An absolute reference frame for nitrous oxide clumped and position-specific isotopes provided by thermal equilibration

Clumped isotopic measurements of nitrous oxide (N₂O) have the potential to offer unique constraints on the processes governing N₂O production and destruction, building on the information provided by δ¹⁵N, δ¹⁸O and ¹⁵N site preference (SP). Extending their application requires a robust absolute reference frame. Here we show that thermal equilibration of N₂O over γ-Al₂O₃ provides such a reference frame for measurements of the isotopologues ¹⁴N¹⁵N¹⁸O and ¹⁵N¹⁴N¹⁸O, as well as for SP. Using a quantum cascade laser absorption spectroscopy (QCLAS) platform, we simultaneously quantify seven isotopologues of N₂O, including ¹⁴N¹⁵N¹⁸O, ¹⁵N¹⁴N¹⁸O, and ¹⁵N¹⁵N¹⁶O. Experiments starting from isotopically-distinct starting materials show convergence to time-invariant compositions that are in agreement with theoretically predicted temperature dependencies. These results demonstrate that γ-Al₂O₃ activated at ≥550 °C catalyzes isotope exchange among isotopologues of N₂O at equilibration temperatures between 153 and 218 °C and thereby define an absolute stochastic reference frame for Δ¹⁴N¹⁵N¹⁸O and Δ¹⁵N¹⁴N¹⁸O. In contrast, ¹⁵N¹⁵N¹⁶O does not equilibrate under these conditions, suggesting selective activation of N–O but not N–N bonds. Comparison of equilibrium SP values with theoretical predictions reveals a systematic offset relative to the current reference scale, which will require future work to reconcile.

1 Introduction

Nitrous oxide (N₂O) is a significant anthropogenically-stimulated greenhouse gas and ozone depleting molecule (Tian et al., 2020; Yu et al., 2020). Mitigation of its emissions is strongly supported by identifying environmental or physiological variations in nitrous oxide production and consumption. The clumped isotopologues ¹⁴N¹⁵N¹⁸O, ¹⁵N¹⁴N¹⁸O, and ¹⁵N¹⁵N¹⁶O show promise as new constraints on these variations, building on the information already available from position-specific and bulk isotopic measurements of N₂O (Kantnerová et al., 2020b, 2022; Magyar et al., 2016; Magyar, 2017; Ostrom and Ostrom, 2017). Two methods have been reported for accurate and precise quantification of these rare clumped isotopologues: high-resolution isotope-ratio mass spectrometry (HR-IRMS) and quantum cascade laser absorption spectroscopy (QCLAS). HR-IRMS utilizes measurements of ¹⁴N¹⁵N¹⁸O+¹⁵N¹⁴N¹⁸O and the fragment ¹⁵N¹⁸O to quantify ¹⁴N¹⁵N¹⁸O and ¹⁵N¹⁴N¹⁸O, but requires significant and variable background and rearrangement corrections and long measurement times of ≥8 h per sample (Kantnerová et al., 2020a; Magyar et al., 2016). QCLAS enables direct measurements of ¹⁴N¹⁵N¹⁸O and ¹⁵N¹⁴N¹⁸O, as well as ¹⁵N¹⁵N¹⁶O (Kantnerová et al., 2020a), but is subject to drift effects and therefore requires relatively fast measurement cycles to take advantage of its intrinsic selectivity and sensitivity (Gonzalez et al., 2019; Nataraj et al., 2022; Tuzson et al., 2008; Waechter et al., 2008; Werle, 2011). This method uses a dual-laser spectrometer to simultaneously quantify three clumped and four singly-substituted stable isotopic constraints and – as long as instrument stability can be assured and fast measurement enabled – appears to be a superior approach for measuring rare N₂O isotopologues (Kantnerová et al., 2020a).

Any approach to measuring the clumped isotopic composition of N₂O requires a reliable reference frame. Clumped isotopes offer a specific advantage because their abundances are directly related to thermodynamically-calculable equilibrium constants (Eiler and Schauble, 2004; Urey and Rittenberg, 1933). This has been used to provide absolute reference frames for clumped isotopic measurements of carbonate minerals, methane, and other molecules (Bernasconi et al., 2018; Stolper et al., 2014). For N₂O, the full equilibrium composition of all twelve isotopologues can be described by a system of eight reactions, each with an associated equilibrium constant, as detailed by Wang et al. (2004). One of these reactions, Reaction (R1), is directly connected to ¹⁵N site preference (SP), and so its variation with temperature at equilibrium can also be predicted (Bigeleisen and Friedman, 1950; Cao and Liu, 2012; Wang et al., 2004; Webb and Miller, 2014).

$$\begin{array}{}\text{(R1)}& {}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{16}}\mathrm{O}\rightleftharpoons {}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{16}}\mathrm{O}\end{array}$$

To date, however, such variations have not been used as a reference for N₂O SP measurements.

The utility of isotope exchange reactions as a reference frame comes not just from the ability to predict but also to experimentally generate equilibrium distributions of isotopes among isotopologues. Unlike CO₂, N₂O cannot readily be brought to an equilibrium isotopic composition by heating alone, as thermal decomposition outpaces equilibrium exchange reactions. However, activated alumina has been identified as a catalyst for driving isotope exchange reactions for ¹⁴N¹⁵N¹⁸O, ¹⁵N¹⁴N¹⁸O, and SP (Kantnerová et al., 2020a; Magyar et al., 2016). In subsequent years, the γ phase of alumina (γ-Al₂O₃) was also identified as a catalyst for the equilibration of methane clumped isotopologues (Eldridge et al., 2019, 2023; Turner et al., 2021; Wang et al., 2020a). Since activated alumina represents a poorly defined mixture of various Al₂O₃ phases including γ-Al₂O₃, we anticipate that γ-Al₂O₃ is also a suitable material for the catalytic activation of N₂O. For methane, γ-Al₂O₃ is activated under vacuum at 567 °C before its use as a catalyst (Eldridge et al., 2019; Robertson et al., 1975). Such activation is likely related to desorption phenomena and surface structural transformations that occur in Al₂O₃ phases before the conversion to α-Al₂O₃ at higher temperatures (Cancellieri et al., 2024; Gramatte et al., 2023; Prins, 2020).

