Matthieu David

Aurélien Prillieux

Daniel Monceau

Damien Connétable
email: damien.connetable@ensiacet.fr

First-principles study of the insertion and diffusion of interstitial atoms (H, C, N and O) in nickel

Keywords: Nickel, interstitial species, DFT, diffusion, solubility

Solubility and diffusivity of interstitials in Ni is of prime importance to understand and quantify reactivity of Ni-base structural alloysat high temperature. We thus present a first-principles study of the insertion and diffusivity of interstitial species in nickel. We put an emphasis on carbon, nitrogen and oxygen atoms and compare our results with those found for hydrogen and in the literature. The interactions with the metal and the relative stability of sites are discussed in detail in terms of phonon, electronic, elastic, magnetic properties and charge transfers. As a result, we identified one new stable interstitial position for C and N atoms, and we also showed that the tetrahedral site can be considered as unstable from an elastic standpoint for carbon. In the light of these new results, diffusion mechanisms were reviewed and diffusion coefficients were calculated. The effects of temperature on enthalpy and migration energies were investigated. We thus showed that C and N atoms on one hand, H and O atoms on the other hand, show the same behavior.

Introduction

Interstitial elements in Ni based alloys play a major role in many physical phenomena such as environmental embrittlement, cementation, precipitation, phase transitions or changes in physical properties. Hydrogen is known to be at the origin of the embrittlement of many metals [START_REF] Birnbaum | Mechanisms of hydrogen related fracture-a review[END_REF]. C atoms can lead to carburation and to metal dusting [START_REF] Young | Recent advances in understanding metal dusting: A review[END_REF], it can be involved in the formation of carbides with many species [START_REF] Connétable | First-principles study of transition metal carbides[END_REF], thereby inducing mechanical reinforcements of nickel alloys, and segregate at grain boundaries and interfaces. Oxygen forms oxides and cause internal and intergranular oxidation [START_REF] Molins | Oxidation effects on the fatigue crack growth behaviour of alloy 718 at high temperature[END_REF][START_REF] Young | High temperature oxidation and corrosion of metals[END_REF] which may lead to stress corrosion cracking [START_REF] Molins | Oxidation effects on the fatigue crack growth behaviour of alloy 718 at high temperature[END_REF]. N atoms are a solid solution strengthening element for steels [START_REF] Iorio | Technical Note: Solubility of nitrogen in experimental Iow-nickel austenitic stainless steels[END_REF] and form Ti and Al nitrides in nickel based super-alloys [START_REF] Nabavi | Nitrogen Effect on the Microstructure and Mechanical Properties of Nickel Alloys[END_REF]. Understanding the behavior of interstitial atoms in metallic systems is a key step in the knowledge of micro-structural evolution. The amount and kinetics of elements insertion and of induced precipitation is controlled by C, H, N, and O permeability [START_REF] Young | High temperature oxidation and corrosion of metals[END_REF]. It is thus crucial to accurately determine the solubility and diffusivity of species responsible for the physical phenomena that are being studied.

For interstitial species, the lattice composed of atoms, here Ni, can be seen as imposing a fixed energy landscape in which these atoms are inserted and diffuse more or less freely. However, contrary to what was done in earlier works [START_REF] Siegel | First principles study of solubility diffusion and clustering ofc in ni[END_REF][START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF], a stronger focus should be put on the stability of interstitial sites by studying the full inter-atomic constants when interstitial sites are filled, as it was recently done in the case of aluminum [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF].

Stable configurations strongly depend on interactions between the metal and interstitial species, leading to different stable positions, as found in earlier literature. This information is a prerequisite for studying diffusion mechanisms at the atomic scale. Indeed, in the case of interstitial elements, the elementary mechanisms, which are supposed to be limited, reveal a complex landscape depending on the specie [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF][START_REF] Wu | Direct diffusion through interpenetrating networks: Oxygen in titanium[END_REF], especially when atoms interact strongly with the metal.

The aim of this work is thus twofold. First, we show that interstitial atoms in fcc systems are not necessarily located in usual interstitial sites and that special attention should be paid to phonon properties. We also discuss the use of elastic modeling for interstitial species. Secondly, we show that the diffusion coefficient formulas classically used do not always apply. Hereinafter, the study is limited to "interstitial" positions, the formation of clusters [START_REF] Wang | Hydrogen influence on diffusion in nickel from first-principles calculations[END_REF][START_REF] Wang | Influence of trap connectivity on h diffusion: Vacancy trapping[END_REF] is not considered in the present work.

The first step consists in the investigation of stable configurations using first-principles approaches. An accurate analysis of stable sites was conducted including a new stable position called M. Previous works (on carbon [START_REF] Siegel | First principles study of solubility diffusion and clustering ofc in ni[END_REF][START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF], oxygen [START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF][START_REF] Fang | First-principles studies on vacancymodified interstitial diffusion mechanism of oxygen in nickel, associated with large-scale atomic simulation techniques[END_REF] or hydrogen [START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF]) have shown that octahedral (o) sites are the most stable positions, that tetrahedral sites (t) are the second most stable, and that mechanisms of diffusion involve o-t and o-o paths. We thus take this opportunity to present an overview of the insertion of single species and of the interactions between the metal and interstitial species. We try to show how interstitial species interact with nickel at the atomic scale, but also at a larger scale by computing elastic dipoles and electronic properties.

Based on these results and structure analyses, we studied the escape rate of different paths, i.e. trajectories, barriers energies and frequencies attempts were investigated. In earlier literature [START_REF] Siegel | First principles study of solubility diffusion and clustering ofc in ni[END_REF][START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF][START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF] simple mechanisms were considered, without discussing in detail the stability of sites. We thus attempt to revisit the diffusion coefficients of C, O and N atoms in nickel considering the M position (see Ref. [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF]) and including phonon energy by means of full vibrations of the systems. The effect of temperature on diffusion coefficients were also evaluated within the quasi-harmonic approximation, at several temperatures based on lattice expansion calculations, as explored by Wimmer [START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF] for H atoms in nickel.

For this purpose, the article is organized as follows. The first part (section 2) summarizes the simulation method. In the section 3, insertion in Ni was investigated: stability of the interstitial, with a particular emphasis on interactions between the metal and interstitial on the different sites. Finally, in section 4, we present a study of the diffusion mechanisms at the atomic scale. The diffusion coefficients based on explicit formulas is therefore presented and compared to experimental findings. The effects of temperature are presented and results are compared to those found in the literature.

Computational details

First-principles calculations were performed with the density functional theory (DFT) using the VASP code (Vienna ab initio simulation package) [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF]. The Perdew-Burke-Ernzerhof [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] generalized gradient approximation of the exchange and correlation functional was used for all calculations, and the PAW method [START_REF] Blöchl | Projector augmented-wave method[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] was adopted. The magnetism was taken into account in all calculations (for bulk and references states as well).

Calculations were performed at zero pressure, lattice parameters and shapes of super-cells were fully relaxed. Atomic forces were always lower than 0.01 eV/Å on atoms. The planewave cut-off energy was set to 600 eV. The study of the insertion (in the dilute solution) were carried out on two supercells (2×2×2 and 3×3×3 super-cells, i.e. 32 and 108 atoms per unit-cell, respectively) with full periodic boundary conditions. The first-Brillouin zones were sampled by 10×10×10 and 6×6×6 Monkhorst-Pack k-mesh grids [START_REF] Monkhorst | Special points for brillouin-zone integrations[END_REF]. The precision reached was about 2-3 meV/atom. The insertion energies of the largest super-cells were considered sufficiently accurate to correspond to the energy of isolated species (dilute limit).

The frequencies and inter-atomic force constants (IFC) were computed using the finite displacements method on 2×2×2 supercells, to limit numerical costs, for the interstitial positions and transitions states. The phonopy package [START_REF] Togo | First-principles calculations of the ferroelastic transition between rutile-type and cacl 2 -type sio 2 at high pressures[END_REF] was used to generate finite displacements according to the symmetry of each structure. It was then used to analyze, to plot phonon properties (phonon dispersion curves and density-of-states, not always shown here) and to compute Helmholtz free energies. A cut-off energy of 600 eV, an electronic convergence criteria of 10 -6 eV and fine k-mesh grids (10×10×10) were used.

Migration paths were investigated on 2×2×2 and 3×3×3 super-cells. We used the climbing-image nudged elastic band method (CI-NEB) with the formulation suggested by Henkelman group [START_REF] Henkelman | Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points[END_REF], using an optimizer [START_REF] Sheppard | A generalized solid-state nudged elastic band method[END_REF] for more efficiency while searching for the minimum energy path (MEP). The main advantage of the CI-NEB method is that it is possible to calculate the transition state (TS) and optimize the volume shapes at the same time. Reaction paths were sampled with five intermediate images. MEP identification was performed through two steps: (i) first, pre-convergence was achieved using a less precise parameter set (3×3×3 k-mesh grids and a cut-off energy of 400 eV); and (ii) energies were then performed with the same convergence parameters as for the study of point defects (cutoff energy of 600 eV and 10×10×10 k-mesh grids, for 2×2×2 super-cells).

Finally, thermal effects were partially incorporated, taking into account the thermal expansion of the lattice within the framework of the quasi-harmonic approximation. We assessed thermal effects on migration and solubility energies. The lattice expansion method was used to include thermal effects. We assumed in this work that interstitial elements do not induce specific lattice distortions with temperature: we therefore used the lattice thermal expansion of the perfect nickel fcc structure. The lattice parameters were calculated using the expression proposed by Wimmer et al. [START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF]:

a(X i , T ) = a o (X i ) + 2.774 × 10 -5 T + 2.282 × 10 -8 T 2 -5.243 × 10 -12 T 3 + 3.629 × 10 -16 T 4 (1)
where a o (X i ) (in Å) is the lattice constant of one super-cell containing a fraction of X atom (X=C, N or O) occupying the i site (o, t or M). 3×3×3 super-cells (108 Ni atoms + 1 X atom) were used to compute formation energies, whereas Helmholtz free energies were computed on 2×2×2 super-cells. The lattice parameters of the intermediate super-cells constituting the NEB were thus interpolated between initial and final configurations as is usually done with this method. Calculations were thus conducted at 600, 1200, 1500 and 1800K.

