs of transitions with nearly identical lower-state rotational energy¹⁹. While these prior proposals and proof-of-principle demonstrations collectively suggest that accurate optical measurements of R(ⁱE/^jE)_sample may be possible without calibrations, each prior work was limited by either the aforementioned temperature biases or by uncertainties in the chosen absorption cross-sections. As a result, the absolute isotope ratios of reference materials inferred from accurate infrared absorption spectroscopy measurements have never been compared with SI-traceable mass spectrometry values.

Here we overcome both experimental hurdles using accurately known absorption cross-sections for carbon dioxide (CO₂) transitions from two distinct wavelength regions—effectively eliminating temperature biases. The results are direct measurements of R(ⁱE/^jE)_sample by infrared absorption spectroscopy with an uncertainty commensurate with calibrated mass spectrometry. By also performing state-of-the-art IRMS on our CO₂-in-air gas samples, we infer R(¹³C/¹²C)_VPDB = 0.011125 ± 0.000043. The value of R(¹³C/¹²C)_VPDB derived from our unified measurement approach is in excellent agreement with other recently reported values⁵ and differs significantly from the historic value of Craig²⁰. Consequently, our results demonstrate the potential for perpetual and direct SI-traceability in stable carbon isotope analysis, thus circumventing limitations associated with curation and further propagation of the new physical artifact standards that underpin the δ ¹³C VPDB scale²¹.

Our accurate isotope ratio infrared spectroscopy (AIR-IS) instrument, illustrated in Fig. 1A, is a robust platform for measuring the mole fraction of both ¹³CO₂ and ¹²CO₂, and therefore the stable carbon isotope ratio of a gas sample. Typical relative measurement precision of R(¹³C/¹²C)_sample for a 3 min spectral acquisition was 0.2 × 10⁻³, with a best recorded relative precision of 0.13 × 10⁻³ [comparable to the World Meteorological Organization / International Atomic Energy Agency (WMO/IAEA) extended compatibility goal for δ ¹³C of ± 0.1 ‰²²]. We simultaneously probed ¹³CO₂ and ¹²CO₂ rotational-vibrational transition pairs with nearly identical lower-state quantum numbers and lower-state energies, and therefore effectively identical Boltzmann population factors (Supplementary Information, Sections S1 and S3). Using diode lasers centered near wavelengths of 2.0 μm (¹³CO₂) and 1.6 μm (¹²CO₂) and high-reflectivity mirrors to form a single sample cell, the AIR-IS instrument created overlapping intra-cavity mode volumes for both wavelengths. This arrangement, combined with synchronous acquisition of the spectra, largely eliminated the influence of spatial-temporal temperature and pressure variations on the measured peak area ratios.

Fig. 1. Accurate isotope ratio infrared spectroscopy (AIR-IS).

A. The spectrometer synchronously interrogated two infrared transitions using diode lasers at wavelengths nominally equal to 2.0 μm (¹³C¹⁶O₂) and 1.6 μm (¹²C¹⁶O₂), respectively. Frequency agile, rapid-scanning was accomplished via two electro-optic modulators (EOM) driven by individual radiofrequency function generators. The length-stabilized optical enhancement cavity and sample cell provided effective path lengths of 12 km and 4.5 km at 2.0 μm and 1.6 μm, respectively. Laser light transmitted by the cavity was split at a dichroic mirror (DM) and then directed onto one of the two photoreceivers (PR). Additional abbreviations: AOM, acousto-optic modulator; M, reflective mirror; L, lens; PZT, piezo-electric transducer; PD, photodetector; PC, personal computer. B-C. Representative high-resolution cavity ring-down absorption spectra of ¹³C¹⁶O₂ (B) and ¹²C¹⁶O₂ (C) for a sample of NIST Standard Reference Material 1720 Northern Continental Air (${\chi }_{{\mathrm{C}\mathrm{O}}_2}$ = 393.23 μmol/mol; δ ¹³C = −8.6 ‰). Measured absorption coefficients (α) in units of inverse length are plotted in the upper panels as a function of laser detuning (B, red dots, 2.0 μm laser; C, blue dots, 1.6 μm laser). Also shown are the individual fitted spectral models (solid lines) and listed are the known transition wave numbers, vibrational band assignments, and rotational transition assignments (Supplementary Information, Section S1). The lower panels show the observed-minus-calculated (O–C) fit residuals for each measured CO₂ transition.