Here, we report the clumped and position-specific isotopic composition of nitrous oxide heated in the presence of γ-Al₂O₃ to drive it towards predicted equilibria. We evaluate the equilibrium nature of these reactions by demonstrating a common, time-invariant isotopic composition matching theoretical predictions for different starting materials, and assess the effect of thermal decomposition as compared to putative equilibrium reactions. We establish conditions for pre-heating γ-Al₂O₃ to activate its catalytic activity for driving internal isotopic equilibrium reactions. To accomplish these measurements and provide a basis for future applications of N₂O clumped isotope measurements, we made improvements to the temperature stability and sample introduction for the QCLAS platform. Finally, we establish an absolute reference frame for the parameters Δ¹⁴N¹⁵N¹⁸O and Δ¹⁵N¹⁴N¹⁸O and discuss implications for measurements and interlaboratory comparisons of ¹⁵N site preference. We expect that our findings will facilitate application of isotopic tools to better constrain natural and anthropogenic source and sink processes for atmospheric N₂O and to monitor ambient N₂O in background air.

2 Methods

2.1 Laser spectroscopy

To measure the seven-dimensional isotopic composition of N₂O, we used an updated version of the QCLAS platform described by Kantnerová et al. (2020a). This instrument consists of a dual-laser spectrometer (QC-TILDAS-DUAL, Aerodyne Research Inc., USA) with quantum cascade laser sources emitting at 2142 and 2182 cm⁻¹ to quantify the eight most abundant isotopologues of N₂O, as shown in Fig. 1. We implemented an astigmatic Herriott multipass cell (AMAC-36LW, Aerodyne Research Inc.) with 36 m optical path length in 350 mL volume to replace the cell (210 m, 2500 mL) originally installed in this instrument (Kantnerová et al., 2020a). This smaller cell improves signal-to-noise, normalized to the sample amount, by substantially reducing the light loss upon reflection and by lowering the interference fringe level. It also enables faster sample gas extraction and sample-reference exchange time. To compensate for the shorter path length, we measure samples as mixtures of ∼8 % N₂O in N₂, instead of the ∼1.65 % mixtures used previously. At a typical working pressure of 4.8 hPa, ∼4 µmol of N₂O are introduced into the multipass cell for each measurement cycle. To reduce the effects of temperature, the QCLAS was housed in a temperature-controlled plexiglass box (IndTec 400, Elinter, Switzerland), supplementing the temperature regulation of the spectrometer (ThermoRack 401, Solid State Cooling, USA). As a net result, variations in the laboratory temperature of ±0.5 K are dampened to ±20 mK in the outer box and ±5 mK or better at the spectrometer cell. This level of temperature stability is necessary to achieve the required measurement precision (Bajnai and Hare, 2025).

Figure 1 Schematic of the inlet system. Samples are introduced through critical orifices at valves 20 or 21 and mixed with N₂ in the intermediate volume between valves 14, 22, 23, and 24, while working reference gas is introduced through the critical orifice at valve 24. From the intermediate volume, samples are expanded into the multipass cell for measurement, and evacuated by vacuum pumps after measurement, as described in Sect. 2.1. Inset: spectral windows covered by the two quantum cascade lasers, showing the absorbance features used to quantify ¹⁴N¹⁵N¹⁶O, ¹⁵N¹⁴N¹⁶O, and ¹⁵N¹⁵N¹⁶O using QCL1 and ¹⁴N¹⁴N¹⁷O, ¹⁴N¹⁴N¹⁸O, ¹⁴N¹⁵N¹⁸O, and ¹⁵N¹⁴N¹⁸O using QCL2.

To enable fast cycling between samples and reference materials, a customized gas inlet system was constructed (Fig. 1), similar to those described previously (Hare et al., 2022; Wang et al., 2020b; Zhang et al., 2025). This system consists of pneumatic valves (Swagelok, USA) connected by stainless-steel tubing, and is evacuated stepwise by a screw (PDV500, Ebara, Japan) and a turbomolecular pump (HiCube Eco 80, Pfeiffer Vacuum, Germany). It allows defined amounts of N₂, 8 % N₂O-in-N₂ working gas, and samples to be delivered to the intermediate gas volume (Fig. 1), from which they can then be expanded into the multipass cell. Gas flows into the inlet system are controlled by critical orifices, as shown on Fig. 1, including two options for sample introduction (30, 50 µm). The operation of the spectrometer and inlet system are coordinated by custom scripts within the spectrometer control software (TDLWintel, Aerodyne). Samples are typically delivered to the spectrometer in 10 mL stainless steel cylinders (Arbor Fluidtec-Swagelok, Switzerland) containing between 30 µmol and 2 mmol of pure N₂O.

A sample-measurement cycle consists of three sample injections, each bracketed by injections of 8 % N₂O-in-N₂ working reference gas (PanGas, Switzerland). For each working reference gas aliquot, the intermediate volume is filled to the target pressure of 27 hPa with 8 % N₂O-in-N₂, while for each sample gas aliquot, pure N₂O is introduced into the intermediate volume first (2.2 hPa) and then topped up to a total pressure of 27.0 hPa with N₂ (6.0 grade). On expansion into the multipass cell, sample and reference gas aliquots yield a pressure of 4.8 hPa. Spectral data are collected for two minutes for each injection, during which time the next sample or working reference gas aliquot is prepared in the inlet system. At the end of each measurement phase the cell is evacuated to <0.02 hPa and the next gas aliquot is introduced within one minute.

Absorption spectra are recorded at 1 s temporal resolution and concentrations of individual isotopologues are quantified in real time by TDLWintel (Aerodyne Research Inc.). Ratios $R_{\text{raw}}^i$ of individual isotopologues i to the main isotopic species ¹⁴N¹⁴N¹⁶O are calculated for both sample and working reference gases. Then, delta values (δⁱ) are calculated for each sample gas aliquot, normalized against the preceding and following working reference gas measurements, according to Eq. (1):

$$\begin{array}{}\text{(1)}& {\mathit{\delta }}^i=(R_{\text{raw, sample}}^i/R_{\text{raw work ref}}^i)-\mathrm{1}+{\mathit{\delta }}_{\text{work ref}}^i\end{array}$$