Solubility

Stability at 0 K

The insertion energy, E s [X i ], is used to quantify the insertion of an atom X in nickel in the site i:

E s [X i ] = E o [n.Ni + X i ] -E o [n.Ni] -E re f [X] (2) E o [n.Ni + X] and E o [n.
Ni] are the DFT values of super-cells (composed of n Ni atoms) with and without an interstitial element, respectively. E re f [X] corresponds to the energy (per X atom) of the reference state of the X atom (molecule or diamond, see Appendix A). A positive (resp. negative) E s [X i ] value corresponds to the energy gained (resp. lost) when the X atom occupies an i site. We considered octahedral (in 4b position in the fcc system, space group 225) and tetrahedral (8c) sites. Results of simulations are listed in Table 1. Details below show that results obtained using the PBE functional at 0K are comparable, quantitatively and qualitatively, with previous works [START_REF] Siegel | First principles study of solubility diffusion and clustering ofc in ni[END_REF][START_REF] Wang | Influence of trap connectivity on h diffusion: Vacancy trapping[END_REF][START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF][START_REF] Metsue | Hydrogen solubility and vacancy concentration in nickel single crystals at thermal equilibrium: New insights from statistical mechanics and ab initio calculations[END_REF][START_REF] Subashiev | Hydrogen trapping at divacancies and impurity-vacancy complexes in nickel: First principles study[END_REF][START_REF] Connétable | Impact of the clusterization on the solubility of oxygen and vacancy concentration in nickel: a multi-scale approach[END_REF]. First, as found in earlier works [START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF][START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF], all interstitial species prefer to occupy the octahedral positions (o sites) of the nickel fcc-lattice. The energy difference, ∆H, (including the zeropoint energy, see section 3.2) between o and t sites is equal to 0.34, 1.54, 1.05 and 0.20 eV for H, C, N and O, respectively. Results are in good agreement with earlier theoretical works: about 0.25 eV in the case of H [START_REF] Wang | Influence of trap connectivity on h diffusion: Vacancy trapping[END_REF][START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF][START_REF] Metsue | Hydrogen solubility and vacancy concentration in nickel single crystals at thermal equilibrium: New insights from statistical mechanics and ab initio calculations[END_REF][START_REF] Subashiev | Hydrogen trapping at divacancies and impurity-vacancy complexes in nickel: First principles study[END_REF], 1.53 eV for carbon [START_REF] Siegel | First principles study of solubility diffusion and clustering ofc in ni[END_REF][START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF] and 0.17 eV for O atoms [START_REF] Connétable | Impact of the clusterization on the solubility of oxygen and vacancy concentration in nickel: a multi-scale approach[END_REF]. From these energies, it is possible to distinguish two groups: on one side, H and O atoms, where ∆H is low (<0.5 eV), and on the other side, C and N atoms, where ∆H is high, >1 eV. This classification will be confirmed considering the following results.

It had been shown in a recent work on aluminum [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF] that there is another possible stable position in fcc systems: the M site (with the Wyckoff position 24d), see Fig. 1. This configuration, located exactly at the midpoint of two first-nearestneighboring Ni atoms, is also located at the midpoint of two o and two t sites. It had been identified as being a (second-order) transition state in the o-o path for hydrogen in many fcc-metals [START_REF] Connétable | Diffusion of interstitial species (h and o atoms) in fcc systems (al, cu, co, ni and pd): Contribution of first and second order transition states[END_REF], although, as seen in Al, M is a stable position for C and B atoms. We therefore investigated this additional site and calcu- lated its insertion energy for all species. DFT values are given in Table 1. Results show that this site is always stable from a geometrical standpoint. The M site is more stable than the t site for C atoms, almost 100 meV lower in energy, while in the case of H, O and N atoms, it is found less stable than t sites (about 0.5 eV). At this level, it is not possible to rule on their stability, i.e it is not possible to know if it is local minimum or maximum. Substitution was also considered, but in all cases, it was found unstable. The atoms move from the ideal position and prefer to form clusters as shown in the case of H [START_REF] Wang | Influence of trap connectivity on h diffusion: Vacancy trapping[END_REF][START_REF] Metsue | Hydrogen solubility and vacancy concentration in nickel single crystals at thermal equilibrium: New insights from statistical mechanics and ab initio calculations[END_REF][START_REF] Nazarov | Ab initio study of h-vacancy interactions in fcc metals: Implications for the formation of superabundant vacancies[END_REF] or of O atoms [START_REF] Connétable | Impact of the clusterization on the solubility of oxygen and vacancy concentration in nickel: a multi-scale approach[END_REF]. In fine, these new results do not change the physics of H, C, N and O insertion in the nickel, since the most stable positions are always the o sites.

DFT values can now be compared against experimental ones. Experimentally, the solubility energy is deduced by fitting the content of species measured with an Arrhenius law or a Sieverts law for dilute solutions. The Sieverts law [START_REF] Sieverts | The absorption of gases by metals[END_REF] relies the amount of species (here hydrogen, C[Hy]) dissolved in the system and the temperature:

C[Hy] ∝ p H 2 p o exp - H s k B T , (3)
where p o is equal to 1atm, p H 2 the pressure of the gas, and H s the solubility energy. It corresponds to the situation where there is a thermal equilibrium between a solid solution (H in insertion) and H 2 gas: Ni + 1/2xH 2 NiH x (x 1). DFT values of H s can thus be directly compared with experimental values. Taking into account the experimental and simulation uncertainties, the result for hydrogen in the o site (+0.07 eV) is in agreement with experimental data (+0.17 eV) [START_REF] Fukai | The metal-hydrogen system: basic bulk properties[END_REF]. This reflects an equilibrium without hydrides or clusters formed in the metal. It should be noticed that the content of V H n clusters is always negligible in Ni, for common experimental conditions, see [START_REF] Tanguy | Stability of vacancy-hydrogen clusters in nickel from first principles calculations[END_REF].

About the O atom, Connétable et al. [START_REF] Connétable | Impact of the clusterization on the solubility of oxygen and vacancy concentration in nickel: a multi-scale approach[END_REF] have already shown that there is a strong disagreement between theoretical and experimental values: at 0K, calculations give 0.46 eV, whereas experimental measurements range between [-2.5; -1.8] eV depending to the experimental conditions [32][START_REF] Park | The diffusion and solubility of oxygen in solid nickel[END_REF][START_REF] Prillieux | Hydrogen and water vapour effects on oxygen solubility and diffusivity in high temperature fe-ni alloys[END_REF]. This discrepancy is in part due to the reference state (here the O 2 molecule) which is badly described in DFT due to an overestimation in the GGA of the O 2 molecule binding. Wang et al. [START_REF] Wang | defauts ce@ y3al5o12[END_REF] have therefore applied an energy shift of 1.4 eV per O 2 molecule to reproduce experimental value of the formation energy of oxides. This value, associated with temperature effects on the chemical potential, could partially reconcile experimental and theoretical values. However, there are other arguments to explain this disagreement. The formation of VO m clusters could also explain this difference. As shown in [START_REF] Connétable | Impact of the clusterization on the solubility of oxygen and vacancy concentration in nickel: a multi-scale approach[END_REF], O atoms strongly interact with nickel to form clusters. A thermodynamic analysis showed that, even at high temperature, O atoms are mainly located in these clusters and are not isolated in nickel. The DFT value was computed in the dilute limit. The Sieverts law is also used in the case of O atoms to determine H s [START_REF] Park | The diffusion and solubility of oxygen in solid nickel[END_REF]. However, measurements were often made under experimental conditions such that the pressure p O 2 is sufficient to form NiO oxide layers on the surface [32,[START_REF] Park | The diffusion and solubility of oxygen in solid nickel[END_REF]. The chemical potential of the O atom is thus not the gas but the oxide, and the Sieverts' law should be thus not valid. These arguments let us think that the link between theory and experimental observations is hard to reconciliate.

In the case of C atoms, there are few reported values of solubility in pure nickel. The solubility of C has been determined by Dunn et al. [START_REF] Dunn | The solubility of carbon in cobalt and nickel[END_REF] and values reported are in the magnitude order of 10 -2 atomic fraction near the melting point. Insertion energies reported are thus in the range of 0.42-0.49 eV (hightemperature measurements of the carbon solubility). These are in good agreement with our results obtained at 0K.

Finally, in the case of nitrogen, there are no experimental data in the nickel crystal. Values found in literature only refer to the solubility of nitrogen in liquid pure nickel and nickel based alloys [START_REF] Abdulrahman | The solubility of nitrogen in liquid pure nickel[END_REF][START_REF] Fedorchenko | Solubility of nitrogen in liquid nickel and in nickel-chromium, nickel-molybdenum, and nickeltungsten melts[END_REF][START_REF] Kojima | Solubility and diffusivity of nitrogen in liquid iron-nickel and iron-cobalt alloys at 1600 • C[END_REF][START_REF] Wada | Solubility of nitrogen in liquid fe-cr-ni alloys containing manganese and molybdenum[END_REF]. In these cases, experimental observations show a low solubility of N in nickel (as for carbon). This must translate into high solubility energy, which is consistent with our result, where the insertion energy in o sites is 0.7 eV.

Vibrational properties

We now discuss the stability of the sites by computing interatomic force constants. Phonon band structures (pBAND) are displayed in Fig. 2. Substituted sites were also studied, but, for simplicity, not represented here. When phonon branches are all positive, the configuration can be considered as stable, otherwise the configuration is unstable. When imaginary frequencies are associated with the interstitial element, with a degeneracy of one, the configuration should probably be near a first-order transition state (for the X atom), and when the degeneracy is double or triple, the configuration does not meet the conditions for being a first-order transition state and is unstable.

In the case of H and O atoms, t and o sites are stable, whereas M sites present two imaginary vibrations, associated with the interstitial elements. It is thus a second-order transition state, see the discussion in Ref. [START_REF] Connétable | Diffusion of interstitial species (h and o atoms) in fcc systems (al, cu, co, ni and pd): Contribution of first and second order transition states[END_REF]. Such configuration plays a negligible effect on the diffusivity, they will be neglected in the following.

