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Research article
Geology

Comparative analysis of nitrogen and carbon isotopic fractionation during diamond formation based on β-factor determination

Authors:
Dmitrii P. Krylov
About authors
  • Ph.D., Dr.Sci. Leading Researcher Institute of Precambrian Geology and Geochronology RAS ▪ Orcid
Date submitted:
2024-03-11
Date accepted:
2024-09-24
Online publication date:
2024-12-06

Abstract

First quantitative estimates are presented for nitrogen isotopic fractionation during diamond crystallization with respect to nitrogen-bearing fluid components using quantum-mechanical (DFT) calculations on the defect (with the substitutional nitrogen) diamond lattice. Provided equilibrium isotopic fractionation, 15N/14N ratio decreases within the sequence of compounds NH4+ > N2 > (diamond, NH3) > CH3N > CN− > NH2. At temperatures of 1,100 to 1,200 °C fractionation among diamond and fluid N-compounds are estimated at –2.23, –0.77, 0.01, 0.44, 1.31 and 2.85 ‰ and substantially (over 1 ‰) exceed the already available estimates based on the modeling diamond C-N bonds by analogy with HCN or CN – molecules. Depending on the dominant nitrogen and carbon substance in the mineral-forming fluid, diamond formation can be accompanied by different isotope compositional trends, as expressed either by zoned patterns within individual diamond grains or by isotopic δ15N vs δ13C covariations during successive crystallization. Provided the dominance of NH3 component (the reduced conditions, high pressures and the cold geotherm) nitrogen isotope fractionation between diamond and fluid does not exceed 0.1-0.2 ‰ and the isotope shifts at temperature ca. 1100 °C Δ15N << Δ13C. In nitrogen depleted reduced mantle fluids possible existence of compounds with low heavy isotope affinity at temperature of diamond formation (especially NH2) implies high isotope fractionation between diamond and the fluid and hence, evolved Δ15N/Δ13C ratios. Oxidized fluids dominated by CO2 or CO3 coupled with N2 component are characterized by close to zero Δ15N/Δ13C ratios as inferred by prevailing carbon isotope fractionation with respect to nitrogen isotopes, the latter change considerably with nitrogen distribution coefficient among diamond and the growth media.

Keywords:
diamond isotope fractionation factors 15N/14N 13C/12C
Online First

Funding

The work was carried out within the framework of the state assignment FMVW-2021-0003.

Introduction

The conditions of diamond growth and transformation are reflected by the morphological properties, spectroscopic features, impurity compositions [1-3], and inclusions [4, 5]. One of the most important indicators of the formation conditions – isotopic composition of carbon in diamond – characterize both the initial substrate (eclogites, peridotites, websterites, with different proportions of sedimentary and mantle components) as well as a mineral-forming fluid (oxidized, reduced, mantle, surface) [6]. Diamond can represent the most ancient isolated parts of the mantle [7] and thus retains carbon isotopic labels from the early stages of the Earth formation. In addition to carbon, diamond is characterized by significant amounts of nitrogen (up to thousands and even tens of thousands ppm in subduction-related microdiamonds), the occurrence of which is driven by mineral-forming fluids [8]. The combination of isotopic systems 12C-13C and 14N-15N is widely used not only to reveal the conditions of formation, but also to characterize the cycles of carbon and nitrogen in deep conditions. Very high retention of carbon and nitrogen in diamonds [9] encourages preservation of isotopic systems both in conditions of formation at great depths (>110 km) and during rise to the surface. Hence, the two elements in diamond taken together provide a unique opportunity to study the Earth's mantle down to depths of about 800 km and ages up to 3.5 billion years [10]. Considering generally low nitrogen content in the upper mantle, diamond 15N/14N isotopic ratio could be a sensitive indicator of mantle and sedimentary reservoirs mixing [11].