Finally, δⁱ are corrected for concentration dependency effects by normalizing to the ratio of ¹⁴N¹⁴N¹⁶O between sample and working reference. For the singly substituted isotopologues ¹⁴N¹⁵N¹⁶O, ¹⁵N¹⁴N¹⁶O, ¹⁴N¹⁴N¹⁸O, and ¹⁴N¹⁴N¹⁷O, delta values δ¹⁵N^α, δ¹⁵N^β, δ¹⁸O, and δ¹⁷O are reported on the international isotope ratio scales, Air-N₂ and VSMOW, respectively. The values for the working reference gas were assigned by measurement against the reference material RM1A (Mohn et al., 2022). For the clumped isotopologues ¹⁴N¹⁵N¹⁸O, ¹⁵N¹⁴N¹⁸O, and ¹⁵N¹⁵N¹⁶O, the parameter Δi describes the abundance of isotopologue i relative to a stochastic reference frame (Eiler, 2007; Magyar et al., 2016). As an intermediate parameter, we also define Δ^∗i, the clumped isotopic abundance calculated as if the working gas has a composition Δi of 0 ‰ for all isotopologues, and so then the measured δⁱ of these isotopologues against the working reference gas can be converted to values of Δ^∗14N¹⁵N¹⁸O, Δ^∗15N¹⁴N¹⁸O, and Δ^∗15N¹⁵N¹⁶O by accounting for the Δi of the working gas.

2.2 Preparation of isotopically-labelled gases

By mixing highly isotopically-enriched ¹⁴N¹⁵N¹⁸O, ¹⁵N¹⁴N¹⁸O, ¹⁵N¹⁵N¹⁶O, and ¹⁴N¹⁴N¹⁸O, prepared previously by thermal decomposition of labelled ammonium nitrate (Kantnerová et al., 2020a; Mohn et al., 2022), with natural-abundance N₂O, we created reservoirs of N₂O “isotopically-spiked” to ∼500 ‰ in each of these isotopologues. Briefly, a defined volume of the isotopologue of interest was injected into an evacuated 2 L high-pressure cylinder using a Vici multi-position valve (Vici Valco, USA) equipped with a sample loop. After flushing the loop with natural abundance N₂O the cylinder was topped off to ∼32 bar. The total amount of added N₂O was determined gravimetrically (PBK987-B60, Mettler-Toledo, Switzerland). The prepared gases were used for the experiments described in the following sections.

2.3 Equilibration of N₂O over γ-Al₂O₃

For equilibration of N₂O over γ-Al₂O₃, we used a vertical tube furnace (Vevor, China) with a custom glass reaction vessel, which was attached to a vacuum line for sample introduction and collection and evacuated by an oil-free rotary (NeoDry 7E, Kashiyama, Japan) and a turbomolecular pump (HiCube Eco 80, Pfeiffer). This system, depicted in Fig. S1 in the Supplement was developed from the equilibrium furnace described in Zhang et al. (2025). The pressure in the vacuum line was monitored with a pair of gauges for pressures below 1 mbar (PKR361, Pfeiffer) and between 0 and 4 bar (Leo 1, Keller Pressure, Switzerland). The bottom of the glass reaction vessel was loaded with 1.0 g of γ-Al₂O₃, provided as cylindrical pellets 1.8” (3.16 mm) in diameter and ∼2–7 mm long (Aluminum oxide, γ-phase, catalyst support, high surface area; ThermoFisher, Germany). The pellets extended to a height of ∼1 cm in the vessel. The same γ-Al₂O₃was used for all experiments without replacement. The temperature of the reactor was monitored with a K-type thermocouple placed at its very bottom, alongside the catalyst pellets, and logged (Hobo, Onset Computer, USA). The uncertainty in equilibration temperature was estimated as ±2 K, dominated by a temperature gradient in the reaction vessel (1.3 K cm⁻¹) and the temporal variation of the furnace temperature (±1.0 K).

In a first series of experiments, we explored the role of temperature in γ-Al₂O₃ activation by sequentially pre-heating the catalyst at increasing temperatures between 338 and 551 °C. In each experiment, the catalyst was heated for ∼1 h while the reaction vessel was evacuated with the turbomolecular pump. The pressure over the catalyst typically rose to between 0.01 and 0.1 mbar during this pre-heating phase, before returning to <0.01 mbar. Then, the equilibration vessel was restored to 200 °C, and once the temperature stabilized, 1.2 mmol of ¹⁴N¹⁵N¹⁸O- or ¹⁵N¹⁴N¹⁸O-spiked N₂O (“458-spike”, “548-spike”) was introduced. After a duration of at least 20 h, N₂O was cryogenically trapped out of the reaction vessel, expanded in a known volume to estimate the N₂O recovery, stored in a 10 mL stainless steel cylinder for storage until analysis, and then analyzed by QCLAS as described in Sect. 2.1.

After the temperature threshold for catalyst activation was established, further experiments, with incubation periods between 6 and 162 h, were conducted to establish the equilibrium nature and evaluate the kinetics of the reactions of N₂O isotopologues over γ-Al₂O₃. For these experiments, the catalyst was periodically re-activated by heating at 550 °C for >1 h. Once activated, the catalyst was kept under vacuum ($\sim {\mathrm{10}}^{-\mathrm{6}}$ mbar) and reactivation was not required for ≥1 month. For each equilibration experiment, the temperature of the reaction vessel was set between 153 and 220 °C, and either 0.3 or 1.2 mmol of N₂O of various isotopic compositions was introduced. At the end of each experiment, N₂O was cryogenically collected and analyzed as described in the Sect. 2.1.

2.4 Thermal decomposition of N₂O over γ-Al₂O₃

To determine the isotopic fractionation effects during thermal decomposition of N₂O over γ-Al₂O₃, a series of experiments was conducted in which 500 µmol of isotopically-labelled or natural-abundance N₂O was exposed to the catalyst at 498 °C. After reaction durations of 15–64 min, residual N₂O was extracted cryogenically and decomposition products, likely N₂, O₂, and NO (Lewis and Hinshelwood, 1938), were removed under vacuum. The collected N₂O was stored in a 10 mL stainless steel cylinder for subsequent isotope analysis. The kinetic isotope effects of thermal decomposition (¹⁵ε, ¹⁸ε, and εSP) were quantified using a closed-system Rayleigh model and the expression $\mathit{\delta }={\mathit{\delta }}_{\mathrm{0}}-\mathit{\epsilon }\ln \left[{\mathrm{N}}_{\mathrm{2}}\mathrm{O}\right]/[{\mathrm{N}}_{\mathrm{2}}\mathrm{O}{]}_{\mathrm{0}}$, where δ₀ is the starting isotopic composition.

