J Labidi

E D Young

The origin and dynamics of nitrogen in the Earth's mantle constrained by 15 N 15 N in hydrothermal gases

The development of high-resolution gas source mass spectrometry has permitted entirely new types of measurements of multiply-substituted isotopologues in gas species of geochemical significance. Here, we present recent advances afforded by measurements of 15 N 15 N in natural samples, together with 14 N 14 N and 15 N 14 N. We show that the abundance of the doubly-substituted 15 N 15 N isotopologue in hydrothermal gases, often mixtures of volatiles of widely different origins, allows tracing the provenance of nitrogen. The approach is based on the recent finding that atmospheric N2 has a substantial enrichment in 15 N 15 N of nearly 20‰ relative to any other source of N2. This is particularly useful for the study of hydrothermal gases, where characterizing the isotopic composition and provenance of volcanic N2 is important for a wide range of applications in high-temperature geochemistry, but where air-derived N2 is unavoidable.

In this review, we summarize the evidence that 15 N 15 N is an unambiguous tracer of air 2 contamination. We compare two sets of published 15 N 15 N data acquired on gases from plume and arc volcanoes. We show how different sources of volcanic N2 may be in plume versus arc environments, and discuss the first-order constraints on the deep N cycle that are provided by the new 15 N 15 N data. Important findings include that the d 15 N tracer, used alone or in conjunction with N2/Ar and N2/He ratios, can be surprisingly deceiving. Isotope fractionation of atmospheric nitrogen occurs within hydrothermal systems, resulting in negative d 15 N values similar to estimates for mantle values, yet with 15 N 15 N values that preclude a mantle origin. The 15 N 15 N data show that the true d 15 N of volcanic components is positive in arcs but near-zero at the Yellowstone plume. In other words, atmospheric N2 can mimic mantle d 15 N, and mantle d 15 N can look like the value of air.

Without 15 N 15 N, the apportioning of mantle and atmospheric N2 in mixed gases can easily be wrong. With 15 N 15 N, we also determine the true N2/ 3 He and N2/ 36 Ar ratios of volcanic components in hydrothermal systems. Results inform our understanding of the deep nitrogen cycle. Plume and arc volcanic endmembers show distinct isotope and elemental ratios, consistent with sub-arc sources being overwhelmed by near-quantitative slab devolatilization, while the Yellowstone plume source is not reflecting volatile subduction.

Introduction

Plate tectonics has shuffled the record of volatiles acquired by Earth's mantle during planetary formation [START_REF] Bekaert | Subduction-Driven Volatile Recycling: A Global Mass Balance[END_REF][START_REF] Hilton | Noble gases and volatile recycling at subduction zones[END_REF][START_REF] Jambon | Earth degassing and large-scale geochemical cycling of volatile elements[END_REF]. Disambiguating the relative importance of subducted and primordial volatiles in the global budget of the deep Earth is crucial for a better understanding of planetary evolution. Information about the N isotopic composition of Earth's mantle comes partly from basalts: quenching during eruption of mid-ocean ridge basalts (MORB) on the seafloor results in the formation of glassy rims preserving magmatic vesicles enriched in volatiles. These vesicles are a potential treasure trove for sampling mantle volatiles. However, because nitrogen is a trace gas in those vesicles, magmatic nitrogen contained within them is commonly contaminated by N2rich air-derived components upon eruption [START_REF] Marty | Nitrogen content of the mantle inferred from N2-Ar correlation in oceanic basalts[END_REF][START_REF] Marty | Volatiles (He, C, N, Ar) in mid-ocean ridge basalts: Assesment of shallow-level fractionation and characterization of source composition[END_REF].

One strategy to see through the contamination by air has been to determine nitrogen isotope ratios together with 40 Ar/ 36 Ar ratios [START_REF] Marty | Nitrogen content of the mantle inferred from N2-Ar correlation in oceanic basalts[END_REF]. The rationale is that air and the mantle have distinct 40 Ar/ 36 Ar ratios, ~ 300 and ~ 25,000 respectively [START_REF] Moreira | Rare gas systematics in popping rock: isotopic and elemental compositions in the upper mantle[END_REF]. Basalts with 40 Ar/ 36 Ar ratios ~ 300 similar to air show evidence for air-derived nitrogen, while lavas with 40 Ar/ 36 Ar ratios greater than ~1,000 appear to exhibit d 15 N 1 values closer to true mantle values [START_REF] Marty | Nitrogen content of the mantle inferred from N2-Ar correlation in oceanic basalts[END_REF][START_REF] Marty | Volatiles (He, C, N, Ar) in mid-ocean ridge basalts: Assesment of shallow-level fractionation and characterization of source composition[END_REF]. Based on basalt studies, a d 15 N value of ~ -4‰ relative to air may be adopted as an estimate for the convective mantle (data in [START_REF] Barry | Release of subducted sedimentary nitrogen throughout Earth's mantle[END_REF][START_REF] Cartigny | Volatile (C, N, Ar) variability in MORB and the respective roles of mantle source heterogeneity and degassing: the case of the Southwest Indian Ridge[END_REF][START_REF] Javoy | The volatiles record of a "popping" rock from the Mid-Atlantic Ridge at 14°N: chemical and isotopic composition of gas trapped in the vesicles[END_REF][START_REF] Marty | Nitrogen content of the mantle inferred from N2-Ar correlation in oceanic basalts[END_REF][START_REF] Marty | Nitrogen and argon isotopes in oceanic basalts[END_REF][START_REF] Marty | Volatiles (He, C, N, Ar) in mid-ocean ridge basalts: Assesment of shallow-level fractionation and characterization of source composition[END_REF]. Nitrogen in Earth's mantle is therefore isotopically distinct from solar (d 15 N ~ -400‰, [START_REF] Hashizume | Solar wind record on the Moon: deciphering presolar from planetary nitrogen[END_REF][START_REF] Marty | A 15N-poor isotopic composition for the Solar System as shown by Genesis solar wind samples[END_REF] and from potential chondritic sources akin to enstatite chondrites (d 15 N ~ -20‰ [START_REF] Grady | Compositional differences in enstatite chondrites based on carbon and nitrogen stable isotope measurements[END_REF]. The clear disagreement between mantle nitrogen and potential primordial signatures has been suggested to reflect subduction of surface derived N with elevated d 15 N values, resulting in the departure from enstatite chondrites and/or solar d 15 N values towards higher d 15 N over time [START_REF] Barry | Release of subducted sedimentary nitrogen throughout Earth's mantle[END_REF][START_REF] Javoy | The birth of the Earth's atmosphere: the behaviour and fate of its major elements[END_REF][START_REF] Marty | The nitrogen record of crust-mantle interaction and mantle convection from Archean to Present[END_REF][START_REF] Williams | Capture of nebular gases during Earth's accretion is preserved in deep-mantle neon[END_REF]. In other words, vestiges of primordial nitrogen 1 d 15 N refers to the per mil deviations in 15 N/ 14 N from the ratio in air.

acquired during Earth's accretion would have been erased with time by ongoing plate tectonics. One requirement of this scenario is that surface nitrogen must have had d 15 N > 20‰ in the Hadean and Archean, so that it can balance extremely low d 15 N for the primordial mantle [START_REF] Javoy | The birth of the Earth's atmosphere: the behaviour and fate of its major elements[END_REF]. The consistently high d 15 N values in deep time are simply not observed [START_REF] Ader | Interpretation of the nitrogen isotopic composition of Precambrian sedimentary rocks: Assumptions and perspectives[END_REF][START_REF] Marty | Nitrogen Isotopic Composition and Density of the Archean Atmosphere[END_REF][START_REF] Pinti | Nitrogen and argon signatures in 3.8 to 2.8 Ga metasediments: Clues on the chemical state of the Archean ocean and the deep biosphere[END_REF], leaving the nitrogen deep cycle unexplained.

Constraining the nitrogen isotopic compositions of the sources of mantle plumes and arc volcanoes should help. Mantle plumes host primordial gases, undisturbed and isolated from mantle convection for nearly 4.4 Gy, mixed with subducted components [START_REF] Mukhopadhyay | Early differentiation and volatile accretion recorded in deep-mantle neon and xenon[END_REF][START_REF] Parai | The emerging portrait of an ancient, heterogeneous and continuously evolving mantle plume source[END_REF]. Plume basaltic glasses are however far less available than MORB, and therefore less well understood. Only seven plume basalts with 3 He/ 4 He ratios > 16 and published d 15 N values pass the tests of having avoided complete volatile overprinting by air contamination (i.e., non-air 40 Ar/ 36 Ar ratios). Those samples have d 15 N values between -2‰ and 0‰ (data in [START_REF] Marty | The nitrogen record of crust-mantle interaction and mantle convection from Archean to Present[END_REF][START_REF] Marty | Nitrogen and argon isotopes in oceanic basalts[END_REF][START_REF] Sano | Volcanic flux of nitrogen from the Earth[END_REF], marginally higher than the convective mantle. Given the paucity of data, the influence of subduction on the N budget of mantle plumes remains unclear. Nitrogen in arc mantle sources is also important to study, as it is likely influenced by recycled surface-derived components [START_REF] Fischer | Subduction and recycling of nitrogen along the Central American margin[END_REF][START_REF] Füri | In situ quantification of the nitrogen content of olivine-hosted melt inclusions from Klyuchevskoy volcano (Kamchatka): Implications for nitrogen recycling at subduction zones[END_REF][START_REF] Taran | Geochemistry of volcanic and hydrothermal fluids and volatile budget of the Kamchatka-Kuril subduction zone[END_REF]. Sub-arc nitrogen may be especially well suited for placing constraints on N isotope signatures that may be transferred to the deep Earth. However, the N isotope compositions of sub-arc sources are even more challenging to constrain because no glasses are available for analysis.

