Isotopic Techniques to Measure N2O, N2 and Their Sources

GHGemissions are usually the result of several simultaneous processes. Furthermore, some gases such as N2 are very difficult to quantify and require special techniques. Therefore, in this chapter, the focus is on stable isotopemethods. Both natural abundance techniques and enrichment techniques are used. Especially in the last decade, a number of methodological advances have been made. Thus, this chapter provides an overview and description of a number of current state-of-the-art techniques, especially techniques using the stable isotope15N. Basic principles and recent advances of the 15N gasflux method are presented to quantify N2 fluxes, but also the latest isotopologue and isotopomermethods to identify pathways for N2O production. The second part of the chapter is devoted to 15N tracing techniques, the theoretical background and recent methodological advances. A range of different methods is presented from analytical to numerical tools to identify and quantify pathway-specific N2O emissions. While this chapter is chiefly concerned with gaseous N emissions, a lot of the techniques can also be applied to other gases such as methane (CH4), as outlined in Sect. 10.1007/978-3-030-55396-8_5#Sec12.

on the origin without the addition of 15 N labelled fertiliser. Note, isotopologues are molecules that differ in their isotopic composition, isotopomers are molecules with the same isotopic atoms but differing in their position, and isotopocules is the generic term for both isotopologues and isotopomers. There is a wealth of information that we can obtain from using diverse isotopic approaches based on 15 N or 18 O labelling but also on natural abundance techniques that take advantage of the different metabolism with which for instance N 2 O is produced. Thus, 15 N provides us with a toolbox to identify emission pathways and in turn provides information on effective mitigation techniques. N 2 O reduction to N 2 is the last step of microbial denitrification, i.e. anoxic reduction of nitrate (NO 3 − ) to dinitrogen (N 2 ) with the intermediates NO 2 − , NO and N 2 O (Firestone and Davidson 1989;Knowles 1982). Commonly applied non-isotopic techniques enable us to quantitatively analyse only the intermediate product of this process including NO and N 2 O, but not the final product, N 2 , a non-greenhouse gas. The challenge to quantify denitrification rates is largely related to the difficulty in measuring N 2 production due to its spatial and temporal heterogeneity and the high N 2 -background of the atmosphere (Groffman et al. 2006). There are three principal ways to overcome this problem: (i) adding NO 3 − with high 15 N enrichment and monitoring 15 N labelled denitrification products ( 15 N gas flux method, 15 N GFM) (e.g. (Siegel et al. 1982)); (ii) adding acetylene to block N 2 O reductase quantitatively and estimating total denitrification from N 2 O production (acetylene inhibition technique, AIT) (Felber et al. 2012); (iii) measuring denitrification gases during incubation of soils in absence of atmospheric N 2 using gas-tight containers and an artificial helium/oxygen atmosphere (HeO 2 method; (Butterbach-Bahl et al. 2002;Scholefield et al. 1997;Senbayram et al. 2018)). Each of the methods to quantify denitrification rates in soils has various limitations with respect to potential analytical bias, applicability at different experimental scales and the necessity of expensive instrumentation that is not available for routine studies. Today the AIT is considered unsuitable to quantify N 2 fluxes under natural atmosphere, since its main limitation among several others is the catalytic decomposition of NO in presence of O 2 (Bollmann and Conrad 1997), resulting in unpredictable underestimation of gross N 2 O production (Nadeem et al. 2012). The 15 N gas flux method requires homogenous 15 N-labelling of the soil (Mulvaney and Vandenheuvel 1988) and under natural atmosphere, it is not sensitive enough to detect small N 2 fluxes (Lewicka-Szczebak et al. 2013). Direct measurement of N 2 fluxes using the HeO 2 method is not subject to the problems associated with 15 N-based methods (Butterbach-Bahl et al. 2013) but the need for sophisticated gas-tight incubation systems limits its use. When applying 15 N GFM in the laboratory, sensitivity can be augmented by incubation under an N 2 -depleted atmosphere Meyer et al. 2010;Spott et al. 2006). In the following, the basic principle, limitations, bias and application examples are presented and discussed.

Principles of the 15 N Gas Flux Method
The 15 N gas flux method consists of quantifying N 2 and or N 2 O emitted from 15 Nlabelled NO 3 − applied to soil in order to quantify fluxes from canonical denitrification (Mulvaney and Vandenheuvel 1988;Stevens et al. 1993), where N 2 and N 2 O are formed from the combination of two NO precursor molecules. Under certain preconditions, it is also possible to identify the production of hybrid N 2 or N 2 O (i.e. molecules formed from the combination of N atoms from one source of oxidised N, e.g. NO 2 − ), and another source of reduced N (e.g. NH 3 or NH 2 OH) via anaerobic ammonia oxidation (annamox) or co-denitrification (Laughlin and Stevens 2002;Spott and Stange 2007;Spott et al. 2011). To quantify canonical denitrification, experimental soil is amended with NO 3 − highly enriched with 15 N. The 15 N gases evolved are collected in closed chambers and 15 N emission is calculated from the abundance of N 2 and N 2 O isotopologues in the chamber gas. 15 N enrichment of N 2 in the gas samples are typically close to natural abundance because the amount of N emitted from the 15 N-labelled soil is small compared to the atmospheric background. Precise techniques of isotope analysis are, therefore, necessary.

The Non-random Distribution of Atoms in the N 2 Molecule
The 15 N gas flux method is based on the assumption that within N 2 or N 2 O from a single source of a given 15 N abundance, the N 2 O isotopologues of a distinct number of 15 N substitutions follow a random (binomial) distribution, as given by the terms in (Eq. 7.1): ( p + q) 2 = p 2 + 2 pq + q 2 (7.1) where p is the atom fraction of 14 N, q the atom fraction of 15 N and p + q is equal to unity (Hauck et al. 1958).
If N 2 O or N 2 from two different N pools, one background pool of natural 15 N abundance (0.3663 atom %) and the second enriched in 15 N are mixed, the distribution deviates from the binomial pattern. Given the distribution of N 2 or N 2 O isotopologues emitted from the first (background) N pool (a bg ) including non-labelled N 2 and N 2 O (derived from the atmosphere and possibly non-labelled N 2 O from non-labelled N sources in soil) and the resulting mixture (a m ), the 15 N abundance in the 15 N-labelled second pool (a p ) and the fraction of N 2 O or N 2 originating from that labelled pool (f p ) can be determined (e.g. Bergsma et al. 2001;Spott et al. 2006). To calculate f p values, the nitrogen isotope ratios 29 R( 29 N 2 / 28 N 2 ) and 30 R( 30 N 2 / 28 N 2 ) are used. In case of N 2 , the three isotopologues 14 N 14 N and 14 N 15 N and 15 N 15 N are detected. For N 2 O, one option is to directly analyse intact N 2 O molecules, consisting of N and oxygen (O) and analysing molecular masses 44, 45 and 46. It has to be taken into account that these molecular masses include not only N-but also O-substituted isotopocules and thus the following 6 species: 14 O-substituted isotopocules ( 14 N 14 N 18 O and 14 N 14 N 17 O) due to their mass overlap with the 15 N-substituted isotopocules (Bergsma et al. 2001). Alternatively, N 2 O can be reduced to N 2 prior to IRMS analysis (Lewicka-Szczebak et al. 2013), thereby allowing direct determination of 29 R and 30 R of N 2 O-N.
There are various calculation procedures that have evolved over time (Hauck et al. 1958;Mulvaney 1984;Arah 1992;Nielsen 1992Spott et al. 2006). In Eqs. 7.2 and 7.3 we show one example (Spott et al. 2006), where the fraction of N 2 or N 2 O evolved from the 15 N-labelled NO 3 − pool (f p ) is calculated: f p = a m − a bg a p − a bgd (7.2) where a m is the 15 N abundance of the total gas mixture a m = 29 R + 2 * 30 R 2 1 + 29 R + 30 R (7.3) and a bg is the 15 N abundance of atmospheric background N 2 . The 15 N abundance of the 15 N-labelled nitrate pool undergoing denitrification is a p = 30 x m − a bgd * a m a m − a bgd (7.4) where 30 x m is the measured fraction of m/z 30 in the total gas mixture: 30 x m = 30 R 1 + 29 R + 30 R (7.5) The same calculations can be used for N 2 and N 2 O, resulting in respective values for fractions of pool-derived N (f p_N2 ; f p_N2O ) and for the respective 15 N abundances of the active N pools (a p_N2 ; a p_N2O ).
If only m/z = 28 and m/z = 29 are determined during isotope analysis of N 2 , then emission of 15 N 2 is underestimated (Hauck et al. 1958). The extent of underestimation is related to the 15 N atom fraction of the NO 3 − pool from which N 2 is emitted ) and f p can thus be calculated if the 15 N enrichment of the denitrified N pool is known (Mulvaney 1984): f p = 29 R sa − 29 R bg / 2 a p 1 − a p (7.6) where lower case sa and bg denote sample and background (typically ambient air), respectively. An alternative equation yielding f p from 29 R that is more complex, but also more precise, is given by Spott et al. (2006). In many studies, a 15 N atom fraction of 0.99 was selected for the 15 N enrichment of applied NO 3 − ( 15 a NO3 ) in order to maximise 30 R (see Fig. 7.1), thus yielding better 30 R signals. However, there are also reasons to keep 15 a NO3 between about 0.6 and 0.4, since 30 R is only detectable with high fluxes due to a typical high IRMS background signal at m/z 30 (see next section), so that f p has to be calculated from 29 R only using Eq. 7.6. But f p calculated from Eq. 7.6 with a given 29 R is relatively insensitive to changes in a p between 0.4 and 0.6 since the nominator yields, e.g. for a p between 0.4 and 0.6, values between 0.48 and 0.5. Hence, uncertainty in the estimation of a p within that range causes minor uncertainty in calculated f p (Well and Myrold 1999). To illustrate how the combination of denitrification rates (i.e. f p ) and homogenous or non-homogenous 15 N enrichment of the soil NO 3 − pool affect instrumental raw data as well as calculated f p and a p values, some theoretical data are shown (Table  7.1). Three cases are represented, (1) the soil is homogenously labelled with 15 N, (2) non-labelled soil-derived NO 3 − dilutes the labelled pool to a different extent in the 0 to 10 and 10 to 20 cm layers, but N 2 and N 2 O production rates in both layers are equal and (3) like case (2) except that production rates of both layers differ. It can be seen that only case (1) calculated using Eq. 7.4 yields results identical to ideal a p and f p . Equation 7.6 gives deviating results when used with 15 a NO3 as this value differs from a p . In the case of (2) and (3), all calculations lead to some deviation due to the non-homogeneity in label distribution. Moreover, isotope ratios show that even at the high denitrification rate assumed (case 2, 542 g N ha −1 20 cm −1 d −1 ), the increase in 29 R ( 29 Rm-29 Ra) and 30 R ( 30 Rm-30 Ra) was 9.2 × 10 −6 and 3 × 10 −6 , respectively, and thus only about one order of magnitude above typical instrumental precision (see Table 7.2).

Identifying the Formation of Hybrid N 2 and/or N 2 O
When N 2 and N 2 O are formed from denitrification, both N atoms are derived from the 15 N labelled pool, and in hybrid N 2 or N 2 O only one N atom comes from the labelled pool (N oxides, i.e. NO 2 − ) and the other one comes from non-labelled reduced N (e.g. NH 3 , NH 2 OH or organic N). Hence, the contribution of hybrid processes is reflected by an increase in 29 R only, while denitrification increases both 29 R and 30 R (Clough et al. 2001). Laughlin and Stevens (2002) derived equations to calculate the fraction of hybrid and non-hybrid N 2 , assuming that the measured 15 N atom fraction of NO 3 − also reflected the enrichment of the NO 2 − that contributed one N atom to the hybrid molecules, and that the 15 N abundance of the non-labelled sources (atmospheric N and non-labelled reduced N) were identical. An extended approach was developed allowing to take into account different 15 N enrichment for all contributing sources, i.e. different values for atmospheric and reduced N Spott et al. 2011). Spott et al. (2011) used those equations to calculate co-denitrification in a soil slurry but pointed out that the approach would be subject to possible bias due to difficulty and inaccuracy when determining the 15 N enrichment of the nitrite (NO 2 − ) pool contributing to the hybrid formation. For N 2 O mixtures consisting of N 2 O from only two sources, i.e. hybrid and non-hybrid N 2 O, the authors, therefore, suggest to use the indicator value R binom to assess the contribution of hybrid N 2 O. R binom reflects the fact that N 2 or N 2 O isotopocules of each non-hybrid source contributing to a gas mixture are following a random (binomial) distribution, whereas this is not the case for the hybrid N 2 O. R binom values >1 indicate a significant hybrid contribution. While fluxes excluding hybrid N 2 O would always yield R binom ≤ 1, respective R binom values would not exclude the possibility of some hybrid contribution. Hence, R binom can only prove the existence (but not the absence) of hybrid fluxes. The limitation of this approach is that it does not work in the presence of additional sources, e.g. if

Analysis of N 2 and N 2 O Isotopologues
Precise quantification of N 2 and N 2 O isotopocules requires isotope ratio mass spectrometry (IRMS) where 29 R and 30 R are obtained from ion current ratios detected at Faraday collectors tuned for m/z 28, 29 and 30 (e.g. Lewicka-Szczebak et al. 2013). A double collector IRMS was used before multi-collector IRMS became available. Double collector IRMS required two measurements with the IRMS so that either 29 N 2 or 30 N 2 is positioned on the first collector (Siegel et al. 1982). Emission spectroscopy has also been used in the past to detect 28 N 2 , 29 N 2 and 30 N 2 (Kjeldby et al. 1987), but its relatively low precision enabled only detection of large N 2 fluxes. While dual inlet IRMS had been used with manual measurement of samples in glass containers that were sealed  or isolated by stopcocks (Siegel et al. 1982), continuous flow IRMS enables automated injection of samples from septum capped vials since the 1990s (Stevens et al. 1993 (Brand et al. 2009, Siegel et al. 1982 due to the omnipresence of oxygen traces. NO + formation can be quantified by the ratio between ideal and measured 30 R of standard gases, giving values of 0.15 to 0.06 for atmospheric N 2 analysed in the instrumentation proposed by Lewicka-Szczebak et al. (2013). NO + formation can be minimised by removal of all O sources (O 2 , H 2 O) from the samples and also from the carrier and reference gases. In some types of IRMS the NO + background is too high and associated with extreme tailing of the m/z 30 peak. This makes it impossible to quantify 30 R . To overcome this limitation, a procedure to quantify 30 R indirectly from 29 R was developed where 29 R had to be analysed twice, (i) in samples where the non-random distribution of N 2 isotopocules was randomised by the temporary splitting up of N 2 molecules during a gas discharge (see change in 29 R due to randomisation in Fig. 7.1). Discharge was actuated using a microwaves source, initially offline in sealed glass tubes, later with online continuous flow IRMS, where the discharge occurred in the gas circuit connecting and IMRS ). An overview of the IRMS precision for 29 R and 30 R in N 2 standard gases is given in Table 7.2, showing that repeatability for 29 R varied significantly between instruments, but 30 R is comparable. However, it is also evident that during the last 35 years (Siegel et al. 1982) there has been no substantial improvement in the measurement precision.