In the case of carbon, while M and o sites are stable, we observe an anomaly for t sites. Indeed, when the t site is filled with carbon, pBAND exhibits a small acoustic instability in the Γ-X direction, see Fig. 2. In order to confirm this result, we conducted additional tests on 3×3×3 super-cells, as well as convergence tests. Again, we found this acoustic instability every time. These results suggest that, at least at low temperature, when a carbon occupies a t site, the structure becomes unstable. From a mechanical standpoint, acoustic branches are directly linked to the speed of sound velocity in crystals. They can thus be used to compute elastic constants, C i j , see for instance [START_REF] Connétable | First-principles calculations of carbon clathrates: Comparison to silicon and germanium clathrates[END_REF]. In cubic systems, C 11 and C 44 must be positive. We thus computed them directly with another approach (using small strains on 3×3×3 super-cells) and found that C 44 is negative. One of the three Born stability criteria (for cubic systems) is therefore not verified, see Mouhat [START_REF] Mouhat | Necessary and sufficient elastic stability conditions in various crystal systems[END_REF]. Additional simulations were conducted to find other stable position(s) around the t sites, without success. When the C atom is pushed a little out the t site, towards either the M site or the o site, it always falls back in these positions. The t site is thus not a local minimum for carbon. This result is not in contradiction with the results on diffusion mechanisms found in the literature [START_REF] Siegel | First principles study of solubility diffusion and clustering ofc in ni[END_REF][START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF], where the transition state for the o-t path was always found to be located very close to the t site. It is not excluded that the effect of temperature could stabilize the t configuration, as it has been shown by Mei [43] in cubic vanadium nitride. However, temperatures should not change the fact that this site is less stable than o sites, and that the t site should be located in the vicinity of a transition state.

In the case of the N atom, the results show that all sites are stable, M sites as well.

At this stage, for H and O atoms, o and t sites are the stable positions, for N atoms, there are three configurations, and for C atoms, only M and o sites are stable positions.

From phonon frequencies, zero-point energies (F 0K [X]) were computed using molecules as reference states for H, N and O atoms, nd diamond for carbon (see Appendix A) as follows:

F 0K [X] = F 0K vib [n.Ni + X] -F 0K vib [n.Ni] -F 0K vib [X re f ] (4)
F vib are the phonon Helmholtz free energy computed using IFC on fine q-mesh grids, here calculated at 0K (labeled 0K).

F vib (T ) = k B T 3N ν=1 n ν (q) ln 2 sinh ω ν,q 2k B T dq (5)
ω ν,q are the frequencies in q and ν of the system. F 0K [X] values are given in Table 1. Except H atoms, the F 0K [X] is low compared to the solubility energy. This does not change the relative stability of sites.

Using phonon excitations, we also calculated the mean square displacements ( dT ) of X atoms in each site as a function of temperature [START_REF] Wallace | Thermodynamics of crystals[END_REF], along the [010] direction for o and t sites and along the [010] and [101] directions for M sites. This quantity is used to evaluate the displacement/oscillations of X as a function of temperature:

dT = |u α ( jl, t)| 2 = 2Nm j q,ν (1 + 2n ν (q, T ))| α ν ( j, q)| 2 ω ν,q (6)
j is the index of the j th atom in the super-cell l, α corresponds to a crystallographic direction (here [010] or [101]). m and N are the atomic mass and the number of super-cells. q is the wave-vector, ν is the index of the phonon mode. ν ( j, q) is the associated polarization vector. ν(q, T ) is the Boson distribution:

n ν (q, T ) = exp( ω ν (q)/k B T ) -1 -1 (7)
Data at 0 and 1200K are listed in Table 2. dT are found smaller in o sites than in t sites; frequencies of X atoms in t sites are indeed always higher in energy than those in o sites. The volumes induced by phonon excitations in o and t sites are thus small (< 0.1 Å 3 ), even at high temperature, smaller than the volume of formation for instance (see below). The most interesting cases are for M sites. Vibrations of N and C atoms at 1200 K in M induce a large displacement, approximately 0.2-0.3 Å, along the directions perpendicular to the Ni-Ni bonds (i.e., [010] for C and N atoms will not stay in M sites at high T. They could be considered as unstable at high T.

We finally evaluated, from the slopes of the acoustic branches around the q = Γ point, elastic constants [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. Three directions were used to compute them: [100], [111] and [110]. Results are given in Table 3 for a high concentration in interstitial specie (here 3% atomic, i.e. one X atom for 32 Ni atoms). From these results, we can have a qualitative idea of the effect of insertion on the elastic properties of nickel. Oxygen, and in to a lesser extent, C and N atoms, lead to a decrease in elastic properties. However, we can see that the effect of H atoms is negligible (in the numerical error). )/2. We used the same approach as the one presented in the Ref [START_REF] Connétable | First-principles calculations of carbon clathrates: Comparison to silicon and germanium clathrates[END_REF]. Results for the bulk are compared to the experimental [START_REF] Kittel | Introduction to Solid State Physics[END_REF] and theoretical literature [START_REF] Connétable | First-principles study of the structural, electronic, vibrational, and elastic properties of orthorhombic nisi[END_REF]. The accuracy on the fit is equal to about ± 3GPa. For the bulk, the value of C 11 is in excellent agreement with literature. We strongly underestimate C [START_REF] Wallace | Thermodynamics of crystals[END_REF]

Charge transfers

The effect of insertion on the electronic and magnetic properties of the metal, as well as charge transfers are now going to be investigated. The Bader charge [START_REF] Henkelman | A fast and robust algorithm for bader decomposition of charge density[END_REF] is the first parameter that was examined. Values are listed in Table 2. Except for the H atom, whose the Bader charge transfer (Be, where B and e are respectively the Bader charge and the atomic charge) is lower and equal to about +0.3e, atoms located in o sites attract part of the charge of 1NN Ni atoms. The charge transfer for C, N and O atoms is thus equal to about +0.9, +1.0 and +1.1 e. The charge decrease of 1NN Ni atoms (not given here) is slightly if carbon atoms are located in o sites rather than in t or M sites, for which the charge transfer is similar. In the case of O atoms occupying o or t sites, this charge decrease was found smaller, and for N atoms occupying o and t sites, it was found identical.

To illustrate this charge transfer, figure 3 displays contour maps associated with charge transfers ∆ρ, when o, t and M sites are filled with an impurity, i.e.:

∆ρ = ρ [bulk + X] -ρ [bulk] -ρ [X] (8)
where ρ[Y] are the DFT charge densities of the Y system. The case of H atom is also depicted for comparison purposes. By comparing the charge density difference between the species for a given site (o or t), it can be noticed that the shapes of ∆ρ are similar. The main difference lies in the range of this charge transfer: stronger and more localized for O atoms than for N and C atoms, in agreement with the Bader charge analysis presented above. Nevertheless, the charge transfer is different for H atoms: it is not only significantly smaller than that of other species, but also highly isotropic and at low scale. The shape is close to a sphere for both o and t sites, which is comparable to the shape of the s orbitals of the hydrogen atom. For M sites, the charge accumulation is slightly higher for C atoms than for N atoms and has the shape of p orbitals. As expected, in M sites, atoms form strong bonds with 1NN Ni atoms. C, N and O atoms thus form bonds with the 1NN Ni atoms. These elements improve the mechanical properties of nickel, as shown above.

Finally, a slightly negative charge ring (blue zone) can be observed around the interstitial site. The charge ring of o site increases with the number of valence electrons from C to O. This is due to the covalent bond between X atoms and the metal, especially in the case of oxygen.

Projected electronic densities-of-states

The electronic density-of-states (eDOS) provides complementary information regarding the relative stability of sites and allows to highlight the nature of the bonds between atoms. The eDOS of the interstitial species projected on s, p and d orbitals are displayed in Figs. 4 and5. Spin up and down contributions were both drawn.

In the case of H atoms, Figs. 4, it can be noted that, (i) there is little change in the eDOS around the Fermi level, ε F , (ii) the hybridization between the H and Ni states is located [-9; -8] eV below ε F . New peaks of the d, p and s Ni states appear where the hydrogen s states are present. This result fits well with charge transfers, Fig. 3. In the case of t sites, we note that the d states of 1NN Ni atoms are highly modified around ε F (small peak), explaining (Hume-Rothery rule) why the configuration is energetically less stable than the o site. For oxygen, the hybridization between 1NN Ni and O atoms modifies the magnetism of one electronic spin only. The p states of O atoms interact with all states of Ni atoms, showing a strong interaction with the metal.

For carbon and nitrogen in t, o and M sites, whose eDOS are displayed in Figs. 4 and5. Their s states (not shown to lighten the graphs) are located at lower energy, around -12.8 and -17.3eV below ε F for C and N atoms, respectively. These species interact mainly with the metal through their p shells. Part of the s-states density of the carbon lies on a broad energy value (between -8.25 and -3eV) that overlays the s-and d-states of 1NN Ni atoms. Peaks in the p-shells of C and N atoms range from [-7; -5.5] eV. These are in interaction with Ni states, essentially with the d shells of 1NN Ni atoms. This can explain the charge transfer between the p and d, and s states.

Magnetic moment

As seen above, when an atom occupies either a t, a o or a M site, a small part of the electrons of first-(and, to a lesser extent, second-) nearest neighbors Ni atoms of the interstitial are transferred towards the interstitial. We can also presume that this leads to a (small) modification of the local atomic magnetic moment (µ B ) when sites are filled. Qualitative values of µ B of X and 1NN Ni atoms and total magnetism (calculations were conducted on 3×3×3 super-cells) are reported in Table 4. In the bulk, Ni atoms have a magnetic moment equal to about 0.62 µ B (in excellent agreement with experimental findings). Insertion always leads to a decrease in µ B of the first neighboring Ni atoms around the interstitial, especially when the system is filled by C and N atoms, whereas the magnetic moment of the interstitial atoms is always close to zero. Regarding the magnetic moment of the 2NN Ni atoms (not given here), the effect is low. For example, even when M sites are occupied by C or N atoms, the magnetism of 2NN Ni atoms remains unchanged. These results reinforce the conclusion that C and N atoms behave similarly in Ni.

In the case of oxygen, the magnetism on O atoms is low (negligible), compensating for the decrease in magnetism of Ni atoms close to O atoms. For H atoms, as seen above through the charge transfer and eDOS analyses, insertion only has a local effect on the lattice.