The study of volcanic gases offers a unique window into plumes and arc volatiles where glasses are rare or absent. A large array of hot springs and fumaroles are vented to the surface and reflect mixtures between surface waters, atmospheric gases, and variable amounts of volcanic volatiles [START_REF] Giggenbach | The composition of gases in geothermal and volcanic systems as a function of tectonic-setting[END_REF]. Instances of vented gases with plume-like 3 He/ 4 He ratios are associated with 40 Ar/ 36 Ar ratios that remain under 1,000 [START_REF] Boudoire | Paroxysmal eruptions tracked by variations of helium isotopes: inferences from Piton de la Fournaise (La Réunion island)[END_REF][START_REF] Caliro | Gas geochemistry of hydrothermal fluids of the S. Miguel and Terceira Islands, Azores[END_REF][START_REF] Chiodini | Insights from fumarole gas geochemistry on the origin of hydrothermal fluids on the Yellowstone Plateau[END_REF][START_REF] Sano | Chemical and isotopic compositions of gases in geothermal fluids in Iceland[END_REF]. This is much lower than the values of ~ 5,000 to 10,000 thought to characterize the deep mantle [START_REF] Mukhopadhyay | Early differentiation and volatile accretion recorded in deep-mantle neon and xenon[END_REF][START_REF] Trieloff | The nature of pristine noble gases in mantle plumes[END_REF]. Low 40 Ar/ 36 Ar in vented gases likely reflects a pattern of even greater air contamination than evidenced in basalts. Gases emitted by arc volcanoes are also heavily studied, but so far have shown 40 Ar/ 36 Ar ratios that are almost exclusively atmospheric [START_REF] Hilton | Noble gases and volatile recycling at subduction zones[END_REF][START_REF] Pedroni | Ar, and C isotope systematics of geothermal emanations in the Lesser Antilles Islands Arc[END_REF][START_REF] Sano | The analysis and interpretation of noble gases in modern hydrothermal systems[END_REF][START_REF] Snyder | Regional variations in volatile composition: Isotopic evidence for carbonate recycling in the Central American volcanic arc[END_REF]. It is unclear whether this reflects even more pronounced contamination directly from air, or instead results from volcanic sources overwhelmed by subducted air-derived argon [START_REF] Staudacher | Recycling of oceanic crust and sediments: the noble gas subduction barrier[END_REF]. In summary, gases from both plumes and arcs appear dominated by atmospheric 40 Ar/ 36 Ar signatures, which is problematic for the identification of volcanic N2 in the gas mixtures.

As an alternative to relying on 40 Ar/ 36 Ar measurements alone as tracers of air contamination, studies of fumaroles have often employed plots of d 15 N values versus bulk N2/Ar and N2/He ratios, in part because these gas species ratios are easy to measure with ordinary gas chromatography and quadrupole mass spectrometry techniques. In principle, N2/Ar and N2/He ratios should allow the identification of mixing between mantle and atmospheric endmembers, since air and various mantle sources are anticipated to be very different [START_REF] Elkins | Tracing nitrogen in volcanic and geothermal volatiles from the Nicaraguan volcanic front[END_REF][START_REF] Fischer | Subduction and recycling of nitrogen along the Central American margin[END_REF][START_REF] Fischer | Fluxes and sources of volatiles discharged from Kudryavy, a subduction zone volcano, Kurile Islands[END_REF][START_REF] Roulleau | He, N and C isotopes and fluxes in Aira caldera: comparative study of hydrothermal activity in Sakurajima volcano and Wakamiko crater, Kyushu, Japan[END_REF][START_REF] Sano | Volcanic flux of nitrogen from the Earth[END_REF][START_REF] Sano | Nitrogen recycling in subduction zones[END_REF][START_REF] Taran | Geochemistry of volcanic and hydrothermal fluids and volatile budget of the Kamchatka-Kuril subduction zone[END_REF][START_REF] Zimmer | Nitrogen systematics and gas fluxes of subduction zones: insights from Costa Rica arc volatiles[END_REF]. By design, the approach relies on the assumption that d 15 N, N2/He and N2/Ar ratios remain unfractionated, and are thus strictly tracers of mixing in hydrothermal systems. Some of the most notable and vexing observations to date using this method include: (1) various sub-arc regions show consistently positive d 15 N at relatively high N2/He and N2/Ar ratios [START_REF] Elkins | Tracing nitrogen in volcanic and geothermal volatiles from the Nicaraguan volcanic front[END_REF][START_REF] Fischer | Subduction and recycling of nitrogen along the Central American margin[END_REF][START_REF] Taran | N 2, Ar, and He as a tool for discriminating sources of volcanic fluids with application to Vulcano, Italy[END_REF][START_REF] Taran | Geochemistry of volcanic and hydrothermal fluids and volatile budget of the Kamchatka-Kuril subduction zone[END_REF]; (2) rare but remarkable occurrences of MORB d 15 N signatures in some arc volcanoes [START_REF] Fischer | Subduction and recycling of nitrogen along the Central American margin[END_REF][START_REF] Zimmer | Nitrogen systematics and gas fluxes of subduction zones: insights from Costa Rica arc volatiles[END_REF]; (3) plume gases at Yellowstone with elevated 3 He/ 4 He ratios and d 15 N values clustered around 0‰, the value of air, even at N2/He ratios five orders of magnitude below the value of the atmosphere [START_REF] Chiodini | Insights from fumarole gas geochemistry on the origin of hydrothermal fluids on the Yellowstone Plateau[END_REF]; and (4) other plume gases, from both Iceland and Azores, also with elevated 3 He/ 4 He but higher N2/He (i.e., closer to air) and negative d 15 N values as low as -10‰, nominally inconsistent with the signature of contaminant air [START_REF] Caliro | Gas geochemistry of hydrothermal fluids of the S. Miguel and Terceira Islands, Azores[END_REF][START_REF] Marty | Gas geochemistry of geothermal fluids, the Hengill area, southwest rift zone of Iceland[END_REF][START_REF] Sano | Chemical and isotopic compositions of gases in geothermal fluids in Iceland[END_REF].

In the latter case, the surprisingly low d 15 N values were interpreted as either reflecting a unique feature of the high 3 He/ 4 He mantle, resembling of the values for enstatite chondrites [START_REF] Caliro | Gas geochemistry of hydrothermal fluids of the S. Miguel and Terceira Islands, Azores[END_REF][START_REF] Marty | Gas geochemistry of geothermal fluids, the Hengill area, southwest rift zone of Iceland[END_REF] or as the result of N isotopic fractionation of atmospheric N2 in the hydrothermal systems [START_REF] Marty | Gas geochemistry of geothermal fluids, the Hengill area, southwest rift zone of Iceland[END_REF]. Overall, it is hard to reconcile d 15 N values with high 3 He/ 4 He from one site to another. What is clear from this brief literature review is that the nitrogen isotope geochemistry of hydrothermal fluids is potentially of great utility for constraining the cycle of nitrogen in the deep Earth, but simple conclusions are hindered by pervasive air contamination.

The burgeoning field commonly referred to as "clumped" isotope geochemistry makes use of the relative abundance of multiply-substituted isotopologues in a given molecule, i.e., the abundance of isotopic molecular species with more than one heavy isotope substitution (hence the term "clumping" of isotopes). A number of molecules have been investigated, although studies of rare isotopologues of CO2 and CH4 dominate the field. The main goal in many of these studies has been to derive temperatures of formation of carbonate minerals (in the case of CO2 liberated from carbonates) or the origins of various gases, with an emphasis on the potential biogenicity of the gases in the case of CH4 [START_REF] Henkes | Temperature evolution and the oxygen isotope composition of Phanerozoic oceans from carbonate clumped isotope thermometry[END_REF][START_REF] Young | The relative abundances of resolved 12CH2D2 and 13CH3D and mechanisms controlling isotopic bond ordering in abiotic and biotic methane gases[END_REF]. Here we describe a somewhat different application in which isotope clumping is a unique tracer of provenance, in this case 15 N 15 N in N2 gas in volcanic systems. This tracer arises because of an extraordinary disequilibrium overabundance of 15 N 15 N in air. In this review, we summarize work to date that illustrates how the 15 N 15 N novel tracer places new constraints on air contamination of hydrothermal samples [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF][START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF][START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF]. We present observations and constraints from our recent studies of the rare isotopologue 15 N 15 N in nitrogen from hydrothermal gases worldwide, representing both plumes and arcs. Recently published 15 N 15 N data offer a distinctive perspective on nitrogen recycling in these various geodynamic settings. We summarize the evidence that the new 15 N 15 N tracer is an unambiguous indicator of air contamination for nitrogen, much more so than d 15 N alone, 40 Ar/ 36 Ar ratios, or even d 15 N associated with N2/He and N2/Ar ratios. In this review, we discuss new evidence that d 15 N and N2/Ar fractionations are at play in hydrothermal systems. We identify the composition of volcanic N2 in the plume and arc environments, and discuss the first-order constraints on the deep N cycle that are provided by the new 15 N 15 N data.