The observed large variations of diamond isotopic composition can be explained by both isotopic heterogeneity (including mixing) of the initial substrate and by fractionation of isotopes during crystallization. Quantitative characterization for processes related to diamond genesis primarily requires estimated isotope fractionation factors between diamond and the crystallization medium. Carbon isotopic fractionation factors depending on temperature have been determined for many systems related to diamond formation, including diamond-CO2, diamond-CaCO3, diamond-CH4, and a number of others [12-14]. Unlike carbon isotopes, fractionation of nitrogen isotopes with diamond are barely studied. There are still no experimental data on the distribution of nitrogen isotopes between diamond and possible nitrogen-containing substances in diamond growth media. The available calculated (theoretical) nitrogen isotopic fractionation factors are restricted to individual molecules (NH4+, NH3, N2) in equilibrium with arbitrary molecular analogues of the C-N bond of diamond, such as HCN or CN[9]. In the absence of more or less reliable determinations of nitrogen fractionation factors between diamond and mantle fluids, it is impossible for example, to distinguish the processes of mixing nitrogen of fluid from different sources, to assess the evolution of the isotopic composition under various mechanisms of interaction between fluids and the nascent diamond, etc.

The present work aims to provide first quantitative estimates of equilibrium isotope fractionation of nitrogen during diamond crystallization relative to nitrogen-containing fluid compounds using quantum-mechanical calculations (within the framework of the density functional theory DFT) for the defective (N-containing) lattice of diamond and N-containing molecules of the mineral-forming fluid. For comparison, calculations were also carried out for diamond without nitrogen (fractionation of carbon isotopes) and the carbon-containing fluid molecules.

Calculation methodology

Isotope fractionation factor between phases A and B, αAB (RA/RB, R being the ratio of the atomic concentrations of the heavier and lighter isotopes in a given compound) at isotopic equilibrium is expressed by

1,000lnαAB=1,000lnβA1,000lnβB,(1)

in which β – values (the reduced partition function ratios of the isotopologues under consideration, or so-called β-factors) without accounting for anharmonicity are calculated from the frequencies of oscillations (phonon spectra) of isotopologues depending on temperature [15],

β=iuiuieui/2eui/21eui1eui,

or, in logarithmic form:

lnβ=ilnsinh0,5uisinh0,5uiilnuiui;(2)

u – the dimensionless frequency, u = hν/kT; ν – the frequency of the harmonic oscillator; T – temperature, K; h and k – are the Planck and Boltzmann constants; superscript * refers to the heavier isotope; the subscript i numbers independent harmonic vibrational frequencies for a compound. Summation is carried out for all vibrational states (3N-6 for nonlinear molecules, 3N-5 for linear molecules, 3N-3 for crystals); N – number of atoms of a compound. Hyperbolic sinus sinh(x) ≡ (exp(x) – exp(–x))/2.

Experimental vibrational frequencies are mostly known only for the prevalent isotopologues (or for their natural mixtures) of individual substances. Therefore, to determine the values of ν for the expression above, calculations “from the first principles” (ab initio) are currently used based on the position of nuclei and the electronic properties of elements. In the present work, the vibrational frequencies of isotopologous are determined by the “frozen phonons” approach of the density functional theory as implemented in the CRYSTAL code (current version CRYSTAL23 [16]) with a set of Gaussian all-electron bases 6-311G(d). The basis sets for carbon and nitrogen are presented at the program site (https://www.crystal.unito.it/Basis_Sets/). Before calculating the phonon spectra, the parameters of the outer orbitals of the basic functions were optimized and the values of the vibration frequencies for the crystal lattice of pure diamond and diamond with different nitrogen impurity contents were compared, as well as structural optimization. Among the tested functionals of interaction between atoms (including LDA, GGA, hybrid), the minimum discrepancy with the experimental values of the calculated vibrational frequencies of pure diamond was obtained for the WC1LYP functional. The energy convergence threshold for self-matched field (SCF) calculations was chosen at the level of 10–11 amu. (Hartree) both during the optimization of the structure and calculation of vibrational frequencies, and the truncation of the summation series was controlled by the parameters of the CRYSTAL (TOLINTEG) program at values of (9; 9; 9; 9; 25).