The absorption spectra of ¹³C¹⁶O₂ and ¹²C¹⁶O₂ in air at mole fractions within the range of natural terrestrial and marine abundances are shown in Fig. 1B–C. These spectra were measured using cavity ring-down spectroscopy and correspond to a reference sample of North American continental air with a value of δ ¹³C = −8.6 ‰ realized by the WMO Global Atmosphere Watch Central Calibration Laboratory at the National Oceanic and Atmospheric Administration (NOAA). We report similar peak absorption coefficients for both isotopologues, and thus similarly precise fits of the individual mole fractions $\chi =\frac{k_{B}T}{cS\left(T\right)p}\int \alpha \left(\nu \right)d\nu$, where c is the speed of light in a vacuum, k_B is the Boltzmann constant, T is the sample temperature, p is the total sample pressure, S(T) is the unweighted temperature-dependent transition intensity (i.e., not weighted by natural terrestrial isotopic relative abundances), and α(ν) is the measured absorption coefficient as a function of frequency ν. Assuming that all stable isotopes of C (¹³C, ¹²C) and O (¹⁸O, ¹⁷O, ¹⁶O) are randomly distributed amongst the corresponding isotopologues of CO₂, then R(¹³C/¹²C)_sample = χ(¹³C¹⁶O₂)/χ(¹²C¹⁶O₂). Taking the ratio of measured mole fractions of the two CO₂ isotopologues leads to the working equation

$$R{\left({}_{}{}^{13}\mathrm{C}/{}_{}{}^{12}\mathrm{C}\right)}_{\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=\frac{a_r}{s_r},$$ [1]

where the subscript r indicates the (¹³C¹⁶O₂)/(¹²C¹⁶O₂) ratio of parameters, $a\equiv \int \alpha \left(\nu \right)d\nu$ are the measured peak areas for each line, and s_r is the line intensity ratio for the transition pair. By concurrent measurements of a for both ¹³C¹⁶O₂ and ¹²C¹⁶O₂ at similar signal-to-noise ratios, we also eliminated biases associated with the relative frequency axis as well as with the choice of line shape profile.

Substituting Eq. [1] into the definition δ ¹³C = R(¹³C/¹²C)_sample/R(¹³C/¹²C)_VPDB − 1 and solving for R(¹³C/¹²C)_VPDB gives the VPDB reference value as

$$R{\left({}_{}{}^{13}\mathrm{C}/{}_{}{}^{12}\mathrm{C}\right)}_{\mathrm{V}\mathrm{P}\mathrm{D}\mathrm{B}}=\frac{a_r/s_r}{1+\delta {}_{}{}^{13}\mathrm{C}},$$ [2]

where all quantities on the right-hand-side are based on values with quantifiable uncertainties.

We emphasize that the line intensities used to evaluate R(¹³C/¹²C)_sample in Eq. [1] depend upon invariant molecular constants and a temperature-dependent function describing Boltzmann population statistics (Supplementary Information, Section S3). As already stated, the temperature dependence of s_r(T) was effectively eliminated by the choice of ¹³C¹⁶O₂ and ¹²C¹⁶O₂ transition pairs (Supplementary Information, Section S1). For ¹³C¹⁶O₂, we used intensities calculated here from an updated, fully ab initio dipole moment surface (DMS) and a semi-empirical potential energy surface (PES). This choice was justified by benchmarking the underlying quantum-chemistry theory against accurate experiments (Supplementary Information, Section S2).