3 Results and Discussion

3.1 Spectrometer performance

Optimal averaging times and calibration intervals for the spectrometer were evaluated by analyzing 8 % N₂O-in-N₂ working reference gas at the typical cell operating pressure of 4.8 hPa and using the Allan–Werle variance technique (Werle et al., 1993) (Fig. 2). We achieved Allan deviation minima of <0.2 ‰ for all isotopologue ratios in less than 100 s, but observed impaired performance for integration times of more than 500 s, due to drift effects. This precision is adequate to resolve differences observed to date among natural or experimental materials (Kantnerová et al., 2020b, 2022; Magyar, 2017; Magyar et al., 2016) or predicted in literature (Schmidt and Johnson, 2015; Wang et al., 2004; Yeung, 2016), and matches or exceeds previous approaches (Kantnerová et al., 2020a; Magyar et al., 2016). To achieve the same precision for samples, rapid sample – reference switching is essential, so we established and optimized a measurement scheme to analyze one “reference–sample–reference” set in 480 s and triplicate sample analyses in less than 20 min. Since all samples are measured against the same internal working reference gas, results can be compared for samples measured in different analytical sessions. In addition, we find that the magnitude and even sign of concentration dependence correction associated with each isotopologue can vary over time. Such variation has been observed in another recent study measuring ¹⁷O in CO₂ by laser spectroscopy (Ellis et al., 2025) and supports our approach of determining a specific concentration correction factor for each sample. To test its performance, we introduced repeated injections of working reference gas into the spectrometer over the course of ∼15 min and processed the data as if half of the pulses were a sample; the standard error for three “sample” pulses, each bracketed by two “reference” pulses, ranged from 0.09 ‰ to 0.18 ‰ for all isotopologue ratios (Fig. S2).

Figure 2 Allan–Werle deviation plots for isotopologues measured by QCL1 (A) and QCL2 (B).

3.2 Catalyst activation

We tested the effect of different pre-treatment conditions, including temperatures, heating sources and duration, and evacuation on the activity of γ-Al₂O₃ for catalyzing isotope exchange reactions in N₂O. Preliminary experiments showed that heating γ-Al₂O₃ with a natural gas torch under vacuum, as described by Magyar et al. (2016) and Kantnerová et al. (2020a), resulted in inconsistent results. Therefore, using the on-line furnace described in Sect. 2.3 we systematically tested the influence of activation temperature on isotope exchange. We evaluated activation by heating ¹⁴N¹⁵N¹⁸O-spiked N₂O, with a Δ^∗14N¹⁵N¹⁸O of 512 ‰, under consistent conditions and found a sharp activation threshold between 467 and 515 °C (Fig. 3). Using a logistic fit to these data we estimate the midpoint temperature of the activation path as 490 ± 4 °C (1 SE). Catalyst activation at temperatures below this threshold led to only subtle changes in Δ^∗14N¹⁵N¹⁸O, while catalyst activated at 515 °C and higher uniformly showed a drop of >400 ‰ towards expected equilibrium values. These results are consistent with the choice of temperatures between 550 and 567 °C to activate γ-Al₂O₃ for catalysis of exchange among methane clumped isotopologues and between CH₄, H₂, and other hydrocarbons, which in turn followed from the temperature required to convert boehmite, γ-AlO(OH), to γ-Al₂O₃ (Eldridge et al., 2019; Robertson et al., 1975; Wang et al., 2020a). Although it has been shown that γ-Al₂O₃ does not serve as a catalyst for methane isotope equilibration in the absence of thermal activation (Turner et al., 2021), the lower threshold of activation has not been explored thoroughly. In a subset of experiments, we also activated γ-Al₂O₃ at 550 °C in the presence of O₂, but did not observe quantifiable effects on catalytic activity; these experiments were again motivated by work with methane and hydrocarbons in which such an oxygen activation step is included to enhance catalytic activity (Eldridge et al., 2019, 2023; Robertson et al., 1975; Turner et al., 2021). Based on the outcome of these experiments, we chose 550 °C and a minimum of 1 h as standard catalyst activation conditions for subsequent experiments.

Figure 3 Threshold temperature for activation of γ-Al₂O₃ for equilibration of N₂O isotopologues. Data points show the abundance of ¹⁴N¹⁵N¹⁸O, as described by the parameter Δ^∗14N¹⁵N¹⁸O, for ¹⁴N¹⁵N¹⁸O-spiked N₂O heated over γ-Al₂O₃ after its activation at the denoted temperature. The blue curve and band represent a logistic fit and associated 95 % confidence interval, yielding estimates of 497 ± 9 ‰, 10 ± 7 ‰, 490 ± 4 °C, and 0.12 ± 0.02 for the isotopic composition before and after activation, the threshold temperature, and the exponential factor k describing the fit, respectively. Error bars (1 SE for three replicate measurements of a given sample gas) are smaller than the marker size.

3.3 Establishing the equilibrium nature of the reaction of N₂O over γ-Al₂O₃

To establish that the reactions undergone by N₂O isotopologues in the presence of γ-Al₂O₃ are of an equilibrium nature, we relied on three lines of evidence: (1) convergence on a common isotopic composition from distinctly different starting materials; (2) time-invariance of isotopic composition for extended time intervals after convergence; and (3) matching theoretical temperature dependence predictions. The first two criteria, the principles of convergence and time-invariance, have a strong basis in isotope geochemistry to identify equilibrium reactions in geochemical or petrological settings (O'Neil, 1986). The materials described in Sect. 2.2, with elevated values of Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O, were exposed to the activated catalyst at 200 °C for durations between 6 and 117 h. We found that after one day these materials had converged on narrow ranges in Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O (Fig. 4), and that rapid progress towards that range was already achieved after 6 h (Fig. 5). For Δ^∗14N¹⁵N¹⁸O, starting materials at 512 ‰, 3.0 ‰, and −0.3 ‰ converged to 10.32 ± 0.60 ‰, apart from one experiment, which displayed a residual Δ^∗14N¹⁵N¹⁸O of 79.7 ‰ after 24 h. This outlier, for the first sample measured after γ-Al₂O₃ activation at the lower temperature threshold (515 °C), suggests as-yet uncharacterized complexity to the activation of γ-Al₂O₃, and the need to verify results by repetitive equilibration experiments. Likewise, for Δ^∗15N¹⁴N¹⁸O, a value of −11.07 ± 0.55 ‰ was converged upon from starting materials at 458.6 ‰, 9.1 ‰, and 0.0 ‰. At the same time, for all equilibration experiments on N₂O with enhanced Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O, SP values increased from ∼ −2 ‰ to 0 ‰ in the starting materials to 20.9 ± 0.4 ‰ after 24 h (Fig. 5). This robust convergence on a putative equilibrium isotopic composition is our first indication of equilibrium reactions at work, for the three parameters SP, Δ^∗14N¹⁵N¹⁸O, and Δ^∗15N¹⁴N¹⁸O.