Nitrogen in diamond occurs at various positions, including in place of carbon (substitutional nitrogen) and is present in various forms of aggregation [17]. Aggregated forms of nitrogen (A and B centers) usually are not associated with the initial defect formation, but are considered as secondary products during a long-term, of the order of billions of years residence (natural annealing) at high temperatures. The aggregation is restricted to the displacement of atoms within separate grains and does not affect the bulk isotope balance of diamond crystals. The present work deals with isotope effects associated with the primary (during diamond growth) entry of nitrogen into carbon sites with the appearance of substitutional nitrogen (C-centers). The possibility of nitrogen atoms being outside the C-center (dislocations, between lattice nodes, etc.) is not considered, as well as the entry of nitrogen in the form ofA-center during crystallization [18]. Thus, carbon atoms were replaced by nitrogen atoms during calculations and the most energetically advantageous configurations of atoms in the diamond structure with C-centers have been determined, followed by the phonon spectra determination for nitrogen isotopologues and calculation of β-factors for substituted nitrogen. To adequately display real nitrogen concentrations, the method of expanded cells (super cells) with the number of carbon atoms n = 16; 32; 54; 128 (replacing one carbon atom with nitrogen atom in the cubic lattice) was used. The convergence of the results is achieved at n = 32-54 (corresponding to 20,000-30,000 ppm and exceeds maximum nitrogen concentrations in natural diamonds). Further volume increase of expanded cells leads to large costs of computing resources without any significant improvement of the results (including values of β-factors).

Calculations of β-factors for molecules representing possible fluid components during diamond formation (the set of molecules such as NH3, NH4+, and N2 has been accomplished by NH2, CH3N, C2H6 in the work) included a sequence of nuclei coordinate optimization followed by vibrational level determination and calculation of β values. The B3LYP hybrid functional and basis vector sets of the TVZP quality in conjunction with the D3 dispersion correction [19], which minimized the discrepancy between calculated and experimentally determined frequencies, were used for molecular calculations. A more accurate presentation of the experimental spectra was achieved using the scaling factor, SF [20], with the reduction of the calculated frequencies of the main isotopologues to the experimental data together with the corresponding change of the frequencies of the heavier isotopologues.

Values of the β-factors were estimated from the expression (2) for temperatures from 0 to 2,500 °C with a step of 10 °C, the corresponding dependencies were approximated by polynomials x = 106/T2 (K–2): 1,000lnβ=aixi. The results (Table 1) for diamond and molecular species are presented in the form of the 5th degree polynomials (i = 6 for CO2 and NH2). The approximation error in all cases does not exceed 0.02 % at temperatures above 100 °C and 0.04 % at temperatures above 0 °C (coefficient of determination r2 > 0.999).

Results

Table 1 shows the obtained temperature dependencies for determinations of diamond 15N/14N β-factors. Estimates are based on calculations for an expanded cell of 54 atoms. As noted above, further increase of the cell volume (with a corresponding decrease in the concentration of nitrogen in diamond) with the substitution of one atom C => N does not yield any noticeable change in isotopic fractionation factors. The Table 1 also lists newly calculated (NH2, CH3N) as well as previously known (N2, NH3, NH4+) results for probable nitrogen-containing components of mantle fluids, together with the results for carbon β-factors of diamond without nitrogen impurities as well as β-factors of fluid-carbon compounds. The presented data can be used to deduce (see expression (1) isotopic fractionation factors for nitrogen (α15/14) and carbon (α13/12) between diamond and fluid components possible during crystallization. Hereafter, isotopic fractionation factors are presented in the form of the conventional values Δ = 1,000 ln(α).

Table 1

Temperature dependences of 1000 lnβ to calculate nitrogen and carbon fractionation between diamond and fluid

Compound

a1

a2

a3

a4

a5

a6

Source

15N/14N

Diamond

13.13525

+0.05589P

–0.454961

–0.00471P

0.023558

+0.00038P

–0.000965

–1.9619E–5P

1.92024E–5

+4.3816E–7P

This work

N2

15.27791

–2.088265

0.248047

–0.015932

0.0004036

[21]

NH3

13.91689

–2.437494

0.339134

–0.023827

0.0006369

[21]

NH4+

18.40726

–3.003117

0.397865

–0.027115

0.0007104

[21]

Diamond (CN)

10.97264

–1.275805

0.138451

–0.008467

0.0002086

[21]