15 N 15 N: an air tracer par excellence

The doubly-substituted isotopologue 15 N 15 N is an example of a "clumped" isotopic species, a term often invoked in the geochemical literature to describe isotopic molecular species (isotopologues) with more than one heavy isotope. The application of 15 N 15 N to tracing the provenance of volcanic gases is based on the recent discovery that atmospheric nitrogen possesses a considerable 15 N 15 N enrichment of nearly 2% [START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF], making air easy to distinguish from any magmatic nitrogen or nitrogen produced by other geochemical or biological processes. In the laboratory, we measure 29 N2/ 28 N2 and 30 N2/ 28 N2 ratios and report them as d 29 N2 and d 30 N2, the per mil deviations from air. In the case of 29 N2/ 28 N2 d 29 N2= 1000×(( 29 Rsample/ 29 Rair -1) (‰) where 29 R=( 14 N 15 N+ 15 N 14 N)/ 14 N 14 N for the gas of interest. Since ( 14 N 15 N+ 15 N 14 N)/ 14 N 14 N equals 2( 15 N/ 14 N) for both the sample and the standard, for all practical purposes, d 29 N2 is equivalent to the bulk isotope ratio expressed as d 15 N = 1000 × ( 15 Rsample/ 15 Rair -1).

For 30 N2/ 28 N2, d 30 N2 = 1000×( 30 Rsample/ 30 Rair -1) (‰) where 30 R= 15 N 15 N/ 14 N 14 N for a given sample. In general, simple fractionation by molecular weight of N2 will result in variations in d 30 N2 that are roughly twice the variations in d 29 N2 (see below).

The D30 tracer is a measure of departures from a purely stochastic distribution of 15 N and 14 N atoms comprising the isotopic molecular species. It is defined such that D30 = 1000×( 30 R/( 15 R) 2 -1) (‰), where 15 R= 15 N/ 14 N for the gas of interest. The value ( 15 R) 2

represents the stochastic relative abundance of 30 N2, and reflects the fact that the probability of forming 30 N2 from a purely random distribution of 14 N and 15 N atoms across all nitrogen isotopologues is the square of the probability of finding molecules with just one 15 N (concentrations are in effect probabilities in the stochastic limit). Of course, the formation of 15 N 15 N is not necessarily purely random. The low vibrational energy of this doublysubstituted molecule results in a thermodynamic driving force for its formation. The thermodynamic favorability for 15 N 15 N formation is strongest at the lower temperatures. At geochemically relevant temperatures ranging from 200 to 1000 °C, equilibrium among N2 isotopologues results in D30 values from 0.5 to 0.1‰, respectively [START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF]. This applies for any N2 molecule made in a geological process, whether it is mantle-derived or inherited from slabs, for example [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF].

In contrast to expectations from equilibrium thermodynamics and normal, purely massdependent fractionation processes, air has a pronounced disequilibrium enrichment in 15 N 15 N, leading to an atmospheric D30 value of 19.1±0.3‰ (2σ) [START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF]. The D30 value of air behaves as an indelible signature that is an isotopologue (molecular) analogue of the three-isotope anomaly that has proven so useful in oxygen isotopes, D 17 O (e.g., [START_REF] Clayton | The oxygen isotope record in Murchison and other carbonaceous chondrites[END_REF] and 12 C 17 O + from 29 N2 + , respectively. Based on the zero-order rule that mass resolving power (instrumental m/Dm) must be ~ 3 x m/Dm between the species, mass resolving powers of at least 7,500, 18,000 and 21,000, respectively, are required to separate N2 from interferences on nominal masses 28 and 29. In the case of mass/charge 30, low backgrounds of nitrogen monoxide (NO + ) are ubiquitous and require a sufficiently high mass resolution to separate 30 NO + from 30 N2 + . The m/Dm between 30 NO + and 30 N2 + is ~13,400, requiring an instrumental mass resolving power of at least ~45,000.

At the time of this writing, and to the best of our knowledge, 30 N2 measurements have been performed only at the University of California, Los Angeles (UCLA) on a Nu Instruments Panorama. Routine measurements at mass resolving powers above 50,000 are achieved with an entrance slit width of about 30 µm. A mass spectrum of the 30 N2 + beam under working conditions is shown in Young et al. (2016). 30 N2 + ions are detected using a secondary electron multiplier, with count rates of between 5,000 and 40,000 cps for most natural samples. Ions with m/z of 28 and 29 are measured using Faraday cups with 10 11 W amplifier resistors.

Natural samples from the field show a broad range of N2 concentrations, typically from 0.01 vol% N2 to 80 vol% N2. These concentrations translate into between 1 and 500 µmoles of N2 for published D30 analyses. Counting times are adjusted to account for the large range in sample N2 concentrations. This results in variable analytical precision from one sample to another. Samples yielding ≥ 5 µmoles of N2 permit conventional dual-inlet measurements, with analysis times of up to 7.5 hours to yield the best precision. For samples with < 5 µmoles of N2, measurements have been performed by concentrating on a liquid-nitrogen "microvolume", with analysis times of 1 hour at most. For analyses lasting 7.5 hours, internal precision for D30 values is typically ≤0.2‰ (1 s.e.). Samples with less than 5 µmoles N2 yield internal uncertainties of between 0.18 and 1.80‰ (1 s.e.).

15 N 15 N composition of N2 in nature

The published data for natural samples for which we have obtained D30 values are from [START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF] The two lines are separated by the ~19‰ offset for air (Fig. 1). Gases heated in the laboratory between 400 and 1,000 °C reflect equilibrium bond-ordering with values of D30 near zero. We take the gases at 1,000 °C that plot on the lower line as useful representatives of the stochastic distribution of N2 isotopologues. Figure 1 underscores the striking nature of the anomalous 30 N2 in air; temperatures ranging from 25 to 1000 °C should result in D30 values from 1.1 to 0.1‰, respectively [START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF]. The ~19‰ disequilibrium 15 N 15 N excess for atmospheric N2 is likely due to nitrogen photochemistry in the modern upper atmosphere [START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF]. The mechanism responsible for the enrichment evidently requires the presence of O2 during N2 photolysis, so predictions could be made for planetary atmospheres containing N2 but no O2, including Earth's atmosphere prior to the great oxygenation event. Most importantly for this review, the 15 N 15 N enrichment in air is an unambiguous measure of air in mixed gases.

Analysis of a variety of natural samples of nitrogen show mass fractionation of molecular species is rife, as evidenced by slope-2 variability in d 30 N2 vs. d 29 N2 space (Fig. 1).

Importantly for this work on volcanic hydrothermal systems, the array of mass-dependent signatures going through air indicates air-derived N2 experiences mass fractionation in nature. This is an important observation; nitrogen derived from air does not have a single These data confirm that magmatic N2 formed by degassing at ~ 1200 °C has D30 ~ 0‰. The mechanism for N2 formation in crustal environments is breakdown of NH4-bearing phyllosilicates at peak metamorphic temperatures [START_REF] Li | N2 in deep subsurface fracture fluids of the Canadian Shield: Source and possible recycling processes[END_REF]. Gases from two deep mines in the Canadian shield have D30 values between 1.1±0.4‰ and 0.5±0.2‰ (2 σ, n=2).

These data confirm that crustal generation of N2 by metamorphism has D30 values < 1‰ as expected for approaches to thermodynamic equilibrium. resides in the gaseous phase [START_REF] Lee | Kinetic nitrogen isotope fractionation between air and dissolved N2 in water: Implications for hydrothermal systems[END_REF]. Linear extrapolation to 100 °C results in a fractionation of +1.0‰. With this fractionation, Rayleigh distillation would push the product and the residue to d 15 N values as low as -5‰ and -6 ‰, for gas and fluid, with ~ 99% N2

The

degassing. The possibility of a higher fractionation would change the exact fractions of degassed nitrogen, but would not fundamentally modify our main hypothesis: N2 isotope fractionations in hydrothermal systems are to be expected as the result of significant degassing in the absence of meteoric, air-saturated water recharge.