Detection Limit for a p and f p
Because f p is calculated from two quantities, 29 R and 30 R, and the relationship between them depends on the 15 N enrichment of the active N pool (a p , see Fig. 7.1), the limit of detection (LOD) for f p at given repeatability of 29 R and 30 R is variable. LOD for f p was thus determined for varying conditions using equations from Spott et al. (2006) using Monte Carlo modelling assuming a normal distribution of 29 R and 30 R errors (Standard deviation of repeated analysis of standard gas samples). The MS-Excel function norm.inv was used to create the normal distribution of values but allowing only a maximum deviation of 3 standard deviations, otherwise unrealistic outlier of 29 R or 30 R yield unrealistically high uncertainty. Different scenarios were tested (f p = 1 to 100 ppm; a p = 0.055 to 0.75 using repeatability for 29 R and 30 R of the first IRMS listed in Table 7.1). LOD is obtained for two cases: 1. Both 29 R and 30 R are taken into account to calculate both a p and f p ; 2. f p is calculated using only 29 R (using Eq. 7.4 in (Spott et al. 2006)) and a p is estimated either from soil extract analysis or from a p of N 2 O (e.g. Stevens and Laughlin 2002). Note that a p of N 2 O is usually much more reliable than a p of N 2 since f p of N 2 O is typically large (often between 0.1 and 1) due to the fact that, in contrast to N 2 , N 2 O is an atmospheric trace gas. Conversely, f p of N 2 is typically very small (usually <10 −5 in ambient atmosphere).
The first calculation is preferable because a p of N 2 and N 2 O can be different (see Fig. 7.3) and a p of N 2 O can only be obtained if N 2 O can be directly measured by IRMS, which is only the case if concentrations are high enough (about 0.3 to 3 ppm necessary, depending on 15 N enrichment of N 2 O). Since incubation under N 2 -depleted atmosphere improves f p sensitivity, LOD is also given for an artificial gas mixture containing 2% N 2 .
LOD results are as follows (Table 7.3): with high f p (i.e. ≥10 ppm) and high a p (i.e. ≥0.5) and ideal IRMS performance (Table 7.2) both calculations yield precise results. Under N 2 -depleted atmosphere, LOD is excellent (2 to 7 ppb N 2 , last columns). With lowering of a p , LOD gets worse if a p has to be calculated using 30 R. But without using  (Siegel et al. 1982) 30 R and assuming an ideal a p value or estimating a p of N 2 from direct determination of a p of N 2 O, LOD of f p is still excellent. This is because with decreasing a p , abundance of 15 N 15 N ( 30 N 2 ), and thus 30 R, decreases exponentially whereas the decrease of 29 R ( 29 N 2 ) is much slower (see Fig. 7.2).

Limitations of the 15 N Gas Flux Method ( 15 N GFM)
The following factors limit the applicability of the 15 N GFM

Inaccurate Definition of the Soil Volume Represented by Denitrification Measurements and Incomplete Recovery of Denitrification Gases
The denitrifying soil volume is clearly defined if soil cores are entirely labelled with 15 N and are incubated in closed systems. However, in situ measurement of denitrification in surface soils or subsoils with approaches other than the core methods do not include complete enclosure of the investigated soil. It is not possible to control the application of 15 NO 3 − accurately. Consequently, the soil volume represented by the detected denitrification gases is not exactly defined, and calculated denitrification rates are associated with uncertainty. Partial enclosure of the investigated soil is typically achieved by driving cylinders into surface soils. This option reduces the problem to a certain extent. Moreover, measuring the spatial distribution of the 15 N label at the end of experiments (Well and Myrold 2002) helps to constrain the soil volume contributing to soil N 2 fluxes that can be "seen" by 15 N analysis of headspace gases. An additional problem of open systems is the difficulty to determine the direction and strength of diffusional gas transport. When chamber methods are used to determine denitrification of surface soils, a significant fraction of the denitrification gases produced in the 15 N-labelled soil diffuses into the subsoil and is thus not recovered in the chambers. Principally, this can be solved by modelling diffusion of 15 N labelled gaseous denitrification products (see Sect. 7.2.7).

Pool
An overview of techniques to supply 15 N-labelled NO 3 to the soil is given in Table  7.4. The 15 N GFM is based on the assumption of an isotopically homogenous NO 3 pool. Because this condition is rarely achieved in soils, underestimation of denitrification rates up to 30% can result (Arah 1992;Mulvaney and VandenHeuvel 1988). An initial homogeneity can be obtained by intensive mixing of the soil, but this is a massive disturbance with huge potential effects on N processes including denitrification dynamics and is only adequate to simulate soil tillage with similar disturbance. But even with initially ideal tracer distribution, non-homogeneity inevitably develops over time, since N transformations including nitrification, denitrification and immobilisation are never homogenous in structured soil where aerobic and anaerobic domains coexist and organic matter fractions of varying reactivity are unevenly distributed. Injection of 15 N tracer solution (Wu et al. 2012) increases moisture and inevitably produces non-homogeneity with maximum label concentration at the injection spots. Saturation and drainage (Nõmmik 1956) or soil water displacement by irrigation of lysimeters  leads to an interim increase in moisture and causes loss of DOC. Labelling with gaseous NO 2 was not a suitable way to achieve high and homogenous enrichment of soil NO 3 − (Stark and Hart 1996). Consequently, non-homogeneity of the label distribution is probably the main source of bias of the 15 N GFM. Often 15 N tracer has been applied to the surface similar to conventional fertilisation (Baily et al. 2012). However, in this case, only fertiliser derived fluxes are detected initially, while during ongoing diffusion and leaching of NO 3 − , the 15 N labelled NO 3 − pool rapidly changes its dimensions and thus non-homogeneity complicates the interpretation of results.
Possible causes and consequences of non-homogenous distribution of the 15 Nlabel and denitrification /nitrification dynamics is illustrated using two conceptual models ). The first model shows how a p of N 2 and N 2 O can differ due to non-homogeneity in 15 N enrichment and also non-homogeneity in N 2 and 3 Model 1 to explain why N 2 and N 2 O from denitrification can originate from different effective 15 N pools: In the lower pool with a higher 15 N enrichment, N 2 fluxes dominate over N 2 O, whereas the opposite is the case for the shallow pool with lower enrichment. Hence, emitted N 2 is more enriched compared to emitted N 2 O N 2 O production rates ( Fig. 7.3). Even if equal amounts of 15 N tracer solution could be applied to each soil layer, 15 N enrichment of NO 3 − would be variable due to the different dilution of the label via soil-derived NO 3 − . Additionally, production rates of N 2 and N 2 O and their ratio are typically spatially variable, which results in differing a p values for N 2 and N 2 O (Fig. 7.4). The development of spatial heterogeneity in 15 N enrichment and the consequences arising from the fact that nitrification and denitrification typically occur in different soil niches is shown with the second conceptual model (Fig. 7.4) that had been used to explain observations (Deppe et al. 2017). In that study, the soil had been mixed with 15 N labelled NO 3 − and non-labelled NH 4 + and isotopic values of initial NO 3 − and final NO 3 − and N 2 O had been compared. Results showed that a p of N 2 O was similar to initial enrichment of soil NO 3 − (13 atom% 15 N), but final NO 3 − enrichment of the bulk soil was much lower (about 3 atom% 15 N) whereas a p of N 2 O did not change significantly. This was postulated to result from the dilution of the label only in aerobic domains where nitrification occurred, whereas in anaerobic microsites there was no nitrification, and hence no dilution of the label. But the undiluted microsites produced all or most of the N 2 O whereas there was negligible N 2 O flux from aerobic domains. While this discrepancy between 15 N enrichment of NO 3 − in the bulk soil and a p of N 2 O was certainly extreme in that study, similar process dynamics can be expected in many cases. Such − in NH 4 + -fertilised soil (Deppe et al. 2017). Colours represent enrichment (blue = nat abundance, red = max. 15 N enrichment). a. Initial enrichment of NO 3 − results from mixing of soil NO 3 − and added 15 N-NO 3 − . b. Initial homogenous distribution of labelled NO 3 − and non-labelled NH 4 + in the soil matrix. c. In anaerobic microsites, nitrification is inhibited and the NO 3 − pool of initial 15 N enrichment is denitrified and produces N 2 O of identical enrichment. In aerobic domains, nitrification of non-labelled NH 4 + produces non-labelled NO 3 − , thus diluting the initial labelled NO 3 − pool and emitting unlabelled N 2 O. Note that the 15 N enrichment of NO 3 − undergoing denitrification is larger than the average 15 N enrichment of extracted NO 3 − and of emitted N 2 O non-homogeneity in label distribution and its dilution as well as N 2 and N 2 O production leads to uncertainties in calculation of f p (see Table 7.1). But these examples also show that comparison of a p of N 2 and a p of N 2 O can be used to identify heterogeneity in labelling and thus stress the importance of using analytical methods including 29 R and 30 R of N 2 and N 2 O-N (Lewicka-Szczebak et al. 2013). Moreover, it shows that calculating f p based on 15 N enrichment of bulk NO 3 − from soil extraction (Eq. 7.6) can lead to severe bias, since the 15 N enrichment of the active pool can strongly deviate from the bulk pool. Moreover, an advantage of the non-random distribution approach with N 2 and N 2 O is that non-homogeneity is indicated by discrepancies between a p of N 2 , a p of N 2 O and 15 a NO3 , which is quite useful (Lewicka-Szczebak and Well 2020). But it also shows that hybrid fluxes are difficult to identify if label distribution is non-homogenous.
Further limitations of the 15 N GFM have been reviewed previously (Aulakh et al. 1991;Groffman et al. 2006;Sgouridis et al. 2016). They include enhancement of denitrification by NO 3 − application in unfertilised systems, gas entrapment in very wet or fully water-saturated soils or sediment, and limited residence time of applied 15 NO 3 − -N due to plant uptake and leaching. Table 7.3 Detection limit of the 15 N GFM determined by Monte-Carlo modelling. Detection limit for the fraction of pool derived N 2 (f p of N 2 ) is given as 1 standard deviation (SD) in dependence of 15 N enrichment of active labelled NO 3 -pool (ap) and magnitude of f p in atmospheres with 100% or 2% N 2 and assuming IRMS precision of 29 R and 30 R according to the first instrument in Table 2 Scenario # Ideal f p (ppm)

Combining the 15 N GFM with Modelling of Gross N Transformation
The current model to analyse data for the 15 N GFM cannot be used to solve situations that include multiple labelled pools and heterogeneity of process activity and thus yield variable results in terms of flux quantification. Therefore, more complex models are needed to fill this gap. A 15 N tracing model had been developed to analyse N 2 O dynamics in terrestrial ecosystems, which builds on previous tracing models for the quantification of the main mineral N transformations and soil nitrite (NO 2 − ) dynamics ). This model is thus a first step in taking more complex dynamics into account. Extending this approach to model heterogeneity of processes and pools might be a promising way to solve current limitations of the 15 N GFM. For more information on the tracing technique see Sect. 7.5 of this chapter.

Evaluation of the 15 N GFM
While quantification of N 2 and N 2 O fluxes from distinct N pools remains a challenge after several decades of method development and improvement, this is even more the case for robust evaluation of methods, as this requires that the reference method is quantitative and is applied under the same conditions as the tested method. From that perspective, all previous tests included some uncertainties to our knowledge and were thus not fully able to evaluate the 15 N GFM. There have been several comparisons between 15 N GFM and AIT with controversial results, i.e. reporting general agreement (Aulakh et al. 1991) and severe underestimation by AIT (Arah et al. 1993;Sgouridis et al. 2016). Aulakh et al. (1991) compared 15 N GFM and AIT in the field and found that 15 N fertiliser derived N 2 + N 2 O fluxes were comparable to total N 2 O fluxes in presence of acetylene (C 2 H 2 ), suggesting that both methods were in general agreement. However, in all comparisons, 15 N fertiliser was surface applied, so only the soil volume reached by the fertiliser contributed to the surface flux, unlike the AIT, were a larger soil volume was reached by the gaseous acetylene supplied by perforated pipes or buried calcium carbide. Hence comparisons did not reflect equal parts of the soil profile. Interestingly, in most comparisons denitrification was enhanced by soil compaction or glucose amendment, to achieve detectable 15 N 2 fluxes against the atmospheric N 2 background. Sgouridis et al. (2016) compared closed chamber 15 N GFM using needle injection to distribute K 15 NO 3 − evenly with the AIT "soil core" variant finding 3 to 5 times higher rates with 15 N GFM. Kulkarni et al. (2014) conducted an extensive comparison of the HeO 2 method using small cores (5 cm diameter × 5 cm height, incubated under HeO 2 in the lab) with in situ measurement using the 15 N GFM where KNO 3 − with 99 atom% was sprayed on the soil surface. Authors discussed difficulties to compare measurements in view of O 2 manipulation in the lab and uneven label distribution in the field as well as variable moisture and temperature conditions in the field, and also that there are N 2 fluxes from sources other than NO 3 − (Butterbach-Bahl et al. 2013). What is still needed for a quantitative evaluation of the 15 N GFM is to incubate 15 N-labelled soil in a HeO 2 setup to allow direct comparison of GC-and IRMS based N 2 fluxes.
If 15 N GFM is conducted under conditions maximising sensitivity and minimising bias, it can be used to evaluate other methods as for example the N 2 O isotopocule approach to determine N 2 O reduction , Buchen et al. 2018) (see Sect. 7.3).

Lab and Field Experiments
Initial application of 15 N GFM in lab incubations was carried out in closed vessels (Melin and Nõmmik 1983;Siegel et al. 1982). Recently, some studies used N 2depleted atmosphere to increase sensitivity in soil incubations Schorpp et al. 2016) achieving sensitivities for pool-derived N 2 of approximately 50 ppb which is thus comparable to GC sensitivity for N 2 O and two order of magnitude more sensitive compared to 15 N GFM under ambient atmosphere. Important to note is that this also improves precision for quantifying a p and thus yields more precise estimates for the dilution of the denitrified pool by soil-derived NO 3 − .
A key feature of 15 N GFM is in situ measurement of denitrification and today it must be considered the only available field method, since AIT has been found unsuitable (Felber et al. 2012;Nadeem et al. 2012;Sgouridis et al. 2016). But 15 N GFM has been used far less compared to the AIT probably due to its low sensitivity and high effort and expense to keep high 15 N labelling in the field for extended periods, and also because of the multiple sources of bias. 15 N GFM has thus been primarily used for soil types and/or conditions with high denitrification potential, e.g. due to abundant organic C (e.g. in organic soils or after soil compaction Arah et al. 1993). Typically, experiments covered only certain phases of the year. Maybe the most extensive study (including an extensive review of past in situ measurements) was by Sgouridis et al. (2016) who conducted 15 N GFM in 4 sites monthly during about 18 months. But it has recently been found that during field application of the 15 N GFM, denitrification is severely underestimated because a large fraction of the labelled N 2 and N 2 O produced is not emitted from into the soil surface but diffuses to the subsoil or accumulates in pore space (Well et al. 2019a). This was confirmed experimentally and production-diffusion modelling showed that under typical experimental conditions, denitrification rates would be underestimated by more than 50%. It was concluded that field surface fluxes of 15 N-labelled N 2 and N 2 O have been severely underestimated in the past, but that diffusion modelling can be used to correct data. Moreover, to overcome the poor sensitivity of in situ 15 N GFM, a new procedure was developed to conduct the 15 N gas flux method using artificial N 2 -depleted atmosphere also for field application (Well et al. 2019b), giving a sensitivity for N 2 + N 2 O fluxes up to 80-fold better compared to the conventional 15 N GFM under ambient atmosphere. Consequently, recent methodical improvements are promising to yield good progress in the study of denitrification control at the field scale. 15 N GFM has been used extensively with water saturated cores of aquatic sediments, e.g. Enrich-Prast et al. (2015), where sensitivity is less critical due to the possibility to measure 15 N-labelled N 2 dissolved in pore water where atmospheric N 2 background is small.

In Situ Measurement in Subsoil and Groundwater
Some modifications of the 15 N GFM for subsurface applications had been proposed and applied. For water saturated subsoil of hydromorphic soils or deeper groundwater, the "push-pull" type experimental setup (Istok et al. 1997) was combined with 15 N tracing (Addy et al. 2002;Well et al. 2003;Well and Myrold 1999), where 15 N tracer solution is injected in groundwater wells and groundwater samples are subsequently extracted over time and analysed for 15 N labelled N 2 and N 2 O. Similar to using 15 N GFM in water saturated sediment in the lab (see above, Enrich-Prast et al. 2015), this approach is quite sensitive since produced N 2 mixes with the small N 2 background of N 2 dissolved in groundwater. The 15 N push-pull approach has been compared to slurry incubations of aquifer samples in the lab Well et al. 2005) finding good agreement between both approaches. It has also been successfully applied for deeper groundwater up to 90 m depth .
In the unsaturated zone, subsoil denitrification has been quantified in situ from the steady-state 15 N 2 + 15 N 2 O concentration within a defined 15 N-labelled soil volume (Well and Myrold 2002). Diffusion-reaction modelling has been used to quantify rates by fitting measured and modelled f p values, but accuracy of this approach was limited by the difficulty to quantify the volume of 15 N-labelled soil, its gas diffusivity and its distribution in 15 N enrichment.

