Caroline Boule

Diana Vaclavkova

Miroslav Bartos

Karol Nogajewski

Lukas Zdražil

Takashi Taniguchi

Kenji Watanabe

Marek Potemski

Jacek Kasprzak
email: jacek.kasprzak@neel.cnrs.fr

Coherent dynamics and mapping of excitons in single-layer MoSe 2 and WSe 2 at the homogeneous limit

I. INTRODUCTION

The interest in two-dimensional (2D) layered semiconductors stems from novel functionalities that can be obtained by varying their thickness, down to the single-layer (SL) limit, or by stacking layers from different materials to create van der Waals heterostructures. Semiconducting transition metal dichalcogenides (TMDs) in the SL form display direct band gap [START_REF] Mak | Atomically thin MoS2: A new direct-gap semiconductor[END_REF] spanning across near infra-red and visible range. They typically show 10 % absorption, which can approach unity, by engineering their photonic environment [START_REF] Horng | Perfect absorption by an atomically thin crystal[END_REF]. They are thus promising for future flexible optoelectronic applications, such as: LEDs, lasers, photovoltaics, photodetectors, electroluminescence or optical modulators [START_REF] Mak | Photonics and optoelectronics of 2d semiconductor transition metal dichalcogenides[END_REF].

On the more fundamental level, SL-TMDs are nowadays a benchmark to study the physics of excitons (EXs), hydrogen-like bound states formed by Coulomb interaction between an electron and a hole. In typical direct band gap bulk semiconductors such as GaAs, due to the large dielectric screening and small effective mass of the carriers, the resulting EX binding energy is barely a few millielectronvolt (meV). This is negligible compared to the thermal fluctuations, unless the sample is studied at low temperatures. In SL-TMDs though, due to large effective masses and reduced dielectric screening, the EX binding energy is increased up to several hundred meV, such that even at room temperature the optical properties are governed by EX transitions. Furthermore, their band structure displays a pair of non-equivalent valleys as regards the spin degree of freedom, establishing original selection rules for the excitation and emission of light, where the EX valley index is locked with the light helicity. SL-TMD have been initially fabricated as bare flakes deposited directly on SiO 2 substrates. In such configuration, they suffer from a direct contact with the environment and its invasive chemistry, thus become unstable and prone to structural aging [START_REF] Gao | Aging of transition metal dichalcogenide monolayers[END_REF] and photodegradation [START_REF] Ahn | Prevention of transition metal dichalcogenide photodegradation by encapsulation with h-BN layers[END_REF]. Due to microscopic and macroscopic disorder factors -such as: strain, wrinkling, flake deformations and cracks, lattice defects and vacancies, variation of charge state and of dielectric constant in the substrate -the optical response was largely dominated by the inhomogeneous broadening [START_REF] Moody | Intrinsic homogeneous linewidth and broadening mechanisms of excitons in monolayer transition metal dichalcogenides[END_REF][START_REF] Jakubczyk | Radiatively limited dephasing and exciton dynamics in MoSe2 monolayers revealed with four-wave mixing microscopy[END_REF], σ. Recently, it was demonstrated that surface protection by encapsulating SL-TMDs between layers of hexagonal boron nitride h-BN [START_REF] Cadiz | Excitonic linewidth approaching the homogeneous limit in MoS2 based van der Waals heterostructures: accessing spin-valley dynamics[END_REF][START_REF] Ajayi | Approaching the intrinsic photoluminescence linewidth in transition metal dichalcogenide monolayers[END_REF][START_REF] Manca | Enabling valley selective exciton scattering in monolayer WSe2 through upconversion[END_REF][START_REF] Jakubczyk | Coherence and density dynamics of excitons in a single-layer MoS2 reaching the homogeneous limit[END_REF] provides stable samples with low disorder, avoiding surface contaminations causing the EX broadening. Encapsulation thus yields step-like improvement of optical properties. The main indicator of such improvement was the narrowing of the EX linewidth and appearance of the Rydberg series of the EX's excited states [START_REF] Manca | Enabling valley selective exciton scattering in monolayer WSe2 through upconversion[END_REF][START_REF] Chernikov | Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2[END_REF][START_REF] Molas | Energy spectrum of twodimensional excitons in a nonuniform dielectric medium[END_REF][START_REF] Goryca | Revealing exciton masses and dielectric properties of monolayer semiconductors with high magnetic fields[END_REF].

II. EXPERIMENT

The question arises if the structural disorder in such heterostructures can be suppressed to the point, where σ is not detectable in the optical response ? If so, how can one demonstrate that the EX transition linewidth indeed attains the homogeneous limit, γ ? Another query regards the spatial extensions, on which such optimal conditions can be maintained. To address these issues one has to use an optical microscopy technique capable to separate σ and γ within the transition spectral lineshape. Four-wave mixing (FWM) spectroscopy, which is the principal tool in our work, has been conceived for this very purpose.

FWM is a nonlinear polarization in the material, proportional to µ 4 E 1 E 2 E 3 , where E 1,2,3 are three short laser pulses generated by a femtosecond laser and µ is the transition oscillator strength. In this approach, σ and γ can be differentiated, by monitoring time-resolved (TR) FWM, while varying the time delay between E 1 and E 2 , τ 12 . In the presence of σ, FWM appears as a photon echo [15]. As depicted in Fig. 1 a, TR-FWM is a Gaussian peak centered at t = τ 12 and displaying the temporal width proportional to 1/σ. Such en echo builds up due to the phase conjugation (labeled with ) between the first order absorption created by E 1 and FWM. Conversely, in the homogeneous limit, FWM transient shows the exponential decay starting at t = 0 for whatever τ 12 , also drawn in Fig. 1 a. This is known as free induction decay, adopting the nomenclature used in the NMR spectroscopy. In general, (σ, γ) pair can be accurately retrieved by a two-dimensional fitting [START_REF] Jakubczyk | Coherence and density dynamics of excitons in a single-layer MoS2 reaching the homogeneous limit[END_REF] of TR FWM for various τ 12 .

To enable microscopy, FWM is here detected through optical heterodyning. In short, i) E 1,2,3 are modulated by distinct radio frequencies Ω 1, 2, 3 around 80 MHz using acousto-optic deflectors. E 1,2,3 propagate co-linearly and are focussed onto the sample down to the diffraction limit using a microscope objective of NA=0.65. ii) in the detection path, the reflected signal is interfered with the reference beam E R modulated at the phase-locked linear combination of radio-frequencies Ω 3 + Ω 2 -Ω 1 , sensitive to the FWM response. iii) background-free spectral interference between E R and FWM is then recorded as DC signal on a CCD camera, which is installed at the output of the imaging spectrometer [START_REF] Langbein | Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems[END_REF]. FWM field is obtained via spectral interferometry, granting access to the amplitude and phase of the signal. Thus, the modeling often requires to apply complex response functions, to properly account for detection of the field and potential interference effects. This setup is particularly suited to perform FWM micro-spectroscopy with a high signal to noise ratio, accessing the EX coherence dynamics with an enhanced temporal (spatial) resolution of 150 fs (500 nm). Further details regarding the experimental technique can be found in Refs. [16 and 17], while sketches of the experimental setup are presented in supplementary material in Refs. [18 and 19].