Figure 4 Convergence of isotopic parameters on a common composition. Starting materials (hollow symbols) that bracket the expected equilibrium composition in Δ^∗14N¹⁵N¹⁸O, Δ^∗15N¹⁴N¹⁸O, and SP were heated over activated γ-Al₂O₃ and are observed to converge on a narrow range in all parameters (filled symbols). Zoomed plots (dashed insets) show that this convergence is observed even over finer scales in Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O for the complementary isotopologues to the ones specifically spiked into the starting materials used.

Figure 5 Time-invariance of Δ^∗14N¹⁵N¹⁸O and SP for extended durations of heating at 200 °C. For all samples heated over γ-Al₂O₃, which wasactivated at 550 °C (black circles), and most samples with activation between 515 and 550 °C (gray diamonds), values of these parameters remain at the value they reach in 24 h for durations up to one week. Error bars represent 1 standard error for replicate measurements of a given sample; when not seen, error bars are smaller than the marker size. The inset shows a zoomed-in view of variation in Δ^∗14N¹⁵N¹⁸O for samples close to the equilibrated value.

Complementary to convergence is time-invariance of isotopic composition after equilibrium has been reached, as long as thermodynamic conditions remain constant. This was confirmed by consistency of SP values at 20.9 ± 0.4 ‰, even when heating for durations of up to one week (162 h) (Fig. 5). Likewise, in all cases where the catalyst was activated at 550 °C, and in all but one case for an activation between 515 and 550 °C, the values of Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O achieved at 24 h were the same as those seen out to one week (162 h) (Fig. 5). In contrast, any kinetic effect associated with N₂O decomposition or rearrangement is expected to lead to continuous evolution in isotopic composition for as long as concentration continues to change. A more specific comparison of isotopic evolution under kinetic and equilibrium conditions is discussed in Sect. 3.4.

The third confirmation that equilibration has been reached is the agreement with theoretical predictions of differences in Δ^∗14N¹⁵N¹⁸O, Δ^∗15N¹⁴N¹⁸O, and SP for different equilibration temperatures. Based on the Urey–Bigeleisen–Mayer quantum statistical mechanical framework using anharmonic potential energy surfaces (Bigeleisen and Friedman, 1950; Bigeleisen and Mayer, 1947; Cao and Liu, 2012; Urey, 1947; Wang et al., 2004), and validated by a Feynman path-integral approach (Webb and Miller, 2014), we have robust estimates for Δ¹⁴N¹⁵N¹⁸O, Δ¹⁵N¹⁴N¹⁸O, and SP at equilibrium (Fig. 6). We introduced N₂O over γ-Al₂O₃ and held it at temperatures of 154, 178, 191 and 218 °C. We compare the results to the experiments conducted for >24 h at 200 °C, discussed in the preceding sections. This temperature range was chosen based on the slow kinetics seen between 25 and 100 °C and enhanced decomposition at temperatures above 200 °C observed previously (Kantnerová et al., 2020a; Magyar et al., 2016). For SP, theoretical predictions indicate differences of 5.23 ± 0.24 ‰ for N₂O equilibrated at 154 °C relative to 218 °C, which agrees within analytical uncertainties with the observed span of 4.92 ± 0.17 ‰ (Fig. 6). Changes seen for the clumped parameters Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O for N₂O equilibrated between 154 and 218 °C also match theoretical predictions, but with more scatter around the expected values (Fig. 6). For instance, for Δ^∗15N¹⁴N¹⁸O, experimental results for the maximum (218 °C) and minimum (154 °C) temperatures show an offset of −2.5 ± 0.3 ‰, matching the expected value of −2.7 ± 0.12 ‰, but deviations from theoretical predictions are seen for measurements at 178 and 191 °C. It is not clear whether these deviations arise from the larger analytical uncertainty of clumped isotope measurements relative to a smaller predicted range in the parameters Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O, or incomplete equilibration of these doubly substituted isotopologues in the time allotted, especially at lower temperatures. Finally, while the variation in SP, Δ^∗14N¹⁵N¹⁸O, and Δ^∗15N¹⁴N¹⁸O between 154 and 218 °C matches equilibrium temperature dependencies well, a systematic offset is observed between the theoretical predictions and observed values for all three parameters. The interpretation and application of this offset is discussed further for the clumped isotopologues in Sect. 3.6, and for SP in Sect. 3.7.

Figure 6 Achievement of expected temperature dependence for SP, Δ^∗14N¹⁵N¹⁸O, and Δ^∗15N¹⁴N¹⁸O. Black circles show measured data, while the black curve shows the predicted temperature variation. Experiments at 218 °C were performed with 300 µmol N₂O, as was one experiment at 200 °C (see Sect. 3.3). All other experiments were performed with 1200 µmol N₂O. All three parameters follow the expected temperature dependence, as indicated by linear regressions and associated 95 % confidence intervals (blue curve and shading), between 154 and 218 °C. All parameters also show an offset from the theoretical predictions, which is discussed in Sect. 3.6 and 3.7.

The behavior of N₂O heated at 218 °C requires further discussion. When the amount of N₂O typical for other experiments, ∼1.2 mmol, was introduced over γ-Al₂O₃ at this temperature, we did not see convergence in SP, Δ^∗14N¹⁵N¹⁸O, or Δ^∗15N¹⁴N¹⁸O (Fig. S3). We posit that above 200 °C, reactions associated with thermal decomposition may start to outpace isotope exchange reactions and impart a kinetic isotope effect. However, when only 0.3 mmol of N₂O was introduced to the catalyst, we found that it converged on values of SP, Δ^∗14N¹⁵N¹⁸O, and Δ^∗15N¹⁴N¹⁸O consistent with outcomes at other temperatures, even with relatively poor yields of 50 %–70 %. Whether this empirical result depends on variations in the relative kinetics of exchange to decomposition at different catalyst-to-N₂O ratios, or differences of the pressure dependencies of these two reactions, remains to be explored further. Subsequently, a 300 µmol N₂O sample was reacted over γ-Al₂O₃ at 200 °C, where its composition was found to be indistinguishable from 1200 µmol samples; its results are reported among those already discussed.