Effect of relaxation on site stability

Interactions between the add-atom and nickel are now discussed. The steric and elastic effects due to insertion were quantified by volume analysis, using (i) Ω f , the volume of formation

(Ω f = V[N • Ni + X] -V[N • Ni],
where V are the volumes of the system with and without impurity) which represents to overall effect of insertion on the lattice, (ii) δV voro which corresponds to the difference in Voronoï volume [START_REF] Rycroft | Voro++: A three-dimensional voronoi cell library in c++[END_REF] of of the i site when filled or when free (local distortion), and (iii) ∆V, which is the difference between Ω f and δV voro . Data are summarized in Table 5.

The volumes of formation (Ω f ) are equal to 0.93, 2.24, 4.87 and 7.31 Å 3 for H, C, N and O atoms in o sites, respectively. As expected, Ω f increases with the size of the interstitial atom. The same effect can also be observed for t and M sites. Ω f is almost twice as large for t sites than for o sites, which illustrates a higher lattice distortion when t sites are filled as compared to o sites. In first-order approximation, it can be explained by the fact that o sites are larger than t sites, the insertion is thus easier. The analysis of steric effects shows that, from C to O, the distortion becomes more and more severe, that which is likely to reduce the stability of the system. We also found that when C or N atoms are in M sites, the volume of formation is smaller than in t sites. This shows that these species strongly interact with the first-nearest-neighboring nickel, to minimize their elastic energy.

The evolution of the volume of Voronoï, V, which illustrates the local distortion, follows the same trend. To analyze it, we defined δV, the difference in Voronoï volume when the site is filled and when it is free. The volumes of Voronoï of free o, t and M sites are respectively equal to 5.44, 5.76 and 5.72 Å 3 . Special attention must be paid to the value of V. Indeed, the largest volume of Voronoï does not correspond to o sites but to t sites: the volumes of Voronoï, which are computed taking into account distances between the nearest atoms, are conducted on anisotropic configurations. δV for o sites is thus found equal to 0.4, 0.9, 0.9 and 1.5 Å 3 for H, C, N and O atoms, respectively. The increase of δV with the size of the interstitial agrees with the fact that the local distortion increases with the size of the interstitial. The same comment can be made for t and M sites. Again, there is only a small difference between C and N atoms regarding volumes, this justifies that they have similar behaviors in nickel.

The difference, ∆V, between the volume of formation Ω f and of Voronoï δV, see Table 6, which corresponds to the nonlocal deformation, or the deformation beyond the 1NN nickel atoms, is of 60, 60, 82 and 79 % of Ω f for H, C, N and O atoms in o sites, respectively. Since this increase can not be explained only by the increase in atomic size, electronic interactions are therefore responsible for this effect.

Elastic dipole

Finally, we discuss the effect of insertion in a continuous elasticity context. As explained by Clouet et al. [START_REF] Clouet | Dislocation interaction with c in alpha-fe: A comparison between atomic simulations and elasticity theory[END_REF] the insertion of an atom (or a point defect) into a lattice in a continuum description of elasticity can be modeled by a dipole tensor, P i j . This makes it possible to characterize elastic interactions between interstitial elements and other lattice defects (vacancies, grain-boundaries, etc.). To compute them, we followed the same approach as described in [START_REF] Connétable | Study of vacancy-(h,b,c,n,o) clusters in al using dft and statistical approaches: Consequences on solubility of solutes[END_REF], but due to the excessive numerical cost (magnetism and many electrons in the pseudo-potential of nickel), the convergence tests of P i j versus the super-cell size were limited to 216 atoms (see figure 6). Converged values are given in Table 6. In the case of t and o sites, and by symmetry, the tensor is diagonal, and all components are equal (P 11 , for instance). As for the analysis of volumes, we note a significant increase in the components between o and t sites. These results can thus be correlated to the formation energies, o sites are always more stable than the others. From C to O atoms, P i j increases with the electro-negativity of guest atoms. One notes that, in the case of the H atom, the elastic distortion on Ni lattice is very low. Not surprisingly, H atoms induce a very low and local distortion of the lattice, which is likely to have little effect on the elastic properties of fcc-Ni. The interaction is short ranged. By comparing hydrogen to other interstitials, we can see that the effect of H insertion is small. The elastic components of the other elements are of higher value, which implies strong longrange steric effects on the lattice. These results are consistent with the fact that C, N and O atoms have a low solubility limit compared to H atoms: this can be explained by evolution of Ω f , as stated above.

On the other hand, interstitials located in M sites induce strong anisotropic elastic strains (especially P 11 ). The small value of P 33 for M sites, as compared to o and t sites, can ex-plain why, in the case of C atoms, this configuration is energetically more stable than t sites. It can be expected that, due to the important difference between in P 11 values in o and t sites, diffusion mechanisms will induce high migration energies.

From these values, the effect of interstitial on lattice parameters can be evaluated using [START_REF] Connétable | Study of vacancy-(h,b,c,n,o) clusters in al using dft and statistical approaches: Consequences on solubility of solutes[END_REF]:

a (C[X]) = a o • (1 + δ • C[X]) (9)
where

δ = 4 a 3 o P 11 3B = u • P 11 ( 10
)
B is the bulk modulus, approximately equal to 198 GPa [START_REF] Connétable | First-principles study of the structural, electronic, vibrational, and elastic properties of orthorhombic nisi[END_REF] and a o the lattice parameter of the bulk, 3.52 Å. This gives u = 0.075 eV -1 , P 11 is thus expressed in eV.

By following the work of Varvenne et al. [START_REF] Varvenne | Point defect modeling in materials: Coupling ab initio and elasticity approaches[END_REF], the relaxation volumes (as well as the tensors V i j ) and the interaction energies with periodic images were computed for 3×3×3 supercells (which are generally used to study the insertion). We used the (theoretical) elastic constants of nickel, i.e. 274, 160 and 130 GPa [START_REF] Connétable | First-principles study of the structural, electronic, vibrational, and elastic properties of orthorhombic nisi[END_REF] for C 11 , C 12 and C 44 , respectively. Results are given in Table 6. Qualitatively and quantitatively, the relaxation volumes are in agreement with the volume of formation calculated on 3×3×3 super-cells. These results confirm the effect of relaxation can be captured by elastic theory. The results also show that the interaction energies due to periodic images are low, especially in the case of the H atom (< 3 meV). The elastic effects are greatest when C and N occupy M sites (about 80 meV). As shown by Varvenne et al. [START_REF] Varvenne | Point defect modeling in materials: Coupling ab initio and elasticity approaches[END_REF], DFT calculations at fixed volume therefore lead to accurate insertion energies taking into account correctly elasticity.

Effect of temperature

We conclude this part by discussing the effect of temperature on formation energies and configuration stability within the quasi-harmonic approximation. So as to take into account the effect of temperature, we considered that insertion does not affect the coefficient of thermal expansion of nickel (dilute limit). Eq. ( 1) was first used to set lattice parameters through lattice expansion of each configuration at different specific temperatures. It corresponds to a homogeneous dilatation of the shape of the super-cells (optimized at 0K) including (or not) an interstitial specie: values are 1.0067, 1.0164, 1.0218 and 1.0276 a o for the temperature of 600, 1200, 1500 and 1800 K, respectively. Energies and frequencies were then calculated using these parameters. Table 7 shows DFT energies (E s ) and vibrational Gibbs energies (which is equal to the Helmholtz free energy). Simulations were conducted at ambient pressure, p = 0 GPa. G s vib was calculated for different temperatures (0, 300, 600, 1200, 1500 and 1800 K):

G s vib [X](T ) = F vib [n • Ni + X](T ) -F vib [n • Ni](T ) -0.5F re f [X](T ) -0.5F re f 0K (11)
This equation includes (i) the effect of T on the free energy of reference states (gas or diamond), F re f [X](T ) (see details summarized in Appendix A), and (ii) the difference in phonon free Table 6: Calculated values of the elastic dipole tensor, P i j (in eV), the relaxation volume tensor, V i j (in Å 3 ), and the insertion energy at fixed lattice, E i (in eV), and its relaxation volume, V r (in Å 3 ), for 3×3×3 super-cells. In parentheses, we give the interaction energy (in meV). I is the matrix identity.

                   
16.4 9.8 0 9.8 16.4 0 0 0 5.2 energies of the super-cell with and without an X atom, calculated at temperature T with the value of lattice parameter for the same T. The Gibbs energy of insertion, G s tot , is thus the sum of E s and G s vib . Up to the melting point, there is no specific change in phonon properties. Octahedral sites are always the most stable sites and M sites still remain stable for C and N atoms. From a quantitative stand-point, we note that internal energy (E s ) slightly increases with T in most cases. Qualitatively, we note that the evolution of G s tot is mainly due to the vibrational component of the Gibbs energy. G s tot varies significantly with temperature due to the translational part of the partition function of the gas (p = 1atm was chosen), see Appendix A.

          - Relaxation volume tensor, V i j o 0.7 I 2.6 I 2.7 I 3.6 I t 1.1 I 3.5 I 3.7 I 4.2 I M -           4.1 2.8 0 2.8 4.1 0 0 0 1.6                     4.4 2.6 0 2.6 4.4 0 0 0 1.6           - E i V r E i V r E i V r E i V r o 0.
It can also be noted that, between simulations performed at 0 and 1800K, the main change is associated with E s (about 0.5 eV). The lattice expansion does not significantly change the value of G s vib . As explained, our approach was inspired by Wimmer's work. We can thus compare our results on H atoms with those of Wimmer [START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF]. He performed two calculations at two temperatures (0 and 1500K). We only notice differences in energies for the calculations at 1500K. When T increases, he sees that the formation energies decrease, unlike us. In his simulations "the lattice parameters of the super-cells (without and with H atoms) were kept at the computed value of a pure nickel crystal". Therefore, they did not include the specific relaxation of the cell when inserting H atoms in o or t sites, before to expand the lattice, contrary to our approach, as explained above. The energies of systems with interstitials are thus smaller in their cases than ours, especially in the case of t site (additional calculations were done with exactly the same approach to confirm it). The change in energy with lattice parameters is thus large, about 200meV, in agreement with the difference observed.

Diffusion

Mechanisms

We will now discuss the diffusion of interstitials in dilute solution; traps (vacancies, precipitates, dislocations, etc.) were neglected in this approach. We took into consideration all possible stable positions and thus all possible paths.