NH2

8.293659

–1.827601

0.351646

–0.038804

0.0021684

–4.751E–5

This work

CH3N

12.69071

–1.384621

0.155546

–0.009723

0.0002388

This work

13C/12C

Diamond

20.931829

+0.10803P

–0.859144

–0.010389P

0.048253

+9.129E–4P

–0.0020401

–4.8886E–5P

4.09635E–5

+1.115E–6P

This work

Diamond

21.649

0.10768P

–0.9790

–0.0097P

0.052834

0.00079P

–0.001879

–3.7385E–5P

0.30747E–4

7.6468E–7P

[22]

Diamond

19.856

–0.4086

[23]

Diamond

20.398968

–0.858690

0.050281

–0.002209

4.56732E–5

[24]

Diamond

19.949068

+0.08177P

–0.622300

–0.00609P

0.014645

+0.00024P

[6]

CO32–

24.74146

–1.08996

0.03178

This work

CO2

31.32194

–3.969173

0.485570

–0.032037

0.0008268

This work

CO2

30.48662

–4.092939

0.620806

–0.061011

0.0032451

–7.027E–5

[13]

CH4

20.23303

–2.915760

0.371371

–0.024974

0.0006513

[9]

CH4

18.81124

–1.818667

0.136254

–0.005159

0.0000743

[25]

CO

17.00663

–1.647067

0.122574

–0.004226

3.76328E–5

[26]

C2H6

20.71712

–2.397034

0.275367

–0.017105

0.0004143

This work

C2H6

19.84650

–1.631380

0.115636

–0.004252

0.0000601

[25]

CH3N

22.33125

–2.555513

0.291656

–0.018280

0.0004487

This work

Notes. The results recalculated in the form of of x polynomials [9, 21]; P, GPadependence is estimated according to [22, (9)]; determinations based on the density of states (DOS) obtained from inelastic neutron scattering [24]; DFT calculations at 0 < P < 80 GPa [6]; experimental data [26]; temperature range: 0 < T < 1,000 °C [23]; 300 < T < 1,200 °C [26]; 0 < T < 800 °C [25]. Hyphen – not calculated.

Fig.1. Isotopic fractionation of nitrogen between diamond and fluid

1-4, 7 – isotope fractionation factors, determined by DFT; 5, 6, 8 – isotope fractionation when modelling CN diamond bonds by CN molecule

In the temperature range within 1,000 and 1,400 °C (probable conditions of the lithospheric diamond formation) isotopic fractionation factors estimated from the calculated values of β are shown in Fig.1 in comparison with previously obtained results of the theoretical calculations. Such calculations of diamond nitrogen isotopic fractionation have been based so far on β-factors of molecules, which are assumed to represent C-N bonds in diamond. The molecules of HCN and CN are considered as the analogues [9]. At 1,127 °C, Δ15N between HCN molecule (diamond) and fluid is –2.3 (for NH3) and –1.1 ‰ (for N2). According to the results of theDFT calculations, isotopic fractionation between diamond and NH3 molecule is nearly absent (–0.01 ‰ at 1127 °C), and between diamond and N2 equals, given the same temperature –0.76 ‰. The results differ by –2.3 and –0.3 ‰. A decrease of 15N/14N occurs in the series of compounds NH4+ > N2 > (diamond, NH3) > CH3N > CN > NH2. At temperature of 1,100 °C, fractionation between diamond and the above N-containing fluid compounds is estimated at –2.23, –0.77, 0.01, 0.44, 1.31 and 2.85 ‰, respectively, and significantly (over 1 ‰) exceeds estimates based on C-N bond in diamond by analogy with CN molecule.