III. RESULTS

A. Photoluminescence and FWM mapping of a MoSe2 heterostructure

We first present results obtained on a h-BN/MoSe 2 /h-BN heterostructure. Its optical image under white light illumination, is shown in Fig. 2 a. The edges of the MoSe 2 flake are contoured with the dashed line. The dark spots correspond to the air blisters between the flake and the top h-BN having the thickness around 20 nm. The bottom h-BN instead shows terraces of different thickness 232 nm (blue area), 240 nm (yellowish), 252 nm (orange) and 290 nm (red). To assess the optical response, the heterostructure is cooled down to T=5 K and we perform confocal photoluminescence (PL) under non-resonant CW excitation at 710 nm, at the position marked with a blue circle in a). As shown in Fig. 2 c, in the PL not only the neutral EX transition is observed, but also the trion (TR) one. For the latter, we note the characteristic low energy tail in the PL signal. While, owing to the residual strain, the spectral position of both transitions fluctuate across the sample by around 15 meV, the TR binding energy is constant and equal to 26 meV. In Fig. 2 d, we present the PL imaging of the sample obtained via raster scanning the excitation and spectrally integrated over TR and EX transitions. It is interesting to notice the correlation between the spectrally integrated PL intensity and the thickness of the bottom h-BN terraces: the thinner the latter, the stronger PL is observed. This result is consistent with the recent observation that the EX radiative lifetime can be tuned through the Purcell effect, when varying the thickness of the surrounding h-BN layers [START_REF] Fang | Control of the exciton radiative lifetime in van der waals heterostructures[END_REF]. To investigate this scenario we propose to perform time-resolved resonant spectroscopy.

To measure the resonant optical response we first perform white light reflectance restricting the signal to the spot marked with a blue circle. The result is shown in Fig. 2 b. Only EX transition appears as a dispersive quasi-Lorentzian line, with the full width at halfmaximum (FWHM) of marely 2.5 meV. The trion line is absent owing to encapsulation and resonant excitation, which both suppress density of free carriers required to form charged excitons. In order to measure coherent nonlinear response FWM microscopy is carried out. In Fig. 2 e we present a typical spectral interferogram between E R (brown line) and the FWM signal at τ 12 = τ 23 = 0. FWM image is constructed by plotting time-integrated (TI) FWM amplitude measured for different points on the sample and is shown in Fig. 2 f. Although the strongest FWM is measured at the flake area covered with the "blue" h-BN stripe, globally there is no clear correlation between the FWM amplitude, reflecting to large extend the EX radiative rate, and the thickness of the encapsulating layers. This is attributed to the underlying disorder, generating σ. We have recently demonstrated [START_REF] Jakubczyk | Coherence and density dynamics of excitons in a single-layer MoS2 reaching the homogeneous limit[END_REF] that there exists correlation between σ and the EX lifetime: the latter is increased at the areas showing larger σ due to the stronger exciton localization. The same effect is present here, as evidenced in the Supplementary Fig. S1. We also note that different interference conditions from the various thicknesses of the encapsulating layers generate different electric field distributions of the excitation laser at a plane of MoSe 2 SL. This could yield different PL intensities measured at different h-BN thicknesses in Fig. 2 d.

B. Dephasing mechanisms of excitons in the MoSe2 heterostructure

It was recently reported that FWM transient in bare SL TMDs is a photon echo [START_REF] Moody | Intrinsic homogeneous linewidth and broadening mechanisms of excitons in monolayer transition metal dichalcogenides[END_REF][START_REF] Jakubczyk | Radiatively limited dephasing and exciton dynamics in MoSe2 monolayers revealed with four-wave mixing microscopy[END_REF][START_REF] Jakubczyk | Impact of environment on dynamics of exciton complexes in a WS2 monolayer[END_REF]. In Fig. 3 a we present TR FWM for different delays τ 12 measured at the same area as in Fig. 2 b. Here, there is virtually no sign of photon echo formation; the maximum amplitude of the transient does not follow the diagonal t = τ 12 . Instead, for τ 12 > 0 FWM is maximal for t = 0, and then evolves as the exponential free induction decay for t > 0. This shows that the optical response is dominated by the homogeneous broadening, i.e., γ σ. It is worth to note that FWM is here also detected for τ 12 < 0. In such configuration, E 1 arrives last and triggers FWM through the local field effect [START_REF] Jakubczyk | Coherence and density dynamics of excitons in a single-layer MoS2 reaching the homogeneous limit[END_REF][START_REF] Wegener | Line shape of time-resolved four-wave mixing[END_REF]. Furthermore, as E 1 -E R delay increases, FWM sets off for increasingly longer times when going into more negative delays, as indeed visible in the color map.

To measure γ, we plot TI FWM versus τ 12 , as shown in Fig. 3 b. The data are modeled by a convolution of a Gaussian, reflecting pulse duration of around 150 fs and exponential decay exp(-γτ 12 /2 ), parameterized with γ (FWHM) or dephasing time T 2 = 2 /γ. At low temperature T=4.6 K we measure γ = (2.5 ± 0.1) meV. This differs from around 1 meV measured in bare MoSe 2 SLs [START_REF] Jakubczyk | Radiatively limited dephasing and exciton dynamics in MoSe2 monolayers revealed with four-wave mixing microscopy[END_REF] and is therein attributed to longer EX lifetime due to localization on disorder. The latter is significantly larger in bare SLs, as monitored by σ.

On top of ideal TMD crystal properties -determin- ing the morphology of the EX wave function, its binding energy and radiative lifetime -there are at least three major environmental parameters influencing γ: temperature, EX density and local disorder, which will be considered in the following. In Fig. 3 c we present γ measured for different temperatures. The apparent increase of γ is modeled by the sum of linear increase and exponential activation part, due to interaction with acoustic and optical phonons, respectively. The data are fitted with the following equation:

γ(T) = γ 0 + aT + b/[exp(E 1 /k B T) -1]
, with E 1 = 37 meV, corresponding to activation energy of optical phonons [START_REF] Molina-Sánchez | Phonons in singlelayer and few-layer MoS2 and WS2[END_REF]. Qualitatively, the phonon dephasing mechanisms are therefore similar as in bare MoSe 2 SLs [START_REF] Jakubczyk | Radiatively limited dephasing and exciton dynamics in MoSe2 monolayers revealed with four-wave mixing microscopy[END_REF]. We note that temperature induced increase of the linewidth can be directly monitored in the FWM spectra, as exemplified in Fig. 3 d. Owing to negligible σ, there is a very good correspondence between γ retrieved from the coherence dynamics and directly measured FWM spectral linewidth. We emphasize that such comparison to be valid, requires prior knowledge that the transition lineshape is homogeneous. Such an independent assessment is here provided by TR FWM data.

An important factor influencing γ is the EX density, n. With its increase, EX-EX interaction increases, causing excitation induced dephasing, as observed in the past in semiconductor quantum wells [START_REF] Wang | Transient nonlinear optical response from excitation induced dephasing in GaAs[END_REF][START_REF] Honold | Collision broadening of two-dimensional excitons in a GaAs single quantum well[END_REF] and more recently for WSe 2 [START_REF] Moody | Intrinsic homogeneous linewidth and broadening mechanisms of excitons in monolayer transition metal dichalcogenides[END_REF] and MoS 2 [START_REF] Jakubczyk | Coherence and density dynamics of excitons in a single-layer MoS2 reaching the homogeneous limit[END_REF] SLs. In Fig. 4 a we present exemplary curves of coherence dynamics, where a steeper slope is visible with increasing E 1 intensity, P 1 . Monitoring the impinging power and assuming absorption of approximately 10%, we can estimate n. In Fig. 4 b we plot the resulting γ(n) dependence: the broadening is well visible, we observe that γ can be doubled with 300-fold increase of the resonantly generated EX density. We propose to describe the effect by the linear dependence γ(n) = γ 0 + α × n, where γ 0 is the zero-density linewidth and α is the interaction parameter, both established for T=5 K. From the linear fit, we obtain γ 0 = (2.07 ± 0.03) meV and α = (5.4 ± 0.2) × 10 -13 meVcm 2 . The interaction strength is here 5 times smaller compared to similar studies performed on WSe 2 SLs [START_REF] Moody | Intrinsic homogeneous linewidth and broadening mechanisms of excitons in monolayer transition metal dichalcogenides[END_REF]. In Fig. 4 b we also show that the broadening is directly visible in the FWM spectra and in a good agreement with the coherence dynamics measurements.