As shown for Al [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF], complex paths could exist in fcc systems, even in the case of interstitials, we must therefore take into account the full energy landscape of the network. The vibrations of the transition states (and the network) were also included in the study. Based on our preliminary results, some direct jumps were ruled out from the description of the diffusion. As shown previously, M is unstable with two imaginary frequencies for H and O atoms, whereas in the case of C and N atoms, M is an additional stable position. The existence of a stable M site thus modified the diffusion paths that should be considered.

To eliminate additional paths, tests were conducted with M-M direct jumps. They were unsuccessful: due to the small distance ( √ 2a o /4), equivalent to the o-M distance, NEB simulations indicate that, in all cases, the interstitial falls into an adjacent o site. Hence, there is no direct path between M sites. In the case of carbon, the NEB along the o-t path does not show a transition state, in agreement with literature data [START_REF] Siegel | First principles study of solubility diffusion and clustering ofc in ni[END_REF][START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF].

In conclusion, for H and O atoms, only one path is necessary to describe atomic diffusion (the one along t-o). In the case of nitrogen, there are three diffusion paths: o-t, o-M and t-M (see Fig. 8) and in the case of C atoms, only one jump has to be considered (along o-M path). As specified in the section 2, CI-NEB calculations were conducted to compute migration energies and the finite displacement method was used to compute inter-atomic forces. NEB simulations are plotted in Fig. 9. From NEB calculations, migration energies and position of transition states were obtained. Inter-atomic forces of transition states were then calculated in order to compute frequency attempts, Γ xy (see details in Appendix Appendix B). Migration energies are listed in Table 8.

The migration energies (E m ) of the diffusion along o-t for H and O atoms are equal to 0.41/0.15 and 0.75/0.53 eV, respectively. These values are relatively low. For hydrogen, the transition state is located in the triangle formed by the three Ni atoms that composed one of the faces of the tetrahedron, which is two-thirds of the distance between o and t sites. The small migration energy value of H atom can be explained by its small size and its weak electronic interaction with nickel atoms (localized charge density, Fig. 3). For oxygen, despite its strong interaction with the metal, E m values are low. Its transition state is located at the midpoint of the o and t sites.

Concerning the migration of C and N atoms along the o-M path, barriers are similar, they are equal to the energy difference between o and M sites: the energy needed to leave the M site is low, less than 10 meV. For N atoms, E m is about the same Oscillations due to temperature lead to rule out M sites as they are unstable. We choose to neglect them hereinafter for simplification purposes. We will further compare our results against experimental data, basing our discussion on the calculation of the diffusion coefficient. These results are consistent with theoretical literature. For H atoms, Wimmer [START_REF] Wimmer | Temperature-dependent diffusion coefficients from ab initio computations: Hydrogen, deuterium, and tritium in nickel[END_REF] and Wang [START_REF] Wang | Hydrogen influence on diffusion in nickel from first-principles calculations[END_REF] found 0.47 and 0.44 eV, respectively; 0.53/0.70 eV [START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF] for oxygen; and 0.28/1.31 eV [START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF] for N atoms. The difference with Fang's results [START_REF] Fang | First-principles studies on vacancymodified interstitial diffusion mechanism of oxygen in nickel, associated with large-scale atomic simulation techniques[END_REF] are prob-ably due to the functional they used (LDA), which reproduces with less precision properties of metallic systems. Regarding the o-o path for carbon, Siegel and Hamilton [START_REF] Siegel | First principles study of solubility diffusion and clustering ofc in ni[END_REF] and Connétable et al. [START_REF] Connétable | First-principles nickel database: energetics of impurities and defects[END_REF] found a value equivalent to ours, i.e. 1.62 and 1.65 eV, respectively.

Effects of temperature

As was done for solubility, we evaluated the effect of temperature on migration energies. Two additional temperatures were considered: 600 and 1200 K. NEB simulations were then performed at fixed shapes, imposed by lattice thermal expansion on 2×2×2 super-cells. Plots of the migration energies for paths of all species at 0, 600 and 1200 K are displayed in Fig. 10 to show the tendency. Results show that the effect on migration energies of the dilatation induced by the temperature is low. From a numerical standpoint, migration energies slightly evolve, the maximum is about 20 meV. Moreover, the positions of transition states are unchanged with T. At this stage of approximation, this effect can be neglected, and we considered that values at 0 K are sufficient to capture the physics qualitatively and quantitatively

Diffusion coefficients

We finally computed diffusion coefficients using DFT values reported previously. Explicit expressions of coefficients, taking into account only first-order transition states, are given in Appendix B. To compute jump rates, Γ, frequencies were fully computed, using the initial states and the transition states. D(T ) were then computed and plotted, see Fig. 11. We also fit it with an Arrhenius law,

D(T ) = D o exp (-E a /k B T ) , (12)
to obtain D o and activation energies (E a ). These parameters are summarized in Table 9. We see that, apart from oxygen, theoretical and experimental values of D are in excellent agreement. Activation energies and D o agree well with the literature (see Table 9). Theory is able to capture atomic scale diffusion mechanisms. The disagreement observed for O atoms is then attributed to the fact that oxygen interacts strongly with nickel and is not in interstitial form in nickel but in the form of clusters. Its diffusion mechanism is therefore highly dependent on the experimental conditions under which diffusivity measurements are performed (see discussion in [START_REF] Connétable | Diffusion of interstitial species (h and o atoms) in fcc systems (al, cu, co, ni and pd): Contribution of first and second order transition states[END_REF]).

Conclusion

In this manuscript, we presented a complete study of the insertion and diffusion of interstitial species: carbon, nitrogen and oxygen. Results were compared to those of hydrogen. We first confirmed that all interstitial elements in nickel prefer to sit in octahedral sites. We also identified that, as in aluminum [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF], C and N atoms admit one additional stable configuration that must be taken into consideration: the M site. However, the existence of M sites does not change the fact that o sites remain the most stable interstitial sites. We also showed that for C atoms, t sites are (elastically) unstable. Based on electronic properties, magnetic properties, charge transfers and elastic dipoles, we explained why o sites are more stable than the others.

We were able to quantify the local distortion effects (volumes of Voronoï) of those with long ranges (volumes of formation and elastic dipoles). Competition between elastic and electronic effects show a strong interaction between interstitials and nickel, especially for C and O atoms. We also showed that O and H atoms on one hand, and C and N atoms on the other hand have similar behaviors. The effects of temperature on migration energies was show to be is relatively limited, as compared to other processes that can take place in the material (cluster formation, interactions with traps, etc.).

When studying the diffusion of interstitial species, we meticulously took into account all different paths and analyzed each possibility. We gave here new explicit expressions of diffusion that takes into account the different configurations. Finally, we showed that DFT simulations and multi-state model reproduce experimental diffusion coefficients with accuracy, except for O atoms.

Table A.10: Θ v and Θ r (in K) are the vibrational and rotational temperatures defined in the equipartition approach of the partition function. A t is in K•m 2 . We also give values of different quantities used in the formula: I r , the moment of inertia (in kg.m 2 ), d, the inter-atomic distance (in Å), σ and q e . ω is the frequency (in meV) of the molecule and F re f 0K is the zero-point energy of the molecule (in meV). Γ xy is the jump rate from x to y computed by using the following equation [START_REF] Eyring | The activated complex in chemical reactions[END_REF]:

Γ xy = k B T h Z T S Z EI exp - ∆E xy m k B T (B.2)
where ∆E m corresponds to the transition migration energy. Z T S is the (vibrational) partition function for the transition state and Z EI is the same quantity for the initial position. Z is expressed by:

F = -k B T ln Z = k B T qν ln 2 sinh ω qν 2k B T (B.3)
where ω qν are the frequencies of the ν mode in q point. Eq. (B.2) is valid regardless of the configuration of the transition state.

With the M position, there are three paths, t-o, t-M and o-M, as described in [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF], and the formula becomes: Formula were derived as described in [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF].

D tom =

Figure 1 :

 1 Figure 1: Schematic representation of interstitial t, o and M sites.

Figure 2 :

 2 Figure 2: Phonon band structures for a system with one atom (H, C, N and O atoms) in an interstitial position, either in o, t or M sites. Negative bands correspond to imaginary frequencies. Energies (frequencies) are expressed in THz.

  X

Figure 3 :

 3 Figure 3: Plot of contour maps of the charge transfer, ∆ρ (in e/Å 3 ), when sites are filled: for H, C, N and O atoms in their different stable configurations. The same scale was used for all plots.

Figure 4 :

 4 Figure 4: eDOS of H and C in o, t and M sites. The blue lines correspond to the eDOS of the interstitial species, the yellow lines of the 1NN Ni atoms of the X atom, and the black lines are the densities of the others Ni atoms. The DOS of the electrons are respectively projected on the s orbitals (plot on the left side), p orbitals (plot in the center) and d orbitals (plot on the right side). The positive (resp. negative) part of the eDOS corresponds to spin up (resp. down).

Figure 5 :

 5 Figure 5: eDOS of O and N atoms in o, t and M sites.

Figure 6 :

 6 Figure 6: Evolution of the elastic tensor P 11 (in eV) as a function of super-cell size.

Figure 7 :

 7 Figure 7: Evolution of the Gibbs energies with temperature, for different values of lattice parameters corresponding to five temperatures, see text.

Figure 8 :Figure 9 :

 89 Figure 8: Schematic representation of possible jumps when t and o sites are stable (a) and when there are t, o and M sites (b).

Figure 10 :

 10 Figure 10: Migration energy evolution as a function of T for interstitial atoms.

Table 1 :

 1 Insertion (E s [X], in eV) and zero-point energies (F 0K [X], in eV), solubility enthalpies (H s = F 0K [X] + E s , in eV). ∆H is the energy difference between i and o sites.

	X		E s	F 0K [X]	H s	∆H
		o 0.063	0.005	0.068	0
	H	t	0.323	0.084	0.407 0.34
		M 0.814		unstable
		o 0.586	0.089	0.675	0
	C	t	2.277 -0.057 2.220 1.54
		M 2.167	0.035	2.201 1.53
		o 0.702	0.011	0.713	0
	N	t	1.762	0.003	1.765 1.05
		M 2.239 -0.032 2.207 1.49
		o 0.468 -0.019 0.449	0
	O	t	0.646	0.004	0.650 0.20
		M 1.265		unstable

Table 3 :

 3

Elastic constants (in GPa) deduced from acoustic branches (calculated on 2×2×2 super-cells): C 11 , C 44 , C 12 and C = (C 11 -C 12

  and C 12 values.