AIR-IS measurements of R(¹³C/¹²C)_sample for five CO₂-in-air samples are plotted in Fig. 2A vs. δ ¹³C values assigned by IRMS (Supplementary Information, Section S4). In Fig. 2A, the fitted black line reveals a high degree of linearity between the two techniques (r² = 0.997), and fitted residuals are plotted in Fig. 2B. Note that the CO₂-in-air samples were chosen to cover a broad range of δ ¹³C values that naturally occur across a wide variety of terrestrial and marine carbon sources²³, as illustrated in Fig. 2C. For example, AIR-IS R(¹³C/¹²C)_sample values are accurate enough to unambiguously differentiate between C3 and C4 plant photosynthetic pathways.

Fig. 2. AIR-IS measurements of R(¹³C/¹²C)_sample vs. IRMS assignments of δ ¹³C.

A. AIR-IS values of R(¹³C/¹²C)_sample were measured for five CO₂-in-air samples with δ ¹³C VPDB value assignments by IRMS. The light-blue shaded region is the interpolated area bounded by the range of measured R(¹³C/¹²C)_sample values, while the mean and standard deviation for all five samples are plotted as gray circles. The black line represents a linear regression using the expression R(¹³C/¹²C)_sample = R(¹³C/¹²C)_VPDB × [(δ ¹³C) + 1]. B. Fitted residuals (O–C), along with their corresponding standard deviations. C. Expanded range of δ ¹³C values for some terrestrial and marine carbon sources²³.

With precise values of δ ¹³C assigned to our CO₂-in-air samples traceability to VPDB, we used Eq. [2] to calculate R(¹³C/¹²C)_VPDB from each individual spectroscopic measurement of R(¹³C/¹²C)_sample. A scatter plot of R(¹³C/¹²C)_VPDB versus an arbitrary measurement number is shown in Fig. 3A. From the ensemble statistics of R(¹³C/¹²C)_VPDB measured using samples with a variety of δ ¹³C values, we assessed long-term reproducibility associated with daily variations in the spectrum signal-to-noise ratio. The scatter in the data points plotted in Fig. 3A included variability associated with spectroscopic interferences, as well as potential optical interference effects (e.g., spurious etalons) and daily variations in laser power and linewidth which slightly affected optical cavity throughput and scan speed. The data was fitted to a normal distribution function (Fig. 3B) with a mean value of R(¹³C/¹²C)_VPDB = 0.011125 and a standard error of the mean of 1.7 × 10⁻⁶. Therefore, our Type A evaluation of the long-term reproducibility gave a relative standard uncertainty of 0.15 × 10⁻³.

Fig. 3. AIR-IS value of R(¹³C/¹²C)_VPDB.

A. Scatter plot of all 301 unique AIR-IS measurements of R(¹³C/¹²C)_VPDB, along with error bars showing relative precision of ≈ 0.2 × 10⁻³. B. Histogram of values fitted to a normal distribution (black line), with mean value of R(¹³C/¹²C)_VPDB = 0.011125. C. Comparison of literature values for R(¹³C/¹²C)_VPDB^5,20,24–29, including this work. Error bars show combined standard uncertainty.

In addition to long-term reproducibility in a_r, we report persistent (systematic) relative standard uncertainties in R(¹³C/¹²C)_VPDB for the transition intensities, cavity ring-down signal digitizer nonideality, choice of distribution function (Supplementary Information, Section S5), IRMS value assignments, and temperature in Table 1. Adding these components in quadrature gave a combined standard uncertainty in R(¹³C/¹²C)_VPDB of 43 × 10⁻⁶.

Table 1. Uncertainty budget for R(¹³C/¹²C)_VPDB.

Summary of individual relative standard uncertainties (u) comprising the combined standard uncertainty for R(¹³C/¹²C)_VPDB of 43 × 10⁻⁶.