3.4 Thermal decomposition can be distinguished from equilibration

We posited in the preceding section that through the expression of a kinetic isotope effect, thermal decomposition would be expected to lead to continuous evolution in the isotopic composition of all isotopologue ratios in the residual fraction as N₂O is destroyed, and that thereby it can readily be distinguishable from equilibrium isotope effects. To provide an explicit comparison, we evaluated the isotopic evolution of Δ^∗14N¹⁵N¹⁸O-spiked and natural-abundance N₂O subjected to heating at 498 °C in the presence of γ-Al₂O₃ (Fig. 7). In both experiments, nearly half of the supplied N₂O was destroyed within 15 min, reaching about 80 % after one hour. Nitrogen isotope effects ¹⁵ε of 3 ‰–5 ‰ and oxygen isotope effects ¹⁸ε of 5 ‰–9 ‰ were observed, reflecting an increased durability of heavy isotopologues relative to ¹⁴N¹⁴N¹⁶O (Bigeleisen, 1949), with higher values for Δ^∗14N¹⁵N¹⁸O-spiked N₂O. Both ¹⁵ε and ¹⁸ε are of smaller magnitude than the isotope effects reported for homogeneous decomposition of N₂O in quartz-glass tubes at 1000 °C (Ogawa and Yoshida, 2004). Surprisingly, ¹⁴N¹⁵N¹⁶O and ¹⁵N¹⁴N¹⁶O are decomposed at similar rates with a slight preference for destruction of ¹⁴N¹⁵N¹⁶O (Fig. 8), resulting in an εSP of $\sim -\mathrm{1}$ ‰ (Fig. S4). In contrast, previous observations of thermal decomposition and of the destruction of N₂O by photolysis or by bacterial reduction all found distinct enrichment in SP, rationalized by a ¹⁵N–¹⁶O bond stronger than the ¹⁴N–¹⁶O alternative (Kantnerová et al., 2020b; Ogawa and Yoshida, 2004; Ostrom et al., 2007). The final element of differentiation between kinetic and equilibrium effects comes from the behavior of Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O. With increasing decomposition, Δ^∗14N¹⁵N¹⁸O decreased in both natural-abundance and spiked N₂O, while Δ^∗15N¹⁴N¹⁸O increased for spiked N₂O and remained constant for natural abundance N₂O. For neither starting N₂O do Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O approach the values of 5.7 ‰ and −5.6 ‰ predicted for equilibrium at 498 °C (Fig. 7). Together these pieces of evidence lead us to conclude that some amount of rearrangement among isotopologues accompanies decomposition.

Figure 7 Evolution of δ¹⁵N, Δ^∗14N¹⁵N¹⁸O, and Δ^∗15N¹⁴N¹⁸O for N₂O subjected to thermal decomposition at 498 °C. Panels (A) and (C) show the outcomes for both Δ^∗14N¹⁵N¹⁸O-spiked (diamonds) and natural-abundance (circles) starting materials for the parameters δ¹⁵N and Δ^∗15N¹⁴N¹⁸O, while panels (B) and (D) show the offset variation in Δ^∗14N¹⁵N¹⁸O seen for starting materials distinctly different in this parameter. Error bars represent 1 standard error for replicate measurements of a given sample.

Figure 8 Coevolution of SP and δ¹⁸O over the course of thermal decomposition. Diamonds represent Δ^∗14N¹⁵N¹⁸O-spiked starting material, while circles represent natural abundance N₂O.

With these results in mind, we revisit the equilibration experiments described in the preceding section. In the ideal situation where only an equilibrium isotope effect is acting on these N₂O samples over γ-Al₂O₃, the amount of N₂O would not change. This was clearly observed for lower equilibration temperatures (154, 178, 191 °C) with yields of 90 %–95 %, while experiments at intermediate (200 °C) and higher temperatures (218 °C) for comparable experimental periods showed reduced yields of 70 %–90 % and 50 %–70 %, respectively. To further evaluate any imprint of kinetic isotope effects associated with this extent of decomposition, we consider the evolution of the bulk isotope composition over the course of equilibration experiments. We find that indeed the subtle variation seen in δ¹⁸O and δ¹⁵N across all equilibrium experiments matches the expression of the kinetic isotope effects determined here, but that SP and the clumped parameters Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O do not appear to be controlled by these kinetic isotope effects (Fig. 9). Nevertheless, we cannot exclude the possibility that some of the variation seen in repeated measurements at 200 °C comes from variable expression of kinetic isotope effects. Indeed, a decomposition fraction of 30 % could drive changes in SP, Δ^∗14N¹⁵N¹⁸O, and Δ^∗15N¹⁴N¹⁸O of −0.36 ‰, −1.59 ‰, and 1.42 ‰, respectively, and so could contribute to the variability seen in repeated equilibrium experiments (Sect. 3.6 and 3.7, Table S1 in the Supplement). Therefore, an important element of any subsequent application of this work will be to choose experimental conditions that minimize decomposition and maximize yield, while still achieving complete exchange.

Figure 9 δ¹⁸O and SP versus δ¹⁵N for N₂O equilibrated over γ-Al₂O₃ at 154–214 °C (denoted by colors). The black arrows indicate the measured kinetic isotope effect of thermal decomposition at 498 °C (50 % destruction) shown for comparison. Error bars show 1 SE for three replicate measurements of a sample; when absent standard errors are smaller than the datapoints.

3.5 The case of ¹⁵N¹⁵N¹⁶O

In a preliminary series of experiments, N₂O spiked in Δ^∗15N¹⁵N¹⁶O was also heated over γ-Al₂O₃. Here we found that even after 94 h at 200 °C the value of Δ^∗15N¹⁵N¹⁶O remained elevated at 558 ‰. Therefore, we conclude that the catalyst γ-Al₂O₃ only activates the N–O bond in N₂O, and not the N–N bond, and accordingly facilitates equilibrium isotope reactions among position-specific and ¹⁸O-bearing isotopologues, but not those involving ¹⁵N¹⁵N¹⁶O. To find alternative catalysts or conditions that activate ¹⁵N¹⁵N¹⁶O remains a challenge for future work. In the meantime, it is impossible to establish an absolute reference frame for Δ¹⁵N¹⁵N¹⁶O and so we will continue to report results on a relative basis as Δ^∗15N¹⁵N¹⁶O; such an approach has also been proposed and applied for ethane and other compounds recalcitrant to equilibration (Clog et al., 2018; Gilbert, 2021).