Significant discrepancies in the substitution of nitrogen bonds in diamond with the molecular bonds are due to substantially different strength and nature of the bonds (which is manifested, for example, in the parameters of vibrational spectra). Isotopic fractionation with condensed phases is different from that of gases with the same composition [27], so that nitrogen in diamond can concentrate the heavy isotope more than gas molecules HCN or CN. Consi-dering CNmolecule as a model of nitrogen isotopic fractionation, the positive value of Δ15N between diamond and fluid corresponds to N-containing fluid compounds which are common for upper mantle, but have not yet been identified by either experimental or theoretical studies. Thus, calculated β-factors of HCN and CN molecules can be considered at best only as very rough approximations for estimating diamond isotopic fractionation [9]. For a number of nitrogen and carbon compounds that may be specific to the sublithospheric mantle, the values of β-factors are already known (Table 1). Our calculations show that nitrogen β-factors of NH2 are significantly smaller than those of other molecules under consi-deration, so that even at temperatures exceeding the lithospheric values, there is a noticeable enrichment of diamond by 15N isotope relative to the equilibrium fluid (about 2 ‰ at equilibrium fractionation diamond-NH2, T = 1,400 °C). Isotopic fractionation of carbon between diamond and C2H6 molecule roughly corresponds to the fractionation with CH4 (differ by ≈ 0.2 ‰, T = 1,400 °C). These compounds can be considered as possible fluid components provided sublithospheric diamond origin.

Isotopic fractionation of nitrogen between diamond and fluid under mantle conditions is considered to exceed significantly carbon isotopic fractionation with most of the known carbon-containing diamond forming fluids [21]. Our results demonstrate that at a temperature of 1,200 °C, diamond nitrogen β-factors are reduced relative to the molecules of N2 and NH4+ by 0.8 and 1.0 ‰, respectively, but exceed β-factors of CH3N and especially, NH2 by 0.3 and 2.4 ‰. Fractionation of carbon isotopes between diamond and fluid species at the same temperature increases from negative values of –3.5 (CO2), –1.8 (CO32–) and –0.4 ‰ (CH3N) to positive 1.1 ‰ (CH4).

Pressure effect

Available calculations of β-factor on pressure dependencies [28, 23] demonstrate significant (about 0, n ‰) increase in diamond β13C at T ≈ 1,000-1,200 °C and P > 10 GPa. Within the framework of the DFT method, pressure effect can be determined by β-factor variations with the lattice volume (within the quasi-harmonic approximation – QHA), including dependence of the volume V on polynomial coefficients 1,000 lnβ with variable x = 106/T2. All polynomial coefficients linearly (r 2 > 0.99) depend on diamond volume with a decrease of V/V0 to 0.88 (which approximately corresponds to a pressure up to 110 GPa):

Δa1V/ΔV24.701;Δa2V/ΔV2.08287;Δa3V/ΔV0.16878;(3)Δa4V/ΔV0.00867;Δa5V/ΔV0.00019(N15/N14);Δa1V/ΔV47.795;Δa2V/ΔV4.5919;Δa3V/ΔV0.40351;(4)Δa4V/ΔV0.021607;Δa5V/ΔV0.0004926(С13/С12).

Thus, the following decomposition is valid

ΔaiΔaiΔVVPΔP+122VP2ΔP2+....(5)

Using the definition of the isothermal bulk modulus of elasticity KT = –V∂P/∂V and confining to the second-order members, the relation (5) can be represented as

ΔaiΔaiVΔV1KΔP+121K2KP+1ΔP2+....

For diamond, V0 = 3.416cm3, K = 442 GPa, ∂K/∂P = 3.61 (our QHA calculation results, which are almost identical to the experimental data [29]: K = 445 GPa, with ∂K/∂P = 4). In general, change in the coefficients ai as a function of pressure (Table 1) is determined by the expression

ΔaiPVΔai/ΔV0.00226ΔP+1.1799105ΔP2.

At T = 1,000 °C, carbon β-factor increases by 0.6 (given P = 10 GPa), and the nitrogen factor increases by 0.33 ‰. At T = 1,700 °C, the change in carbon β-factor is 0.25 (10 GPa), and that of nitrogen 0.14 ‰. Thus, the dependence of β-factors on pressure for nitrogen isotopes is approximately 2 times lower than that for carbon. For control, the data on the change in the lattice constant of diamond during the substitution of carbon isotopes can be used. In particular, at T = 25 °C (∂β/∂P)T = 0.619 (GPa–1), and the calculated value of the change in the coefficient of thermal expansion, α at isotopic substitution (Δα)/α = ΔV/3V = –(∂β/∂P)TRT/(3V) is 1.50E–4, which is equal to the experimental value [23]. The obtained diamond β13C or β15N dependencies on pressure are insufficient however, to determine the isotope fractionation between diamond and fluid, since for high-density fluids the assessment of the effect of P on β-factors of mineral-forming solutions is still a challenge.