There is also evidence that the fractionations of d 15 N in air-derived N2 are variable on human timescales. In Iceland, fractionated air was observed with d 15 N values as low as -6‰

for gases sampled in 2018 [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF]. Fumaroles sampled elsewhere in Iceland in 1982 [START_REF] Sano | Chemical and isotopic compositions of gases in geothermal fluids in Iceland[END_REF] had d 15 N values down to -10‰ [START_REF] Marty | Gas geochemistry of geothermal fluids, the Hengill area, southwest rift zone of Iceland[END_REF]. At Poás, fumaroles sampled between 1998 and 2001 produced unfractionated air with d 15 N = 0 ‰ [START_REF] Vaselli | Fumarole migration and fluid geochemistry at Poás volcano (Costa Rica) from 1998 to 2001[END_REF], while field expeditions a few months later in 2001, and in 2006-2007, the same fumaroles yielded negative d 15 N values of -3‰ [START_REF] Fischer | Temporal variations in fumarole gas chemistry at Poás volcano, Costa Rica[END_REF][START_REF] Fischer | Subduction and recycling of nitrogen along the Central American margin[END_REF][START_REF] Zimmer | Nitrogen systematics and gas fluxes of subduction zones: insights from Costa Rica arc volatiles[END_REF]. The latter samples tend to have atmospheric D30 values [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF], demonstrating the negative d 15 N are due to fractionation of atmospheric N2.

Similarly, at Yellowstone, a field expedition in September 2007 [START_REF] Chiodini | Insights from fumarole gas geochemistry on the origin of hydrothermal fluids on the Yellowstone Plateau[END_REF]

Identifying high-temperature N2

Gases from Yellowstone, Eifel, and Central America all incorporate some hightemperature N2 with D30 ~ 0, resulting in variable D30 values due to mixing with air.

Extrapolation of trends in data to D30 = 0 ‰ constrains the volcanic endmembers for d 15 N (Fig. 3), N2/ 3 He (Fig. 4), N2/ 36 Ar (Fig. 5) and 40 Ar/ 36 Ar (Fig. 6). Some of the main conclusions on high-temperature volatiles are summarized in a cartoon on Fig. 7.

Eifel gases happen to be the simplest to interpret of the entire D30 dataset because they have D30 values as low as 0.3±0.7‰, i.e., within uncertainty of 0‰, requiring no extrapolation whatsoever to determine the d 15 N of high-temperature N2. For those gases, the high-temperature d 15 N value is -1.2±0.5‰, a value that is slightly higher than estimates for the average for MORB mantle of ~-4±2‰ [START_REF] Javoy | The volatiles record of a "popping" rock from the Mid-Atlantic Ridge at 14°N: chemical and isotopic composition of gas trapped in the vesicles[END_REF][START_REF] Marty | Volatiles (He, C, N, Ar) in mid-ocean ridge basalts: Assesment of shallow-level fractionation and characterization of source composition[END_REF]. This marginally higher value could indicate the addition of recycled nitrogen with positive d 15 N in the Eifel mantle source [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF]. respectively [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF]. It is unclear whether the Poás endmember reflects a unique volcanic endmember across central America, or a fractionated value, disconnected from the magmatic signature [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF]. Overall, in view of the expectation that slab-derived nitrogen should have d 15 N values of near + 5‰ [START_REF] Busigny | Nitrogen isotopes in ophiolitic metagabbros: A re-evaluation of modern nitrogen fluxes in subduction zones and implication for the early Earth atmosphere[END_REF][START_REF] Li | Carbon and nitrogen geochemistry of sediments in the Central American convergent margin: Insights regarding subduction input fluxes, diagenesis, and paleoproductivity[END_REF], these high-T estimates are consistent with the addition of recycled N in the central American sub-arc mantle.

For Yellowstone, the D30 values range down to 3.8±1.7‰, but the d 15 N-D30 scatter is substantial (Fig. 3). An extrapolation of the d 15 N-D30 data to D30 = 0 was carried out by Labidi et al., (2020) using only the data comprising the steep negative sloping array, resulting in an estimate for volcanic d 15 N of +3±2‰ [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF]. Given the magnitude of the d 15 N-D30 scatter, it is unclear whether this extrapolation is valid. The alternative, more conservative approach is to use the sample with the lowest D30 to define a minimum d 15 N of~+0.2‰. Higher values suggested by [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF] are possible, but are not required by the data as a whole on Fig. 3. We note that a high-temperature endmember with d 15 N~ 0‰ is consistent with the data from [START_REF] Chiodini | Insights from fumarole gas geochemistry on the origin of hydrothermal fluids on the Yellowstone Plateau[END_REF] for Yellowstone, perhaps lending support for the more conservative approach.

Overall, based on our D30 values from the fumaroles sampled thus far, we find high d 15 N in high-temperature components compared to MORB nitrogen (Fig. 7). Elevated

15 N/ 14 N isotope signatures may reflect the addition of nitrogen recycled from the downgoing slab, as suggested in the literature [START_REF] Barry | Release of subducted sedimentary nitrogen throughout Earth's mantle[END_REF][START_REF] Bebout | Nitrogen-isotope compositions of metasedimentary rocks in the Catalina Schist, California: implications for metamorphic devolatilization history[END_REF][START_REF] Busigny | Massive recycling of nitrogen and other fluid-mobile elements (K, Rb, Cs, H) in a cold slab environment: evidence from HP to UHP oceanic metasediments of the Schistes Lustrés nappe (western Alps, Europe)[END_REF][START_REF] Dauphas | Heavy nitrogen in carbonatites of the Kola Peninsula: A possible signature of the deep mantle[END_REF]. To constrain whether subducted nitrogen is added to the sources of plumes and/or arc source, we make use of N2/ 36 Ar and N2/ 3 He ratios. values are shown on Fig. 5 and the significance of the potential correlations are discussed in the following section. We also present combinations of D30 data with 40 Ar/ 36 Ar (Fig. 6) and discuss the intricacies of the 36 Ar-N2 geochemical pair, as revealed by 15 N 15 N data.

Coupling

7.1 Using 15 N 15 N to determine N2/ 36 Ar ratios

The first important observation on Fig. 5 is that at atmospheric D30, N2/ 36 Ar ratios are variable. They range from the nominal value for air of ~2.5x10 4 to~0.4x10 4 . The low end of the range is three times below the anticipated N2/ 36 Ar for air-saturated waters of ~1.2x10 4 . This is likely the result of hydrothermal degassing described in section 6. The solubilities of argon and nitrogen are distinct by about a factor of 2 in geothermal waters, at temperatures between 20 and 100 °C [START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF]. Assuming hydrothermal degassing occurs as a Rayleigh distillation process, and using the solubility relationships reviewed in [START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF], 95% N2 degassing would be associated with 75% Ar degassing, causing N2/ 36 Ar of the residual fluids to be fractionated (lowered) by a factor of ~3 relative to the starting composition [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF]. The discovery that degassing fractionates N2/Ar ratios is important with respect to the use of nitrogen excesses, N2*, introduced by notion, used to pro-rate the bulk flux of N2 to obtain the volcanic fraction only, in entire arcs [START_REF] Fischer | Subduction and recycling of nitrogen along the Central American margin[END_REF]. Any N2/Ar > 40 is thought to reflect the contribution of volcanic nitrogen with high N2/Ar added to unfractionated air-saturated waters. The prospect that atmospheric components in fact have variable N2/Ar likely warrants revisiting the notion of N2*.

The relationships between N2/ 36 Ar and D30 can be projected to high-temperature endmembers with D30 = 0‰, to estimate the N2/ 36 Ar ratio of the volcanic gases. Yellowstone fumaroles characterized by air-like N2/ 36 Ar span nearly the entire range of D30 values (Fig. 5A). However, having taken into account the variable degrees of elemental fractionation of air prior to mixing, the Yellowstone dataset is reasonably consistent with a high N2/ 36 Ar of ~10 6 , similar to MORB (Fig. 5A). Fumaroles from the Central America subduction zone are far more ambiguous. The data distribution can be fit with a high-temperature endmember with N2/ 36 Ar anywhere between ~10 6 (similar to MORB, Fig. 5A) and ~10 4 (similar to air, fig. 5B). These ratios have vastly different implications for the volatile budget and fluxes in a subduction zone. The ambiguity results from the curvature in D30 -N2/ 36 Ar space. For mixtures between air and MORB-like gases, mixing hyperbolae are associated with considerable curvature. Most of the curvature occurs where D30 < ~ 4 (Fig. 5A), such that for any D30 > 4, gas mixtures will inevitably be biased towards the N2/ 36 Ar of air. This means that unless D30 of a mixed gas is almost indistinguishable from the high-temperature endmember, which is a rare occurrence, the N2/ 36 Ar ratio remains poorly constrained by D30 -N2/ 36 Ar extrapolations.