C. Reflectance and FWM response in a WSe2 heterostructure

The second part of this work is devoted to investigations of a WSe 2 heterostructure. Its optical image is presented in Fig. 8 a. A large WSe 2 SL of around (60 × 60) µm was transferred at the bottom h-BN (90 ± 10) nm thick layer and then partially covered with the top one of nominally the same thickness, resulting in the elongated heterostructure, with a shape marked with a dashed contour in Fig. 8 a. Let us first investigate the area of the flake marked with a cross, which is free from wrinkling (occurring at the edges) and air blisters (visible as dark spots on the neighboring surface).

The reflectance spectrum measured at T=5 K is presented in Fig. 5 a. The fundamental EX 1S transition is detected as an emissive quasi-Lorentzian line, with FWHM only 2.6 meV. Interestingly, the second resonance is clearly observed around 130 meV above EX 1S, with even a sharper line-shape of 1.7 meV (FWHM). It is attributed to the EX excited 2S state. An additional spectral wiggle is seen another 10 meV higher, however its contrast is not sufficient for the unequivocal assignment of the underlying transitions. FWM spectral interferograms at τ 12 = 0 are shown as blue traces in Fig. 5 a. To drive FWM of the 1S EX we still employ a Ti:Shapphire laser. However, to study the excited states occurring well below 700 nm, an Optical Parametric Oscillator must be used. As the technique is background free and detection is at the shot-noise limit, apart from 1S and 2S states, FWM reveals further resonances attributed to 3S and 4S states. When plotting the energy versus the quantum number (see inset) we observe non-hydrogenic series, in line with previous findings [START_REF] Chernikov | Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2[END_REF] and the EX binding energy [START_REF] Molas | Energy spectrum of twodimensional excitons in a nonuniform dielectric medium[END_REF][START_REF] Goryca | Revealing exciton masses and dielectric properties of monolayer semiconductors with high magnetic fields[END_REF] of E b =161 meV.

D. Exciton coherence dynamics in the WSe2 heterostructue: dephasing and coherent coupling

FWM transients versus τ 12 measured for 1S and 2S states are shown as color maps in Fig. 5 b. With increasing τ 12 , FWM is emitted as an exponentially decaying transient with a maximum centered around t = 0. We see strictly no indication of the photon echo formation. FWM takes the form of the free induction decay, proving that at the probed position, the EX transition is homogeneously broadened. To determine FWHM of γ = 2 /T 2 , in Fig. 5 c we plot time-integrated FWM amplitude as a function of τ 12 . Resulting coherence dynamics displays a mono-exponential decay over three (six) orders of magnitude in a FWM field (intensity). From these decays we extract γ, similarly as for the MoSe 2 heterostructure (Fig. 3 b).

Let us note that the excitation bandwidth of the OPO laser is sufficient to simultaneously cover the excited states 2S, 3S and 4S, as shown in Fig. 6 a. Because these transitions share the same ground state, they are expected to be coherently coupled (absorption on one of the transitions, mutually generates FWM on the other ones). Such coupling should therefore induce quantum beating in the corresponding coherence dynamics. In Fig. 6 b, we present such dynamics measured via τ 12 -dependence. Indeed, we observe that FWM is here modulated with the period of ∆ = 2π /δ = 0.26 meV, well corresponding to the 2S-3S splitting δ 16 meV. We can resolve two oscillation periods within the initial 0 < τ 12 < 0.5 ps. For subsequent delays τ 12 > 0.5 ps the beating is smeared out owing to fast dephasing of higher excited states. This measurement thus indicates that one can induce couplings within the Rydberg states of TMD EXs.

Focussing back on the dephasing mechanisms, in Fig. 5 c we observe that for increasing temperature, the FWM decay gets more pronounced, owing to the increase of γ (and correspondingly, shortening of dephasing time T 2 ). Such temperature dependence of γ, measured at 1S and 2S states [25] is summarized in Fig. 5 d.

The exciton 1S EX transition is once again modeled with a linear increase, with γ 0 = (2.96 ± 0.05) meV and a = (0.02 ± 0.002) meV/K, attributed to the interaction with acoustic phonons, and the exponential activation term with E 1 = 44 meV, due to the onset of interaction with optical phonons [START_REF] Molina-Sánchez | Phonons in singlelayer and few-layer MoS2 and WS2[END_REF].

As regards the 2S EX transition, let us remind that for an ideal two-dimensional Wannier exciton, the oscillator strength µ scales with the quantum number n as: µ ∝ (n -1/2) -3 . The radiatively limited broadening of the 2S EX could be therefore estimated to be 3 3 = 27 times weaker than the one of the 1S EX, thus only around 0.1 meV at low temperature. It is interesting to note that for T < 20 K, we indeed measure γ(2S) < γ(1S). However γ(2S) is around 20 times larger than the radiative limit, indicating that even low temperature dephasing is dominated by a non-radiative decay, plausibly through the scattering into 1S EX at high center-of-mass in-plane momenta. With increasing temperature, we observe much stronger EX dephasing for the 2S state with respect to 1S. However, recent theoretical work [START_REF] Brem | Intrinsic lifetime of higher excitonic states in tungsten diselenide monolayers[END_REF] predicts that 2S transition is less prone to interaction with phonons than the 1S. We here therefore refrain from the modeling of the γ(T ) dependence measured for the 2S EX.

E. Population dynamics of 1S and 2S excitons

To further elucidate the interplay between radiative and non-radiative decay mechanisms of EX 1S and 2S transitions, we infer the dynamics of resonantly excited EX density. We set τ 12 = 0 and employ E 1 and E 2 to drive modulation of EX's population, slowly varying with the frequency Ω 2 -Ω 1 = 1 MHz. TI-FWM, triggered via E 3 delayed from E 2 by τ 23 , reflects the amplitude of this oscillating EX density, as depiced in Fig. 1 b. Exemplary data reflecting such dynamics of 1S EX measured at T=5 K for increasing P 1 and fixed P 2 and P 3 are given in Fig. 7 a. While the EX coherence, inspected in previous sections, vanishes after a few ps, the EX density evolves here on a timescale of at least one nanosecond. This is consistent with recent FWM experiments on SL MoSe 2 [START_REF] Scarpelli | Resonantly excited exciton dynamics in two-dimensional mose2 monolayers[END_REF] and MoS 2 heterostructure [START_REF] Jakubczyk | Coherence and density dynamics of excitons in a single-layer MoS2 reaching the homogeneous limit[END_REF], and also perovskite nanocrystals [START_REF] Becker | Long exciton dephasing time and coherent phonon coupling in CsPbBr2Cl perovskite nanocrystals[END_REF] and nanoplatlets [START_REF] Naeem | Giant exciton oscillator strength and radiatively limited dephasing in twodimensional platelets[END_REF].

Qualitatively, in the dynamics we distinguish: i) the rise from negative τ 23 , owing to the finite duration of the laser pulses, ii) initial decay on a few hundred fs scale, attributed to the radiative decay and EX scattering to dark EX states. iii) followed after τ 23 > 1 ps by a much longer FWM decay, due to populating the dark EX states and their subsequent relaxation and interaction with the bright ones, i.e., EXs within the light cone, obeying spin selection rules for radiative recombination.