Symbol	Value (10−3)	Notes
u 636	3.8	Line intensity, 13C16O2
u 626	0.6	Line intensity, 12C16O2
u ADC	0.6	Digitizer nonideality
uf	0.16	Choice of distribution function
uar	0.15	Standard error in ar
u IRMS	0.06	IRMS δ 13C value assignments
uT	0.04	Temperature-dependence of S

The value of R(¹³C/¹²C)_VPDB inferred from the unified AIR-IS and IRMS approach reported here is plotted as a gray circle at the top of Fig. 3C and labeled “This work.” The error bar illustrates the combined standard uncertainty in our measurement, dominated by the uncertainty in the ¹³C¹⁶O₂ transition intensities. For comparison, several literature values are also plotted in Fig. 3C. The blue diamond corresponds to the most widely cited value of ¹³R_VPDB = 0.0112372 (standard uncertainty, 30 × 10⁻⁶), measured by calibrated mass spectrometry of the original Pee Dee belemnite sample performed by Craig in 1957²⁰. More recently, in a 2010 technical report²⁴, the International Union of Pure and Applied Chemistry (IUPAC) recommended a value of R(¹³C/¹²C)_VPDB = 0.011180 (standard uncertainty, 14 × 10⁻⁶) based on mass spectrometry measurements by Chang & Li in 1990 (C&L, red triangle)²⁵. Additional experiments performed over the last 20 years^5,26–29 (open black squares) collectively allude to an even lower absolute value of R(¹³C/¹²C)_VPDB = 0.011117 (standard uncertainty, 15 × 10⁻⁶) show as the black square⁵, with those literature values deviating by more than −14 ‰ from Craig²⁰.

The AIR-IS result is in good agreement with the 2019 recommendation of Malinovsky et al.⁵, suggesting from that a reevaluation of the internationally accepted value of R(¹³C/¹²C)_VPDB is appropriate. Recently, Skrzypek & Dunn³⁰ highlighted an immediate motivation for the adoption of an SI traceable, consensus value of R(¹³C/¹²C)_VPDB. They report that three different reference values are currently in use by commercial optical stable carbon isotope analyzers, resulting in potential differences between calibrated analyzer δ values of ≈ 0.1 ‰. Therefore, the addition of our independent AIR-IS value to the VPDB discussion will aid in establishing consistency and traceability in stable carbon isotope analysis.

By directly interrogating highly homogenous gas-phase samples of CO₂ in air, AIR-IS avoids biases common to the general practice of CO₂ extraction from a VPDB traceable carbonate. Furthermore, AIR-IS also eliminates the need for O-isotope calibrations, potentially allowing the technique to accurately quantify R(¹³C/¹²C)_sample values even in the event of anomalous oxygen isotopic compositions. For example, careful attention must be paid to ¹⁷O corrections when analyzing CO₂ samples with oxygen from different sources using mass spectrometry (e.g., samples originating from reactions with ¹⁷O-enriched stratospheric ozone)²⁴. Oxygen isotope anomalies are also observed in extraterrestrial samples and may explain early solar system evolution³¹. Specifically, the stable carbon isotopic composition of meteorites with potentially anomalous oxygen compositions reveal ¹³C-enrichment in the organic molecules carried to early Earth³². These O-isotope anomalies, unless known a priori, present a unique challenge for accurate mass spectrometry where oxygen corrections are explicitly required^24,31.

The accurate isotope ratio infrared spectroscopy methodology reported herein is suitable for the possible optical realization of various artifact-based scales, including ¹⁴C, ¹⁸O, ¹⁷O, ¹⁵N, ²H, and ³⁴S. For stable carbon isotope ratios, our own efforts to ascertain more accurate values for ¹³C¹⁶O₂ transition intensities in the 2 μm wavelength region using gravimetrically prepared, potentially isotopically enriched gas samples with traceability to the kilogram are just beginning. Accurate measurements of analogous ¹²C¹⁶O₂ transitions could also reduce uncertainty in the ¹³C¹⁶O₂ ab initio predictions (Supplementary Information, Section S2). Those results are expected to reduce our relative combined standard uncertainty associated R(¹³C/¹²C)_VPDB to ≤ 1 × 10⁻³. Over time, a consensus determination of R(¹³C/¹²C)_VPDB through comparisons amongst national metrology institutes and their partners could ultimately obviate the current cumbersome and time-consuming task of artifact-based scale management and long-term scale propagation. Combined with recent advances in the optical detection of ¹⁴C with sensitivity below the modern mole fraction of 1.176 pmol/mol^33,34, as well as emerging optical techniques for precision clumped isotope analysis³⁵, the possibility exists for finally realizing primary laser-based carbon and oxygen isotope metrology with traceability to the revised, “quantum” SI.