β-factors considered determine the values of isotope fractionation during diamond formation for the volume (bulk) properties of the crystal lattice under conditions of equilibrium with the growth medium. The values of β can decrease significantly (up to several permille) in natural nanocrystalline diamonds (the size effect). For example, based on phonon density of states (PDOS) from the results of inelastic neutron scattering on diamonds with different size, isotopic shifts of carbon β-factors 103lnβbulk-103lnβnano relative to the bulk values were estimated [24, Fig.6]. For nanodiamonds, the change of the coefficient Δa1 was estimated from the approximate digitization of the lines [24, Fig.6], corrected for the deviation of the phonon DOS from the parabolic ratio due to surface inclusions. Interpolation of Δa1 value for nanodiamonds by linear dependence on size:
Δa1 » 1.7768 – 0.0092×size, nm with the coefficient of determination r2≈ 0.99. Shifts of 1,000 lnβ13С for nanodiamonds at 5 nm Δa1 – 1.799; at 40 nm Δa1 – 1.321; at 170 nm Δa1 – 0.227 ‰. Given the diamond size of 5 nm, isotopic ratio shifts due to surface effects can reach –1 ‰.

It is possible to assess the surface effects due to the preferential development of certain facets during diamond growth and the formation of sectorial (including isotopic) zoning [30]. The composition of the growing crystal is a consequence of the competition of growth and diffusion at the surface (for example, GEM is a model of growth capture [31]). The crystal inherits the composition of the growth surface at low diffusion at the surface, which can be characterized by the Peclet number:

Pe=r·0.5L/D,

where r – is the growth rate, mm/s; L – is the thickness of the surface layer, nm; D is the diffusion coefficient of the element (or isotope) under consideration. The growth face is represented by a periodic plate with the number of layers LS. The inner layer is determined by the structure of the gross diamond, the outer LS-1 layers are reconstructed [32]. The parameters of the CRYSTAL calculation and sets of basis vectors are the same as during calculations on the bulk (gross) structure. Change of 1,000 lnβ relative to the gross values of iai0aixi; ai0 – are given in Table 1.

Table 2

1000 lnβ for diamond growth facets

Edge Index*

LS

a1

a2

a3

a4

a5

15N/14N

111

2

12.14579

–0.59010

0.03920

–0.00176

4.190E–5

4

12.93791

–0.60624

0.04139

–0.00204

4.621E–5

110

2

10.99022

–0.52717

0.02365

–0.00097

1.948E–5

4

12.70019

–0.69832

0.04544

–0.00304

3.966E–5

100

2

8.66005

–0.20426

0.01380

–0.00129

9.954E–7

4

11.30632

–0.41044

0.02322

–0.00102

2.129E–5

6

12.67306

–0.44731

0.02390

–0.00100

2.015E–5

13C/12C

111

2

19.98622

–1.11557

0.08240

–0.00415

9.234E–5

4

20.50007

–1.17320

0.08803

–0.00447

9.997E–5

6

20.82774

–1.18265

0.08748

–0.00432

9.623E–5

110

2

18.23049

–0.93891

0.05031

–0.00205

4.197E–5

4

20.55262

–1.14874

0.08495

–0.00428

9.542E–5

100

2

14.19463

–0.44483

0.02006

–0.00074

1.364E–5

4

18.91818

–0.81383

0.04992

–0.00227

4.770E–5

6

20.02506

–0.85221

0.04975

–0.00216

4.402E–5

* Miller's index.

The key parameters of the model can be controlled by experimental observations but in most cases, they are not determined under natural conditions [32]. Since the volume β-factors characterize growth at equilibrium (Pe << 1), the values of β-factors for individual faces (Table 2) provide possibilities to estimate deviations from ideal conditions.