	X	C 11 C 44 C 12 C
	bulk (sound velocity) 274 116 106 75
	GGA [46]	274 130 160 57
	Exp. [45]	261 131 150 55
	H	269 117 108 73
	C	267 118 100 70
	N	266 118 100 70
	O	261 111 89 69

Table 4 :

 4 Atomic magnetic moment (in µ B ) of the interstitial atom and the 1NN Ni atom of the interstitial. The total magnetic moments of the super-cells, µ tot B , were reported too. For information, the total magnetic moment of the free super-cell is equal to 68.15µ B .

		config. µ B [X] µ 1NN B	[Ni]	µ tot B
	H	o t	-0.05 -0.07	0.47 0.54	67.44 67.36
		o	-0.02	0.18	64.94
	C	t	-0.03	0.14	65.14
		M	-0.02	0.12	65.07
		o	0.00	0.21	65.02
	N	t	-0.02	0.15	65.26
		M	0.00	0.14	65.56
	O	o t	0.15 0.12	0.57 0.53	67.96 67.90

Table 5 :

 5 Calculated valuers of the volume of formation (Ω f in Å 3 ) and of Voronoï (V voro in Å 3 ), and difference between the volume of formation and of Voronoï ∆V (in Å 3 ). The free o, t and M volumes of Voronoï are equal to 5.44, 5.76 and 5.72 Å 3 , respectively.

		conf.	Ω f	δV voro	∆V
	H	o t	0.93 2.43	0.37 0.91	0.56 1.52
		o	2.24	0.90	1.34
	C	t	6.34	2.21	4.13
		M	6.11	1.28	4.83
		o	4.87	0.87	4.00
	N	t	7.55	2.15	5.40
		M	6.31	1.28	5.03
	O	o t	7.31 14.08	1.54 2.70	5.77 11.38

Table 7 :

 7 Internal energies (E s , in eV) and vibrational enthalpies (G s vib , in eV) for different temperatures: 0, 600, 1200, 1500 and 1800 K.

	atom site		0 K	600 K	1200 K	1500 K	1800 K
			E s	G s vib	E s	G s vib	E s	G s vib	E s	G s vib	E s	G s vib
	H	o	0.06 0.01 0.08 0.31 0.11 0.61 0.14 0.89 0.17 1.05
		t	0.32 0.08 0.35 0.41 0.39 0.70 0.43 0.97 0.46 1.16
	C	o	0.59 0.09 0.59 -0.05 0.59 -0.31 0.61 -0.38 0.63 -0.52
		M 2.17 0.03 2.28 -0.30 2.44 -0.80 2.54 -1.06 2.62 -1.32
	N	o	0.70 0.01 0.81 0.46 0.95 0.91 1.04 1.22 1.13 1.46
		t	1.76 0.00 1.88 0.35 2.02 0.67 2.12 0.86 2.20 1.02
		M 2.24 -0.03 2.25 0.23 2.27 0.52 2.30 0.55 2.32 0.67
	O	o	0.47 -0.02 0.58 0.30 0.73 0.55 0.82 0.78 0.91 0.90
		t	0.65 0.00 0.75 0.30 0.88 0.55 0.97 0.75 1.05 0.87

Table 8 :

 8 Migration energies (in eV) for paths.

		H	C	N	O
	o ⇒ t	0.41	-	1.34 0.75
	t ⇒ o	0.15	-	0.32 0.53
	o ⇒ M	-	1.42 1.31	-
	M ⇒ o	-	0.02 0.02	-
	t ⇒ M	-	-	0.29	-
	M ⇒ t	-	-	0.02	-
	along o-t and o-M paths. Therefore, all paths should be taken
	into account in order to calculate diffusion coefficients. The po-
	sition of the saddle points along the o-o path are smaller than
	the mean-square displacement of atoms, dT , presented above.

Table 9 :

 9 Activation energies (E a , in eV) and diffusion coefficient (D o , in m 2 /s) obtained from an Arrhenius fit: theoretical-experimental data comparison.

		H	C	N	O
			E a (in eV)
	This Work	0.42	1.40	1.28	0.94
	Theo.	0.47 [15], 0.41 [9]	1.72 [8], 1.68 [9]	1.33 [9]	0.71 [9]
	Exp.	0.42 [58]	1.54-1.71 [59-61]	1.30 [62]
			D o (in m 2 /s)
	This work	9.5 10 -7	2.0 10 -5	2.1 10 -5	2.5 10 -6
	Theo.	38.4 10 -7 [15], 13.1 10 -7 [9]	0.63 10 -5 [9]	0.18 10 -5 [9] 1.15 10 -6 [9]
	Exp.	1.8-6.9 10 -7 [58, 63]	-	4.6 10 -5 [62]

  Γ mo Γ om Γ tm + 3 Γ mt Γ om Γ tm + 6 Γ mo Γ ot Γ tm +2 Γ mt Γ ot Γ tm + 4 Γ mo Γ om Γ to + 6 Γ mt Γ om Γ to + 4 Γ mo Γ ot Γ to + 4 Γ mt Γ ot Γ to ]/[6 Γ mt Γ om +4 Γ mo Γ ot + 4 Γ mt Γ ot + 3 Γ mo Γ tm + 18 Γ om Γ tm +12 Γ ot Γ tm + 2 Γ mo Γ to + 2 Γ mt Γ to + 12 Γ om Γ to ] (B.4)If we now consider three stable sites (t, o and M) but only t-M and o-M jumps, D is thus expressed as:Γ mo + Γ mt ) Γ om Γ tm (2 Γ mt Γ om + Γ mo Γ tm + 6 Γ om Γ tm ) (B.5) Γ mo Γ om Γ mo + 6 Γ om (B.6)if we do not consider t sites as stable, but only o and M sites.

	a 2 o 4 [6 D = (2 Finally, the diffusion coefficient becomes: a 2 o 4
	D =	a 2 o 2

Preprint submitted to JALCOM, june 2019 December 18, 2019

Acknowledgments

This work was performed using HPC resources from CALMIP (Grant 2017-p0749, 2017-p0912 and 2018-p0912) and GENCI-CINES (Grant 2016-c2016097722 and A0020907722). Authors acknowledge M.-L. Doublet (Institute Charles-Gerhardt of Montpellier) for his remarks.

Appendix A. Statistical model and parameter of the Helmholtz energy of the gas

The Helmholtz energy of a X 2 gas (without the ZPE contribution) can be expressed by [START_REF] Helrich | Modern Thermodynamics with Statistical Mechanics[END_REF]:

where

The first terms correspond to the translational components of the partition function of the molecule X 2 , ln 1e -Θ v /T corresponds to the vibrational component, ln T σΘ r to the rotational one and the last to the electronic one. Parameters used in the present work are listed in Table A. [START_REF] David | Diffusion of interstitials in metallic systems, illustration of a complex study case: aluminum[END_REF], where m = 2M (M atomic mass), I r = Md 2 /2 the moment of inertia. q e is the electronic partition function considered here as constant (see Helrich [START_REF] Helrich | Modern Thermodynamics with Statistical Mechanics[END_REF]). Values are computed for the current work. In all calculations, the gas pressure, p, is set to 1 atm.

Appendix B. Diffusion coefficients

We indicate here different expressions of the diffusion coefficient regardless of stable positions and possible paths. These expressions cover all configurations. For H and O atoms, we used the eq. (B.1), for C atom, we used eq. (B.6) and for N atoms eq. (B.4).

In the case where there are two stable positions (t and o) and only one path (for H and O atoms), the diffusion coefficient is thus expressed by:


Evangelia Adamou

Denis Costaouec

Cipriano Ramirez Guzmán

Rufina Robles

Juliana Salazar Bautista

Research report: The gap strategy in Ixcatec (Otomanguean, Mexico) *

Keywords:

The present research report provides an overview of the gap strategy in Ixcatec, a criticallyendangered and under-described Otomanguean language of Mexico. Ixcatec RCs using the gap strategy are postnominal. They are built with an uninflected complementizer and involve a finite verb or a predicate adjective. The head of the RC can be a noun, but in free-speech RCs are frequently constructed with a null nominal.

Introduction

This article presents an overview of the gap relativization strategy in Ixcatec (ISO code: ixc). Ixcatec is a language of the Otomanguean stock and the Popolocan branch, together with Chocho (or Ngigua and Ngiba), Popoloc, and Mazatec (Fernández de Miranda 1951[START_REF] Fernández De Miranda | Glotocronología de la familia popoloca[END_REF][START_REF] Hamp | Chocho-Popoloca Innovations[END_REF][START_REF] Gudschinsky | Proto-Popotecan. A Comparative Study of Popolocan and Mixtecan[END_REF], Swadesh 1960: 83). The Ixcatec speakers reside in Santa María Ixcatlán in the State of Oaxaca, in Mexico (see map in Figure 1). Nowadays Santa María Ixcatlán has roughly 400 inhabitants, but it was an important centre when the Spaniards arrived in 1522 knowing that it was estimated to have had a population of 10,000 to 30,000 inhabitants [START_REF] Hironymous | Santa Maria Ixcatlan, Oaxaca: From Colonial Cacicazgo to Modern Municipio[END_REF].

Ixcatec is a critically endangered language as it is spoken by less than ten speakers who do not use the language in their daily lives (see endangerment scale in [START_REF] Krauss | Classification and terminology for degrees of languages endangerment[END_REF]. This situation is the result of a long process of shift to Spanish, starting with Spanish colonization in the 16th century. Although contact with Spanish speakers was most likely scarce in the past centuries, intensive and extensive shift to Spanish took place during the early twentieth century and is nowadays fully completed [START_REF] Adamou | A corpus-driven approach to language contact[END_REF].