7.2 Constraints from 40 Ar/ 36 Ar ratios A novel way to independently constrain N2/ 36 Ar ratios of high-temperature endmembers is to plot D30 directly against 40 Ar/ 36 Ar (Fig. 6). In this space, samples from intra-plate volcanism (Eifel and Yellowstone) show correlations which confirm that argon and N2 act essentially as geochemical pairs in hydrothermal fluids. Mixing curvatures are defined by [ 14 N 14 N/ 36 Ar]Mantle/[ 14 N 14 N/ 36 Ar]Air. Taking the N2/ 36 Ar ratio for air-saturated water and 40 Ar/ 36 Ar of both air (~300) and the plume endmembers (~30,000 for Eifel, ~10,000 for Yellowstone, see detail in [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF], a simple data fit returns the N2/ 36 Ar for the volcanic endmember. Overall, Yellowstone data are fit with a N2/ 36 Ar between 0.3 !".$ &".' × 10 ( and 1.0 !".) &".( × 10 ( for the plume endmember (Fig. 6A). This is striking because again, N2/ 36 Ar of individual samples are much closer to air (section 7.1). This exercise would be valuable for fumaroles in a subduction zone, since it could distinguish whether N2/ 36 Ar ratios for sub-arc source is 10 6 or 10 4 (Fig. 5A versus 5B). However, arc fumaroles from Central America have atmospheric 40 Ar/ 36 Ar values even for near-zero D30 values, so a spectrum of scenarios may account for the data. An endmember possibility is that the sub-arc mantle sources have 40 Ar/ 36 Ar ~ 25,000, i.e. the value of the upper mantle away from direct slab influences [START_REF] Moreira | Rare gas systematics in popping rock: isotopic and elemental compositions in the upper mantle[END_REF]. To fit the vertical relationship in figure 6, the required N2/ 36 Ar would have to be ~10 8 (Fig. 6B). In this case, the sub-arc mantle would have to receive subducted nitrogen but no slab-derived argon at all, hence a high N2/ 36 Ar and a MORB-like 40 Ar/ 36 Ar. This may appear as unreasonable. In addition, arguments based on 3 He/ 36 Ar suggest the addition of 36 Ar to sub-arc sources is actually significant, which argues against a N2/ 36 Ar ratio of ~10 8 [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF]. Instead, the sub-arc mantle may be overwhelmed with atmospheric argon, with 40 Ar/ 36 Ar ~ 300. In this scenario, the N2/ 36 Ar of the arc source can be anything, but the vertical relationship on Fig.

6 is explained by volcanic argon having atmospheric 40 Ar/ 36 Ar ratios. This scenario is consistent with the notion of subduction barrier for light noble gases [START_REF] Moreira | Noble gas constraints on the origin and evolution of Earth's volatiles[END_REF][START_REF] Moreira | The origin of rare gases on Earth: The noble gas 'subduction barrier'revisited[END_REF][START_REF] Staudacher | Recycling of oceanic crust and sediments: the noble gas subduction barrier[END_REF]. It would incidentally confirm that as opposed to D30 data, the Ar isotope systematic is not a strong tracer of air contamination in arc regions, since high-temperature volatiles would have argon isotope compositions similar to air.

Determining the sources of N in the mantle

A quantification of the flux of slab-derived N2 to mantle sources is necessary for understanding Earth's nitrogen cycle. Assuming that 3 He subduction is negligible [START_REF] Porcelli | An Overview of Noble Gas Geochemistry and Cosmochemistry[END_REF], the combination of N2/ 3 He ratios and D30 values provides a particularly powerful approach. For a discussion on 3 He/ 22 Ne and 3 He/ 36 Ar ratios, see [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF]. At Yellowstone, the data require a N2/ 3 He endmember of 3 x 10 6 (Fig. 4), indistinguishable from MORB. This N2/ 3 He estimate derived from D30 is the same as the lowest N2/ 3 He observed at Yellowstone by [START_REF] Chiodini | Insights from fumarole gas geochemistry on the origin of hydrothermal fluids on the Yellowstone Plateau[END_REF]. The MORB-like N2/ 3 He ratio for the high 3 He/ 4 He Yellowstone plume, confirmed using D30, is an important observation. Every 3 He-rich mantle source may reflect a mixture of subducted and primordial reservoirs [START_REF] Jackson | Ancient helium and tungsten isotopic signatures preserved in mantle domains least modified by crustal recycling[END_REF] especially for heavy noble gases [START_REF] Ballentine | Neon isotopes constrain convection and volatile origin in the Earth's mantle[END_REF][START_REF] Parai | Xenon isotopic constraints on the history of volatile recycling into the mantle[END_REF].

Based on 15 N/ 14 N arguments, some have suggested the subducted components dominate the N budget of plume sources [START_REF] Barry | Release of subducted sedimentary nitrogen throughout Earth's mantle[END_REF][START_REF] Bekaert | Subduction-Driven Volatile Recycling: A Global Mass Balance[END_REF][START_REF] Marty | The nitrogen record of crust-mantle interaction and mantle convection from Archean to Present[END_REF]. Following [START_REF] Sano | Volcanic flux of nitrogen from the Earth[END_REF], the subducted components would likely show extremely elevated N2/ 3 He, perhaps as high as ~10 11 . For subducted nitrogen to occur in the Yellowstone source but still account for a MORB-like N2/ 3 He ratio of the mantle plume, the primordial component would have to be characterized by an even lower N2/ 3 He ratios prior to slab addition. In [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF], mixtures involving slabs with N2/ 3 He of 10 11 were shown to require a primordial reservoir with N2/ 3 He ≤ 2 × 10 * . A potential problem with this view is that the N2/ 3 He of Yellowstone gases of ~10 6 would result from mixing two extreme components in just the right proportions to yield convective mantle values. The homogeneous N2/ 3 He of ~10 6 between MORB and a plume would have to result from chance.

A potentially simpler solution suggests that indistinguishable N2/ 3 He ratios for the convective and deep mantle reflect that subduction is not a major player at all for nitrogen in these reservoirs. The implications of the distinctive behaviors of nitrogen and heavy noble gases during subduction are discussed elsewhere (Labidi, in revision in this issue). As described above, we can use the sample with the lowest D30 reported by [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF] to provide a minimum d 15 N of ~+0.2‰ for Yellowstone mantle, similar to the d 15 N of ~0‰

for the Yellowstone endmember reported by [START_REF] Chiodini | Insights from fumarole gas geochemistry on the origin of hydrothermal fluids on the Yellowstone Plateau[END_REF]. Given that the d 15 N of 0.2 ‰ is a minimum value, it becomes clear that the high 3 He/ 4 He mantle source of Yellowstone features elevated d 15 N values yet with no resolvable N addition to the source (compared to a MORB mantle). We tentatively suggest that d 15 N ~ 0‰ in plumes may be indigenous to the pristine mantle source rather than being the result of additions from descending slab material (Fig. 7).

Literature data from Iceland and the Azores fumaroles may have to be revisited in light of these findings. Fumaroles from these localities have N2/ 3 He values as low as ~10 7

that are associated with markedly negative d 15 N values [START_REF] Marty | Gas geochemistry of geothermal fluids, the Hengill area, southwest rift zone of Iceland[END_REF][START_REF] Caliro | Gas geochemistry of hydrothermal fluids of the S. Miguel and Terceira Islands, Azores[END_REF]. It is tempting to attribute d 15 N ≤-10‰ to a hint of primordial nitrogen in plumes, akin to the -20 ‰ to -47 ‰ d 15 N values exhibited by enstatite chondrites [START_REF] Grady | Compositional differences in enstatite chondrites based on carbon and nitrogen stable isotope measurements[END_REF]. However, the N2/ 3 He ratios of ~10 7 are still far removed from the MORB-like mantle endmember N2/ 3 He of ~3x10 6 that we obtain from D30-d 15 N systematics for the high 3 He/ 4 He mantle at Yellowstone. Mixing between a mantle gas (N2/ 3 He ~10 6 ) and a relatively minute amount of fractionated air (N2/ 3 He ~10 11 ) can easily yield N2/ 3 He values ~10 7 . A simple mass balance shows that any mixing proportion involving more than 1% of air increases the N2/ 3 He ratios to ~10 7 and overwhelms both D30 and d 15 N of the gas mixture air d 15 N values (e.g., Figure 4). Should d 15 N of that particular air component be mass fractionated, as we have shown is likely for air-derived nitrogen, N2/ 3 He ~10 7 will be observed together with low d 15 N, leading us astray. We suggest this is the simplest explanation for the negative d 15 N signatures observed in Iceland [START_REF] Marty | Gas geochemistry of geothermal fluids, the Hengill area, southwest rift zone of Iceland[END_REF] and the Azores [START_REF] Caliro | Gas geochemistry of hydrothermal fluids of the S. Miguel and Terceira Islands, Azores[END_REF].

Instead of explaining very low d 15 N values for the high 3 He/ 4 He mantle as vestiges of an enstatite chondrite source, we tentatively suggest that the high 3 He/ 4 He mantle may have a d 15 N slightly elevated compared to MORB, with values around ~0‰ or slightly higher (Fig. 7).

Eifel gases display N2/ 3 He ratios of 1.2 (±0.1) x10 7 at near-zero D30 values (Fig. 4).

This is about a factor of 2-4 times the value for MORB, supporting the notion of nitrogen addition to the Eifel mantle source. For gases from Central America, two-component mixing curves define mixing between air with N2/ 3 He ~10 11 and high-temperature gases with N2/ 3 He ~10 8 (Fig. 4). A N2/ 3 He ~10 8 , higher than MORB by two orders of magnitude, is consistent with considerable nitrogen addition to sub-arc mantle sources [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF]. Samples at Poás have negative d 15 N, values associated with low N2/He ratios [START_REF] Fischer | Subduction and recycling of nitrogen along the Central American margin[END_REF][START_REF] Zimmer | Nitrogen systematics and gas fluxes of subduction zones: insights from Costa Rica arc volatiles[END_REF]. This combination of geochemical features has been interpreted as resulting from contributions of gases from the upper mantle, suggested to account for ~80% of the vented N2 [START_REF] Fischer | Subduction and recycling of nitrogen along the Central American margin[END_REF]. However, these particular fumaroles have atmospheric D30 signatures [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF], which argues against a magmatic origin for N2. Here again, the simplest explanation for the low N2/He ratios may involve mixing between a mantle gas and air with fractionated d 15 N values. Importantly, the D30 data mitigates the need to invoke substantial MORB-like nitrogen within the Central American arc.