Comparison with natural 15N/14N fractionation

In the conditions of isotopic equilibrium between growth medium (homogeneous fluid) and the surface of the produced diamond (according to the Rayleigh mechanism), linear relationships can arise between ratios 13C/12C, 15N/14N and the logarithm of nitrogen concentration ln(N) which are noted in numerous works [21, 33, 34]. The observed covariances both within individual diamond grains and among genetically related diamonds enable estimates “natural” isotopic nitrogen fractionation. The regression slope between genetically related diamonds in coordinates is determined by the equality

d13Cvsd15N=Δ13C/(Δ15NKN),

where KN – is the nitrogen partition coefficient between diamond and fluid; Δ = 1,000 lnα, α – being factors of equilibrium isotope fractionation of carbon 13C/12C or nitrogen 15N/14N between diamond surface and growth medium [21]:

δ15Nδ15N0=Δ15lnfN=Δ15KNlnf,

f – fraction of fluid consumed; fN – fraction of fluid nitrogen species consumed. Provided f = fC (the fraction of fluid consumed equals the fraction of carbon species in the fluid), the following expression is valid

δ13C=δ15Nδ15N0Δ13/Δ15KN+δ13C0,

where δ13C0 or δ15N – diamond isotope composition at the initial growth stage. The value of KN is defined by the dependence of δ13C on nitrogen content in diamond ln(N),

δ13C=lnNlnN0Δ13/KN1+δ13C0.

The above expressions are restricted to fluids dominated by carbon species although can be genera-lized for fluids of mixed composition. For example, 15N/14N fractionation between diamond and growth media was estimated at −4.0 ‰ (given T = 1,100 °C for a fluid dominated by CO32− carbonate ion, Δ13C » −1.7 ‰, KN = 4.4, Jericho kimberlite [33]). Data on the bulk isotopic composition of carbon and nitrogen in diamonds from a lherzolite xenolith (representing the lithospheric mantle) were used to estimate Δ15N value at +1.2 ‰ (T = 1,200 °C, provided the methane-rich fluid [35]). Data on ultra-deep diamonds from Guinea, with a crystallization temperature of about 1,725 °C, were applied to determine Δ15N between diamond and fluid (–4.5 ‰) in the conditions of crystallization from a fluid dominated by CO32–, Δ13C = –0.9 ‰ [36].

Isotopic fractionation factors of nitrogen in diamonds, Δ15N estimated from natural covariances δ13C-δ15N-ln(N) depend, in addition to the choice of Δ13C values, on the balance of fluid components as well as other parameters, including the fractionation model and the accepted crystallization temperature. Despite numerous assumptions and speculations especially regarding the values of KN, in the absence of other calibrations, this method has so far been practically the only way to estimate Δ15N between diamond and fluid.

The results of Δ15N/Δ13C estimates by covariances from the natural genetically related samples can be compared with empirical dependencies and fractionation factors determined within the work. 64 series of conventionally cogenetic diamonds have been identified among the available dataset [9, 37] with the significant correlation Δ15N vs Δ13C (at a confidence level of 1σ), with the derived distribution of Δ15N/Δ13C ratio (Fig.2, solid red line). At 2σ confidence level, the number of the series decreases (57 series), with no significant change in the distribution (the main maximum of Δ1513 of about 0 and local peaks around 1 and 2). The representativeness of each series was assessed using the Student’s t-distribution. The lines represent theoretical (see Table 1) Δ15N/Δ13C values among diamond and fluid of different composition calculated at temperatures of 800 to 1,800 °C. Composition of nitrogen species shown as the numerator, carbon species – as the denominator. Solid lines – assuming independent fractionation from KN = 1, dashed KN = 5, double dashed line KN = 2.5. Double dashed green line – NH3/CO2, purple – N2/CO2, blue – N2/CO3. The figure was constructed with no use of the “empirical” fractionation factors estimated from natural nitrogen and carbon isotopic distributions. Most of the Δ15N/Δ13C ratios fall within the values of –1.5 to 0.5 (with a maximum of about –0.3). Close to zero Δ15N/Δ13C ratios may reflect the predominance of NH3 as nitrogen-containing fluid compound during diamond genesis (Δ15N close to zero) and/or the dominance of carbon-containing component with relatively high fractionation of 13C/12C during diamond growth (CO2, CO3). CO2+N2 or CO3+N2 dominated fluids have positive Δ15N/Δ13C ratios. Provided two-component carbon source, the maximum distribution of Δ15N/Δ13C may represent isotopic exchange of diamond with CO2 or carbonate ion mixed with CH4, with N2 dominating as the nitrogen source. Evolved Δ15N/Δ13C ratios (local maxima of the Δ15N/Δ13C ≈ ratio at 1.5 and 2.3) may correspond to diamond crystallization from a fluid dominated by NH4/CO2, N2/CO3 or NH2/CH4. C2H6 or CH3N species are possible as a source. Negative Δ15N/Δ13C correspond to a combination of such components as NH4/CH4, N2/CH4 or NH3/CH4 (the latter, at temperatures above 1,250-1,300 °C). It should be noted that temperature increase shifts the equilibrium isotopic ratios of 15N/13C. Empirically estimated values of the Δ15N/Δ13C ratio beyond the range of –4 to +4 may indicate extreme nitrogen partitions between diamond and the growth medium, or the presence of some unconsidered fluid components enriched in either 15N or 13C relative to diamond. Depending on the Δ15N/Δ13C ratio, the Rayleigh crystallization within the cogenetic series of diamond may be accompanied by the increase of d15N, by the decrease of this value or by the lack of any trend.