Because the density dynamics is here measured via the interference with E R , which is phase-sensitive, it should be described with a complex response function R. The latter is here proposed in a form of a coherent superposition of three exponential decay components [START_REF] Jakubczyk | Coherence and density dynamics of excitons in a single-layer MoS2 reaching the homogeneous limit[END_REF][START_REF] Scarpelli | Resonantly excited exciton dynamics in two-dimensional mose2 monolayers[END_REF]:

R(τ 23 ) = 3 j=1 A j Θ(τ 23 ) exp(iφ j -τ 23 /τ j ) (1)
where (A j , φ j , τ j ) with j = (1, 2, 3) are amplitude, phase and decay constant of each of the three processes. The pulse width is taken into account by convoluting R with the Gaussian of τ 0 = 150 fs. Furthermore, owing to the repetition period T R 13 ns, there is a pileup effect building up throughout many repetitions of the heterodyne experiment equal to (exp( T R τj ) -1) -1 . The response function therefore takes the following form: R(τ 23 ) =

3 j=1 A j {[exp( T R τj ) -1] -1 + 1 2 [(1 + erf( τ23 τ0 - τ0 2τj )]} × exp (iφ j -τ23 τj + τ 2 0 4τ 2 j
). Finally, to fit the FWM amplitude we take absolute value of R(τ 23 ). The result of the modeling is given by red traces and retrieved set of decay constants for different P 1 is given in Fig. 7 c. For the initial decay we consistently find τ 1 = T 1 (1S) 0.18 ps. The second and third process decay on a scale of a few tens and a few hundred ps, respectively. They describe EX evolution in the dark state reservoirs, namely EXs with the in-plane momentum out of the light cone and the spin-forbidden ones. Note that the wiggling observed around τ 23 = 100 ps, building up when increasing P 1 , is due to interference between all three processes. The interference becomes more pronounced as the ratios A 2 /A 1 and A 3 /A 1 increase with P 1 , as reported in Fig. 6 d. In other words, increasing the excitation results in higher occupancy of dark EX states and their enhanced interaction with the bright EX, which is probed with FWM τ 23 -dependence.

In Fig. 7 b we present the density dynamics measured on the 2S EX transition. The modeling includes now only two decay processes. Interestingly the initial decay of τ 1 = T 1 (2S) 0.75 ps is measured to be more than four times longer than for the 1S EX, indicating longer radiative decay of the 2S state. It is however still far off from the prediction based on the relative oscillator strength, again indicating dominance of the non-radiative decay processes of the 2S states. We note that dephasing of the higher states is much faster than our temporal resolution of 150 fs, showing prominently non-radiative decay of the higher excited EX states. An important conclusion emerging from our experiments is that EX's excited states can be observed uniquely at the areas with eradicated disorder, where the optical response is dominated by γ. To substantiate this claim, we perform FWM spatial imaging across the flake. In Fig. 8 b we present a color map where FWM amplitude and transition energy of 1S EX are encoded as brightness and hue level, respectively. Even though locally (on the areas up to (5 × 5) µm 2 ) the transition energy can be quasi-constant (spectrally fluctuating an order of magnitude less than γ), globally the EX energy changes by around 25 meV. This variation is attributed to residual strain generated during the processing of the heterostructure -other sources of disorder, like variation of dielectric permittivity or doping were shown to be suppressed [START_REF] Raja | Dielectric disorder in two-dimensional materials[END_REF] in h-BN encapsulated TMD SLs.

To illustrate the interplay between strain-induced disorder and coherent response, we perform mapping of the EX coherence dynamics: we therefore spatially scan within the part of the heterostructure enclosed by a white square, while at each position the delay τ 12 is varied. In Fig. 8 c we provide four examples of the time-and τ 12resolved FWM. In the first pair of panels, the response clearly displays the photon echo. When cross-correlating with b), one assigns these to the areas exhibiting a strong radial strain gradient of the EX energy, generating σ. Conversely, the second pair of panels, measured on a spot showing a quasi-constant energy level, yields FWM as a free-induction decay signal. We note that at some positions, we observe FWM temporal response generated by a mixture of inhomogeneous contributions. In other words, even within the sub-µm excitation area, several domains characterized by different γ and σ can contribute to the signal, rendering quantitative analysis of these transients challenging.

Finally, we have carried out FWM spatial imaging of the 2S EX transition. As in panel b), we encode the amplitude and transition energy as brightness and hue, and present the result in Fig. 8 d. One can notice that at areas with suppressed disorder -here identified by FWM displaying free-induction decay instead of photon echo -the 2S EX transition can be observed. Interestingly, FWM of the 2S EX is suppressed at the areas exhibiting a pronounced photon echo. This indicates that absorption of EX's excited states can only be induced at locations with low disorder on both microscopic and macroscopic scales.

IV. CONCLUSIONS

We investigated coherent non-linear response of the EX transitions in high quality h-BN/MoSe 2 /h-BN and h-BN/WSe 2 /h-BN heterostructures. Using FWM microscopy and imaging we showed that encapsulating SLs of MoSe 2 and WSe 2 generally provides low disorder samples, which on areas reaching (5 × 5) µm 2 host homogeneously broadened EX transitions. At such locations, we: i) investigated EX dephasing mechanisms, ii) revealed coherent nonlinear response of EX excited states up to 4S, iii) provided indication of coherent coupling within excited states. We compared population lifetimes of 1S and 2S states, notably finding longer dephasing time (smaller homogeneous broadening) of the 2S state. This is assigned to its larger spatial extent and correspondingly longer radiative lifetime with respect to the EX 1S, as indicated by investigation of the EX population dynamics. Even though locally optical response of these TMD heterostructures is nowadays brought close to perfection, our work shows that further efforts in processing are necessary, to attain such performance across the entire heterostructure. Eliminating spatially varying disorder would, for instance, permit to quantitatively evaluate the impact of dielectric surrounding on the coherent response of TMD EXs. Furthermore, the resulting weak EX local-ization would enable to investigate propagative aspects of EX coherence and density (diffusion) by implementing FWM with a spatially separated excitation beams.

V. METHODS: SAMPLE FABRICATION

The heterostructures were obtained by a bottom-up method using the gel-film assisted exfoliation technique. As a substrate an ultra-flat silicon wafers were used, provided by Ted Pella. The WSe 2 and MoSe 2 crystals were bought from HQ Graphene and the high purity h-BN was prepared in NIST (Japan). A bottom layer of h-BN was exfoliated directly to the silicon substrate. The flakes of required thickness were identified using the optical microscope and some of them verified using AFM. For the heterostructures h-BN/WSe 2 /h-BN the thickness of bottom h-BN is (90 ± 10) nm. The heterostructures h-BN/MoSe 2 /h-BN possess sequence of different thicknesses in the range from 200 to 250 nm for the bottom h-BN. SLs of TMDs and top h-BN were transferred directly from gel-film using all-dry transfer method. Heterostructures were annealed before each transfer (180 • C, 5 min) to remove as much residues as possible. After completing the stacking process all samples were annealed for another 5 hours at 200 • C. All heterostructures contain some air trapped between TMD and top h-BN flake (easily visible in form of bubbles in Fig. 2 a and Fig. 8 a).

Figure 1 .

 1 Figure 1. Schema of pulse sequences used in the experiments and resulting forms of the four-wave mixing transients a) Exciton coherence dynamics is measured by varying τ12 and keeping τ23 = 0. In a presence of inhomogeneous broadening TR-FWM is a photon echo. When homogeneous broadening dominates, FWM is a free induction decay. b) Exciton density dynamics is probed when varying τ23 and fixing τ12 = 0.

Figure 2 .

 2 Figure 2. Characterization of a h-BN/MoSe2/h-BN heterostructure via optical microspectroscopy, T=5 K. a) Optical image of the sample. Different colors correspond to different thicknesses of the bottom h-BN layer. b) Measured micro-reflectance from the area marked with a blue circle in a), showing EX linear response. c) PL spectrum measured at the same location as in b) showing neutral (EX) and charged exciton (TR) transitions. d) Imaging of the spectrally integrated PL. e) Typical spectral interference (black) between FWM at zero delay and ER (brown trace). f) Imaging of the time-integrated FWM amplitude.

2 F

 2 W M a m p . p h o t o n e n e r g y ( m e V ) a y τ 1 2 ( p s ) T i m e -I n t e g r a t e d F W M a m p .

Figure 3 .

 3 Figure 3. Temperature dependent EX dephasing measured at the area of MoSe2 heterostructure dominated by the homogeneous broadening. a) Timeresolved FWM amplitude measured for various delays τ12. FWM transient appears as free-induction decay indicating that the response is dominated by γ. b) Time-integrated FWM amplitudes versus τ12 presented for several temperatures along with the fits (red lines). c) Homogeneous broadening γ as a function of temperature, indicating phononinduced dephasing mechanism, as modeled with the red line. d) Exemplary spectra of FWM amplitude for different temperatures, τ12 = 0.6 ps.