Conclusions and Outlook
The 15 N GFM is a powerful approach to quantify soil denitrification and its N 2 O/(N 2 + N 2 O) mole ratio, to distinguish N 2 O fluxes derived from NO 3 − and other N sources and, under certain conditions, also to identify the formation of hybrid N 2 and N 2 O fluxes. It is applicable in the lab as well as in the field. But it is based on a variety of assumptions and prerequisites that are not always easy or possible to validate or to fulfil. Therefore, and because of its high expense for isotope tracers, IRMS analysis and demanding experimental setups, it has until now rarely been used routinely to study denitrification. Moreover, systematic evaluation using independent methods, e.g. using the HeO 2 method, is still pending. Progress has been made in automated IRMS approaches that can be established using commercially available devices with some custom-made modifications. While sensitivity was clearly improved in the lab by incubation under N 2 depleted atmosphere, this has not yet been fully realised for field conditions. These are good reasons to intensify the use of 15 N GFM in future N cycle research, since despite large efforts during preceding decades, the magnitude of denitrification is still the big unknown of the N cycle (Butterbach-Bahl et al. 2013;Müller and Clough 2014 (Toyoda and Yoshida 1999). Natural abundance isotopic signatures can be used as an alternative approach to 15 N tracing to constrain N 2 O transformations in the environment. Variations in stable isotope abundances are due to the fact that for many biotic and abiotic processes, the reaction rates differ between isotopic species, e.g. reduction of 15 NO 2 − versus 14 NO 2 − , leading to a so-called isotopic fractionation. As the isotopic fractionation is distinct for certain reaction pathways, isotopic signatures of particular production pathways and reduction fractionation factors determined in laboratory pure culture studies can be used to differentiate processes from each other. Distinct process information is provided by the difference in 15 N substitution between the central and terminal position within the N 2 O molecule (SP), which is independent of the precursor's isotopic composition and characteristic of specific reaction mechanisms or enzymatic pathways. The most common interpretation strategy used to date is the dual isotope plot, also known as "mapping" approach, presenting the relationship between two isotopic parameters-commonly δ 18 O/ δ 15 N bulk , δ 15 N SP / δ 15 N bulk or δ 15 N SP / δ 18 O. From such figures, estimates can be made about trends, probable dominance of particular pathways, or reduction progress (Toyoda and Yoshida 1999;Lewicka et al. 2017;Koba et al. 2009;Ibraim et al. 2019) (Fig. 7.5). N 2 O isotopocules at natural abundance levels can be analysed by isotope ratio mass spectrometry (IRMS) (Toyoda and Yoshida 1999) and more recently midinfrared (MIR) laser spectroscopic techniques.
With N 2 O isotopic analysis, the qualitative information can be added to the quantitative information gained from the concentration measurements. This is to naturally occurring differences between N 2 O from various origins as a result of isotopic fractionation, which causes enrichment or depletion of the reaction product in heavy isotope. Typically, for biochemical reactions we deal with the product depleted in heavy isotopes, but different biochemical pathways show characteristic isotope fractionation, which results in larger or smaller isotope effects (ε, difference between substrate and product (Eq. 7.7)), including also possible inverse isotope effects (product enriched in heavy isotopes, negative ε).
Isotope effect is often expressed as Δ values, representing the difference between δ values of product and substrate. The values of ε should be used for a particular chemical reaction or physical transformation and describe the characteristic isotopic fractionation for this process (so-called intrinsic isotope effects), whereas Δ values may also be applied to describe an isotopic change between initial substrate and the final product, which may be due to a chain of following reactions and diffusion. This is the case e.g. for denitrification where we can mostly only determine the overall observed isotope effect between NO 3 − and N 2 O (also called apparent or net isotope effect, Δ 15 N bulk N2O/ NO3 − ) but without insight into intermediate products (NO 2 − , NO) we cannot determine the ε values of individual reduction steps.
Due to distinct isotopic fractionation for various biochemical reactions, the N 2 O isotopic studies have been often used to distinguish between different N 2 O production pathways, e.g., nitrification and denitrification (Cardenas et al. 2017;Deppe et al. 2017;Köster et al. 2015;Toyoda et al. 2011;Wolf et al. 2015), or between different microorganisms involved in N 2 O production, e.g. fungal and bacterial denitrification (Kato et al. 2013;Schorpp et al. 2016;Zou et al. 2014). Moreover, also N 2 O reduction can be potentially monitored by N 2 O isotopic data. The possible reduction of N 2 O to N 2 during denitrification is associated with isotopic fractionation, which changes the isotopic signature of the residual N 2 O. Therefore, isotopic analyses of residual N 2 O can be used to estimate the magnitude of its reduction and thereby the N 2 production (Kato et al. 2013;Lewicka-Szczebak et al. 2017;Toyoda et al. 2011). Comprehensive reviews on the use of N 2 O isotopocules to estimate N 2 O dynamics are given by Ostrom and Ostrom (2011), Decock and Six (2013), Toyoda et al. (2017) and Yu et al. (2020). The main problem in the interpretation of isotopocule analysis of emitted N 2 O is the parallel production, possibly from various pathways, and consumption due to reduction to N 2 .

Principles
For a proper interpretation of the analysed isotopic values of emitted N 2 O, both the possible production pathways and consumption due to N 2 O reduction to N 2 must be taken into account.
To be able to identify potential production pathways, we need the basic data of the characteristic isotopic signatures for particular pathways, so called endmember values. These are obtained from the pure culture studies, where specific microorganisms are incubated separately and N 2 O is collected and analysed. Numerous pure culture studies are summarised in detail in the recent review papers (Denk et al. 2017;Toyoda et al. 2017). N 2 O isotopic signatures for specific pathways were also determined in controlled incubation of the whole soil by applying conditions favouring specific pathways. Such experiments were also summarised before (Denk et al. 2017;Toyoda et al. 2017). Here we present an overview of the most common pathways including results from pure culture studies and controlled soil incubations with some necessary critical selection explained below (after (Denk et al. 2017;Lewicka-Szczebak et al. 2017;Toyoda et al. 2017). For each isotopic signature (δ 15 N sp , δ 18 O, and δ 15 N bulk ) the rules how to properly use endmember values are explained and for each N 2 O production process the range of values (minimal and maximal literature reported value), the mean (of all literature reported values) and the median (of all literature reported values) is given.
δ 15 N bulk of the produced N 2 O depends on the precursor isotopic signature, i.e. on soil NO 3 − for denitrification and soil ammonium for nitrification. Therefore, to compare any results with literature endmember values we need to calculate the N isotopic signature of the N 2 O in relation to the precursor, i.e. Δ 15 N bulk N2O/ NO3− for denitrification and Δ 15 N bulk NH4+ for nitrification. Some pure culture denitrification studies also reported the isotope effect between nitrite and N 2 O (Δ 15 N bulk N2O/ NO3− ), especially for fungal denitrification, but for field studies, we usually analyse soil NO 3 − . By calculating isotope effects between N 2 O and N precursors one should be aware that the reaction progress changes the isotopic signature of the precursor: the more substrate is consumed, the more 15 N enriched gets its residual pool. Therefore, the precursor N isotopic signature at the beginning and at the end of an experiment may differ depending on the reaction progress. Moreover, the δ 15 N of the measurable bulk N pools (by soil extraction) may deviate from the δ 15 N of the active N 2 O producing pools if the fractionating processes are heterogeneously distributed. This is especially the case in unsaturated soils where NO 3 − in anoxic microsites is denitrified and thus progressively enriched in 15 N, while in aerobic domains nitrification adds NO 3 − at a lower 15 N bulk NO 3− enrichment. Substantial deviation between bulk soil and active pool enrichment has been recently shown in tracer studies in the laboratory (Deppe et al. 2017) and in the field (Buchen et al. 2016). This indicates that the interpretation based on δ 15 N bulk values is very complex and requires a good understanding of N transformation processes in the soil (see also Sect. 7.5).
δ 15 N sp of the produced N 2 O is independent of the precursor isotopic signature. Hence, unlike δ 15 N bulk , the endmember values are identical in δ 15 N sp of the produced N 2 O. Therefore, the measured N 2 O δ 15 N sp values can be directly compared with the following endmember values: • heterotrophic bacterial denitrification: determined in pure culture studies from − 7.5 to +3.7‰, mean −1.9‰, median −1.9‰ (Sutka et al. 2006;Toyoda et al. 2005). The values obtained in the controlled soil incubation experiments targeted for bacterial denitrification from −4.7 to +1.7‰ fit within the range given by pure culture studies (Lewicka-Szczebak et al. 2014); • nitrifier denitrification: from −13.6 to +1.9‰, mean −5.9‰, median −5.9‰ (Frame and Casciotti 2010;Sutka et al. 2006); • fungal denitrification: from 27.2 to 39.9‰, mean 33.5‰, median 33.6‰ (Maeda et al. 2015;Rohe et al. 2014Rohe et al. , 2017Sutka et al. 2008). A recent study indicated also a lower δ 15 N sp value for one individual fungal species, which was disregarded here due to its very low N 2 O production: C. funicola showed δ 15 N sp of 21.9‰ but less than 100 times lower N 2 O production with nitrite compared to other species, and no N 2 O production with NO 3 − (Rohe et al. 2014). Similarly, from the study of Maeda et al. (2015), only the values of strains with higher N 2 O production were accepted for this summary (>10 mg N 2 O-N g −1 biomass).
• nitrification: from 32.0 to 38.7‰, mean 35.0‰, median 34.6‰ (Frame and Casciotti 2010;Heil et al. 2014;Sutka et al. 2006 (Kool et al. 2007). The extent of this exchange differs for various bacterial and fungal species (Rohe et al. 2017), but it has been shown recently that for soil incubations it is rather high  The most common way of identifying various N 2 O producing pathways is a graphical presentation of the measured values together with the literature endmember values. From the graphs, we can often identify the dominant pathway.
To obtain more precise quantitative information, the contribution of a pathway (A) can be calculated based on the measured N 2 O isotopic signature (δ N2O ) using the isotope mass balance: It must be noted that for this calculation (Eq. 7.8), the δ N2O value may not be changed due to N 2 O reduction. This is only fulfilled if reduction is inhibited, measured to be negligible or included in calculations as described below. Using one isotope signature (δ 15 N bulk , δ 15 N sp or δ 18 O), we are able to determine the mixing ratios of two pathways. Applying more isotopic signatures can theoretically enable quantification of more pathways. However, the results are not very exact due to the sometimes wide ranges of possible isotopic values for different pathways and overlapping of these ranges for more pathways. For both, δ 15 N sp and δ 18 O, the ranges for heterotrophic bacterial denitrification and nitrifier denitrification. Additional interpretation of δ 15 N bulk can further help but is often problematic due to lacking information on precursor isotope values (Lewicka-Szczebak and Well 2020). To increase precision of such calculations, controlled soil incubations with the soil under study may help to determine more narrow ranges of endmember values characteristic for the particular soil .
But besides the mixing processes also isotopic fractionation during N 2 O reduction changes the final isotopic value of the residual N 2 O. During N 2 O reduction to N 2 (the last step of bacterial denitrification) preferentially the N-O bonds between light isotopes ( 14 N and 16 O) are broken and as a result the residual unreduced N 2 O is enriched in 15 N α and 18 O. In consequence, δ 15 N sp , δ 18 O and δ 15 N bulk values of residual N 2 O increase with progressing reduction. The magnitude of the shift towards higher values depends on the amount of reduced N 2 O and the isotopic fractionation factor associated with the N 2 O reduction. Hence, if we know the fractionation factor and the δ value of initially produced N 2 O before reduction (δ 0 ), we can calculate the amount of reduced N 2 O and thereby determine the magnitude of N 2 flux based on the measured δ value of the residual N 2 O after reduction (δ r ). This is calculated according to the following isotopic fractionation Eqs. 7.9 to 7.11 by applying Rayleigh model that is valid for closed systems, either in its exact form (Mariotti et al. 1981): (7.9) or in simplified, approximated form: where δ r is the residual N 2 O isotopic signature, after reduction, δ o is the initial N 2 O isotopic signature, before reduction, ε N2-N2O is the isotopic fractionation factor associated with N 2 O reduction and r N2O is the residual unreduced N 2 O fraction (r N2O = y N2O /(y N2 + y N2O ); (y: mole fraction)) The application of the closed system model has been confirmed by several studies (Köster et al. 2015;Lewicka-Szczebak et al. 2017. However, it was also suggested that an isotopic fractionation model for open systems could be suitable (Decock and Six 2013), which is associated with smaller apparent isotope effects during N 2 O reduction: To be able to determine r N2O from N 2 O isotopic values of individual samples according to the above equations, isotopic fractionation factor associated with N 2 O reduction to N 2 (ε N2-N2O ) must be known. They were determined in numerous studies in controlled soil incubations (Jinuntuya-Nortman et al. 2008;Lewicka-Szczebak et al. 2014;Menyailo and Hungate 2006;Ostrom et al. 2007;Well and Flessa 2009) and the following ranges were obtained: In the summary, we disregarded one study which provided an inverse isotope effect for ε 15 N bulk N2-N2O and ε 18 O N2-N2O . These values might have been a result of untypical reduction conditions in the experiment or an experimental artefact (Denk et al. 2017), therefore, they are neglected here. From the study of Lewicka-Szczebak et al. (2015) only the data of moderate reduction (from Pool1) were summarised here, because it was shown that by very intensive reduction the results can be strongly affected by N 2 O diffusion. This depends on the balance between diffusive and enzymatic fractionation during N 2 O reduction ). By nearly complete N 2 O reduction, we observe a relatively large impact of diffusive N 2 O fractionation, resulting in residual N 2 O more depleted in heavy isotopes, hence the apparent isotope effects are significantly lower, i.e. −2.7‰, −1.5‰, and −2.0‰ for ε 15 N bulk N2-N2O , ε 15 N sp N2-N2O , and ε 18 O N2-N2O , respectively .
It is often problematic to separate the impact on the final N 2 O isotopic values by the mixing endmember for the produced N 2 O and by the isotopic fractionation due to N 2 O reduction. The interpretations and calculations based on N 2 O isotopic studies are difficult when we deal with the simultaneous variations in r N2O and δ 0 values. Usually, to calculate r N2O a stable δ 0 is assumed  and to precisely determine temporal changes in δ 0 , we need independent data on r N2O (Köster et al. 2015). In field studies, both r N2O and δ 0 cannot be determined precisely, but the possible ranges for each parameter can be given (Zou et al. 2014).
It is often attempted to distinguish between mixing and fractionation processes by using the changes in the isotopic signatures and their relations: Although the range of possible ε N2-N2O variations is quite large, it has been shown recently that the mean values and typical ε 15 N sp N2-N2O / ε 18 O N2-N2O ratios are well applicable for oxic or anoxic conditions unless N 2 O reduction is almost complete, i.e. the ratio N 2 O/(N 2 + N 2 O) < 0.1, meaning more than 90% of N 2 O was reduced .
For comparison, here are the relations between isotopic signatures of emitted N 2 O resulting from mixing processes calculated based on literature ranges for mixing endmembers given above. Because of the overlapping endmember ranges, we cannot distinguish between all individual pathways, and we determine the slopes of mixing lines between selected endmember values (Figs. 7.5 and 7.6) as follows: • mixing between heterotrophic bacterial denitrification and nitrification: Fungal denitrification cannot be distinguished from relations including δ 15 N bulk because of the overlapping range with bacterial denitrification (see Fig. 7.5). Anyway, relations including δ 15 N bulk are difficult to use due to dependence of this isotope value on the precursor, which differ for nitrification and denitrification. Here   Fig. 7.5). Often the isotopic signatures of the precursors are not known, which make the interpretation of δ 15 N bulk values rather ambiguous. Nevertheless, some studies apply the δ 15 N sp / δ 15 N bulk isotope maps for distinction of mixing and fractionation processes, but for such isotope maps, systematic changes in δ 15 N bulk induced by systematic changes in the N isotopic composition of one of the precursors NH 4 + or NO 3 − could be misinterpreted as reduction events Wolf et al. 2015). Hence, the careful monitoring of precursor isotopic signatures is needed (Zou et al. 2014).
A δ 15 N sp / δ 15 N bulk isotope mapping approach allowing for assessment of minimal and maximal reduced N 2 O fraction and nitrification and denitrification mixing ratios was proposed by Zou et al. (2014) (Fig. 7.5). Such an approach is most often used for distinguishing between nitrification and bacterial denitrification only. However, other cases have also been analysed (Zou et al. 2014). The calculation method presented ( Fig. 7.5) assumes first mixing of N 2 O from different endmembers and afterwards its partial reduction. Two mixing lines are defined-for the minimum and maximum values for both endmembers as well as two reduction lines-with maximal and minimal slope. From the intercept 1 the maximal denitrification contribution is determined whereas from the intercept 2 the minimal one. Based on the difference between the sample point and intercept 1 or 2 the reduction contribution, respectively, maximal and minimal, is also determined. However, it must be noted that in case of significant admixture of fungal denitrification or nitrifier denitrification the results may be biased.
The application of δ 15 N sp / δ 18 O isotope mapping approach may be easier since Isotopic values of the samples analysed are typically located between these two, reduction and mixing, lines. Here we defined only one mixing line for the median values of bacterial and fungal denitrification and one reduction line with a mean slope. From sample's, location, we can estimate the impact of fractionation associated with N 2 O reduction and admixture of N 2 O originating from fungal denitrification. We can deal with two scenarios: While both scenarios yield identical results for the admixture of N 2 O from fungal denitrification, the resulting reduction shift, and hence the calculated r N2O value, is higher when using Scenario 2. It is still not clear which scenario is more realistic. The uncertainty analysis of this method has been recently presented by Wu et al. (2019) and this approach has been successfully applied in the field case studies (Buchen et al. 2018;Ibraim et al. 2019;Verhoeven et al. 2019). However, after the appearance of those publications, it has been found that other δ 18 O values should be applied for nitrification . This summary reports the most current choice of endmember ranges, which differ from those presented recently (Buchen et al. 2018;Ibraim et al. 2019;Lewicka-Szczebak et al. 2017;Verhoeven et al. 2019;Wu et al. 2019).