Ixcatec is an under-described language. Among the earliest studies, we note a phonological analysis, a dictionary, and a study of possessives [START_REF] Fernández De Miranda | Las formas posesivas del ixcateco[END_REF][START_REF] Gudschinsky | Proto-Popotecan. A Comparative Study of Popolocan and Mixtecan[END_REF][START_REF] Fernández De Miranda | Diccionario ixcateco[END_REF]. In the early 2000s there has been some preliminary work on Ixcatec nominal phrases (Veerman-Leichsenring 2001a) and the Ixcatec data were discussed in publications on the Popolocan branch (e.g., [START_REF] Veerman-Leichsenring | Popolocan Independent Personal Pronouns: Comparison and Reconstruction[END_REF]. More recently, work has been conducted on Ixcatec phonetics and phonology (Alarcón Montero 2010), on syntax and morphology [START_REF] Adamou | El complementante la en ixcateco: marcador de clausula relativa, completiva y adverbial[END_REF][START_REF] Adamou | L'antipassif en ixcatèque[END_REF][START_REF] Costaouec | Classification nominale en ixcatèque[END_REF]Adamou 2017b), on spatial language and cognition (Adamou 2017a), and on information structure (Adamou, Gordon & Gries in press). Ixcatec is phonologically distinguishing between three level tones, a high tone (noted ¹), a mid tone (noted ²), and a low tone (noted ³). Verbs and nouns are clearly distinct. Verbs are the syntactic nucleus of independent or dependant clauses and receive tense, aspect, and modality markers (perfective, imperfective, progressive, evidential, and anterior; the latter is also used with predicate adjectives), valency-changing suffixes (e.g., antipassive, applicative, causative), and person suffixes. Nouns are determined by the definite and the indefinite articles, demonstratives, possessives, quantifiers, and numerals. Animate nouns are formed with the noun classifiers di²-'man', k w a²-'woman', ʔu²-'animal' (glossed CLF) and agree with the cross-reference morphemes -ða² 'male', -k w a² 'female', and -βa³ 'animal' which are suffixed on verbs, possessives, and predicate adjectives only with S arguments and only when those are in a syntactically marked position. Noun classifiers are distinct from class terms (glossed CLS) which partake in word formation for certain categories of nouns (designation of plants, body parts, etc.) but do not exhibit agreement.

Ixcatec is a head-marking language, i.e., grammatical relations are marked on the verb [START_REF] Nichols | Head-marking and dependent-marking grammar[END_REF]). Core arguments are not coded through case morphemes or adpositions. Ixcatec has accusative alignment in indexing (A = S ≠ P) (Malchukov et al. 2010), i.e., only the single argument of intransitive verbs (S) and the agent-like argument of transitive verbs (A) are indexed on the verb through a series of suffixes, with third person being zero. A dozen experience predicates take a different coding for person, namely through possessive suffixes.

Ixcatec is a pro-drop language, i.e., free pronouns are optionally used for all functions, and noun phrases (NPs) are frequently omitted. In main clauses, Ixcatec has a VS unmarked order and monotransitive main clauses have an SVO order. Ixcatec yes/no questions are introduced by an optional interrogative particle ʔa². Wh-words (e.g., ja¹ra² 'who', nda¹ra² 'what', ndi¹ra² 'where'), always precede verbs in direct and indirect questions.

The present paper is organized as follows. Section 2 presents the methodology. Section 3 illustrates the gap strategy in Ixcatec. Finally, section 4 offers some concluding remarks and future directions for research.

Method

A free-speech Ixcatec corpus was collected from 2010 to 2013 as part of the Ixcatec language documentation programme.1 50 hours of video recordings are now available on the Endangered Languages Archive (ELAR). 2 30 minutes of annotated speech, synchronized with the videos, are available at the Pangloss Collection. 3 Part of the Ixcatec corpus is transcribed using ELAN (Max Planck Institute for Psycholinguistics, Nijmegen, Netherlands), part of it is in Toolbox, and part of it in field notes. For this study, four hours were analysed: relative clauses (RCs) were identified and coded as intransitive subject RCs, agent RCs, and object RCs. Examples that were unclear were discarded from the analysis. Moreover, we glossed and analysed the RCs in the Ixcatec texts that were collected, transcribed, and translated in Spanish by [START_REF] Fernández De Miranda | Diccionario ixcateco[END_REF]. In addition, elicited data were used whenever there were no instances of relevant RCs in the corpus.

The four most fluent speakers of Ixcatec participated in this study. The last Ixcatec speakers learned Ixcatec from having spent more time with their grandparents who were monolingual in Ixcatec and from their parents who were bilingual, L1-Ixcatec and L2-Spanish. One of the speakers was raised monolingual in Ixcatec and learned Spanish from school. All four speakers have not been using Ixcatec in their daily lives for the last 50 years. They have participated at the Ixcatec language documentation programme and on this occasion started using Ixcatec again for the recording sessions. Moreover, three of the four speakers also gave classes of Ixcatec at the kindergarten, primary, and secondary school, once per week, roughly at the same period as the Ixcatec language documentation program.

The two 80-year-old female speakers, who are close friends, were brought together at the house of one of the speakers. They vividly discussed their daily life and their conversations are characterized by frequent overlaps and back-channelling and quick turn-taking, 220 ms in average (Adamou 2016: 10). The two 80-year-old men, who are family members, would generally discuss local traditions and recollections. Their conversations showed few overlaps and back-channelling and were characterized by slow turn-taking, i.e., 1000 ms in average, which can be considered as part of a more formal communication register (Adamou 2016: 10).

All Ixcatec speakers were informed about the goals of this research and signed formal authorizations. In parallel, the Ixcatec language documentation programme was approved by the community's general assembly. Participants were paid for their participation in this study.

3.

Gap strategy

A relative clause can be defined as "a subordinate clause which delimits the reference of an NP by specifying the role of the referent of that NP in the situation described by the RC" (Andrews 2007: 206). For example, subordination of the clause 'The cat catches the rat', in (1b), to the clause 'I see the cat', in (1a), results in the complex clause 'I see the cat that catches the rat', in (1c).

(1) a. I see the cat. b. The cat catches the rat. c. I see the cat [that Ø catches the rat].

More specifically, (1c) is composed of a matrix clause and a relative clause (noted in brackets). The head of the RC, 'cat' in this example, is underscored. When the co-referential NP is not overtly expressed in the RC, it is referred to as a gap and it is noted as Ø; see (1c). When the gap has the grammatical function of a subject, as in (1c), the RC is dubbed subject relative clause, and when it has the function of an object it is dubbed object relative clause. The RC in (1c) is postnominal, i.e., it follows the head NP.

To account for gaps at a theoretical level, three influential analyses were extensively discussed in the literature: the traditional base-generated head (or head external) analysis, according to which the head NP originates outside the RC (Chomsky 1977 among many others); the head raising analysis, according to which the head NP raises from inside the RC by syntactic movement (Kayne 1994 among others); and the matching analysis, according to which there is both an external and an internal head, but the internal head is phonologically deleted through matching with the external head (see [START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF]. However, in the functional-typological literature, the term 'gap' is adopted without any discussion as to where the gap originates from. The use of the term 'gap strategy' is merely one referring to a RC where no overt NP expresses the relative function (Andrews 2007: 206). Following the latter tradition, the theoretical discussion concerning the origin of the gaps remains beyond the scope of this paper.

Ixcatec RCs are generally built with the gap strategy. Example in (2) shows how two main clauses, in (2a) and (2b), form a complex clause consisting of a matrix clause and a RC, in (2c).

(2) a. ʔi²na¹na³ k w -i²ʃkũ²-na³ sa¹ ʔu²-tʃi¹tu¹ 1SG Adamou (2017) observes the syntactic and morphological ambiguity of the Ixcatec RCs. Indeed, even though the semantics and pragmatics favour the subject interpretation of (2c), 'I see the cat that catches the rat', the Ixcatec RC has no case marking, no agreement and no word order that could indicate whether this is a subject RC or an object RC: la² is uninflected, there is no verb morphology, and the word order of the NP in the RC is postverbal in both subject and object RCs. As illustrated in (2c), Ixcatec RCs are generally constructed with a finite verb, marked for tense-mood-aspect and receiving the personal suffixes (third person being zero as in this example). The head of the RC can be a noun, as in (2c), but it can also be a personal pronoun, a demonstrative pronoun, a definite article, a numeral, or a quantifier.

The Ixcatec RCs are typically introduced by the uninflected complementizer la². Albeit frequent, the use of the complementizer la² is optional as shown in (3) with an example from the conversational corpus. Such uses were also noted during systematic data collection with translation tasks (Adamou 2017).

Noun headed RCs

Example in (6) illustrates a subject RC headed by a NP. It also illustrates the use of the complementizer la² to introduce a complement clause with the verb 'to think'. In addition, it can be seen that the head noun 'people' is only used in the first subject RC, but is omitted in the two following subject RCs, i.e., 'that worked'. ( 6 12) and ( 13) illustrate RCs with an oblique instrumental gap and in ( 14) with a comitative. Note the absence of an applicative instrumental marker on the verb or the instrumental ku² 'with'.

RCs with null nominal head

The analysis of the free-speech Ixcatec corpus revealed a high number of RCs in which the nominal head is not overtly expressed [START_REF] Lehmann | Der Relativsatz. Typologie seiner Strukturen, Theorie seiner Funktionen, Kompendium seiner Grammatik[END_REF]. In ( 19) and ( 20) the head is a personal pronoun. These examples also illustrate non-restrictive RCs, i.e., RCs that only comment on the NP. Non-restrictive RCs may involve a pause between the head NP and the RC (Andrews 2007: 207), and this is the case in (20) where a small pause precedes the complementizer la². The RC may also have a definite article as a head. See an example of a subject RC in ( 22) and ( 23), a direct object in (24), and a locative in (25). ( 22 '(The one) which is spicy, is (the one) that is tasty, that is, with (the one) that isn't spicy, it isn't tasty.' {RRM-conv-2011-Caldo de res_465} Similar cases have been reported in the literature (see Andrews 2007: 214), but they are not discussed extensively. In the Ixcatec examples, it appears that the referents are well identified in the discourse and can therefore be omitted.

Accessibility Hierarchy

This section summarizes the various RCs with respect to the Noun Phrase Accessibility Hierarchy (Keenan & Comrie 1977), presented in (24). According to this hierarchy, if a language can relativize a position in the Accessibility Hierarchy, it can relativize any higher position. Subjects and agents are considered as more accessible to relativization than direct objects, indirect objects, oblique objects, possessors, and objects of comparison.