6 FFigure 4 .

 64 Figure 4. EX dephasing measured as a function of the resonantly generated density at T=5 K at the area of MoSe2 heterostructure entirely dominated by homogeneous broadening. a) τ12-dependence of the timeintegrated FWM amplitude measured for different E1 excitation powers, P1. b) Homogeneous broadening γ versus EX density (black squares), compared with FWM spectral FWHM (green circles).

Figure 5 .

 5 Figure 5. Four-wave mixing microspectroscopy of the exciton transitions in a WSe2 heterostructure. a) Measured micro-reflectance showing linear response of 1s and 2s states (bottom) and FWM-reference spectral interference revealing exciton nS series up to n=4. Inset: energy versus quantum number n. b) Transients of the FWM amplitude versus τ12 showing free-induction decay (directly exemplified with orange traces for τ12 = 0.2 ps and 1 ps) for 1S and 2S states. c) Time-integrated FWM amplitudes as a function of τ12 for temperatures as indicated. d) Temperature dependence of the homogeneous broadening γ for 1S and 2S states. The modeling is indicated with a gray trace.

Figure 6 .

 6 Figure 6. Coherent coupling in the EX excited states measured in the WSe2 heterostructure. a) FWM spectral interferogram showing EX excited states 2S, 3S and 4S (black), simultaneously excited by the OPO laser pulse (brown). b) The beating measured in the coherence dynamics via TI FWM(τ12) indicates the presence of coherent coupling between the excites states, especially between 2S and 3S, which dominate the spectrum.

Figure 7 .

 7 Figure 7. Time-integrated FWM as a function of delay τ23 reflecting resonantly excited EX density dynamics in the WSe2 heterostructure a) Density dynamics of EX 1S measured for different E1 excitation powers, as indicated.Modeling is given with red traces. b) as in a), but measured for EX 2S. Longer radiative decay of the 2S state with respect to the 1S has its impact on the initial population loss, which is measured to be more than four times longer for the 2S state. Both measurements were performed at the same location of the flake. c) Decay constant τ1, τ2 and τ3 retrieved from the modeling of 1S EX density (a) and 2S EX density dynamics (b). d) Amplitude ratios A2/A1 and A3/A1 versus P1 obtained from the modeling.

F

  . FWM mapping of exciton 1S and 2S transitions in the WSe2 heterostructure

Figure 8 .

 8 Figure 8. Four-wave mixing imaging of 1S and 2S EXs in the WSe2 heterostructure. a) Optical image of the sample. The heterostructure area is enclosed with a dashed line. b) FWM imaging of the 1S EX, τ12 = 0.1 ps. Amplitude and transition energy (with respect to central energy of 1720 meV) are encoded as brightness and hue level, as indicated. Note that WSe2 not covered with the h-BN also generates FWM signal. c) Exemplary time-resolved FWM versus τ12. The areas displaying strong energy gradients, owing to varying strain, generate photon echo. Free induction decay is measured on the locations where the transition energy is quasi-constant. d) FWM imaging of the 2S EX, τ12 = 0.1 ps. Scale as in b), central energy is 1845 meV.

VI. ACKNOWLEDGMENTS JK acknowledges support by the European Research Council (ERC) Starting Grant PICSEN (grant no. 306387) and he is grateful for discussions with D. Wigger, M. Richter, M. Selig and J. Renard and W. Langbein. LZ acknowledges the support from the Student Project IGA Pr F2019 031 of Palacký. MB acknowledges the support from the Ministry of Education, Youth and Sports of the Czech Republic under the project CEITEC 2020 (LQ1601) University. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. Finally, we acknowledge support by the EU Graphene Flagship project (no. 785219).


domain: sdu sde

text:
Anastassia Y Borisova

Nail R Zagrtdenov
email: nailzagrtdenov@gmail.com

Michael J Toplis

Jérémy Guignard

New model of chromite and magnesiochromite solubility in silicate melts

Chromite is a key magmatic mineral frequently used as petrogenetic indicator of physicochemical conditions of mafic magma crystallization. In this work, magnesiochromite and chromite solubility in a natural basalt and an iron-free haplobasalt was investigated at 1440°C and atmospheric pressure under controlled CO-CO2 gas mixtures corresponding to the range two log units below to two log units above the fayalite-magnetite-quartz buffer (FMQ-2 to FMQ+2). The source of chromium was either natural chromite or synthetic Cr2O3, the latter reacting with the basaltic liquids to form a magnesiochromite. The highest concentrations of Cr in haplobasaltic melts are found in equilibrium with magnesiochromite, ranging from 3500 to 6800 ppm depending on redox conditions. In detail, at low fO2, liquids have high Cr contents, but the variation of log [Cr, ppm] is not a linear function of log fO2. Using our new data and data from the literature a model for Cr concentrations at chromite/magnesiochromite saturation in silicate melts has been developed based upon a thermodynamic formalism. This model has the form:

where X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 is mole fraction of chromium in the liquid, X 𝐶𝑟 2 𝑂 3 𝐶ℎ𝑟 is mole fraction of chromium oxide in the chromite, T is temperature in Kelvins, 𝑓 𝑜2 is oxygen fugacity in bars, ⋋ is optical basicity, and a (-7.01 ± 2.10), b (13.72 ± 2.73), c (-12405 ± 1253), d (24.46 ± 3.13), k (-24395 ± 2037), g (-23.59 ± 4.20) are constants. This model may be used to assess the effect of melt composition on chromite and magnesiochromite solubility in silicate melts during peridotite melting and assimilation. At moderate and high oxygen fugacities (for example, >FMQ-1 at 1200 C), concentration levels of Cr at chromite saturation are higher for ultramafic than for felsic rocks. Our data imply that assimilation of magnesiochromite-bearing serpentinite lithosphere could result in high Cr contents in mafic melts, triggering massive crystallization of chromite, especially upon the system hydridization, cooling, oxidation and magma degassing. Our model may be applied for quantitative prediction of chromite crystallization and formation of the terrestrial mantle-crust transition zones and mantle represented by chromitites and dunites.

Introduction

Crystallization of chromite from fluids and silicate melts is the most probable origin of large concentrations of chromite in igneous rocks such as disseminated, stratiform, nodular and podiform chromitites [START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF]Borisova et al., 2012;[START_REF] Martin | Fluids are bound to be involved in the formation of ophiolitic chromite deposits[END_REF].

Experimentally determined chromite solubilities in silicate melts are variable (Maurel and Maurel, 1982b;[START_REF] Murck | The effects of temperature, oxygen fugacity and melt composition on the behaviour of chromium in basic and ultrabasic melts[END_REF][START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF][START_REF] Forsythe | Comparison of experimentally crystallized and natural spinels from LEG-135[END_REF][START_REF] Poustovetov | The distribution of Cr between basaltic melt and chromian spinel as an oxygen geobarometer[END_REF] ranging from 50 ppm at 1150 C and fO2 ~ -6.9 [START_REF] Forsythe | Comparison of experimentally crystallized and natural spinels from LEG-135[END_REF] to 9800 ppm at 1300 C and fO2 ~ 12.8 [START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF], suggesting an importance of such physical-chemical parameters as temperature, oxygen fugacity and the melt composition controlling the chromite saturation in the silicate melts. High concentration of chromium at chromite saturation (~1220 ppm Cr) of basaltic melt at FMQ and 1350°C contrasts with the fact that natural basaltic to boninitic melts have lower concentrations of Cr (Fig. 1), with typical values in the range from 20 to 700 ppm [START_REF] Allison | The mean composition of ocean ridge basalts[END_REF]); yet, the real Cr concentrations may be affected by the secondary fluorescence effect in the glasses (Borisova et al., 2018), although the effect is absent in the glasses without any traces of the Cr-rich minerals.