Analysis of N 2 O Isotopocules by IRMS
The most common method for N 2 O isotopocule analysis is isotope ratio mass spectrometry (IRMS). In order to perform N 2 O isotopic analysis the gas samples need to be purified, and N 2 O must be separated and pre-concentrated. First, water and CO 2 are removed by chemical traps, and then N 2 O is concentrated with liquid N traps. Afterwards, the gases are separated with gas chromatography and finally introduced in the isotope ratio mass spectrometer.
In the mass spectrometer, N 2 O isotopocule values are determined by measuring m/ z 44, 45 and 46 of the intact N 2 O + ions as well as m/ z 30 and 31 of NO + fragment ions. This allows the determination of average δ 15 N (δ 15 N bulk ), δ 15 N α (δ 15 N of the central N position of the N 2 O molecule), and δ 18 O (Toyoda and Yoshida 1999). δ 15 N β (δ 15 N of the peripheral N position of the N 2 O molecule) is calculated from δ 15 N bulk = (δ 15 N α + δ 15 N β ) / 2 and 15 N site preference (δ 15 N sp ) from δ 15 N sp = δ 15 N α -δ 15 N β . Since the IRMS approach was developed simultaneously by two groups (Brenninkmeijer and Röckmann 1999;Toyoda and Yoshida 1999), two different nomenclatures had been introduced for the two positions of N 2 O-N. Hence, in some studies, the peripheral (β) position is referred to as 1-and the central (α) as 2-position (Brenninkmeijer and Röckmann 1999). The scrambling factor resulting from the exchange of 15 N atoms on the ion source must be taken into account. The magnitude of the scrambling factor should be determined individually for each mass spectrometer (Röckmann et al. 2003). Also, 17 O-correction should be taken into account, because 17 O substitution is indistinguishable from 15 N, therefore typical terrestrial 17 O content (0.528) is assumed (Kaiser and Röckmann 2008).
Up to now, there are still no internationally agreed gaseous N 2 O reference materials for N 2 O isotopocule analyses. Usually, the laboratories calibrated pure N 2 O gas for isotopocule analyses in the laboratory of the Tokyo Institute of Technology according to the method of Toyoda and Yoshida (1999). Recently, the first interlaboratory comparison has been performed and now the standards from this study (REF1, REF2) are available for the laboratories and allow the performing of two-point calibration for δ 15 N sp values ). This intercalibration study has shown that the two-point calibration method is necessary to obtain accurate δ 15 N sp values. Recently, two N 2 O standards had been tested in a further interlaboratory comparison (Ostrom et al. 2018) and is available from United States Geological Survey (USGS).
The sample volume needed for the N 2 O isotopocule depends on the concentration and is about 100 ml for ambient N 2 O concentration samples (about 300 ppb) and about 10 ml for N 2 O concentration of above two ppm.

Laser Spectroscopic Analysis of N 2 O Isotopomers to Differentiate Pathways
The invention and availability of non-cryogenic light sources in the mid-infrared (MIR) spectral range (Brewer et al. 2019) coupled with different detection schemes such as direct absorption quantum cascade laser absorption spectroscopy (QCLAS) (Mohn et al. 2010, Mohn et al. 2012, Wächter et al. 2008, cavity ring down spectroscopy (CRDS) (Erler et al. 2015) and off-axis integrated-cavity-output spectroscopy (OA-ICOS) (Wassenaar et al. 2018) has provided sensitive and fielddeployable laser spectroscopic analysers for N 2 O isotopocule analysis. These instruments can analyse the N 2 O isotopic composition in gaseous mixtures (e.g. ambient air) in a flow-through mode, providing real-time data with minimal or no sample pre-treatment, which is highly attractive to better resolve the temporal complexity of N 2 O production and consumption processes. Most importantly, MIR laser spectroscopy is selective for 17 O, 18 O and position-specific 15 N substitution due to the existence of characteristic rotational-vibrational spectra (Gordon et al. 2017). Therefore, laser spectroscopy has the potential to open a new field of research in the N 2 O biogeochemical cycle, but, applications remain challenging and are still scarce for the following main reasons: (1) laser spectrometers as any analytical instrument are subject to drift effects, in particular under fluctuating environmental conditions, limiting their performance (Werle et al. 1993); (2) changes in N 2 O concentration affect N 2 O isotope results when using the δ-calibration approach (Griffith 2018); (3) laser spectroscopic results are affected by mole fraction changes of atmospheric background gases (N 2 , O 2 , and Ar), called gas matrix effects, due to the difference of pressure-broadening coefficients, and potentially by spectral interferences from other atmospheric constituents (H 2 O, CO 2 , CH 4 , and CO, etc.), called trace gas effects, depending on the wavelength region used in an instrument. Spectral interferences are particularly pronounced for N 2 O due to its low atmospheric abundance in comparison to other trace gases; (4) only since recently two pure N 2 O isotopocule reference materials (USGS51, USGS52) have been made available through the United States Geological Survey (USGS) (Ostrom et al. 2018), which was identified as a major reason limiting interlaboratory compatibility (Köster et al. 2013;Mohn et al. 2014Mohn et al. , 2016. In a recent study, the most common commercially available N 2 O isotope laser spectrometers were carefully characterised for their dependence on N 2 O concentration, gas matrix composition (O 2 , Ar) and spectral interferences caused by H 2 O, CO 2 , CH 4 and CO to develop analyser-specific correction functions. In addition, the authors suggest a step-by-step workflow that should be followed (Fig. 7.7) by researchers to acquire trustworthy N 2 O isotopocule results using laser spectroscopy ).

Introduction
As an example, the Picarro G5101-i analyser can be used to determine N 2 O concentration, 15 N bulk isotope ratios and isotopomer values ( 15 N α and 15 N β ) by continuous or discrete sample measurement. Small volume discrete samples (≤20 ml) can be measured using the SSIM (small sample isotope module) (see also Sect. 5.3.) peripheral unit in conjunction with the Picarro G5101-i analyser. The G5101-i analyzer is the predecessor of the current G5131-i analyzer which also measures δ 18 O in addition to δ 15 N bulk , δ 15 N α and δ 15 N β . The SSIM can also be used to dilute samples. Larger volume samples (e.g. Tedlar bags) can be measured by direct input into the G5101-i analyser or through the 16-port distribution manifold. The 16-port distribution manifold allows for partial automation of measurement and can be used in conjunction with the SSIM for smaller volume samples (see also Fig. 5.5 that illustrates the coupling of a 16-port manifold and a SSIM). The SSIM can also be used to dilute samples.

Principle
Samples are measured using mid-IR laser by CRDS (cavity ring down spectroscopy). Measurement precision increases with measurement time. Several options are available for delivery of N 2 O samples into the analyser and how long measurements take. Sample volume and the required precision of measurements should be considered to decide which operational set up is the most appropriate.
• N 2 O working standards.
• Septa capped vials for discrete gas samples.
• Tedlar bags for larger volume gas samples.
• Side port needles for sample injection to SSIM.

Sampling
For discrete gas samples follow a suitable sampling procedure as outlined by De Klein and Harvey (2012). Small volume samples (≤20 ml) should be stored in septum capped vials, ensuring to overpressure when filling to prevent inward contamination by ambient air. Vials can be stored in a cool dry place. Larger volume samples in Tedlar bags should be measured ASAP as storage reliability decreases greatly after 24-48 h.

Operational Procedure
• To start the analyser, ensure the power switches are on for the pump, analyser and monitor. Turn the power switch at the rear of the analyser from O to I. NB: the power switch on the pump should always be in the on position, the pump will power up when the analyser is turned on. To turn on the analyser press the button on its front. Windows will load on the monitor and the analyser software will run through the system checks. • When the analyzer is in startup mode, monitor the liquid coolant at the back of the analyzer. You should observe little to no bubbles and the fluid should be flowing. If the bubbles have not disappeared after a few minutes or the liquid is not flowing, refer to the troubleshooting section in this document. • After the system checks are complete, the GUI (Graphical User Interface) will appear. It will begin by measuring the Cavity Pressure, DAS (Data Acquisition System, i.e. the analyser) temperature and Etalon temperature. Once the correct temperatures and pressures are reached a message will appear on the bottom of the GUI screen; e.g. "Pressure locked", "Cool Box Temperature locked", "Preparing to measure", Measuring…". The GUI will then begin to show the continuous N 2 O measurements in real time. It may take up to 1 h for the analyser to begin N 2 O measurements. Before measuring samples, allow the laser to stabilise for up to 24 h by measuring room air. • Continuous samples (e.g. incubation experiments) can be measured by directly connecting a piece of tubing from the sampling container to the inlet at the back of the analyser and segmenting the data into the respective time periods.
Discrete samples (≤20 ml) • To measure small volume discrete samples (≤20 ml) allow the laser to run continuously for 24 h to ensure that the laser has been given sufficient time to stabilise.
• Before operating the SSIM check that the "Valve Seqeuncer MPV" is turned off.
To do this click Shutdown on the GUI and select "Software Only". Double click on the Picarro Utilities icon located on the desktop and double click on "Setup Tool". Under the "Port Manager" tab check that the "Valve Sequencer MPV" is turned off. If necessary change this setting to off and close the Picarro Utilities folder. • Restart the GUI software by double clicking on the Picarro Switcher Mode icon located on the desktop and select the Isotopic N 2 O option followed by clicking Launch. • In the standard GUI mode, the H 2 O parameter is not available from the Data Key drop-down menus. This is necessary to check for pressure leaks in the system. To access this, log into the service GUI mode under the settings tab of the GUI. The password is "picarro". line to the SSIM and adjust slowly to 1.5 bar. This will rise to 2 bar when it meets the resistance of the SSIM. • The second pressure regulator has been set to 3 psi (following Picarro's recommendations), check to ensure this is the case and only adjust if necessary. Never allow the final pressure into the SSIM to go above 8 psi. The indicator on the valve may flicker during operation due to valve switching within the SSIM. • Connect the stainless steel tubing from the SSIM to the analyser. Finger tighten and then apply a ¼ turn using the adjustable spanner. • Connect the grey (valve switching controls) and black (pressure detector) cables from the analyser to the SSIM. This will power on the SSIM, indicated by the green light on the front of the unit. • NB Turn on the SSIM vacuum pump. This must be done before launching the SSIM software. • Launch the SSIM software by double clicking on the "SSIM Pressure Detector" icon. This locates the COM port (COM 7) of the analyser that the SSIM is connected to. Leave this software window open while using the SSIM. • Measure the Zero Grade Air only for 30 min before beginning sample analysis. This is to obtain the average N 2 O concentration, 15 N bulk , 15 N α and 15 N β of the Zero Grade Air, necessary for correcting concentration dilution and isotope mixing. • Double click on the "SSIM Coordinator" from the desktop and select G5101-i.
Configure the settings to suit the measurement procedure required. There are nine parameters (1-9) to be set. • Click OK. Select G5101-i for reference standard.
• SSIM pressure measurements should be available in the GUI data key dropdown tabs. Select this parameter to monitor SSIM pressure visually on the left side of the GUI.
Note: Under vacuum, the SSIM pressure should be~8 Torr or below. When a sample is injected the max pressure is reached upon filling the cavity with sample/ ZA. The max pressure should read between 980 and 1000 Torr. If the pressure is too high down-regulate the second pressure regulator. If the pressure is too low up-regulate the second pressure regulator being very careful not to exceed 8 psi.
• Overlay the SSIM Coordinator screen on to the bottom right corner of the GUI screen. This allows the user to monitor the parameters on the left side of the GUI while following the prompts of the SSIM Coordinator. • Follow the steps indicated on the SSIM Coordinator screen to process each sample. • The first prompt requires the operator to inject the sample syringe with the valve closed and to click "Resume" under "Control". The SSIM coordinator will then run through several valve sequencing steps. • The next prompt to the operator is to open the syringe valve and to click "Resume" under "Control". The SSIM coordinator will then run through several valve sequencing steps. • The operator will then be prompted to inject the sample. The sample will begin to draw itself in but the operator may be required to manually complete the injection depending on the sample volume. Once the sample is fully injected, close the syringe valve. Allow the SSIM pressure reading to settle and record this pressure value followed by clicking "Resume" under "Control". NB-Always manually record the SSIM pressure as it settles after sample injection, and record the max SSIM pressure when the ZA dilution is carried out. This is used to work out the actual volume of a sample using the pressure vs volume calibration curve. • The SSIM coordinator will then begin the dilution process. NB-watch the SSIM pressure readings and record the maximum pressure reached during the dilution step.
• The SSIM will then begin the sample measurement. At this stage, the syringe can be removed from the injector nut to prepare for the next sample injection. • Before each measurement day, complete a pressure vs volume calibration curve.
Use room air injected at the following volumes: 0 ml, 5 ml, 10 ml, 15 ml and 20 ml. To complete the 0 ml point do not inject the syringe, instead allow the zero air to fill 20 ml (cavity volume) into the SSIM.
Note: The calibration curve should be almost perfectly linear with a R 2 = 0.99 +. Deviations from the curve or lower R 2 values may indicate a leak. Check the injector nut and septum, change septum if necessary. Check the ZA line connections from the SSIM unit to the analyser. Tighten loose connections if necessary by finger tightening + ¼ turn with an adjustable spanner. Never over tighten as this can lead to leaks.
• To check the instrument precision and to avoid measurement drift, it is recommended that a room air/zero blank is run after every 10 samples. A reference standard or working standard may also be used if available. • To discontinue SSIM use and return to continuous measurement reverse the order of the SSIM setup steps. Close the SSIM Coordinator window. (Note: a system alarm will appear on the GUI, this is normal) Close the SSIM pressure detector window. Turn off the SSIM vacuum pump. Disconnect the grey and black cables from the SSIM. Disconnect the stainless steel tubing from the SSIM output. Close the black valve on the first pressure regulator at the lab bench and close this regulator by turning in the decrease direction. Close the valve and the ZA cylinder in the cylinder cabinet.