The new D30 data have been used to revisit the fluxes of nitrogen in arcs. Taking 3 He fluxes from the literature and a newly-defined N2/ 3 He ratios of ~ 10 + , [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF] suggested N2 outgassing fluxes are between 4.0 x 10 8 and 1.0 x10 9 mol N2/y. The revised nitrogen outgassing fluxes are comparable to the subduction flux of nitrogen in this subduction zone (Busigny et al., 2019). This suggests that in the Central American subduction zone, the subduction and volcanic degassing fluxes of nitrogen are in balance [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF], consistent with recent experimental work on the nitrogen partition coefficient between slabs and fluids [START_REF] Jackson | Warm and oxidizing slabs limit ingassing efficiency of nitrogen to the mantle[END_REF][START_REF] Mallik | Nitrogen evolution within the Earth's atmosphere-mantle system assessed by recycling in subduction zones[END_REF]. This in turn allows for the prospect that N2 is quantitatively returned to the surface by degassing in subduction zones, precluding a net delivery of nitrogen to the deep mantle (Fig. 7). If this circumstance was typical globally and over geological timescales, it would explain the apparent contradiction that air has a higher 15 N/ 14 N than the mantle whereas delivery of isotopically heavy nitrogen in slabs to the mantle would predict the opposite over time.

Relative isolation of the mantle nitrogen from the surface seems to be required over geological time scales [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF].

Concluding remarks

New D30 data, acquired with high-mass-resolution mass spectrometry, are a valuable tool to constrain the behavior and sources of nitrogen in natural gas mixtures. In where for convenience we have omitted the factor of 10 3 that puts the values in per mil. As an illustration, we consider the extreme case where the isotopic compositions of the two nitrogen pools differ by 100 ‰, a difference much greater than found among natural geochemical or biogeochemical reservoirs. In this example, 15 R1=1.100 × 15 R2, or d 15 N(1)= 100‰ and d 15 N(2)= 0‰. In addition, for convenience, we stipulate that the "clumping" in this gas is 0, meaning that there is no excess in 15 N 15 N relative to stochastic. Using the average natural abundances of 15 N and 14 N to define 15 R2, 0.00364 and 0.99636, respectively, we have 15 R2 = 0.003653, and the stochastic abundance of 15 N 15 N in the N2 composed of one atom from reservoir 1 and the other from reservoir 2 is x 15 N 15 N/x 14 N 14 N= 15 R1× 15 R2 = 1.46812×10 -5 . Using Equation ( 7) to calculate D30, we obtain 0, as expected. However, the analyst cannot apply Equation ( 8) because there is nothing to indicate that the two nitrogen atoms represent two distinct isotopic pools; the values of 15 R1 and 15 R2 are unknowable. Forced to use Equation ( 6), the analyst obtains a D30 value of -2.27 ‰ for this stochastic gas. This is the spurious negative D value caused by the combinatorial effect. For more typical differences in d 15 N among relevant reservoirs of tens of per mil at most, these effects are on the order of a tenth of ‰ or less, and then only if N2 is constructed from nitrogen atoms from two distinct isotopic pools. The combinatorial effect is not a factor in the applications described in this review, and it cannot explain the ~ 19 ‰ enrichment in 15 N 15 N in air.

Appendix C: evidence against 15 N 15 N bond re-ordering Atmospheric N2, with D30 ~19‰, may be kept at relatively high temperatures within heated hydrothermal systems.

This could re-order nitrogen to a near-stochastic distribution of the doubly-substituted isotopologue 15 N 15 N, leading to N2 with D30 =0. Partial or total bond re-ordering of atmospheric signatures is however considered unlikely, on the basis of experimental evidence and circumstancial observation in nature.

Pure N2 at a pressure of 0.1 bar was heated at 800 °C for between 1 and 69 days. The heated N2 yielded average D30 of 19.2±0.2 indistinguishable from the starting D30 composition of air. In a second set of experiments, N2 was in contact with 2 grams of basalt powder (< 68 microns mesh), as a ways to increase the potential reaction surfaces between gases and solids with rock material that may be present at depths in natural hydrothermal systems. The gas+powder mixture was heated at 800 °C for up to 38 days. In these experiments N2 yielded average D30 of 19.0±0.2.

Δ 30 = x 15 N 15 N / x 14 N 14 N 15 R 1 + 15 R 2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 -1 Δ 30 = x 15 N 15 N / x 14 N 14 N 15 R 1 15 R 2 - 1
The evidence against re-ordering in heating experiments is corroborated by circumstantial observations in nature.

Gases vented at 400 °C in el Salvador, up to 212 °C in Iceland, and 200 °C in Hawaii are among the samples analyzed for D30 with the highest venting temperatures. At these temperature, bond re-ordering would result equilibrium D30 value below 0.3‰ [START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF]. However, nitrogen from these samples have high D30 value of 15.5±0.3‰, 18.6±0.8‰ and 18.2±0.4‰ respectively. Those are near atmospheric signatures which argues against the erasure of 15 N 15 N atmospheric excesses. A number of other gases vented at ~100-150 °C show strictly atmospheric D30 values.

Last, we note that re-ordering would cause vertical trends in plots of D30 vs d 15 N on Figure 4, as it would only affect D30 values, not the bulk 15 N/ 14 N ratios. This is not observed; 15 N/ 14 N ratios correlate with D30 values. These correlations require mixing processes. The absence of D30 re-ordering on timescales relevant to hydrothermal systems allow extrapolation to D30 ~ 0‰ to distinguish melt-derived volatiles from atmospheric components. [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF][START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF]. Detail on heated gases can be found in [START_REF] Yeung | Extreme enrichment in atmospheric 15N15[END_REF]. Briefly, heated gases in the absence of a catalyst show un-equilibrated D30 values, so they plot near air. Only heated gases in the presence of strontium nitride show re-ordered, stochastic distributions of 15 N 15 N. Most data on natural samples are from Labidi et al., (2020Labidi et al., ( , 2021)). Natural samples are dominated by hydrothermal vents, but also include mid-ocean ridge basalts and mine gases.

Captions

δ N (‰)

δ N (=δ N) (‰) et al., (2012). No enstatite-like d 15 N signatures appear to be required by the data in the Yellowstone plume source. The plume source has a N2/ 3 He similar to the MORB mantle, which could challenge the notion of nitrogen addition to plume sources with elevated 3 He/ 4 He ratios [START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF]. The arc sources underneath the central American arc are even more enriched in d 15 N, with values up to +5‰ according to 15 N 15 N data. There, elevated N2/ 3 He across the entire arc require the overwhelming addition of surface-derived N to sub-arc sources. Recalculated fluxes suggest that, in that one subduction zone, ingassing and outgassing fluxes are matching within uncertainties [START_REF] Labidi | Recycling of nitrogen and light noble gases in the Central American subduction zone: constraints from 15N15[END_REF].

  figure1). Magmatic samples were extracted from mid-ocean ridge basalts with high volatile

d

  29 N (or d 15 N) because N2 can be fractionated by molecular mass. The mass fractionation array defined by gases with equilibrium, or nearly equilibrium, D30 values result from fractionation during geological and/or biological processing (Fig. 1). In figure 2, D30 values are plotted against d 29 N2. Because d 29 N2 is numerically equal to d 15 N, all plots going forward directly use the d 15 N notation instead of d 29 N2. N2 may be produced during volcanic degassing, but also during metamorphism or thermogenic cracking. D30 values of ~ 0‰ are expected for N2 molecules from magmas because at high temperatures, thermodynamic equilibrium approaches the random distribution of 15 N 15 N relative to other N2 isotopologues. This was verified by the study of mid-ocean ridge basalts with extraordinarily high volatile concentrations. Crushing experiments under vacuum released N2 with D30 values between +1.2±2.6‰ and +3.0±3.6‰ (Fig. 2, Labidi et al., 2020).