The fact of wide variation of Cr in the natural mafic melts implies that natural melts of mantle origin have lost their initial Cr concentrations upon chromite crystallization during partial melting of peridotites and the melt percolation in the lithosphere and asthenosphere, magma mixing and possible hydridization (e.g., Borisova et al., 2020a,b).

The available experimental data suggest that the Cr content of melts saturated in chromite strongly depends on temperature and oxygen fugacity and, to a lesser extent, melt composition. For example, chromium concentrations in mafic melts increase with increasing temperature and rise dramatically when oxygen fugacity decreases (Maurel and Maurel, 1982b;[START_REF] Murck | The effects of temperature, oxygen fugacity and melt composition on the behaviour of chromium in basic and ultrabasic melts[END_REF][START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF]. The high Cr concentrations in silicate melts have long been postulated to be the result of assumed Cr 2+ which prevails in basaltic melts compared to Cr 3+ [START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF], as subsequently confirmed by [START_REF] Berry | A XANES determination of the oxidation state of chromium in silicate glasses[END_REF] and [START_REF] Berry | The effect of composition on Cr2+/Cr3+ in silicate melts[END_REF] by X-ray Absorption Near Edge Structure spectroscopy (XANES). Concerning melt composition, basalts with higher MgO contents and lower Fe2O3 and Al2O3 contents have higher Cr contents at the chromite saturation, while more total alkalis and more FeO lead to lower Cr content [START_REF] Murck | The effects of temperature, oxygen fugacity and melt composition on the behaviour of chromium in basic and ultrabasic melts[END_REF][START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF].

The latter observation was taken to indicate that Cr 2+ and Cr 3+ occupy similar structural positions in the melt as Fe 2+ and Fe 3+ leading to competition between these cations. The composition of chromites coexisting with silicate melts is also controlled by physico-chemical conditions, but the compositional Cr 3+ to Al 3+ ratio of the melt has the principal control (e.g. Maurel and Maurel, 1982b;[START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF]. For this reason, at higher temperature, chromite has higher content of chromium oxide and MgO (Maurel and Maurel, 1982b;[START_REF] Murck | The effects of temperature, oxygen fugacity and melt composition on the behaviour of chromium in basic and ultrabasic melts[END_REF][START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF]. From a thermodynamic standpoint, an alternative way to look at chromite saturation is to consider the distribution coefficient of Cr between spinel and liquid (DCr). Maurel & Maurel (1982b) showed that this parameter increases when temperature decreases and oxygen fugacity increases. [START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF] have showed that the distribution coefficient ( (Cr+Al+Fe 3+ ) in the chromite (Cr+Al+Fe 3+ ) in the coexisting melt

) reaches a maximum of 140 at log fO2= -5.0. The authors suggested that this point probably corresponds to the maximum of Cr 3+ content in the melt, more oxidized conditions leading to the formation of Cr 6+ in the melt [START_REF] Schreiber | Chromium in basalts: Experimental de-termination of redox states and partitioning among synthetic silicate phases[END_REF]. In other words, a mixture of Cr 2+ and Cr 3+ is inferred to dominate at log fO2 lower than -5.0 [START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF], consistent with direct measurement of chromium speciation (Cr 2+ /Cr) in synthetic silicate glasses as a function of oxygen fugacity [START_REF] Berry | A XANES determination of the oxidation state of chromium in silicate glasses[END_REF]. Indeed, at 1400 C in the interval of oxygen fugacities characteristic for most mafic magmas (between nickel-nickel oxide and quartz-fayalite-magnetite buffers), Cr 2+ /Cr ratio ranges from 0.3 to 0.8 as a function of melt composition, indicating a significant proportion of Cr 2+ in the melt [START_REF] Berry | A XANES determination of the oxidation state of chromium in silicate glasses[END_REF]. In terms of compositional dependence, [START_REF] Berry | The effect of composition on Cr2+/Cr3+ in silicate melts[END_REF] found a linear correlation between log[Cr 2+ /Cr 3+ ] and theoretical optical basicity (a compositional parameter, calculated from composition of the melt) [START_REF] Duffy | A review of optical basicity and its applications to oxidic systems[END_REF].

Optical basicity is useful for ranking a wide variety of glasses in order of increasing basicity (high optical basicity in more mafic liquids). It correlates such properties as refractive index or redox equilibria irrespective of the anionic composition of the glass [START_REF] Duffy | A common optical basicity scale for oxide and fluoride glasses[END_REF]. Higher optical basicity corresponds to lower Cr 2+ /Cr ratio and iron-bearing and iron-free melts appear to follow the same trend.

To extend this previous work to wider range of silicate melts, which are saturated in chromite ((Fe,Mg,Mn) 2+ (Cr,Al) 3+ 2O4) or magnesiochromite (MgCr2O4), we have performed additional experiments of chromite saturation in anhydrous iron-containing and iron-free melts as a function of oxygen fugacity, also covering a range of Cr/Al ratios. These new data and those of the literature are used to develop a predictive thermodynamically based model of chromite saturation that may be applied to model the geochemistry of chromium in natural melts generated and evolved in the mantle and the mantle-crust transition zone.

Experimental and Analytical Methods

Starting materials

The starting material compositions are shown in Table 1. Two crystalline starting materials were used for this study. The first was natural chromitite from the Silesia ophiolite in Poland (S2 chromite, [START_REF] Wojtulek | Podiform chromitites from the Variscan ophiolite serpentinites of Lower Silesia (SW Poland) -petrologic and tectonic setting implications[END_REF]. This sample was crushed; and a fraction of 300 -500 µmsize was separated by sieving. The chromite fraction was then separated from secondary silicates (e.g., serpentine) manually. After extraction of the chromite fraction, the chromite grains were leached using 6.2 N HCl and 5% HF, following the method of [START_REF] Snow | Nd and Sr isotope evidence linking mid-oceanridge basalts and abyssal peridotites[END_REF].

First, chromite was maintained in the leaching solution for 10 minutes, then the solution was removed. Second, chromite was kept in a new leaching solution in an ultrasonic bath for 10 minutes. Third, it was placed in a drying oven at 125 C for 10 minutes in a closed Savillex container. After cooling, the solution was removed. After leaching, the chromite grains were rinsed with deionized water and dried. Minor residual salts were visible on the chromite surface, but their influence is considered negligible on the experimental results at high temperature. The second crystalline starting material was crystals of synthetic Cr2O3, extracted from an electromelted furnace refractory with a measured starting composition of 98.5 wt% Cr2O3, and 1.5 wt% TiO2. This material was crushed; and the 300 -500 µm-fraction was separated.

These crystalline materials were mixed with either a natural mid-ocean ridge basaltic glass (from Mid-Atlantic Ridge, Knipovich ridge, № 3786/3 in the 38 th cruise of scientific research vessel Akademik Mstislav Keldysh; [START_REF] Sushchevskaya | Magmatism of Mona and Knipovich ridges from spreading zones of polar Atlantic Ocean[END_REF] that had been crushed to powder in an agate mortar to produce particles <100 m in size or a synthetic iron-free haplobasaltic glass corresponding to the eutectic composition of the anorthite-diopside system (Di63An37). This material was made from reagent grade oxides (SiO2, Al2O3) and carbonates (CaCO3 and MgCO3) that were melted at 1500°C, quenched and then crushed. This procedure was repeated twice to ensure homogeneity of the final glass powder.

Experimental method

Experiments were performed in a vertical gas-mixing furnace at one atmosphere pressure, 1440°C and controlled oxygen fugacity [START_REF] Toplis | The role of phosphorus in crystallisation processes of basalt: An experimental study[END_REF]. Redox conditions in the furnace were controlled by mixtures of CO and CO2 spanning the range from two log units below to two log units above the fayalite-magnetite-quartz (FMQ) buffer (FMQ-2 to FMQ+2). The samples were held on Pt wire loops (the wire 0.3 mm in diameter and the loop typically 1 to 2 mm in diameter). For experiments using MORB at or below the FMQ buffer, the wire loops were presaturated with iron at the relevant fO2 for 24 hours at 1400°C using an Fe-rich basalt powder [START_REF] Toplis | The role of phosphorus in crystallisation processes of basalt: An experimental study[END_REF]. These wire loops were cleaned using hydrofluoric acid and rinsed with MQ water to get rid of residual salts.