Note:
A system alarm will probably appear on the top left of the GUI and a message stating "Pressure unlocked". This results from the SSIM being disconnected. To resolve: click Shutdown and select "Stop Analyser Software Only". Wait a couple of minutes and relaunch the analyser software by double clicking on the Picarro Switcher Mode icon on the desktop and selecting G5101-i Isotopic N 2 O and click launch. Monitor the system as it relaunches and wait until it begins measuring N 2 O parameters.
• To turn off the instrument completely click shutdown on the GUI. NB: Never leave the analyser measuring ZA overnight, this will lead to drift.

Expression of Results
• N 2 O concentration is expressed as ppb.

Quality Assurance
• Prior to taking gas samples in the field (i.e. from static chamber) ensure vials are properly sealed and that they have been flushed and evacuated three times.
• Ensure samples are injected with slight overpressure (e.g. 20 ml into 12 ml vial) to avoid inward contamination that would dilute the sample concentration. • Ensure samples are stored in a cool dry place. Process samples as quickly as possible. Vials lose pressure over time. Avoid storing in direct sunlight. • The laser should be given sufficient time to stabilise. 24 h is recommended prior to measuring samples. • Before each measurement day, complete a pressure vs volume calibration curve as described above. Check for leaks based on any variation detected. • Ensure the septum in the injector nut is replaced approximately every 100 injections. • Use side bore needles to reduce the damage caused to the septum.
• For acceptable precision and accuracy ensure that sample concentrations are within the stated operating range of the analyser (300-1500 ppb N 2 O). • Minimise moisture (H 2 O) in samples. Use drying tubes to introduce samples to the analyser if necessary. • Use a gas-tight syringe to inject discrete samples into the SSIM. If using a plastic syringe and with a three-way valve, replace when necessary due to wear and tear. • Never leave the analyser measuring ZA overnight. This will cause measurement drift.

Reporting of Results
Raw data files are automatically generated by the analyser and are stored on the instrument's computer as a DataLog_User file. These raw data files can be found by following the file path: C:\UserData\DataLog_User\Year\Month\Day. An example of the file naming convention is JBDS5030-20170331-140739Z-DataLog_User. JBDS5030 refers to the instrument serial number. 20170331 is the Year, Month and Date the file was started. 140739 is the Hour, Minute and Second of when the file was started. There are a number of values available for the N 2 O parameters measured. The dry corrected values are the appropriate values to select for analysis. When measuring discrete samples using the SSIM there is sufficient time between samples to record the real-time values on a separate spreadsheet that has been premade with sample reference numbers included.

Safety
• When using syringes and needles for sampling and analysis, take extra care to avoid needle stick injuries. • Regularly check the pressure reading of the instrument and the pressure regulators on the ZA line. • Never handle pressurised gas cylinders without the appropriate safety training and certification. • If moving the instrument, always ensure it is shut down so that the cavity returns to ambient pressure and does not remain under vacuum. • There are a number of valve sequences during operation of the SSIM. Ensure to follow the prompts carefully to avoid loss of sample or pressure build ups.
• Read and follow the information in the Risk Assessments for the Stable Isotope Analysis lab.

Trouble Shooting
• Start-up: If the chiller line contains large air bubbles this may stop the circulation of water in the line. This can lead to the baseplate temperature being exceeded which causes the analyser to enter safe mode (error message appears in GUI). This problem should be avoided by keeping the cooling agent LIQ-702 (propylene glycol) in the buffer tank (externally mounted on the chiller cover) topped up to 90% of its full volume with deionised water. To do this unscrew the black cover and use a wash bottle to add in fresh deionised water. This can be done while the analyser is running. If the error message does appear this may require the instrument to be shut down and for the chiller line be flushed following the instructions provided in the installation manual for the installation of the water buffer tank.

Accuracy, Precision and Bias
The analytical precision for IRMS measurements determined as standard deviation (1σ ) of the internal standards for measurements of δ 15 N bulk , δ 18 O and δ 15 N sp is typically 0.1, 0.1 and 0.5‰, respectively. Commercially available laser spectrometers at ambient N 2 O concentrations offer a precision of 0.2 to 1 ‰ for δ 15 N α , δ 15 N β and δ 18 O, which can be reduced to 0.1 ‰ at higher concentrations, or by using a preconcentration device. However, from the inter-comparison study, we see that the bias may be much larger, up to: for δ 15 N bulk 0.8 and 2.8‰, and for δ 15 N sp 4.3 and 3.7‰ for mass spectrometry and for laser spectroscopy, respectively . But these potentially large errors can be minimised by a proper data calibration using two points standardisation with the reference gases that bracket the measurement range. Care must be also given when samples with high concentrations are diluted as the dilution matrix (typically Helium or N 2 ) may apparently have an impact on the final result. The rule of identical treatment of standards and samples should be held, including identical dilution matrix and similar concentration range ). Possible bias is also associated with calculations applied for data interpretation. Due to large ranges of literature data, the N 2 O source partitioning cannot be done precisely, but rather the ranges of possible results can be given. To increase precision of such methods controlled soil incubation can be applied to determine the soil specific endmember isotopic values or fractionation factors (Lewicka-Szczebak et al. 2017). Köster et al. (2015) This experiment applied an N 2 O isotopocule approach combined with conventional N 2 O and N 2 flux measurements to study microbial pathways after different organic fertiliser applications. The direct determination of emitted N 2 was used to take isotope effects during N 2 O reduction to N 2 into account. The measured isotope signatures were corrected for isotope effects during N 2 O reduction with Eq. 7.10 using previously determined fractionation factor ranges. Based on the corrected values the isotope mass balance equations (Eq. 7.8) for δ 15 N sp and δ 18 O were applied. The ranges for different pathways contribution were given for δ 15 N sp -and δ 18 Obased results and the common area for both was accepted as most probable. Two mixing scenarios were considered: bacterial denitrification and nitrification or bacterial and fungal denitrification. Although the range of possible results for endmembers contribution varied up to 30%, a clear increase in nitrification contribution with the incubation time has been documented. Schorpp et al. (2016) In this experiment, incubations with soil fauna were applied to check the impact on N 2 O and N 2 emission of anecic earthworms and euedaphic collembola. Isotopocule approach was applied together with 15 N tracing. Interpretation of the isotopocule results based on the δ 18 O-δ 15 N sp isotope map, similar as presented in Fig. 7.6, including three possible mixing endmembers: bacterial and fungal denitrification and nitrification (hydroxylamine oxidation) and taking N 2 O reduction into account. Isotope data allowed concluding that the presence of collembolans shifted the process pathways towards bacterial denitrification although no change in N 2 O concentration could be noted. Deppe et al. (2017) In this incubation experiment high NH 4 + concentrations in soil were established to check the supposed inhibition of nitrification. An isotopocule approach, together with 15 N tracing and acetylene inhibition approach, was applied to gain insight into N 2 O production processes. Interpretation of the isotopocule results based on the δ 18 O-δ 15 N sp isotope map, similar as presented in Fig. 7.6, including two mixing endmembers: denitrification and nitrification (hydroxylamine oxidation) and N 2 O reduction. This assumption of the mixing conditions appeared incorrect, since some data were located outside of the mixing and reduction lines. This indicated a substantial contribution of nitrifier denitrification and/or coupled nitrification-denitrification (10-40%) to total N 2 O production. Cardenas et al. (2017) Laboratory incubation was carried out at different saturation levels for a grassland soil and emissions of N 2 O and N 2 were measured as well as the N 2 O isotopocules. Thanks to direct measurements of N 2 flux, the extent of N 2 O reduction was known.

Examples of Laboratory Applications
Hence, the measured δ values were mathematically corrected to obtain the δ values of the produced N 2 O before reduction applying Eq. 7.9. An endmember mixing model (Eq. 7.8) was then used to calculate the percentage of bacterial N 2 O in the total N 2 O flux based on δ 15 N sp and δ 18 O. To assess the uncertainty of this approach the ranges of possible endmembers isotopic signatures and reduction fractionation factors were taken into account. The variations of the bacterial N 2 O contribution due to assumed ranges of input values reached up to 40%. But still it allowed to distinguish the dominant pathways for different water saturation levels and indicated that only when the micropores become partially dry, the more aerobic soil conditions allow a higher contribution of nitrification. The dryer conditions in soil macropores did not result in significant changes in bacterial denitrification contribution. Toyoda et al. (2011) N 2 O emitted from agricultural soils planted with rice, wheat, soybean, and vegetables, and treated with synthetic (urea or ammonium) and organic (poultry manure) fertilisers was analysed. The observed isotopic values for Δ 15 N and δ 15 N sp were compared with literature endmembers of nitrifying and denitrifying bacteria. A characteristic relationship between δ 15 N bulk and δ 15 N sp during N 2 O reduction by denitrifying bacteria was used to quantify N 2 O reduction. The relative fraction of N 2 O derived from nitrification and the approximate progress of N 2 O reduction were calculated by a Monte Carlo method. Different scenarios for pairs of mixing endmembers were tested (nitrification and denitrification; nitrification and nitrifier-denitrification; fungal denitrification and denitrification; fungal denitrification versus nitrifier-denitrification) but due to overlapping ranges for δ 15 N sp values it was chosen to consider only the mixing between bacterial nitrification and denitrification. It was found that the contribution from nitrification was relatively high (40%-70%) in soils amended with synthetic ammonium fertiliser, while denitrification was dominant (50%-90%) in the same soils amended with poultry manure. Kato et al. (2013) In this study, field samples from static flux chambers located on alpine meadow, shrub and wetlands were collected and analysed. Interpretation of results based on the relationship between δ 15 N bulk and δ 15 N sp (similar as presented in Fig. 7.5). A mixing of two endmembers was assumed: bacterial and fungal denitrification and subsequent N 2 O reduction. Applying literature values for endmembers and fractionation during reduction the contribution of fungal denitrification (from 23 to 41%) and degree of reduced N 2 O (from 83 to 93%) was calculated. The calculations were performed with Monte Carlo simulations and the assessed uncertainty of the results ranged from 17 to 23% for contribution of mixing endmembers and from 10 to 19% for degree of reduced N 2 O. Zou et al. (2014) Soil gas was collected from a highly fertilised tea field at 10-50 cm depths using a silicone tube. δ 15 N sp -Δ 15 N bulk isotope maps (Fig. 7.5) were applied for interpretations. The precursor isotopic signatures were determined, and the endmember ranges have been recalculated according to the measured precursor values for bacterial and fungal denitrification, nitrification and nitrifier denitrification. For the N 2 O reduction two scenarios were taken into account: assuming reduction after mixing and applying closed system dynamics and assuming reduction preceding mixing and applying open system dynamics. Predictions of δ 15 N sp values for different scenarios, reduction degrees and mixing ratios were presented and compared to the measured results. The study identified the bacterial denitrification as the dominant process and allowed for indication of the particular events when the contribution of nitrification or fungal denitrification increased pronouncedly. Wolf et al. (2015). These isotope maps allowed concluding that N 2 O was predominately formed by bacterial denitrification and that variations in isotopic composition may have been caused predominately by N 2 O reduction to N 2 . The study did not attempt to quantify the mixing ratios or N 2 O reduction. The high-frequency isotope data was combined with a biogeochemical model Landscape DNDC with a stable isotope model for nutrient cycles (SIMONE) to identify and address weaknesses in N cycling of the model (Denk et al. 2019

Outlook
N 2 O isotopocule analyses provide a unique possibility to get insight into processes contributing to N 2 O production as well as to assess the magnitude of N 2 O reduction and thereby also N 2 flux. However, the information is still rather indicative than strongly quantitative. The calculation methods presented allow estimates of ranges of possible mixing ratios and reduction contribution rather than precise numbers. However, such information is also quite precious hence often not attainable by any other methods. 15 N tracing, which is often a more precise tool, is much more expensive and laborious, moreover applicable only on a very limited space and time scale, hence much more constrained in application potential.
A promising perspective is to apply the N 2 O isotopocule analyses in combination with other methods, like with 15 N tracing (Deppe et al. 2017;Schorpp et al. 2016) (see also Sect. 7.5.) or with process modelling (Bai and Houlton 2009;Denk et al. 2017) which vastly increases the interpretation potential of such studies. Moreover, more quantitative estimates can be expected if the isotopocule approach is calibrated using controlled incubations where endmember values and isotopic fractionation factors are determined for specific conditions using independent estimates of contributing processes, e.g. by direct measurement of N 2 production or 15 N tracing Wu et al. 2019). The most recent idea for interpretation of N 2 O isotope data is the application of a N 2 O isotopocule model which incorporates all three measured isotopic signatures (δ 15 N bulk , δ 15 N sp and δ 18 O) (Lewicka-Szczebak and Well 2020).