  Fig.3). The negative d 15 N values are deceptively similar to the MORB value of -4±2‰[START_REF] Javoy | The birth of the Earth's atmosphere: the behaviour and fate of its major elements[END_REF] Pineau, 1991, Marty et al., 1999), but the D30 values preclude a magmatic origin. The negative d 15 N signatures must result from isotopic fractionation of air. The distribution of d 29 N2 and d 30 N2 along a slope of ~2 in Fig. 1 indicates mass fractionation of the air-derived nitrogen. The fractionation likely occurs in the sub-surface, after air-saturated meteoric waters feed atmospheric nitrogen into hydrothermal systems. Upon heating, fluids undergo water/gas partitioning, eventually resulting in volatile losses. If water/gas exchange is associated with a 15 N/ 14 N fractionation, the degassed fluids would have 15 N/ 14 N ratios (d 15 N values) distinct from that of the residual N2 left in the geothermal waters. Air with d 15 N values between -5 and -10‰ indicate two alternative possibilities: (1) N isotope fractionation factors are similar to Graham's law, resulting in the instantaneous release of N2 greatly depleted in 15 N, while residual N2 is enriched in 15 N, depending on the extent of degassing; or (2) N isotope fractionations are smaller than suggested by Graham's law, but amplified by open-system Rayleigh distillations, in which both the product and the residue of the water/gas fractionation are drawn to low d 15 N as the proportion of N held by the aqueous phase decreases.In case 1, where fractionation occurs as a single-step process, the fractionation factor for isotopologue partitioning could be as great as the inverse of the square root of the inverse of the isotopologue mass ratio, which would result in d 29 N2 (or d 15 N) for the effusing gas that is -17 ‰ lower than that of the aqueous phase. The residual nitrogen in the fluid would have a complimentary enriched d 29 N2 (or d 15 N) value. By simply propagating a 17‰ fractionation between dissolved and exsolved N2, one expects the low d 15 N of -6‰ seen in the Iceland dataset from[START_REF] Labidi | Hydrothermal 15 N 15 N abundances constrain the origins of mantle nitrogen[END_REF] to eventually produce the complimentary high d 15 N values of +11‰, but these are not seen. The most positive d 15 N at an atmospheric D30 is only +0.5‰ in Icelandic fumaroles (Fig.3). A single-step process therefore seems unlikely, and this is supported by experiments. The 15 N/ 14 N isotope fractionation between N2 gas and N2 dissolved in aqueous fluid, a = ( 29 R)gas/( 29 R)fluid, when expressed as d 15 Ngas -d 15 Nfluid, is found to range from -0.9 to +0.4 ‰ from 6 to 60 °C[START_REF] Lee | Kinetic nitrogen isotope fractionation between air and dissolved N2 in water: Implications for hydrothermal systems[END_REF]. The reversal of the fractionations at 40 °C indicates the N2 isotope fractionation factors result from a kinetic rather than an equilibrium process, and that the kinetic fractionation is fundamentally different from the simple square-root of the inverse mass ratio. Above 40 °C, isotopically light N2 is preferentially partitioned into the aqueous fluid phase, and isotopically heavy N2

  yielded no measurable d 15 N fractionation. Samples taken in August 2018 in the same hydrothermal vents resulted in greatly fractionated air signatures with d 15 N as low as ~ -10‰ (Labidi et al., 2020). The d 15 N variability of air-derived N2 should be a subject of future dedicated studies. The time variability of d 15 N in N2 derived from air in hydrothermal systems is particularly problematical for using d 15 N as a tracer for the provenance of N2. In the interim, very negative d 15 N in hydrothermal systems should be regarded as a natural consequence of degassing in the context of intermittent meteoric (air-saturated) water recharge. In all cases, d 15 N alone in hydrothermal gases is a poor tracer of the origin of N2. In the absence of 15 N 15 N data, d 15 N alone may lead to erroneous conclusions regarding the origin of volatiles in hydrothermal gases.

At

  Poás and Momotombo, two volcanoes from Central America, D30 values range down to +3.4±1.0 and 1.5±0.3‰. Extrapolation of the linear fits to D30 values of zero suggests that the volcanic N2 at Poás and Momotombo have d 15 N values of ~+1±0.5‰ and ~+5±0.5‰

  15 N 15 N and 36 Ar data: important but challenging The ratios N2/Ar and N2/ 36 Ar are used routinely as tracers of volatile origin, and are also important data for calculating fluxes of N2 outgassing. Published N2/ 36 Ar data with D30

Fischer

  et al., (1998). Excess nitrogen is defined as [N2*] = [N2]measured -40 x [Ar]measured. The numerical value 40 corresponds to the N2/Ar ratio of air-saturated water. It is an important

  hydrothermal fluids from various locations, the data are dominated by air-derived components rather than by high-temperature volcanic volatiles. Using these data, we find that air-derived components undergo isotope mass fractionations in hydrothermal systems. This leads to the emanation of air-derived gases with negative d 15 N values and anomalously low N2/Ar and He/Ar ratios. This has consequence for the use of d 15 N and N2/Ar as source tracers in hydrothermal gases. The new D30 data have allowed determinations of mantle d 15 N, N2/ 36 Ar, and N2/ 3 He ratios beneath Eifel and Yellowstone. For Eifel, the mantle N2/ 36 Ar and N2/ 3 He values are slightly greater than the values for the MORB mantle. These values may reflect the addition of nitrogen via subduction in the Eifel source. At Yellowstone, the mantle d 15 N, N2/ 36 Ar, and N2/ 3 He values determined with D30 data yield important information about the high 3 He/ 4 He mantle. The N2/ 3 He elemental ratio is indistinguishable from the MORB mantle, and thus inconsistent with N addition to the high 3 He/ 4 He mantle. The d 15 N value is indistinguishable from 0‰, after appropriate correction for air contamination. The new D30 data show that suggestions of a high 3 He/ 4 He mantle with very negative d 15 N values are not warranted. Instead, the observed low d 15 N values are the result of 15 N/ 14 N fractionation in hydrothermal systems. Air-like d 15 N values with MORB-like N2/ 3 He ratios appear to be a fundamental observation that constrains the origin of volatiles in the high 3 He/ 4 He mantle. More work is needed on 3 He rich hotspots. Iceland hosts many fumaroles and geothermal wells where light noble gases have features associated with the plume mantle[START_REF] Füri | Apparent decoupling of the He and Ne isotope systematics of the Icelandic mantle: The role of He depletion, melt mixing, degassing fractionation and air interaction[END_REF]. Acquisition of D30 measurements for these fumaroles will be important. The only D30 measurements of nitrogen from Icelandic fluids, reviewed here, have identified no non-atmospheric N2. Future attempts to identify high-temperature N2 there, if successful, would be useful for determining whether the low N2/ 3 He that characterizes the Yellowstone plume is a global signature of the primordial mantle.Plots featuring 40 Ar/ 36 Ar versus D30 are helpful when mantle endmembers have a known, elevated, 40 Ar/ 36 Ar ratio, as in the cases of plumes. For those cases, measured N2/ 36 Ar on individual samples are ambiguous regarding the source, but the approach of fitting mixing curves to the data (e.g., Fig.6) affords a determination of N2/ 36 Ar. The approach is sensitive to the N2/ 36 Ar ratios used for air components. When the 40 Ar/ 36 Ar of a mantle source is unknown (e.g., a subduction zone), the D30 data support the concept of sub-arc mantle sources entirely dominated by air-derived argon.New D30 data provide novel constraints on the fate of nitrogen in subduction zones.The sub-arc source from the warm subduction zone in Central America is shown to have elevated N2/ 3 He by orders of magnitude compared to the MORB mantle. This shows considerable addition of N2 to sub-arc sources. Revised fluxes in the Central American subduction zone are based on N2/ 3 He that in turn derive from the analysis of D30 data. The derived fluxes suggest that this margin is a subduction barrier for N2. Using fluxes determined on warm subduction zones may be more relevant to the deep time than those characterizing cold subduction zones. The true influence of subduction on mantle volatiles through time warrants revision based on these results.

Figure 1 :

 1 Figure 1: Isotopic composition of N2 from natural samples and laboratory experiments for d 30 N2 versus d 29 N2. Massdependent fractionation curves for air and high-temperature stochastic N2 are shown. Data are from Yeung et al. (2017),

Figure 2 :Figure 3 :

 23 Figure 2: Isotopic composition of N2 from natural samples and laboratory experiments for D30 versus d 15 N. A ~19‰ offset is observed between air and stochastic gases. Hydrothermal samples show variable d 15 N and D30. The D30 variation reflects mixing between stochastic and air-derived N2. Importantly, negative d 15 N are almost exclusively associated with air that underwent N isotope fractionation, as shown by atmospheric D30 values.

Figure 4 :Figure 5 :

 45 Figure 4: Measured N2/ 3 He ratios versus D 30 values. Mixing lines are shown for mixtures between air and high temperature

Figure 6

 6 Figure 6 The relationship between nitrogen D30 and argon isotopes in volcanic gases D30 and 40 Ar/ 36 Ar ratios are shown for Iceland, Yellowstone, Eifel and central America gases. Panel A and B are identical geochemical spaces with distinct mixing scenarios. Panel A aims at accounting for the Yellowstone data. Mixing lines are between ASW and a hypothetical plume endmember with 40 Ar/ 36 Ar ratio of 10,000. Variable N2/ 36 Ar ratios for the plume component cause variable curvatures in the mixing relationships. Panel B aims at account for the Central American dataset. Mixing lines are between ASW and a hypothetical mantle wedge endmembers with variable 40 Ar/ 36 Ar ratios.