METHODS

Absolute isotope ratio infrared spectroscopy.

We performed frequency agile, rapid-scanning cavity ring-down spectroscopy³⁶ using two distributed feedback diode lasers simultaneously coupled to a single high-finesse enhancement cavity and sample cell. Long-term stability and reproducibility were achieved in part by actively stabilizing the uniform grid of optical frequencies transmitted by the enhancement cavity to a frequency-stabilized HeNe laser³⁷. The enhancement cavity and sample cell comprised two triple-coated high-reflectivity mirrors at opposite ends of a stainless-steel tube. One of the mirrors was mounted to a piezo-electric transducer which was used as a slow actuator to adjust the cavity length to maintain resonance with the HeNe laser. The optical cavity was nominally 75 cm in length, with a free spectral range of 200.07 MHz (standard uncertainty, 30 kHz).

Before introduction of CO₂-in-air samples, the sample cell was evacuated under high vacuum for approximately 1 h and then flushed with a continuous flow of high-purity N₂ gas for approximately 2 h. The procedure effectively eliminated sample cell memory of the previous CO₂-in-air sample, as well as minimized spectroscopic interferences from previously adsorbed molecules within the gas delivery system. CO₂-in-air samples were then introduced and maintained at a constant pressure near 8 kPa (as measured by calibrated microbolometer gauges) for a total time of 2 h. After 2 h, the sample was evacuated, and the cavity prepared to receive another sample. Temperature stabilization was enhanced using an insulating box around the AIR-IS sample cell and was monitored during spectral acquisition using calibrated platinum resistance thermometers in good thermal contact with the outside of the enhancement cavity. A comprehensive summary of the NIST technical approach to accurate gas metrology by cavity ring-down spectroscopy can be found in Fleurbaey et al.³⁸.

Digitizer linearity.

Linearity of the analog-to-digital acquisition cards was tested using synthetic exponential decay signals and a metrology-grade reference digitizer as a transfer standard³⁹. The synthetic decays of 1 ms in length were constructed using an arbitrary waveform generator with 14-bit vertical resolution and a sampling rate of 128 MS/s. Their time constants spanned the relevant range for each digitizer; 26 μs to 45 μs for the 2.0 μm digitizer (¹³C¹⁶O₂), and 10 μs to 20 μs for the 1.6 μm digitizer (¹²C¹⁶O₂). Linear fits of observed time constants (τ_obs) vs. programmed time constants (τ_pro) yielded a transformation function between observed and true round-trip losses in the enhancement cavity. The linear transformations were used to estimate the relative shift in measured integrated absorption ${\delta }_a=\frac{a_{\mathrm{o}\mathrm{b}\mathrm{s}}}{a_{\mathrm{p}\mathrm{r}\mathrm{o}}}-1$ for simulated lines at both 2.0 μm and 1.6 μm. The digitizer analysis yielded the following δ_a, which were then applied to all the measured spectra: δ_a = 4.7 × 10⁻³ for ¹³C¹⁶O₂ at 2.0 μm, and δ_a = −4.8 × 10⁻³ for ¹²C¹⁶O₂ at 1.6 μm. We estimated the uncertainty in δ_a to be equal to $\sqrt{2}\times 0.4\times {10}^{-3}\approx 0.6$ × 10⁻³ by fitting a generalized extreme value distribution function to the observed distribution in τ_obs over a broad range of synthetic exponential decay signals. We note that the limit in the uncertainty in δ_a is calculated from the arbitrary waveform generator vertical resolution (14 bits) to be $1/\left(2^{\mathrm{\Delta }+1}\sqrt{3}\right)\approx 0.02$ × 10⁻³. The measured values of δ_a estimated all potential non-linearities, impedance mismatches, radiofrequency back reflections, dissimilar metallic connectors, and other issues associated with all electronics and cable after the photoreceivers, including the digitizer.

Gas samples and δ ¹³C value assignments.