Dual Isotope Method for Distinguishing Among Sources of N 2 O
Various microbial processes can produce N 2 O (for a simplified overview, see Fig. 7.8). These may occur simultaneously in distinct soil microhabitats or take  place temporally separated with fluctuating soil conditions. Often, only nitrification and denitrification are considered to be the main sources. However, the methods often applied cannot distinguish among all sources. For example, using 15 N tracing with labelled ammonium or NO 3 − does not allow a distinction among N 2 O produced by nitrifiers either via hydroxylamine (termed here nitrification, N) or via nitrite reduction (nitrifier denitrification, ND), or by denitrifiers using NO 3 − produced by nitrifiers (nitrification-coupled denitrification, NcD). All N 2 O produced by these sources is summarised as 'nitrification' by authors using this method. No method based on 15 N alone can so far separate the sources shown in Fig. 7.5. However, a distinction is important, as ND can under certain conditions produce all N 2 O derived from NH 4 + and has been reported to cause up to 90% of total N 2 O emissions (Kool et al. 2010).
A distinction between ND and other sources of N 2 O is possible if 18 O labelling is used in addition to 15 N labelling (Kool et al. 2011). As seen in Fig. 7 The preparation of soil samples proceeds in a similar way as for other stable isotope methods. However, one has to keep in mind that water needs to be added as tracer, so that the water content during conditioning needs to be a bit less than intended for the incubation. So far, conditioning has been done at 40% water-filled pore space (of samples dried at 40°C), and incubation at 80%, but this is adaptable as long as the requirements for tracer additions are kept. Soil samples of 75-100 g have been incubated in glass jars of about 300 ml for 24-28 h. These ratios and times may be adapted, but care must be taken to ensure linear N 2 O production over the incubation period, as well as stable concentrations of substrates ( The treatments (TR) are established (Table 7.5) with proper replication (at least five times) after conditioning of the soil as needed. So far, added label has been − pool may help to overcome this. Immediately after establishing the treatments, the jars are closed, and samples are taken for N 2 O content and isotopic signature as explained in Chap. 3 and above. At the end of the incubation, soil samples are taken for analysis of mineral N and its isotopic signature (the latter only in TR 3 and 4), as well as the soil moisture content to verify that this did not change during incubation. Consider that the label added with 18 O-H 2 O is diluted when mixed with moist soil.
For quantifying the O-exchange between N oxides and 18 O-H 2 O, the ERR approach is used. It is assumed that the O-exchange is similar for denitrifiers and nitrifiers. This need not be true, as O-exchange by nitrifiers has often been found to be less than in denitrifiers. Such a discrepancy would lead to an underestimation of the N 2 O produced by ND and NcD. No O-exchange is assumed to affect N 2 O derived from N. The ERR is calculated in Eqs. 7.12 to 7.17 as follows: This AOI may come from O ex quantified as shown above and the reaction stoichiometry of the different pathways as shown in Table 7.6.
A large AOI may thus be caused either by a larger contribution of pathways with a larger incorporation of 18 O-H 2 O (ND or NcD) or by a larger O ex . For further evaluation, O ex is maximised, i.e. assumed to take place in the NH 4 + -derived pathways to the same extent as in FD (Scenario A) or minimised, i.e. assumed to be absent in nitrification pathways (Scenario B). Furthermore, in Scenario A, the contributions of N and NcD are maximised, while in Scenario B, ND is maximised. Under both scenarios, a theoretical O incorporation (TOI) is calculated and compared to the AOI.
Under Scenario A, the TOI (TOI A ) is calculated (Eq. 7.18) as  Kool et al. 2009). If TOI A ≥ AOI, no contribution by ND is necessary to explain the AOI. The minimal contribution of ND, ND min , is then set to zero, and the maximum contribution of N, N max = N 2 O NH + 4 −NcD max . If not, ND must have contributed to N 2 O production (ND min > 0), which implies at the same time a maximum contribution of N, N max N max < N 2 O NH + 4 −NcD max . In this case, we can calculate the contribution of ND min (Eq. 7.19) as follows: (7.19) N max is then equal to N 2 O NH + 4 −NcD max −ND min . Under Scenario B, ND is maximised by assigning N 2 O NH + 4 to ND and assuming no O ex during this pathway, and in Eq. 7.20 TOI B is calculated as and the contribution of N min was zero. A larger AOI (TOI B < AOI) may either come from a contribution of NcD or O ex during ND, which was assumed not to take place under this scenario. As both may equally well explain the numbers, NcD min is set to zero in this case and O ex assumed to have occurred during ND.
If N min was found to be larger than zero, we can calculate ND max from this scenario as follows (Eq. 7.21): In that case, N min = N 2 O NH + 4 −ND max . Thus, in the dual isotope method, the contribution of NcD is always maximised, and consequently minimum and maximum contributions of N and ND are estimated based on Scenario A and B. Applying this method allows insight into these three potential sources of N 2 O plus fertiliser denitrification. However, in soils, further microbial processes can lead to N 2 O production. In the following, we will briefly discuss potential effects of nitrification by heterotrophs and archaea, fungal denitrification, as well as DNRA and co-denitrification.
If N 2 O of nitrification by heterotrophs and archaea is produced by the same sources and similar pathways as in autotrophic nitrifiers, this should not interfere with the calculations. The contribution of N would then comprise that of other nitrifiers. However, archaea have also been suggested to produce N 2 O in a pathway similar to ND (Jung et al. 2014). If so, this would be included in the contribution of ND. However, the pathway of N 2 O production by archaea is not clear yet and needs further study (Stieglmeier et al. 2014), the outcome of which will also affect the calculations presented here. In soils where fungal denitrification occurs, this is counted as FD using the dual isotope method, if the fungi use added NO 3 − as a source in a reaction similar to denitrification. Fungal denitrification may be quantified using the isotopomer method (Sect. 7.3), calling for a combination with the dual isotope method.
The occurrence of DNRA should be tested for as explained above. Should it lead to N 2 O production (Stevens et al. 1998), this would lead to an overestimation of N 2 O from FD. As DNRA may be important in soils (Rütting et al. 2011), this pathway should always be considered by checking for enriched NH 4 + in incubations with added 15 N-NO 3 − . Co-denitrifiers combine NO 3 − or NO 2 − with other nitrogenous compounds to produce N 2 O or N 2 . The occurrence of such a process could be quantified using the triple labelling 15 N tracing model  in combination with nonrandom 15 N distribution (Laughlin and Stevens 2002). Incorporating this would be an improvement of the dual isotope method, as co-denitrification could interfere with the source estimations presented above.
The dual isotope method could be further developed by incorporating better rates of O ex for the pathways starting from NH 4 + . Despite potential for improvements, however, this method allows an estimation of the contributions of N, ND, NcD and FD to N 2 O production and should be applied to a range of soils to further our understanding of these sources of N 2 O and potential mitigation strategies.

Background
The N cycle is a conceptual model that illustrates where and in which form N is present in the environment and how N is transformed and exchanged between organic, mineral and gaseous N forms. Since the N cycle is a dynamic system not only the sizes of the different N pools, e.g. NH 4 + , NO 3 − or organic N but also the rates between the pools provide an understanding of the dynamic nature of this important elemental cycle in soils and aquatic systems (Ryabenko 2013). The most common and easiest approach to understand the dynamic nature of the N pools is the determination of net process rates, such as net mineralisation rates by calculating the difference in the size of the mineral N pool between two time points. If this rate is positive, we refer to a net mineralisation, if it is negative then we call it net immobilisation. Thus, a net rate always refers to the difference between the production and consumption of the N pools in question. It can easily be shown that different pairs of production and consumption rates will lead to exactly the same net result. Thus, the analyses of net rates do not provide a measure of the individual rates that are contributing to the observed net rate. The individual rates associated with the N pool in question are called gross transformation rates. However, the quantification of these individual rates is not trivial because they cannot be measured directly. The most commonly used method to quantify the gross rates is the isotopic dilution technique (Stark 2000). The principle of this technique relies on the 15 N labelling of a certain N pool so that Illustration of the dilution technique by considering pool size and 15 N abundance at two time points according to Kirkham and Bartholomew (1954) the gross rate entering this pool can be quantified by taking into account the change in pool size and 15 N enrichment of at least two times after label addition (Fig. 7.9). The example in Fig. 7.9 shows that the pool size is decreasing which means that a net immobilisation of N occurred. However, the decline in 15 N abundance of the pool N during the same period also shows that N at natural abundance or low 15 N abundance must have entered the pool N. Thus, via visual inspection of the data we can say that N must have entered but also left the pool and that the rate leaving the pool must have been faster than the rate entering the pool. To quantify the individual rates requires a numerical analysis via a suitable N cycle model. Based on a simple twopool N model, Kirkham and Bartholomew (1954) were the first to derive analytical equations that allowed the calculations of the two rates between two-time points, i.e. the gross mineralisation and immobilisation rates. The underlying assumptions are (i) 15 N is homogeneously labelled and no preferential usage of either 15 N or 14 N occurs in the soil, (ii) immobilised N will not re-mineralise and (iii) N transformation rates follow zero-order kinetics (constant rates). The conceptual model of the Kirkham and Bartholomew approach and the equations derived for their model are illustrated in Fig. 7.10.
Since Kirkham and Bartholomew's pioneering work in the 1950s, analyses techniques have been developed which are based on more realistic conceptual N models. These include the division of the mineral N pool into NH 4 + and NO 3 − pools with separate immobilisation rates, the consideration of more than one organic N pool and additional N loss rates such as ammonia volatilisation and denitrification (Myrold and Tiedje 1986). The dilution technique works well in simple systems where the inflow into a pool occurs via a single gross N transformation rate. However, in reality, often more than one pathway contributes to the buildup of a pool size. This can be illustrated by the NO 3 − pool in soil. Production of NO 3 − can occur via oxidation Fig. 7.10 The conceptual model, the differential equations of the various pools and the closed-form analytical solutions for the individual gross rates (m and i) according to Kirkham and Bartholomew (1954). Note, N org (assumed to contain only 14 N) depicts the organic N pool which mineralises into mineral N (M) which consist of H ( 15 N) and N ( 14 N), M = H +N. The subscript 0 refers to the concentrations of the pools at time zero of NH 4 + to NO 3 − (usually termed autotrophic nitrification), and via oxidation of organic N to NO 3 − (usually termed heterotrophic nitrification) (Fig. 7.11). Following the principles of the dilution technique, the total gross rate of NO 3 − production can be quantified by labelling the NO 3 − pool and following the concentrations and 15 N dilution of this pool over time. This total NO 3 − production rate includes both, autotrophic (oxidation of NH 4 + ) and heterotrophic nitrification (oxidation of N org ). To separate the two processes, in addition to the 15 NO 3 − label also the NH 4 + pool should be labelled in a separate 15 N labelling treatment. To keep the conditions in the two 15 N labels the same, it is important to also apply NH 4 + in the soil that has received the 15  will provide a measure of autotrophic nitrification while heterotrophic nitrification can be calculated by difference: N h = N tot -N a where N tot , N a and N h refer to total, autotrophic and heterotrophic nitrification, respectively. In practice, to quantify N tot the dilution of the 15 N labelled NO 3 − pool (Fig. 7.10) can be used while N h and N a can only be estimated via a simulation model that takes into account both nitrification rates (Barraclough and Puri 1995). A parameter optimisation technique can also be used to estimate N a or N h (Myrold and Tiedje 1986). Thus, in modern 15 N tracing applications dilution-enrichment principles will be taken into account which we refer to as tracing.
In models with several simultaneous N transformations, it is impossible to derive analytical solutions. Therefore, the development of 15 N tracing models which use numerical solutions has become the state-of-the-art approach (Mary et al. 1998). These models rely on a set of differential equations for example of a simple system that describes the N cycle. Transformation rates between the various pools can be constant (zero-order kinetics) or are dependent on the pool size where the rate is originating from (first-order kinetics) or follow enzyme kinetics (i.e. Michaelis-Menten kinetics). While zero and first-order kinetics are described by one parameter, rates calculated via Michaelis-Menten kinetics are dependent on two parameters, i.e. the maximum velocity of the reaction rate and the half-saturation constant (Müller 2000). The determination of the parameters in such equation systems rely on parameter optimisation tools. A whole range of parameter optimisation tools are available and different algorithms have been used in 15 N tracing models (Mary et al. 1998;Myrold and Tiedje 1986). More recently, parameter optimisation tools based on Bayesian probability have become more common because they allow the simultaneous optimisation of a large number of parameters (for more details see Müller et al. 2007). It should be noted that the sole purpose of tracing models is to quantify gross transformation rates and are therefore data analysis tools and should not be confused with simulation models.
In the following sections, current 15 N tracing techniques are illustrated. This includes the description of experimental requirements to obtain suitable data sets and the subsequent model analysis. A number of 15 N tracing models have been developed (e.g. FLUAZ, Mary et al. 1998), and here, the data analysis will be illustrated by the Ntrace model. This model is based on the tool presented by Müller et al. (2007) and has since been developed further to analyse data sets from a range of differently complex setups, including dynamics of nitrite, gaseous N emissions, soil-plant interactions, biochar, etc. An advantage of Ntrace is its flexibility to adapt to various conditions and models because it is programmed in MatLab with code that can easily be changed and amended.

Stable Isotope Tracing Technique
A stable isotope tracing study consists of two parts, an experimental study where one or more pools are isotopically labelled and a data analysis tool (e.g. Ntrace) to quantify individual gross transformation rates. The technique can be regarded as a calculation procedure to quantify gross rates which cannot be quantified via any other means. Thus, both the tracing experiment and the numerical tool are building an analysis unit and it is important that the experimental approach is taking into account the requirements of the numerical analysis and vice versa. What is also important is that the quality of the final results critically depends on the data quality and therefore on the careful execution of the experimental part of the tracing study. For instance, data with high uncertainties may also result in gross N rates that are associated with large errors.

Setup of Tracing Experiments
To be able to analyse experimental data with the Ntrace model, the experiment needs to be set up in a certain way. Based on the research questions both field and laboratory experiments can be carried out. The research question usually requires the setup of several treatments (e.g. effect of various soil amendments). To quantify the individual gross N transformation rates in each treatment usually a set of at least two 15 N labels should be employed per treatment (i.e. 15 N-labelled NH 4 + and 15 N-labelled NO 3 − , typically applied as NH 4 NO 3 to ensure the application of equal quantities of each N species). However, often multiple labels are used (e.g. very common is a triple labelling approach with NH 4 NO 3 where either NH 4 + , NO 3 − or both moieties are 15 N labelled).
Ideally the 15 N label should be applied without enhancing the concentration because this will also have an impact on the N transformations. Thus, in ecosystems that are not used to receive large N concentrations often a high 15 N enrichment (e.g. 99 atom% 15 N) is applied at a very low application rate. However, in agricultural soils which receive N in the form of fertiliser, this is less of a problem. The advantage of applying a reasonable, but not unrealistically high, N concentration is that it can more homogenously be applied to the soil. In most cases, a 15 N enrichment of a few percent (e.g. 10 atom% 15 N) is sufficient to determine gross rates. However, in situations where, for instance, the nitrite or gaseous N species such as N 2 O are analysed, the labelled N pool (e.g. NO 3 − ) should ideally be enriched by approximately 50 atom% 15 N which allows most precise analysis based on the expected 29/28 iron current (Stevens et al. 1993). The 15 N solutions are made up according to standard calculations which are, for instance, summarised by Cabrera and Kissel (1989). To homogeneously label the soil a variety of application techniques are described in the literature ranging from application via side port needles in different depth, multiple needle applicators and automated techniques (Buchen et al. 2016) (Table 7.4). In field tracing studies often an application via a watering can is preferred, simply, because under field conditions when large plots of several m 2 have to be treated, it is critical that the solutions are applied within a short time window to ensure the same starting conditions (Plate 7.1). This is particularly important if dynamically changing N species such as N 2 O should be compared among treatments (Moser et al. 2018). However, the application rate should be slow enough to avoid by-pass in cracks and fissures down the soil profile because this would cause uneven distribution of the 15 N.
The time of labelling should be carefully noted because the difference between 15 N application and soil analysis during the experiment provides the time after N supply which is required for the model analysis. If both, soil extractions and gaseous measurements are planned then in the field an area for the gas sampling and an adjacent soil sampling area should be setup (in Plate 7.1, gas measurement are in the forefront, and the area for soil sampling is at the top). In laboratory incubations usually one set of jars is reserved for gaseous measurements (which will be extracted at the end of the experiment) while for each analysis day, separate sets of jars are prepared for destructive sampling. The question arises for how long we need to carry out a typical incubation study. Since the application of N may cause a stimulation of microbial activity resulting in faster gross N rates shortly after N application, the duration of a typical tracing study should be continued until after this initial stimulation has subsided. A typical duration of such a study is approximately 14 days. To characterise the non-linear dynamics of the gross N rates over time it is necessary to determine the N pool sizes and their 15 N enrichment at least 5 times throughout that period. Gas analysis should be carried out more frequently but at least at the times when soils are extracted.
Soil incubations have typically been carried out under controlled conditions at a pre-defined moisture content (set to a certain water filled pore space (WFPS) or water holding capacity (WHC)) and temperatures in a climate chamber (Plate 7.1).

Case study
To investigate the effect of a nitrification inhibitor in two soils on gross N transformations the following setup is realistic (using a triple 15 N labelling approach, numbers in brackets refer to the number of entities): Soils (2) × Inhibitors (2) × 15 N labels (3) × Replicates (3) × Time of soil extraction (5) = 180 jars.
Thus, a total of 180 jars (i.e. 36 jars per extraction day) need to be prepared. The label needs to be applied with minimal disturbance while providing an equal distribution in the soil. This can be done using a long needle with side ports. If 150 g of dry soil equivalent should be used per jar, then approximately 14 kg of soil is required from each soil.
The extraction times should be timed in such a way that the first extraction happens as soon as possible after 15 N labelling (typically after 2 h), then on day 1, 3, 7 and 15. Note, soils can react quite differently, therefore, the times and duration of the experiment should be adjusted accordingly.

Soil Extraction
If nitrite concentrations should be investigated it is recommendable to carry out the blending procedure of Stevens and Laughlin (1995). They discovered that nitrite is chemically reduced to N 2 in the KCl extract at pH below 5.5. They recommended a soil extraction at pH 7 and fast soil extraction. The blending procedure of Stevens and Laughlin (1995) is typically carried out at a soil: solution ratio of 1:1 in a blender for 90 s (Plate 7.2).
Immediately after the blending, the soil suspension needs to be centrifuged at 2000 × g for 5 min, and the supernatant filtered sequentially through a GF/D and GF/F (Plate 7.2).
The extracts have to be analysed for NO 3 − and NH 4 + and possibly for NO 2 . Based on the concentration a certain μmol of N of each N species will then be converted to 15 N-N 2 O or via a diffusion approach.