Four different mixtures were prepared for the equilibrium experiments. For the first two, 5 wt% of chromite crystals were mixed with the 95 wt% of mid-ocean ridge basalt MORB powder or Di63An37 powder. For the second two, 3 wt% of Cr2O3 crystals were mixed with 97 wt% of MORB or Di63An37 powder.

Droplets of these mixtures were attached to the wire loops during a 30 s heat treatment at 1400°C in air in a muffle furnace. Once samples of the four different starting mixtures were ready they were mounted together on a Pt-basket and then introduced into the hot-spot of the gas mixing-furnace, the basket being suspended at the end of a ceramic rod on a thin (0.2 mm Pt wire. The experimental run duration was estimated according to a minimal distance necessary for Cr to diffuse through the whole volume of thee reacting basaltic melt (e.g., [START_REF] Zagrtdenov | Preliminary experimental investigation of chromite dissolution in mid-ocean ridge basalt melt[END_REF]. At the end of the experiments with duration of 24h at 1440°C, an electric current was passed through the thin wire causing it to melt and the basket to fall into the cold zone of the furnace, still under CO-CO2 mixture. Quench rate with this method is estimated at 1000°C/s. After evacuation of the CO, the samples were recovered and mounted in epoxy and then polished by SiC grinding paper. The attainment of equilibrium was controlled by: i) homogeneity of the real Cr concentrations obtained by EPMA and LA-HR-ICP-MS and ii) the euhedral morphology of the magnesiochromite crystals in contact with the glass.

Electron Probe Microanalysis (EPMA)

Major elements in the crystals and glasses (and trace Cr in glasses) were analyzed using a CAMECA SX-Five electron microprobe at the Centre de Microcaractérisation Raimond Castaing (Toulouse, France). The glass analyses were performed at the distance more than 200 microns from the chromite grains in order to avoid the secondary fluorescence effect of the high-concentrated chromium-bearing phase [START_REF] Borisova | Secondary fluorescence effects in microbeam analysis and their impacts on geospeedometry and geothermometry[END_REF]. An accelerating voltage of 15 kV and current of 20 nA were used. Synthetic and natural standards were used for calibration: albite (for Na), corundum (Al), wollastonite (Si, Ca), sanidine (K), pyrophanite (Mn, Ti), hematite (Fe), periclase (Mg), Cr2O3 (Cr). Element and background counting times for most elements were 10 s, whereas the counting time for Na and K was 5 s to avoid volatilization under the electron beam. The peak counting time for Cr was 120 s with a resulting detection limit of 70 ppm. MPI-DING mafic and ultramafic glasses (KL2-G, ML3B-G, GOR132-G, GOR128-G, [START_REF] Jochum | MPI-DING reference glasses for in situ microanalysis: New reference values for element concentrations and isotope ratios[END_REF] were analyzed as unknown samples to monitor the accuracy of the major and trace element analyses. The accuracy estimated on the reference glasses ranges from 0.5 to 3 % (1σ RSD = relative standard deviation), depending on the element contents in the glasses. Error bars (one standard deviation, characterizing homogeneity of the sample) are below 5 RSD% (relative standard deviation).

Method of laser ablation inductively coupled plasma mass spectrometry (LA-HR-ICP-MS)

To confirm no influence of the secondary fluorescence effect on the obtained Cr contents in the glasses analyzed by EPMA, additional major and trace (Cr, Co and Ni) element analyses were performed using a New Wave Research UP213 system (Fremont, CA, USA). This is a nanosecond laser system equipped with a Nd:YAG, Q-switched laser operating at 213 nm and a pulse width of 4 ns coupled with a high-resolution ICP-MS Element-XR. A 10 µm spot size was applied with 10 Hz and 30 J/cm 2 conditions. The background was measured during 30 s, ablation for 30 s and wash-out after ablation for 15 s. To quantify the elemental composition of the glasses, average calcium concentrations (based on 43 Ca) measured by electron microprobe were used as an internal standard. SILLS 1.2.0. software was used for the elemental concentration quantification. For external calibration in bracketing mode, we used NIST SRM 610 reference material [START_REF] Jochum | Determination of Reference Values for NIST SRM 610-617 Glasses Following ISO Guidelines[END_REF]. Furthermore, to control the accuracy, we used the MPI-DING mafic and ultramafic glasses (KL2-G, ML3B-G, GOR132-G, GOR128-G) [START_REF] Jochum | Determination of Reference Values for NIST SRM 610-617 Glasses Following ISO Guidelines[END_REF]Borisova et al., 2012) as secondary standards. The detection limit for major elements varied between 1 and 20 ppm, whereas it was 0.21 ppm for 53 Cr, 0.04 ppm for 59 Co, and 1.65 ppm for 60 Ni. The relative accuracy for Cr estimated from the reference glasses ranges from 3.1 to 8.6 % (1σ RSD = relative standard deviation), depending on the element contents in the reference glasses.

Results

The data on composition of chromite crystals after experiments and chromium distribution coefficients between the chromite crystals and glass are shown in Table 2. The glass compositions after experiments are provided in Table 3. The description of the four systems is given below.

Chromite-MORB melt system

After dissolution of chromite in MORB, we find that after heat treatment, residual chromite crystals form a spongy texture (Fig. 2). The formation of this texture indicates that the basaltic melt partly dissolved the chromite crystals. Of note is the fact that during interaction with the melt, chromite changes in composition, becoming more Cr-rich. In detail, chromium oxide (Cr2O3) contents in the final chromite range from 49.67 wt% at FMQ+2 to 55.42 wt% at FMQ-2, compared to the initial average of 41 wt% Cr2O3. Total iron oxide content (FeOtot) in the chromite is 12 wt% at FMQ-2. At higher oxygen fugacities (at FMQ and FMQ+2), it is significantly higher (16 and 17 wt%, respectively). Magnesium content (MgO) is 14.5 wt% at FMQ-2, 13 wt% at FMQ and 15 wt% at FMQ+2. Titanium oxide concentration is constant around 0.4 wt% for all redox conditions. Aluminium oxide contents range from 14.7 to 16.4 wt% Al2O3 that is dramatically lower than that of initial chromite (26 wt% Al2O3).

Alumina concentration is 16 ~wt% Al2O3 at all redox conditions, slightly higher than the initial 15.3 wt%, consistent with the reequilibration and dissolution of Al-rich chromite.

The concentration of TiO2 is close to 1.5 wt% (which is slightly higher than the starting 1.45 wt% content) with no dependence on oxygen fugacity. MgO content is around 9 wt% with no significant variation as a function of oxygen fugacity. Iron oxide content shows an irregular variation as a function of fO2 (6.04 wt% at FMQ-2, 8.9 wt% at FMQ, and 7.2 wt% at FMQ+2), these complex variations possibly being the result of competing effects of chromite dissolution/reequilibration. As illustrated in Figure 3 the highest concentration of Cr (6300 ppm) is observed at the most reducing conditions (FMQ-2, logfO2= -7.9) decreasing to 2856 ppm at the FMQ buffer and 1987 ppm at FMQ+2. This variation is not a linear function of log fO2. Figure 4 shows how the distribution coefficient of chromium (ratio of Cr concentration in the crystal to that in the liquid) varies as a function of oxygen fugacity.