Five CO₂-in-air samples were measured by AIR-IS. Mentioned in the main text is NIST Standard Reference Material (SRM^®) 1720 Northern Continental Air with WMO/NOAA-assigned δ ¹³C = −8.6 ‰ (cylinder number CC324315 and sample number 1720-A-25)^40,41. The remaining four CO₂-in-air samples, having nominal CO₂ molar fractions approximately equal to atmospheric natural abundances (≈400 μmol/mol), were prepared at NIST. Briefly, the preparation involved mixing (diluting) isotopically distinct pure CO₂ (>99.996 %) samples with a balance of synthetic air (N₂ = 78.1 %, O₂ = 20.89 %, CO₂ < 0.3 μmol/mol, N₂O < 0.2 nmol/mol) using gravimetric and volumetric standard preparation methods for compressed gas mixtures^40,42.

The VPDB-CO₂ δ ¹³C and δ ¹⁸O value assignments of these four samples were made at NIST on the parent CO₂ by dual inlet isotope ratio mass spectrometry (DI-IRMS)⁴³. The NIST VPDB-CO₂ scale realization was achieved using NIST pure CO₂ reference materials 8562, 8563, and 8564⁴⁴. A summary of NIST DI-IRMS results and related gas sample information is available as Table S4 of the Supplementary Information. Importantly, the uncertainty of 0.06 ‰ associated with the NIST IRMS value assignments had a minimal impact on the absolute isotope ratio R(¹³C/¹²C)_VPDB estimated in this work. Further details regarding IRMS value assignment uncertainties can be found in Srivastava & Verkouteren⁴³ and in Section S4 of the Supplementary Information.

Isotope ratio notation and nomenclature.

Here we use notation for isotope ratios recommended by the Commission on Isotopic Abundances and Atomic Weights of the International Union of Pure and Applied Chemistry (IUPAC)⁴⁵.

Quantum chemistry calculations.

Rotational-vibrational transition intensities were calculated for ¹³C¹⁶O₂ using an updated ab initio dipole moment surface (DMS) and semi-empirical potential energy surface (PES). Here we improved upon the ab initio DMS of Polyansky et al.⁴⁶ by increasing the number of ab initio data points by 50 % (3000 total points). Also, the semi-empirical PES of Huang et al.⁴⁷, previously used by Polyansky et al.⁴⁶ without modification, was fitted to known CO₂ energy levels using the DVR3D program⁴⁸ resulting in a standard deviation of 0.02 cm⁻¹. Analogous quantum chemistry methods recently applied to water (H₂O) showed that both an increased number of ab initio DMS data points⁴⁹ and an accurate semi-empirical PES⁵⁰ will improve accuracy in predicted transition intensities.

For our denser grid of ab initio points we applied the same level of quantum chemistry theory as Polyansky et al.⁴⁶: an all-electron multireference configuration interaction (MRCI) calculation with the aug-cc-pwCVQZ basis set, inclusive of relativistic corrections determined from separately fitted one-electron mass-velocity-Darwin (MVD1) terms. Calculations used the MOLPRO package⁵¹, and the wave functions—required to predict the transition intensities—were solved numerically from the rotational-vibrational Schrödinger equation using the DVR3D program⁴⁸.

To evaluate uncertainty in the predicted ¹³C¹⁶O₂ transition intensities, we also calculated transition intensities for ¹²C¹⁶O₂ and compared with highly accurate experiments^38,52–54. For ground-state rotational quantum numbers of J″ = 12 and J″ = 18 we find that, for five near-infrared vibrational bands of ¹²C¹⁶O₂, the standard deviation of a uniform distribution chosen to represent the experimental measurements relative to the predictions of our updated ab initio DMS is equal to 3.8 × 10⁻³. This value, listed in Table 1 of the main text as the uncertainty in the ¹³C¹⁶O₂ transition intensities, is consistent with upper-bound estimates from quantum chemistry methods^46,55. Further details, including numerical results from the updated ab initio DMS and their comparison with the experimental literature, can be found in Sections S1 and S2 of the Supplementary Information.

Data availability

Source data are available for this paper at doi:10.18434/mds2-2369. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.