Chemical Conversion of Mineral N to 15 N-N 2 O
A precise method to determine the 15 N content of ammonium, NO 3 − and nitrite is via a method that converts the N species to nitrous oxide (N 2 O). The reduction of NO 3 − to N 2 O is described by Stevens and Laughlin (1994). For this, sulphamic acid (2.5 ml of 0.2 M solution) is added to 50 ml soil extract and shaken by hand for 5 s to ensure conversion of NO 2 − to N 2 . Then 5 ml of 1 M sodium acetate-1 M acetic Plate 7.2 Extraction procedure for quick soil extraction (a) and glass filter unit for glass fibre filter papers (b) acid buffer has to be added to increase the pH to 4.7. Then a CD-Cu reductor has to be placed in the bottle (Plate 7.3). The flask, capped, has to be laid flat in an orbital incubator at 20°C and shaken at 120 rpm with an orbit diameter of 50 mm for 2 h. A gas sample of the headspace is analysed with an IRMS for the 15 N content of the N 2 O. The 15 N content of the NO 3 − is considered to be the same as that of the N 2 O. The production of N 2 O from NH 4 + is described in Laughlin et al. (1997). Firstly, the ammonium must be diffused into (NH 4 ) 2 SO 4 . For this, 50 ml of the soil extract has to be pipetted in the diffusion unit (Plate 7.4). Above this liquid, a small flask containing 3 ml of H 2 SO 4 has to be placed. Before the diffusion jar is closed, 0.2 g of heavy MgO must be added. The MgO has to be brought into suspension by gentle swirling for 30 s. After this, the diffusion jar has to be left for 4 days. After diffusion of the NH 3 , the (NH 4 ) 2 SO 4 -H 2 SO 4 has to be poured into a 12 ml glass exetainer and evaporated to dryness in a 150°C oven, before cooling it in a desiccator and sealing it with a septum and cap. Then the vial has to be evacuated and filled with He to atmospheric pressure. One ml of NaOBr, with the molarity of NaOH adjusted to 10 M has to be injected through the septum. The vial has to be tilted and the solution gently swirled to ensure that the NaOBr reacts with as much of the (NH 4 ) 2 SO 4 as possible. The concentration and 15 N content of the N 2 O in each vial has to be determined by an IRMS system. For the Ntrace Nitrite model, data on NO 2 − concentration and 15 N content are also necessary. The NO 2 − concentrations can be determined by a manual photometer method.
The 15 N content of the NO 2 − extracts can also be determined by a method based on conversion to N 2 O as described by Stevens and Laughlin (1994). For this 50 ml of the soil extract has to be pipetted into a bottle. One ml of 1 M HCl and 0.5 ml of 0.04 M NH 2 OH has to be added to the bottle. The bottle should then be capped and laid flat in an orbital incubator and shaken at 120 rpm with an orbit of 50 mm for 16 h. A 12 ml sample of the headspace has to be transferred to an evacuated septum-capped glass vial, and the 15 N content of the N 2 O in each vial can be determined by IRMS. The atom% excess in 15 N in NO 2 − is calculated as two times the 15 N atom% excess in N 2 O minus the 15 N atom% excess in NH 2 OH.
The specific steps of the conversion method to N 2 O are summarised below.
1. NH 4 + -N: NH 4 + -N is first oxidised to NO 2 − -N by BrO − in a vacuum with N 2 O being the by-product (Eq. 7.22). The production of N 2 O can be catalysed by Cu + at the appropriate pH .
− -N is reduced to NO 2 − -N and NH 2 OH by copper-plating cadmium grains at a pH of 4.7. Then NO 2 − -N reacts with NH 2 OH to produce N 2 O, and the production of N 2 O is positively correlated to the production of NO 3 − -N (Eq. 7.23). The ammonium and N from other sources has no effect on the determination of NO 3 − -N; (Stevens and Laughlin 1994).
3. NO 2 − -N: NO 2 − -N reacts with NH 2 OH to produce N 2 O (Eq. 7.24) and the reaction is pH-dependent. When pH < 4, the reaction rate increases rapidly, and the reaction time should be at least 16 h. Because N 2 O is formatted via an asymmetric intermediate (N-nitroso-hydroxyl-amine) under acidic condition, the reaction requires at least 10 μmol of NH 2 OH (Stevens and Laughlin 1994).
The amount of N 2 O produced is about half of the theoretical yield. According to the isotopic distribution, the two N atoms in N 2 O are formed from NO 2 − -N and NH 2 OH, respectively. Hence, the atom% in NO 2 − -15 N needs to be calculated with Eq. 7.25 (assume the atom% 15 N in NH 2 OH is 0.365 atom%)  (a) Pipet 15-20 ml (about 20 μg N) soil extract into a semi-micro steam distiller. Carry out steam distillation immediately after adding 0.2 g MgO. The NH 3 is absorbed by 5 ml 0.01 M H 2 SO 4 . After 5 min of steam distillation, the distillate is concentrated to 2-3 ml. Transfer part of the concentrate into a 50 ml reaction vial, and evaporate to dryness at 90°C; (b) Evacuate the vials and fill them with He. Then inject 1 ml NaBrO together with 10 M NaOH through the septum, and swirl the solution around the sides of the vial to ensure that NaOBr reacts with as much of the (NH 4 ) 2 SO 4 as possible; (c) Transfer a known amount of sample to an evacuated septum-capped glass vial. The 15 N content of the N 2 O is then determined by IRMS.

Inorganic Nitrogen Isotopic Analysis in Soil Extracts via the Diffusion Method
An alternative method for the determination of 15 N in NO 3 − and NH 4 + is the diffusion method. The diffusion method is easier to apply and has the advantage that only a solid analysis on an IRMS is required rather than a gas measurement. These Mass spectrometers are more readily available. However, it should also be pointed out that chemical conversion method described above is quicker and is free from contamination by atmospheric N. It has very low detection limits, which are 20 μg N for NH 4 + -N, 5 μg N for NO 3 − -N and 0.5 μg N for NO 2 − -N.

Principle
During diffusion, ammonium in the soil samples is converted to ammonia by the use of MgO (Eq. 7.26). Then the ammonia is absorbed by using a filter paper containing a weakly acidic absorbent liquid during the volatilisation process. For determination of NO 3 − -N, titrate some alkaline reagent to remove NH 4 + -N in the sample then add some Devarda's alloy to reduce the NO 3 − -N into NH 4 + -N. Procedures 1. Put clips on the perforated silicon film and place it in the cap of a flask. Then place two pieces of 6 mm-diameter filter paper (Whatman #41 ashless filter paper) which are perforated by a needle on the clip; 2. For soil extracts > 2 mg l −1 in inorganic N concentration, only 20 ml of soil extract is needed. Put the 20 ml of soil extract into the container and add 3 glass beads before adding the MgO and Devarda's alloy. Onto each piece of filter paper pipette 10 μl of 1 M H 2 C 2 O 4 solution; 3. Add 0.3 g MgO and close the container quickly. Swirl the container carefully for 15 s. Then incubate the sample for 24 h at 25°C in a shaker running at 140 rpm to complete the diffusion and recovery of NH 4 + -N; 4. To determine the 15 N enrichment of NO 3 − -N from the same sample, replace the used filter paper with two new pieces also spiked with H 2 C 2 O 4 . Incubate the sample in a shaker running at 140 rpm for 48 h to remove the remaining NH 4 + -N. Then replace the used filter paper with two new acid-spiked pieces again, add 0.3 g Devarda's alloy, and incubate it for 24 h to complete the processes of diffusion and recovery of NO 3 − -N; 5. Remove the filter papers from the clips by forceps and dry them in a desiccator containing an open container of concentrated H 2 SO 4 (to remove traces of NH 3 ) and silica gel. Then wrap the filter papers in tin capsules and analyse them for 15 N enrichment by using a coupled elemental analyser-isotope ratio mass spectrometer (EA-IRMS); 6. Use the amount of N measured in diffusion blanks to calculate the corrected 15 N enrichment of the sample (Eq. 7.27): where E s is the corrected abundance 15 N enrichment of the sample, E m is the enrichment of the sample + blank measured by mass spectrometry, M s+b is the mass of N (sample + blank) recovered in the acid trap, M b is the mass of N recovered in the acid trap from the diffusion blank, and E b is the enrichment in the blank (assumed to be 0.3663 atom%). Note, for soil extracts < 2 mg l −1 in inorganic N concentration, 50 ml of extract is needed to ensure accurate determination. When the abundances of NH 4 + -N and NO 3 − -N are very different, it is better to diffuse NH 4 + -N and NO 3 − -N separately (do not use the same extract).

Inorganic Nitrogen Isotopic Analysis in Soil Extracts at Natural Abundance
The diffusion method and chemical conversion method described above are both suitable for N at high abundance, but not for N at natural abundance. They both have a high demand for N and high levels of background N can interfere with the reaction. There are two modified chemical methods for N isotopic analysis at natural abundance. These simplify the preparation procedures, reduce the preparation time and do not require large amounts of N. The chemical method for ammonium requires 2.5 μg N in a 4 ml sample volume for analysis, and its accuracy of δ 15 N measurements is less than 0.3‰. The method for NO 3 − needs only 4.5 μg N, and its accuracy of δ 15 N and δ 18 O reaches 0.31‰ and 0.55‰, respectively.

Principle
The method is to oxidise NH 4 + -N to NO 2 − -N by BrO − instead of extraction of NH 4 + -N in solution. Subsequently, the NO 2 − -N is reduced to N 2 O by NH 2 OH-HCl, thus replacing HN 3 (Liu et al. 2014;Stedman 1959) (Eq. 7.28).

PT-IRMS
Shaker 20 ml headspace glass vials: Acid rinsed and combusted at 450°C for 4 h Reagents 10 M NaOH: Evaporate 100 ml of 5 M NaOH to 50 ml of 10 M NaOH NaBrO: (a) Bromate-bromide stock solution: Mix 0.6 g NaBrO 3 and BrNa into 250 ml DIW (deionised water) (can be stored ≥ 6 months); (b) Take 1 ml bromate/bromide stock solution into 50 ml water, and place the solution in the dark with 3 ml 6 M HCl added for 5 min to produce Br 2 ; (c) Add 50 ml of 10 M NaOH quickly to produce BrO − . NaAsO 2 : Mix5.1 g NaAsO 2 and 100 ml DIW NH 2 OH, HCl: (a) NH 2 OH-HCl stock solution: Add 0.2778 g NH 2 OH-HCl in 100 ml DIW (can be stored ≤ 7 d) (b) Take 3 ml NH 2 OH-HCl stock solution and dilute in 500 ml DIW 6 M HCl 5 M NaOH Procedures 1. Take samples (1:10 (v:v) sample to NaBrO, e.g. 4 ml) and place it into 20 ml headspace glass vials. Dilute the sample to 10-20 μM to maximise oxidation yield. NO 2 − -N must be removed by NH 2 SO 3 H earlier to ensure the accurate determination of NH 4 + -N; 2. Add NaBrO (e.g. 0.4 ml) into the vial, shake the vial vigorously, and then let it stand for 30 min; 3. Pipet 0.05 ml NaAsO 2 to remove excess BrO − and terminate oxidation; 4. Add 6 M HCl to lower pH (pH < 1) and seal the vials; 5. Inject NH 2 OH-HCl with a gas-tight syringe (n(NH 4 + ): n(NH 2 OH) = 1:2). Put the samples in a shaker running at 120 rpm at 37°C for 16 h; 6. Inject 0.5 ml 5 M NaOH to absorb CO 2 in the vials and terminate the reaction; 7. Transfer a known amount of gas to a PT-IRMS for analysis; 8. Treat 3 international NH 4 + -N standards (IAEA N1, +0.4‰; USGS 25, −30.4‰; USGS 26, +53.7‰) using the same protocol for calibration (Eq. 7.29): δ 15 N NH + 4 sample = (δ 15 N N 2 Osample − intercept)/slope (7.29) where the intercept and slope are obtained from the linear regression of the δ 15 N measured and the δ 15 N assigned from N 2 O produced by the standards.

Conversion of NO − 3 at natural abundance Principle
The NO 3 − -N is reduced to NO 2 − -N by copper-plated cadmium granules in a weakly alkaline environment (Eq. 7.30): When there is a large number of halogen ions present the reaction will be accelerated (Stedman 1959) (Eqs. 7.34, 7.35): (7.35) N 2 O is an asymmetric molecule with a molecular structure of N-N-O. The δ 15 N Air of N 2 O is the mean value of δ 15 N Air in two N atoms (Eq. 7.36): (7.36) and the N 2 O produced is composed of a N atom and an oxygen atom provided by the NO 2 − -N and a N atom provided by N 3 − , a N source. The isotope ratio of N and oxygen of the NO 2 − is identical with that of NO 3 − in the original solution. So, the following relationships hold (Eqs. 7.37, 7.38): Therefore, when the isotope ratio of N 3 − is constant, the N and oxygen isotope ratios of N 2 O produced is linear with the N and oxygen isotope ratios of NO 3 − , and the theoretical slopes of their correlation curves are 0.5 (N) and 1.0 (O), respectively.

M NaOH
Copper-plated cadmium granules Cadmium reduction column Procedures 1. Place at least 40 ml of sample into the vials to ensure that there is still 16 ml sample with 4.5 μg N for the reaction after any loss. If the concentration of NO 3 − -N in sample is above 20 μM, dilute the sample with 0.5 M NaCl. If the concentration of Cl − is below 0.5 M, add solid NaCl to ensure that the concentration of Cl − is 0.5 M. 2. The blank must be analysed during each analysis to test the seal of vials and the reagent blank. The signal value of blank should be below 0.6 nA. 3. Add 0.5 M HCl into the samples to adjust pH to 2-3. The blank will only need one drop of 0.5 M HCl. Then add 1 M C 3 H 4 N 2 to adjust the pH to 7.8-8.0. 4. (a) Connect the cadmium reduction column to a peristaltic pump of which the flow rate is 5 ml min −1 . Plug the end of the column with foam sponge. After filling the column with copper-plated cadmium granules, also plug the other end. Rinse the pipeline with 0.5 M NaCl (pH = 8) to activate the column. Then transfer 20 ml of adjusted sample into a 25 ml beaker, and place the inflow and outflow ends of the column in the beaker. After 80 min of continuous reduction, rinse the pipeline with 40 ml 0.5 M NaCl again. When moving the column air must not enter the column to prevent oxidation of the cadmium column.
(b) Copper-plated cadmium granules can be added directly to the samples. Place the samples in a shaker running at 200 rpm at 30°C for 3 h to reduce NO 3 − -N. Filter the sample into another vial. 5. Take a 16 ml sample and place it into a 50 ml headspace glass vial with the cap sealed. Evacuate the vial and fill it with He gas.
6. Inject 0.8 ml NaN 3 -CH 3 COOH into the vial (pH = 4.5) and shake the solution vigorously for 1.0 min. Then place the samples in a water-thermostat at 30°C for 30 min, or in a shaker running at 200 rpm at 35°C for 30 min. 7. Inject 0.5 ml 6 M NaOH (pH ≥ 10) to terminate the reaction. 8. Transfer a known amount of the gas to a PT-IRMS for determination of δ 15 N and δ 18 O of N 2 O in the sample. 9. Mix 2 international NO 3 − -N standards (USGS 32, δ 15 N Air ‰ = 180‰, δ 18 O SMOW ‰ = 25.7‰; USGS 34, δ 15 N Air ‰ = −1.8‰, δ 18 O SMOW ‰ = − 27.9‰) in different proportions (e.g. 6:0, 4:2, 0:6), then treat them with the same protocol for calibration. The calibration equation shown below is where the intercept and slope are obtained from the linear regression of the δ 15 N and δ 18 O measured from N 2 O produced by the standards and the δ 15 N and δ 18 O are assigned from the standards.