Fig.2. Frequency distribution of Δ15N/Δ13C ratio in a series analysis of natural diamonds

Conclusion

Isotopic fractionation factors have been relatively well established for carbon in many systems related to diamond genesis. The distribution of nitrogen isotopes between diamond and the mineral-forming fluid is, however much less studied and is mainly based on empirically observed trends from natural diamonds, yielding significant uncertainties in estimates and the need for more rigorous constrains [11]. The reliable values of nitrogen isotope fractionation factors should be a prerequisite for revealing the causes of the observed isotopic heterogeneity of diamond, the nature of covariances in δ13C-δ15N-N values, the composition of fluids and, possibly, other conditions during diamond formation. In this paper, an attempt is made to quantify the fractionation of nitrogen isotopes “from first principles” based on crystallo-chemical properties of diamond with nitrogen substitution and nitrogen-containing fluid species.

In particular, the “frozen phonon” method within the density functional theory using the full-electron bases was adopted to determine the sets of β-factors (15N/14N, 13C/12C) of diamond in harmonic and quasi-harmonic approximations at temperatures from 0 to 2,500 °C. To achieve representativeness of the calculations, the method of expanded cells (with an increase in volume by 8, 16, 27 times) was utilized. In addition, some β-factors of fluid components (including carbon C2H6, CH2NH and nitrogen CH2NH) have been deduced to estimate isotopic fractionations during diamond formation. The results of the ab initio calculations differ significantly from the previously obtained theoretical results based on the representation of the CN bonds of diamond by the molecular compounds HCN or CN. In the conditions of diamond formation, with a volume decrease under pressure, factors of isotopic fractionation of nitrogen and carbon can increase by tenths of permille for carbon. Pressure effects diamond nitrogen isotope fractionation more than carbon isotope fractionation, which in extreme conditions (super cold subduction) can lead to an increase in fractionation factors up to significant values. The volume increase in diamond lattice due to thermal expansion however, partially reduces the effect of pressure, so that the shifts in isotopic ratios are generally determined by the value of the ΔPT gradient. In the “standard” lithospheric conditions of diamond genesis (about 6 GPa and 1,100 °C), factors of carbon and nitrogen isotopic fractionation do not change significantly.

Depending on the dominant nitrogen and carbon substances, fractionation of isotopes during diamond formation can result either in various zonation of isotopic composition within individual grains and different variations within cogenetic series. In most cases (maximum in the Δ15N/Δ13C distribution, Fig.2), nitrogen isotope fractionation is negligible (Δ15N << Δ13C), which is consistent with calculations of the fractionation factors between diamond and NH3-CO2 or N2-CO2 fluids. Shifts in the Δ15N/Δ13C ratio to the negative values may indicate reduced conditions at high pressures and the “cold” geotherm [38]. In nitrogen-poor reduced mantle fluids [39], amines (NH2) or methanimine (CH3N) may play an important role and their dominance may also correspond to the occurrence of local maxima within the Δ15N/Δ13C distribution. Oxidized fluids dominated by CO2 or CO3 in combination with the N2 component yield positive Δ15N/Δ13C between diamond and fluid and significantly depend on the nitrogen partition coefficient ratio.

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