Figure 7

 7 Figure 7 Cartoon representing various processes and reservoirs constrained by 15 N 15 N data. Not to scale. A section of Earth with the core, the mantle, the continental and oceanic crust, and the atmosphere are shown. A slab is shown to enter subduction. Air is the only reservoir with N2 associated with a 15 N 15 N anomaly of ~19‰. All other sources of N2 have D30 ~ 0‰. The slab hosts nitrogen fixed as NH4 + (Busigny et al., 2019 and references therein) so no D30 values are defined; D30 is only relevant for N2 molecules. Upon partial melting of a mantle source, magmatic N2 would form with a D30 of 0‰. The high-T N2 is then contributed to hydrothermal systems in the sub-surface by near-quantitative magmatic degassing. Air circulation in the sub-surface allows air-saturated waters to undergo degassing, and subsequent gas release with fractionated compositions. Plume data require a mantle source with a d 15 N ~0‰, from the study of Yellowstone, in agreement with Chiodini

Measurements of isotope ratios at high mass resolution: the challenge of 15 N 15 N

  Measurements of nitrogen isotopologues involve high-mass-resolution mass spectrometry. The abundances of 14 N 14 N + , 14 N 15 N + (representing both isotopomers, 14 N 15 N and 15 N 14 N) and 15 N 15 N + ions are measured with high mass resolution to avoid mass-tocharge interferences. Interferences for 28 N2 ( 14 N 14 N + ) and 29 N2 ( 14 N 15 N + ) are from carbon monoxide, 28 CO ( 12 C 16 O + ) and 29 CO ( 13 C 16 O + and 12 C 17 O + ). The m/Dm separating 28 CO ( 12 C 16 O + ) from 28 N2 ( 14 N 14 N

	of both nitrogen and argon, especially when associated with conventional noble gases
	measurements.
	3.
	: it is not erased by mass-dependent fractionation of N2 and is not re-
	ordered in heated hydrothermal systems (see appendix C). The use of D30 as a tracer of air
	does not require any assumptions about the relative abundances of other tracers, as is the
	case for 40 Ar/ 36 Ar or N2/Ar ratios. Instead, we use the isotopic compositions of the N2
	molecules themselves to trace the origin of the nitrogen in a manner that is more diagnostic
	than d 15 N. Based on the application of this tracer to volcanic hydrothermal systems, we show
	that both d 15 N and N2/Ar experience fractionations in natural hydrothermal systems, and
	that atmospheric-like 40 Ar/ 36 Ar ratios may be genuine features of the sub-arc mantle.
	Ultimately, previously unrecognized fractionation of 15 N/ 14 N (expressed as shifts in d 15 N)
	and N2/Ar appear to have conspired to produce deceptive indicators for the origin of N in
	gas mixtures that D30 measurements can help rectify.
	In what follows we illustrate the potency of 15 N 15 N to constrain the true d 15 N, N2/ 36 Ar,
	N2/ 3 He, 40 Ar/ 36 Ar, and even 3 He/ 36 Ar ratios of volcanic components contributing to
	hydrothermal gases. The new 15 N 15 N data offer valuable constraints on the deep-Earth cycles

+ ) is ~2,500, while values of ~6,000 and ~7,000 separate 13 C 16 O +

  not all) hot spring data. This likely reflects the recharge of surficial hydrothermal systems by air-saturated meteoric fluids, causing atmospheric N2 to dominate over hightemperature N2. Gases from Iceland, the Yellowstone caldera, and the Poás volcano all show atmospheric D30 values but negative d 15 N down to -10‰ (

D 30 -d 15 N evidence that air is ubiquitous and fractionated By way of summary, we can say that high-temperature N2 is characterized by D30 of ~0‰ and variable d 15 N, and that air-derived N2 has a D30 value of 19‰ and can also have variable d 15 N (Fig.

2

). A number of samples exhibit D30 values intermediate between air and stochastic. Because there is no primary mechanism for producing intermediate values, these samples must be mixtures between air and high-temperature, volcanic N2 (see appendix C for evidence against D30 re-ordering in nature). In principle, mixing between two N2 gases with very different d 15 N values but similar, near-equilibrium D30 values can produce anomalously high D30 values (i.e., mixing relationships are curves, not straight lines). In practice however, the differences in d 15 N values required to produce mixing curves with sufficient curvature to cause discernible aberrations in D30 are on the order of hundreds of permil

(Yeung et al., 2017 -also, see appendix)

.

Because mixing in D30 -d 15 N space is effectively linear, it is straightforward to identify endmember N2 compositions for mixtures. For hydrothermal samples with D30 values of 19‰, virtually all N2 in the gas mixture is derived from air. Atmospheric D30 values dominate most (but

Appendix A: Mixing loops

The concept of isotope clumping uses the stochastic distribution of isotopologues as a reference frame. Mixing in this reference frame can be highly non-linear. We illustrate mixing with N2 gases with different bulk isotopic composition, gas 1 with d 15 N of -200‰, and gas 2 with a d 15 N of +100‰. For this exercise, both gases will be assigned a stochastic distribution of isotopologues, i.e. D30 = 0‰. The stochastic relative abundance of 15 N 15 N in gas 1 is the square of the singlysubstituted value, treating relative abundances as probabilities:

where x( 15 N 15 N)gas 1 is the mole fraction of the 15 N 15 N isotopologue in gas 1, and so forth. Similarly, the stochastic relative abundance of 15 N 15 N in gas 2 is:

x( 15 N 15 N)gas2 = x( 15 N) 2 = 1.60x 10 -5 .

(2)

We calculate the expected stochastic abundance of 15 N 15 N for a 50:50 mixture of these two gases from the bulk 15 N/ 15 N ratio in the usual way, as the square of the bulk isotopic concentration

x( 15 N 15 N)50:50,stochastic = x( 15 N) 2 = 1.20 x 10 -5 .

(3) However, simple mixing between these two gases does not involve bond rupture. Since molecules retain their original bond ordering, the physical mixture has an observed fractional abundance of 15 N 15 N different from that in equation 3.

The relative concentration of 15 N 15 N for the physical mixture is

Comparing the measured value from Equation ( 4) with the predicted stochastic value from Equation (3) leads to an apparent D30 value greater than zero:

The mixing between these two gases results in a D30 value even higher than air. This effect may prove important for the interpretation of data on other planets or in the interstellar medium where cold chemistry will cause 15 N/ 14 N ratios to vary by extreme magnitudes in terrestrial terms [START_REF] Füri | Nitrogen isotope variations in the Solar System[END_REF]. However on Earth, mixing between 15 N/ 14 N ratios differing by hundreds of per mil is unrealistic. Mixing between two gases with d 15 N of -10‰ and +10‰ encompasses the vast majority of likely scenarios on Earth. In this case, the mixing produces negligible effects. At a 50:50 mixture of gases with d 15 N values of -10‰ and +10‰ and D30 of 0‰, results in D30 value of 0.1‰. In other words, in mixing scenarios where 15 N/ 14 N ratios vary by less than ~ 40‰, the induced D30 values deviate from endmember values by at most tenths of per mil. We conclude that mixing in the D30 -d 15 N space where variations in d 15 N are less than many tens of per mil (e.g., Fig. 2) produces linear trends.

Appendix B: Combinatorial effects

The combinatorial effect is an artifact of the use of the stochastic reference frame to characterize enrichments and depletions of multiply-substituted isotopologues signatures [START_REF] Taenzer | Low Δ12CH2D2 values in microbialgenic methane result from combinatorial isotope effects[END_REF][START_REF] Yeung | Combinatorial effects on clumped isotopes and their significance in biogeochemistry[END_REF][START_REF] Yeung | Biological signatures in clumped isotopes of O2[END_REF][START_REF] Young | The relative abundances of resolved 12CH2D2 and 13CH3D and mechanisms controlling isotopic bond ordering in abiotic and biotic methane gases[END_REF]. It can arise where more than one isotopic pool of an element contributes to the formation of the multiplysubstituted molecules of interest. The most succinct description of the effect is that the analyst can only measure the arithmetic mean of those distinct pools, in the form of the product molecules, whereas the actual abundance of the multiplysubstituted, or "clumped", species is the result of the geometric mean. Here we dispense with the possibility that the difference between air and other reservoirs of N2 might be the result of this effect.

We might imagine that N2 molecules are constructed from two different sources of nitrogen with distinct 15 N/ 14 N ratios, 15 R1 and 15 R2. Such a circumstances may obtain in the construction of O2 molecules during photosynthesis, for example [START_REF] Yeung | Biological signatures in clumped isotopes of O2[END_REF][START_REF] Yeung | Combinatorial effects on clumped isotopes and their significance in biogeochemistry[END_REF], or when adding H to CH3 to form methane [START_REF] Taenzer | Low Δ12CH2D2 values in microbialgenic methane result from combinatorial isotope effects[END_REF]. In the case of the formation of a molecule with a stochastic abundance of 15 N 15 N, but in which 15 R1 and 15 R2 contribute to the two different positions in the homonuclear diatomic N2, the ratio of 15 N 15 N to 14 N 14 N, given as the ratio of their mole fractions,

x 15 N 15 N/x 14 N 14 N, must be the probability of 15 R1 and 15 R2 occurring in the same molecule, or 15 R1× 15 R2, so in reality

x 15 N 15 N/x 14 N 14 N = 15 R1× 15 R2. However, the analyst has no a prior knowledge about the ratios 15 R1 and 15 R2. Therefore, when reporting D30 value for this gas, the only recourse available is to use the bulk 15 N/ 14 N of the molecules themselves to