Data Requirements for the Ntrace Model
Data obtained through the 15 N tracing experiment will be further analysed by the Ntrace model to quantify gross N transformation rates and pathway-specific N 2 O emissions. The various Ntrace model versions differ in their data requirements. The Ntrace Basic model has the fewest data requirements. The other models require more data on top of the data required for the Ntrace Basic model. The Ntrace Basic requires the fertiliser application rate (in μmol N g −1 ) and its 15 N excess (in atom%). It also requires the average NO 3 − and NH 4 + concentration and 15 N excess (in atom% excess) including their standard deviations for each data point in time. The NO 3 − and NH 4 + concentrations should be given in the same unit as the fertiliser application rate. Next to this, a one-time measurement of total organic N (in %) is required. This measurement can be taken from basic soil characteristics. The Ntrace Plant model also requires plant N and plant 15 N data, and total plant biomass data, at each time when destructive sampling was carried out, i.e. preferably the same time points when NO 3 − and NH 4 + were determined. Ideally, above-and belowground biomass (roots) should be determined. The Ntrace Urea model requires the urea application rate and its excess, and if plants are included, it also has the additional requirements of the Ntrace Plant model. The Ntrace Nitrite model requires measurements at multiple time points (preferably the same as for NO 3 − and NH 4 + ) of the average NO 2 − concentration and its 15 N excess (in atom%) including standard deviations. Transformation can follow zero-order, first-order, or Michaelis-Menten kinetics. The type of kinetics used needs to be specified for each transformation separately. If the transformation uses N from a large pool, it is generally appropriate to use zeroorder kinetics. For N transformations coming from pools that change rapidly it is generally more realistic to use first-order or Michaelis-Menten kinetics. Especially the transformations associated with the NH 4 + consumption (e.g. nitrification) may be most realistically represented by Michaelis-Menten kinetics. However, under conditions when microbial activities may be affected by conditions other than substrate, the N transformation rate may also follow first-order kinetics. For example, when the rate is governed by temperature or soil moisture.

The Ntrace Model System
The 15 N tracing model Ntrace described by Müller et al. (2007) is a tool to quantify gross soil N transformations; this model considers five N pools and twelve simultaneously occurring N transformations ( Fig. 7.12 ); dissimilatory NO 3 − reduction to NH 4 + (D NO3 ); adsorption of NH 4 + on cation exchange sites (A NH4 ); and release of adsorbed ammonia (R NH4a ), adsorption and release of NO 3 − on/from stored NO 3 − , i.e. A NO3s and R NO3s , respectively. Ntrace is a family of 15 N tracing models to quantify gross N transformations in soils and sediments. The model consists of a N transformation model that is programmed in Simulink, a graphical programming language associated to Matlab, and a parameter optimisation routine based on a Markov Chain Monte Carlo routine in combination with the Metropolis algorithm (Müller et al. 2007).
Several extensions exist of the Ntrace Basic model, namely Ntrace Plant (Fig. 7.13a), Ntrace Urea (Fig. 7.13b), Ntrace Nitrite (Fig. 7.14) and Ntrace Gas (Fig. 7.15). The boxes represent the different N pools, and the transformations are represented by the arrows between the boxes. For each model all transformations are quantified simultaneously (Fig. 7.12).
The Matlab-Simulink files (m-files and mdl-files) alongside their description that are part of the Ntrace model are presented in Table 7.7.
Currently, a new optimisation routine for the Ntrace model is being implemented. This will further improve optimisation speed, and more importantly be quicker to find a global minimum as opposed to a local one. The method used for determining optimal parameters will be Matlab's GlobalSearch algorithm (Ugray et al., 2007).   (2006) and Rütting and Müller (2008) Standard deviations of the optimised parameters will be determined as described by Gavin (2019). After filling out an input file for the model (DataNtrace.xlsx) that contains all the required data specified in data requirements including kinetics for each transformation, initial parameters and minimum and maximum values for the parameters, the model can be run (Fig. 7.16).

Procedure
The first step of the optimisation is generally done by hand. The model is run with the initial parameter set, and graphs of modelled versus measured data are inspected. From this, parameters are adjusted until a visually reasonable fit is obtained. Thereafter, all parameters are optimised simultaneously using a Markov chain Monte Carlo (MCMC) method, and this is explained in more detail by Müller et al. (2007). For this, the parameters are slightly adjusted, and the model is run. At the end of the run, the misfit is calculated based on the difference between the modelled and measured values. If the misfit is lower compared to the previous run, i.e. a better fit between modelled and measured data, the new parameter set is accepted, and it will start again by adjusting the newly accepted parameter set and execute the next iteration. If the misfit is higher, so a worse fit, there is also a chance the parameter set is accepted via a likelihood function. By this, it is possible that the algorithm moves out of a local minimum and enters the global minimum. If a new parameter set is not accepted, the last accepted parameter set will be used for the parameter adjustment. This iterative procedure of parameter adjustment and running the model should go on till the probability density functions (PDFs) are well characterised for all parameters. If the initial parameter set is fairly close to the optimal set, PDFs are generally well characterised after 50,000 to 100,000 iterations. So, for the final run, the maximum number of iterations is generally set between 50,000 and 100,000. During the optimisation, generally three parallel sequences, each with different starting parameters, are calculated. The number of parallel sequences should be defined before running the model, but three is generally appropriate. From these parallel sequences, a reduction factor is determined, which determines the accuracy of the sampling (Gelman et al. 2003). If the reduction factor is below a pre-specified number (default 1.1) for all parameters, the optimisation will also be stopped, regardless of reaching the maximum number of iterations. The reduction factor near 1 indicates that all parallel sequences resulted in statistically the same parameter set. Inspection of the PDFs can show that for certain parameters the peak is close to zero. This would indicate that this particular parameter can be neglected. The parameter can then be set to zero, and excluded from optimisation when the model is re-run.

Ntrace Model Output
At the end of the optimisation, the model output will be exported to an Excel file. This output contains the initial parameter value, the optimised parameter value, the standard deviation of the optimised parameter, the average N flow for each transformation, the overall R 2 and the AIC. The standard deviation (SD) of the transformation rate is based on the SD and average (AVG) parameter values as shown in Eqs. 7.41 to 7.43: S D transformation rate = AV G transformation rate · S D parameter AV G parameter (7.41) Response ratios between transformation rates can be calculated by McGeough et al. (2016) where X E and X C are the average N transformations of the elevated and control group, sd E and sd C the associated standard deviations and n E and n C the repetitions. To compare the effect of different treatments on transformation rates, the individual treatments have to be run separately. After this, the rates can be compared using a one-way ANOVA based on the averages and standard deviations. Pairwise comparisons can be calculated with the Holm-Šídák test.
Another way to compare gross N transformation is via the determination of least significant difference (LSD) as described by Müller et al. (2011).
Output of the correlation matrix can be used to find parameters that tend to be strongly constrained together. A correlation value of above 0.8 indicates that it is constrained. There is also another output file that gives the pool sizes and transformation rates for each time step. This output can be used to create graphs.

Ntrace Approach to Quantify N 2 O Pathways
Nitrous oxide (N 2 O) can be emitted via a number of pathways including inorganic and also organic pathways, involving a range of microbes (e.g. bacteria, fungi) (Butterbach-Bahl et al. 2013). The Ntrace Gas model can be applied to quantify N 2 O pathways based on the underlying N transformations and especially based on the nitrite dynamics (Ntrace Nitrite ). To accurately estimate N 2 O pathways via Ntrace Gas also the N 2 production is calculated. The two predominant biological processes for N 2 O production in soil are traditionally considered to be autotrophic nitrification and heterotrophic denitrification (Ambus 1998;Wrage-Mönnig et al. 2018;Wrage et al. 2001). In both processes NO 2 − is the key precursor to N 2 O production. In nitrifier nitrification, it is rather NH 2 OH or at least something before nitrite. In nitrifier denitrification, it is nitrite.
Assuming that the N 2 O production is derived from a single NO 2 − pool, the 15 N enrichments of the N 2 O and the NO 2 − should be similar. However, experimental data show that the enrichment of the N 2 O is deviating from the theoretical 1:1 line ( Fig. 7.17) leading to the conclusion that the N 2 O originated from various NO 2 − sub-pools and also from sources which were at or close to natural abundance. Based on the experimental setup, the only common unlabeled N pool in all 15 N treatments is organic N. Therefore, two possible processes were included in the Ntrace Gas model to account for such a dilution effect: (a) reduction of NO 2 − originating from oxidation of organic N derived (NO 2 − org ) and (b) hybrid-reaction for N 2 O production whereby one atom of the N 2 O is derived from an enriched NO 2 − pool and another from organic N at natural abundance.

Fig. 7.17
Relationships between nitrite and nitrous oxide a as well as the 15 N enrichment b  Heterotrophic nitrifiers can also denitrify (Blagodatsky et al. 2006;Papen et al. 1989) and it is, therefore, possible that NO 2 − org in the Ntrace Nitrite model (Rütting and Müller 2008) originating from N org oxidation could be further reduced to gaseous N products. A hybrid-reaction between NO 2 − and organic N is also possible which occurs, for instance, in the fungus Fusarium oxysporum (Kurakov et al. 2000;Tanimoto et al. 1992) and possibly in other heterotrophic organisms (Kumon et al. 2002).
Based on the above considerations the Ntrace Gas model analyses four N 2 O processes. The entire model includes all the previous 15 N tracing models (see Ntrace family above). In Ntrace Gas NO 2 − sub-pools are reduced to associated N 2 O pools which may further be reduced to N 2 (Eq. 7.44). (7.44) (x = nit, den or org) In addition, a hybrid-reaction between denitrification derived NO 2 − (NO 2 − den ) and recalcitrant organic N (N rec ) was introduced (Eq. 7.45).

NO 2
− den + N rec → N 2 O cod → N 2 (7.45) Each soil N 2 O sub-pool can be further reduced to N 2 via specific N 2 O reduction rates or emitted to the atmosphere, which is governed by gas diffusion parameters. For N 2 O emission a first-order notation has been implemented (Cho and Mills 1979;p. 97 in Müller 2000).
The total N 2 O emission is calculated (Eq. 7.46) by E x = k x · N 2 O x (7.46) where E x is the emission rate (μmol N g −1 h −1 ), k x is the emission rate constant (h −1 ) and N 2 O x the soil N 2 O pool concentration (μmol N g −1 ). The symbol x stands for the process specific pools, i.e. nit, den, org and cod. In the following section, two simplified approaches are presented that are based on the abundance of NH 4 + , NO 3 − , N org and N 2 O. The methods are based on the assumption that N 2 O is derived from three uniformly labelled pools, i.e. NH 4 + , NO 3 − and N org . and bases the analysis on the 15 N enrichments of the different N species.

Three-Pool Model
Based on the two-pool source-partitioning model by Stevens et al. (1997) a three-pool solver method (Rütting et al. 2010) was developed. The solver method (Microsoft Excel 2007) calculates the N 2 O fractions associated with NH 4 + (n) and NO 3 − (d) by minimisation of the absolute difference between observed and calculated 15 N enrichments of N 2 O according to the equation: a N 2O = d * a d + n * a n + (1 − d − n) * a o (7.47) where n and d are the fractions related to the NH 4 + and NO 3 − pools, respectively, and a d , a n and a o represent the 15 N abundance of the NO 3 − , NH 4 + and N org (assumed to be at natural abundance) respectively. The data are setup in an Excel spreadsheet and the Excel Solver is used to minimise the difference between measured and calculated 15 N N 2 O enrichments (Rütting et al. 2010) (Fig. 7.18).
With this method, it is possible to subdivide total N 2 O emission into the three sources, autotrophic nitrification, denitrification and heterotrophic nitrification.

Four-Pool Model
The three-pool model has been developed further by Jansen-Willems et al. (2016) to analyse four simultaneous processes (nitrification, denitrification, co-denitrification and oxidation of organic matter). The assumption is that isotopic discrimination is negligible. The conceptual model for this approach is illustrated in Fig. 7.19.

Background and development of the four-pool model
Each N pool contains both 15 N and 14 N atoms. If one N atom would be randomly selected from a pool, the chance it would be a 15 N atom is equal to the 15 N atom fraction of that pool. So for the NH 4 + pool this would be a n , for the NO 3 − pool a d , and for the organic-N pool a o . The chance it would be a 14 N atom equals 1 minus the 15 N atom fraction of that pool. So, for the NH 4 + pool it would be 1-a n , for the NO 3 − pool 1-a d , and for the organic-N pool it would be 1-a o . N 2 O consists of two N atoms. For nitrification, denitrification and oxidation of organic N, both N atoms come from the same pool. For these three processes: • The chance that N 2 O contains no 15 N atoms is the chance that the first atom is a 14 N atom multiplied by the chance that the second atom is a 14 N atom (Eq. 7.48). • The chance that N 2 O contains one 15 N atom is the chance that the first atom is a 15 N atom, multiplied by the chance that the second atom is a 14 N atom, plus the chance that the first atom is a 14 N atom and the second is a 15 N atom (Eq. 7.49) • The chance that N 2 O contains two 15 N atoms is the chance that the first atom is a 15 N atom, multiplied by the chance that the second atom is a 15 N atom (Eq. 7.50) In Eqs. 7.48 to 7.50, a x would be a n for nitrification, a d for denitrification and a o for oxidation of organic N. For co-denitrification, one atom comes from the NO 3 − , and one comes from the organic N pool. So, for co-denitrification, the chance that N 2 O contains • No 15 N atoms, is the chance of a 14 N atom from the NO 3 − pool, multiplied by the chance of a 14 N atom from the organic N pool (Eq. 7.51).
• One 15 N atom, is the chance of a 15 N atom from the NO 3 − pool, multiplied by the chance of a 14 N atom from the organic N pool plus the chance of a 14 N atom from the NO 3 -pool multiplied by the chance of a 15 N atom from the organic N pool (Eq. 7.52).
• Two 15 N atoms is the chance of a 15 N atom from the NO 3 − pool, multiplied by the chance of a 15 N atom from the organic N pool (Eq. 7.53). The N 2 O in the gas sample is assumed to come from one of four processes. The fraction that comes from nitrification is written as n, the fraction that comes from denitrification is written as d and the fraction that comes from oxidation of organic matter is written as o. The fraction that comes from co-denitrification is written as c. As these are the only four processes considered, the four fractions should add up to one. Therefore, the following two equations apply: a + d + o + c = 1 (7.54) c = 1−a−d−o (7.55) The fraction of N 2 O in the gas sample that is expected to contain zero 15 N atoms can be calculated by multiplying the fraction of that sample from a specific process by the chance that the N 2 O from that process contains zero 15 N atoms. So for nitrification, this would be n(1-a n ) 2 , and for co-denitrification this would be (1-n-d−o) (1-a d ) (1-a 0 ). This should be done for all four processes, and then should be added together (Eq. 7.56). The fraction of the N 2 O in the gas sample that is expected to contain one 15 N atom is calculated in the same way (Eq. 7.57), and the expected fraction containing two 15 N atoms as well (Eq. 7.58) Chance of zero 15 N atoms: n(1 − a n ) 2 + d(1 − a d ) 2  2n(1−a n )a n + 2d(1 − a d )a d + 2o(1 − a o )a o + (1−n−d−o)(a d (1−a 0 ) + a 0 (1−a d )) (7.57) Chance of two 15 N atoms: a d a 0 (7.58)

Mass Spectrometer Measurements and Calculation of Fractions
To determine the fractions of the different processes 45 R and 46 R measurements are needed. These need to be corrected for the presence of 18 O. Therefore, this means that 45 R is the fraction of N 2 O molecules containing one 15 N atom divided by the fraction of N 2 O molecules containing zero 15 N atoms, and 46 R is the fraction of N 2 O molecules containing two 15 N atoms divided by the fraction of N 2 O molecules containing zero 15 N atoms. The expected fractions for N 2 O containing zero, one or two 15 N atoms are given in Eqs. 7.56−7.58. In the study published by Jansen-Willems et al. (2016), a 0 was set to 0.003663 (natural abundance), and a n and a d were considered to be the 15 N abundance of NH 4 + and NO 3 − . Using these values, n, d and o were quantified using the fminsearchbnd function in MatLab (The MathWorks Inc, Natick, MA). Thus, from this, c could be calculated according to Eq. 7.55.
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