(29) subject > direct object > indirect object > oblique object > possessor > object of comparison

In the previous sections it was shown that Ixcatec allows relativization of subjects, direct objects, indirect objects, oblique objects (instrumental, comitative), adjuncts (locative, time, reason), and possessors through the gap strategy and the use of the complementizer la². Table 1 summarises these possibilities.

4.

Concluding remarks

Ixcatec being an under-described language, this work is the first comprehensive study of the most common relativization strategy, the gap strategy. More research remains to be done on other types of relativization strategies and RCs which have not been discussed in the present paper. For example, we have found some rare instances of internally headed relative clauses (IHRCs) both in the texts by Fernández de [START_REF] Fernández De Miranda | Diccionario ixcateco[END_REF], exemplified in (30), and in the contemporary corpus, shown in (31). In these examples, we note that the RC involves an existential predicate.

[ steam_bath-DEM.PROX 'Thus when the steam bath was built, the one that built the late Alberto, this steam bath that is here... ' {PSG-conv-2008-Temascal_35} In addition, Adamou (2017) draws attention to the use of the cross-reference morphemes (-k w a² 'woman', -ða² 'man' and -βa² 'animal') as resumptive pronouns. The theoretical question of whether such uses are alternative realizations of gaps or ordinary pronouns is left for future research (see Asudeh 2012 for an overview of analyses of resumption).

In conclusion, this paper draws mainly on free-speech data, which have the advantage of showing what speakers are doing in natural communication. Elicited data were used to illustrate RCs that were not found in the analysed parts of the corpus. As larger portions of the Ixcatec corpus are being annotated, we hope to gain a better understanding of how relativization works in Ixcatec. In the meantime, this research report contributes to the typology of RCs in the languages of the Mesoamerica presented in this volume.

Figure 1 .

 1 Figure 1. The localities in which the Popolocan languages are spoken (Mexico). Ixcatec is only spoken at Santa María Ixcatlán.

  all the people are going to inhale, that's why.' {RRM-conv-2010-Everyday life_2_86}In addition, the Ixcatec complementizer la² introduces adverbial clauses, in (4), and completive clauses, in (5). (4) k w a²tu¹=βi²hi²-ʔa²-na² [la² mã¹hũ¹ ra²] PFV.PL-arrive-NEG-FOC-FOC COMP sweep there?? 'Won't they come to sweep there?' {RRM-conv-2010-Everyday life_3_11} (5) ku¹-tʃe²-ku¹-na³ sa² k w a¹-e²nɸe²rme¹ra² ku¹-tʃe²-ku¹-na³ PFV-say-ANT-1SG DEF CLF.F-nurse PFV-say-ANTto the nurse, I said how ugly it is!' {JSB-conv-2010-Everyday life_3_90} Adamou & Costaouec (2013) noted the similarity of the Ixcatec la² and the Popoloc complementizer la³/na³ (Veerman-Leichsenring 1991).

('

  as says the Word of God) that [with] the ruler with which you measure, you will be measured.' (de Miranda 1961: 86) 6 of mine with whom I work.' {JSB-elic-2015} Examples in (15) and (16) illustrate RCs with time and (17) with reason. (15) nda²tʃi²ka²-na² ʃi²nga² [la² ku²-tse² ʃa³ Ø TIME ] how_many-FOC year COMP PFV-do work 'How many years during which they worked!' {PSG-conv-2010The reason for which he/she got out.' {JSB-elic-2015} Example (18) illustrates a RC with a possessive function (also known as genitive). No examples were found in the spontaneous corpus. whose work is fine.' {JSB-elic-2015}

(

  We, who are doing the tasks…' {RRM-conv-2010-Everyday life_2_202}In some rare examples, the RC is introduced by a demonstrative pronoun, as in (21).

	19) ʔi²na¹na³	[la²	mi²ʔe²-na³ Øs] ki¹=ʃte²nga²		ki¹=ʃte²nga²
	1SG	COMP	hear-1SG	PROG.3SG-thunder	PROG.3SG-thunder
	'I, who have not heard that it is thundering, thundering...' 7
	{JSB-conv-2010-Everyday life_6_459}
	(20) ʔi²na³ri² /	[la²	tu¹=tse²-ri²	Ø A ɸa²e¹na²]
	1PL.EXCL	COMP	PROG.PL-do-HON/2PL.EXCL	task
	'(21) me¹ra²	la²	/ [la²	βi²-e²ʔe²	Ø A	sa²βo¹r	ndra²]	//
	DEM.DIST	COMP	/ COMP	IPFV-give		taste	like_this	//
	'That one, that gives taste, like this.' {RRM-conv-2011-Caldo de res_385}

  The matrix clause of a RC is not always straightforward in conversations as speakers frequently replan their speech and negotiate turns.

		{PSG-conv-2011-Milpa_153}
	(24) tsu²-ku¹		/	k w -e²na²-na³	//	sa¹	/
		want/say-ANT / /	PFV-buy-1SG	//	DEF	/
		sa¹	[la²		k w a²-ja¹	Ø DO	ka²	ɾha²-na³]
		DEF	COMP		PFV-POSS.PRED	all	hand-POSS.BP.1SG
		'One says I buy the one that I have on my hands.' {RRM-conv-2010-Everyday life_1_240}
	(25)	ki¹i²-ke²			sa¹	[la²	β-i²ti¹ʔ-i²	Ø LOC ]
		LOC.PRED-ITER	DEF	COMP	IPFV-enter-1PL.INCL
		'There is again the one (the Monte Alban archaeological site) in which we enter.'
		{CRG-conv-2010-Monte Alban_96}
		RCs may also have an indefinite article as head, in (26), and a quantifier, in (27).
	(26)	me¹ra²		ʃe²			h ŋgu²	[la²	ʧa²ʃĩ²ʔẽ² Ø A	ka² lĩ²-ʔĩ¹]
		DEM.DIST	PRST.PRED one	COMP	teach	all CLS-little
		'That is the one that teaches all the children.' {CRG-conv-2010-Monte Alban_730}
	(27) la²		k w e²nda²		ɸi¹	ka²	[la²	lĩ²-ʔĩ¹ Øs]
		COMP		for			go	all	COMP	CLS-little
		[la²		mĩ¹ʔẽ²	Øs]
		COMP		sick
		'It is in order for all those who are children, who are sick, to go.'
		{PSG-conv-2008-Temascal_16}
	In some cases, there is no nominal head and no determiner either; see (28).
	(28) [la²		tʃe³]			[la²	ʰni¹] /
		COMP		spicy		COMP	tasty /
	) k w -i²te¹ʔẽ² me¹	ku²	ja²te³ k w e²nda² la² [la² tʃe³-ʔa²-na²]	// ʰni¹-ʔa²-na²
		PFV-enter therefore COORD/INS COMP spicy-NEG-FOC // tasty-NEG-FOC wood for COMP
		k w -i²hi² ndju²ja¹	ʃta¹nda² la²	ʔa²ku²	ʃti²nde¹e¹
		arrive	steam_bath after	COMP inside	bathe
		ka²	sa¹	[la²		k w a²tu²-ɸi²	Ø S ndju²ja¹]
		all	DEF	COMP	PFV.3PL-go	steam_bath
		'One enters the wood in order for all those who go to the steam bath to bathe.'
		{PSG-conv-2008-Temascal_6}
	(23) me¹	/	sa¹	[la²	tu¹=ka²tse¹ngi²	Ø S ] i¹ɲa³	β-a²ʧa²
		therefore	DEF	COMP PROG.PL-earth_up/lift	fine	IPFV-earn
		'So… the ones who are lifting (the maize plants) earn good money.'

7

Table 1 .

 1 Accessibility Hierarchy of the gap strategy in Ixcatec

		Subject Direct	Indirect	Instrumental Comitative Locative Time Reason Possessor
			object	object
	Gap									
	strategy

  Preceding sentence: 'How is the world formed?']

	(30)	ʔmee¹ndrii²	si¹	fo²rma¹du¹	ʧi²k²a ku¹
		like_this		EXS.PRED formed		like COORD/INS??
		[la²	tʸu²si¹ʃe¹ S	si¹-nga²		ɾha²-na³]
		COMP	orange	EXS.PRED-inside	hand-POSS.BP.1SG
		'In this way, it is formed like the orange that is in my hand.'
		{Fernández de Miranda 1961: 193}
	(31)	ka²ndi²		me¹ndi²	la²	tse¹-ʃi²	ndju²ja¹
		when		for_this	COMP	do-APPL.INS	steam_bath
		me¹ra²		tsi¹-tse²-na²	sa¹		a²ni¹ma³	βe¹tu²
		DEM.DIST	EVD-do-FOC	DEF	deceased	NP
		[la²		si¹	sa¹	ndju²ja¹-ri² S ]
		COMP		EXS.PRED	DEF

Endangered Languages Documentation Program (ELDP): Lexical Documentation of Ixcatec, a highly endangered Otomanguean language of Oaxaca (HRELP, MDP 0214),

[START_REF] Lehmann | Der Relativsatz. Typologie seiner Strukturen, Theorie seiner Funktionen, Kompendium seiner Grammatik[END_REF]-2013, PI. D. Costaouec. 2 Accessed at http://www.elar-archive.org/index.php 3 Accessed at http://lacito.vjf.cnrs.fr/pangloss/corpus/list_rsc.php?lg=Ixcatec

The examples code the speaker through her/his initials, the type of data (conv: conversational, elic: elicited), the year of recording, the name of the text and the number of the sentence when appropriate.

We note that a problem with the annotation of the gaps arises for Ixcatec. The subject gap in this example is postverbal because the unmarked position of subjects in the Ixcatec RCs is postverbal. However, the position of the subject gap before the object is rather arbitrary; the unmarked word order in the Ixcatec transitive main clauses is SVO while the two arguments are not expressed simultaneously in the RCs.

We have respected the transcription and segmentation provided by the author and have added glosses and a translation from Spanish to English.

, and Pedro Salazar Gutierrez. This work has been supported by the Endangered Languages Documentation Programmes of the Hans Rausing Foundation for data collection and analysis. It benefited greatly from discussions within the Programme of International Cooperation (PICS) between the CNRS and CIESAS

Abbreviations

Glosses follow the Leipzig Glossing Rules.