3.6 Establishing absolute reference frames for Δ¹⁴N¹⁵N¹⁸O and Δ¹⁵N¹⁴N¹⁸O

Having found that the abundances of the isotopologues ¹⁴N¹⁵N¹⁸O and ¹⁵N¹⁴N¹⁸O can be brought to equilibrium isotopic distributions between 153 and 218 °C, we use this outcome to establish an absolute reference frame for future measurements of Δ¹⁴N¹⁵N¹⁸O and Δ¹⁵N¹⁴N¹⁸O. Since 200 °C is the temperature where we evaluated the conditions of convergence and time-invariance and have the most robust replicated dataset, we use these measurements to determine the values of Δ¹⁴N¹⁵N¹⁸O and Δ¹⁵N¹⁴N¹⁸O of our working gas against a true stochastic distribution of rare isotopes among isotopologues. These values, reported in Table 1, can then be used to convert values of Δ^∗14N¹⁵N¹⁸O and Δ^∗15N¹⁴N¹⁸O, as reported here or in future studies, to Δ¹⁴N¹⁵N¹⁸O and Δ¹⁵N¹⁴N¹⁸O. Previous N₂O clumped isotope studies also sought to achieve such an absolute reference frame. Early mass spectrometric work was limited by the need to use measurements of Δ¹⁴N¹⁵N¹⁸O in equilibrated samples to account for ion source scrambling in the generation of the ¹⁵N¹⁸O⁺ ion used to measure this isotopologue (Magyar et al., 2016), while first QCLAS studies lacked the full suite of convergence, time-invariance, and temperature dependency results used here to confirm the equilibrium nature of the reactions of N₂O isotopologues over γ-Al₂O₃ (Kantnerová et al., 2020a). By establishing this absolute reference scale and reporting future results on it, we enable their interpretation in the context of theoretical predictions or natural phenomena where the true distribution of isotopes among isotopologues is relevant, and we also facilitate comparisons among different labs and analytical approaches for measuring these parameters.

Table 1 Comparison of theoretical predictions and measured values for SP, Δ¹⁴N¹⁵N¹⁸O, and Δ¹⁵N¹⁴N¹⁸O for equilibration at 200 °C. Measured Δ¹⁴N¹⁵N¹⁸O, and Δ¹⁵N¹⁴N¹⁸O values are reported relative to the working reference gas being 0 ‰, while measured SP values are reported using the value assigned by comparison with Science Tokyo (Mohn et al., 2022; Toyoda and Yoshida, 1999). Uncertainties represent 1 standard deviation for replicate experiments comprising independent equilibration and analysis.

3.7 Implications for the absolute determination of SP

For N₂O equilibrated at 200 °C, we observed an offset of −3.43 ± 0.36 ‰ between predicted SP values and our repeated equilibration experiments (Table 1). This was not expected since SP is defined as the difference of two parameters, δ¹⁵N^α and δ¹⁵N^β, both connected through a chain of reference materials to the international Air-N₂ scale. Therefore, SP should reflect an absolute determination of the relationship of ¹⁴N¹⁵N¹⁶O and ¹⁵N¹⁴N¹⁶O. The key point of connection is through the thermal decomposition of ammonium nitrate to N₂O, whereby δ¹⁵N^α and δ¹⁵N^β values are assigned to N₂O by comparison to the precursor δ¹⁵N-${\mathrm{NO}}_{\mathrm{3}}^{-}$ and δ¹⁵N-${\mathrm{NH}}_{\mathrm{4}}^{+}$,determined using established means and reference materials. This procedure has been repeated in a number of laboratories (Harris et al., 2014; Mohn et al., 2016, 2022; Toyoda and Yoshida, 1999; Westley et al., 2007). Currently, to maximize comparability among labs, the ammonium nitrate decomposition conducted at Tokyo Tech (Toyoda and Yoshida, 1999), now Science Tokyo, is commonly used as the basis for setting the δ¹⁵N^α and δ¹⁵N^β for other laboratories and reference materials (Mohn et al., 2022; Ostrom et al., 2018). However, the decomposition of ammonium nitrate has been found to exhibit significant variation in assigned δ¹⁵N^α and δ¹⁵N^β values, controlled by yield and isotopic fractionation (Mohn et al., 2016). Mohn et al. (2022) found an offset of −1.51 ‰ in SP between the outcome of ammonium nitrate decomposition according to current best practices and the reference frame based on Toyoda and Yoshida (1999). This deviation is in the same direction, although of smaller magnitude, as that estimated here from thermodynamic equilibrium; together these provide two lines of evidence to suggest an offset between the current international reference frame for δ¹⁵N^α and δ¹⁵N^β and the true ratio of ¹⁴N¹⁵N¹⁶O versus ¹⁵N¹⁴N¹⁶O. To follow up from this conclusion will require further equilibration experiments – best would be to subject international reference materials with well-characterized δ¹⁵N and δ¹⁸O to equilibration over γ-Al₂O₃ at optimal conditions for maximum yield. Thereby we can determine if the uncertainty in the results reported here is tied to the equilibration experiments themselves, including possible imprints of a kinetic isotope effect from thermal decomposition. This work can be coupled to further repeat measurements of equilibrated reference materials to evaluate any room for improvement in measurement or gas handling procedures. The outcome of such equilibrium experiments can also be compared to other approaches that provide an estimate of the absolute relationship between ¹⁴N¹⁵N¹⁶O and ¹⁵N¹⁴N¹⁶O, whether novel spectroscopic methods (Griffith et al., 2009; Łapiński et al., 2001; Schlagin et al., 2025) or alternative experimental frameworks (Sadiek et al., 2025). If such experiments and measurements confirm the conclusions drawn here, the international N₂O isotopomer community will need to determine how to harmonize these latest results with the existing reference frame for δ¹⁵N^α and δ¹⁵N^β.

Finally, it is necessary to note that SP, defined as the difference in δ¹⁵N between α and β nitrogen atoms, is only an approximation of the equilibrium constant for Reaction (R1), which is the ratio ${}^{\mathrm{15}}{R}^{\mathit{\alpha }}{/}^{\mathrm{15}}{R}^{\mathit{\beta }}$; for an equilibrated sample. As discussed in Magyar et al. (2016) and Sadiek et al. (2025), for some combinations of δ¹⁵N^α and δ¹⁵N^β this difference becomes significant, but for the set of samples described in this study, and for many natural samples, SP is remarkably close as an approximation to the ratio ${}^{\mathrm{15}}{R}^{\mathit{\alpha }}{/}^{\mathrm{15}}{R}^{\mathit{\beta }}$. Here, the difference between this ratio and SP was between 0.001 ‰ and 0.3 ‰ across all samples. Therefore, we can quantitatively compare SP to predicted equilibrium constants.

4 Conclusion

This study provides a firm foundation for applying clumped isotopic analysis to investigate the reactions responsible for producing and destroying N₂O in natural systems. We present a comprehensive analytical toolkit for repeatably measuring seven isotopic parameters, including the clumped isotopologues ¹⁴N¹⁵N¹⁸O, ¹⁵N¹⁴N¹⁸O, and ¹⁵N¹⁵N¹⁶O, in N₂O samples as small as 20 µmol. Our measurements achieve a precision of 0.3 ‰ or better by triplicate analysis completed within 20 min using quantum cascade laser absorption spectroscopy.

Furthermore, we established reaction conditions for equilibration among N₂O isotopologues catalyzed by γ-Al₂O₃, thereby reliably producing N₂O with predicted SP, Δ¹⁴N¹⁵N¹⁸O, and Δ¹⁵N¹⁴N¹⁸O. The results of these equilibration experiments provide an absolute reference frame for the parameters Δ¹⁴N¹⁵N¹⁸O and Δ¹⁵N¹⁴N¹⁸O. This reference frame allows robust evaluations of these parameters in natural samples, comparison to theoretical predications, and intercomparison of measurements among laboratories and across various analytical techniques.

For SP, the offset we observe between our results and theoretical predictions suggests that current measurements using established gas reference materials provide too low values. Given the high number of laboratories linked to the current realization of the scale, we recommend further work to establish a community-wide absolute reference frame. Over the past decade, great progress has been made towards reconciling and unifying the diverse analytical approaches used to determine the position-specific isotopic composition of N₂O. This progress has been driven both by methodological intercomparison studies (Harris et al., 2020; Mohn et al., 2014; Ostrom et al., 2018) and by improvement and harmonization of corrections for ion-source rearrangement in mass spectrometric analyses (Frame et al., 2014; Harris et al., 2014; Kelly et al., 2023). Collectively, these advances have helped the community to improve its internal consistency and enable intercomparability among diverse techniques and laboratories, thereby facilitating the comparison of laboratory and field measurements of N₂O. But to leverage the full potential of position-specific measurements to understand N₂O cycling in nature requires accurate knowledge of the absolute relationship between δ¹⁵N^α and δ¹⁵N^β. We propose that the equilibrium experiments described in this study provide the foundations for an improved understanding of position-specific ¹⁵N incorporation in N₂O.

Appendix A: Isotopologue parameters and equilibrium constants

As described in Sect. 2.1, we describe the clumped isotopic composition of N₂O with Δi values. The full isotopic composition of N₂O is best accounted for as a system of equilibrium reactions (Wang et al., 2004), but individual Δi values are calculated from measured isotopologues. Practically, Δ¹⁴N¹⁵N¹⁸O is calculated as

$$\begin{array}{}\text{(A1)}& {\mathrm{\Delta }}^{\ast }{}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{18}}\mathrm{O}={\displaystyle \frac{R^{\mathrm{458}}}{\left(\frac{\mathrm{1}}{\mathrm{2}}(R^{\mathrm{456}}+R^{\mathrm{546}})\right)R^{\mathrm{448}}}}-\mathrm{1}\end{array}$$

relative to the working reference gas and then adjusted for its clumped isotope composition as described in Sect. 3.6. This equation for Δ¹⁴N¹⁵N¹⁸O is closely related to the equilibrium reaction:

$$\begin{array}{}\text{(AR1)}& \begin{array}{rl}& {}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{18}}\mathrm{O}+{}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{16}}\mathrm{O}\rightleftharpoons \\ & {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{16}}\mathrm{O}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{16}}\mathrm{O}+{}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{18}}\mathrm{O}.\end{array}\end{array}$$

Analogous expressions are used for Δ¹⁵N¹⁴N¹⁸O and Δ¹⁵N¹⁵N¹⁶O.

For N₂O, it is also possible to define the ¹⁵N site preference for ¹⁸O containing isotopologues, SP¹⁸. By analogy to SP, which is related to the equilibrium Reaction (R1), SP¹⁸ is related to the reaction

$$\begin{array}{}\text{(AR2)}& {}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{18}}\mathrm{O}\rightleftharpoons {}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{18}}\mathrm{O}\end{array}$$

SP¹⁸ was defined by Kantnerová et al. (2020a) as

$$\begin{array}{}\text{(A2)}& {\text{SP}}^{\mathrm{18}}=\mathit{\delta }\left({}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{18}}\mathrm{O}\right)-\mathit{\delta }\left({}^{\mathrm{15}}{\mathrm{N}}^{\mathrm{14}}{\mathrm{N}}^{\mathrm{18}}\mathrm{O}\right)\end{array}$$

and by Magyar et al. (2016) as the ratio of abundances of the two isotopologues ¹⁴N¹⁵N¹⁸O and ¹⁵N¹⁴N¹⁸O. As discussed in Sect. 3.6, these definitions will converge for most isotopic compositions. Since the individual clumped isotopologues ¹⁴N¹⁵N¹⁸O and ¹⁵N¹⁴N¹⁸O are already each directly related to an equilibrium reaction and can be expressed as a Δi value, we find it most insightful, as well as the best application of analytical capacities, to consider these on their own. Nevertheless, it is possible to calculate SP¹⁸ values from measured abundances of ¹⁴N¹⁵N¹⁸O and ¹⁵N¹⁴N¹⁸O, and the parameter may prove to be useful in future studies of N₂O clumped isotope effects imparted by certain processes.