Chromite-haplobasaltic melt system

Chromites dissolved in the haplobasaltic glass (Di63An37) also have a spongy texture similar to that described in the MORB glass. The chromium oxide content in final chromite is 63.8 wt% at FMQ-2 and 65.4 wt% at FMQ+2. These concentrations are slightly higher than those of chromite observed in the MORB melt, and much higher than for those of the initial chromite (41 wt%). Total iron oxide content (FeOtot) in the residual chromite dissolved in haplobasalt (Di63An37) is 1.73 wt% at FMQ-2 and 1.06 wt% at FMQ+2 that is dramatically lower than the initial chromite. Magnesia concentration at both oxygen fugacities is around 21 wt% MgO, which is more than in the initial chromite (15 wt% MgO). Alumina content is 11.5 wt% Al2O3 at FMQ-2 and 10.3 wt% at FMQ+2. These values are much lower than the initial value of 26 wt% Al2O3.

Magnesium content is constant at all fO2, with a value of ~15 wt% MgO (compared to the starting value of 11.3 wt%). Total iron oxide content varies from 0.22 wt% to 1 wt% in the final glass (that was initially iron-free). Chromium content in the haplobasaltic glass is depicted in Figure 3. A maximum concentration of 6023 ppm is observed at FMQ-2. The Cr concentration in the glass drops to 4095 ppm at FMQ and 3260 ppm at FMQ+2. The value at FMQ-2 is thus similar to that of the MORB melt, but values do not decrease to the same extent with an increase in fO2.

Cr2O3-MORB melt system

After dissolution of chromium oxide (Cr2O3) in the MORB melt, a chromite was formed. At low oxygen fugacity (FMQ-2), crystals of Cr2O3 were completely transformed to chromite. At higher oxygen fugacity (FMQ and FMQ+2), crystals are zoned with residual chromium oxide surrounded by rims of chromite with average thickness ~40 m. A spongy texture was not observed. At FMQ-2, chromite contains an average of 61.2 wt% of Cr2O3 in the central zone of the crystals and 58.7 wt% of Cr2O3 at the rim. At the FMQ buffer the rim chromite has 66.6 wt% of Cr2O3, while at ∆FMQ+2, rim chromite has an average of 63.2 wt% Cr2O3.

Magnesium content in the chromite is 13 wt% of MgO at ∆FMQ-2, whereas it is 10.5 wt% at FMQ and 10.9 wt% at FMQ+2 in the chromite rims. Titanium oxide content in the chromite varies from 0.34 to 0.41 wt%, which is close to values in the system of chromite-MORB (0.36-0.41 wt% of TiO2). Total iron oxide concentration in the chromite changes with oxygen fugacity increasing with increasing fO2 (13.3 wt% at FMQ-2, 16.5 wt% at FMQ, 18.6 wt% at FMQ+2 of FeO). Al2O3 concentrations are highly variable, both between and within experiments. For example, at FMQ-2 Al2O3 concentration is 11.7±0.5 wt%, whereas it is 4.2±3.3 and 5.1±3.1 wt% at FMQ and FMQ+2, respectively.

Al2O3 varies from 15 to 16 wt% for all oxygen fugacities and it is similar to the initial 15 wt% value. Total iron oxide (FeO) content is a function of oxygen fugacity (6.3 wt% at FMQ-2, 8.5 wt% at FMQ and 9.7 wt% at FMQ+2). Magnesium (MgO) content of 8 wt% is constant and does not change with fO2. TiO2 ranges from 1.54 to 1.63 wt%, which is slightly higher than the initial (1.5 wt%) content. Chromium oxide content, as illustrated in Figure 3, reaches a maximum of 6415 ppm at the most reduced conditions (FMQ-2). The Cr content drops to 3058 ppm at FMQ and 1948 ppm at FMQ+2 with increasing oxygen fugacity.

Cr2O3 -haplobasalt system

Chromium oxide dissolved in haplobasaltic melt (Di63An37) was partly replaced by magnesiochromite. When the oxygen fugacity is low (FMQ-2), crystals of Cr2O3 were completely transformed to magnesiochromite with little zonation (71.78 of Cr2O3 in the center and 68.16 at the rim). At more oxidized conditions (FMQ and FMQ+2), residual crystals have a central zone of almost pure Cr2O3 (98 wt%), with rims of magnesiochromite (73.32 wt% of Cr2O3 at FMQ and 73.35 wt% of Cr2O3 at FMQ+2) with a thickness of ~40 m (Figure 5). The rim consists of euhedral crystals, indicating that the magnesiochromite is in equilibrium with the surrounding glass. Magnesia (MgO) content in magnesiochromites is stable and varies slightly from 19.3 to 20.7 wt%. Titanium oxide is quite low in the magnesiochromite (at the level of the detection limit of 0.15 wt%). Al2O3 in the magnesiochromite varies from 3.1 to 7.7 wt%.

Aluminium oxide content in the glasses is around 8.5 wt% for all three experiments. This value is lower than the original (15 wt% of Al2O3). Magnesia is close to 14.5 wt% for all three glasses that is higher than starting 11.3 wt% content. As shown in Figure 3, the highest concentrations of chromium were found in the Cr2O3haplobasalt system. The maximum value of 6824 ppm Cr corresponds to the system saturation with magnesiochromite at FMQ-2. The concentration of chromium in the glass decreases when conditions become more oxidized. At FMQ the chromium content is 4498 ppm, furthermore, at FMQ+2, the Cr concentration is 3500 ppm. The initial system is iron free, but iron has been detected in the final products. The total iron oxide content in crystals varies from the detection limit to 2.72 wt%. In the glass, FeO varies from detection limit to 1.59 wt%.

Model of chromite solubility in mafic and ultramafic melts

Based on our new data on magnesiochromite and chromite solubility and previous data on chromite solubility in the mafic and ultramafic melts, we are able to develop a new predictive model of Cr solubility in mafic and ultramafic melts. The framework for this model is based upon the oxidation of chromium (II) to chromium (III) in the melt described by the reaction:

Cr 2+ liq + 1 4 O2 = Cr 3+ liq + 1 2 O 2- liq (eqn. 1).
The equilibrium constant of this reaction is:

K = 𝑎 𝐶𝑟3+ 𝑙𝑖𝑞 𝑎 𝐶𝑟2+ 𝑙𝑖𝑞 * 1 𝑓 𝑂 2 1/4 , (eqn. 2),
where K is the equilibrium constant, 𝑎 𝐶𝑟3+ 𝑙𝑖𝑞 is activity of the Cr 3+ in the liquid and 𝑎 Cr2+ 𝑙𝑖𝑞 is activity of Cr 2+ in the liquid and 𝑎 O2- 𝑙𝑖𝑞 1/2 = 1 is activity of O 2-in the silicate liquids. It is known that at equilibrium:

G = G + R T lnK = 0, (eqn. 3),
where G is the change in Gibbs free energy, G is the standard Gibbs free energy change, T is temperature in Kelvin, and R is the gas constant (in J*K -1 *mol -1 ). Equations 2 and 3 may thus be combined to give:

G = -R T ln( 𝑎 Cr3+ 𝑙𝑖𝑞 𝑎 𝐶𝑟2+ 𝑙𝑖𝑞 * 1 𝑓 𝑂 2 1/4 ) (eqn. 4),
which may be rewritten as :

𝑒 -Δ𝐺 0 RT = 𝑎 Cr3+ 𝑙𝑖𝑞 𝑎 𝐶𝑟2+ 𝑙𝑖𝑞 * 1 𝑓 𝑂 2 1/4
(eqn. 5).

At equilibrium, the activity of Cr 3+ in the chromite (Chr) should be equal to the activity of Cr 3+

in the liquid such that:

𝑎 𝐶𝑟3+ 𝑙𝑖𝑞 = 𝑎 Cr3+ 𝐶ℎ𝑟 (eqn. 6),
Thus:

𝑒 -Δ𝐺 0 RT = 𝑎 𝐶𝑟3+ 𝐶ℎ𝑟 𝑎 𝐶𝑟2+ 𝑙𝑖𝑞 * 1 𝑓 𝑂 2 1/4
(eqn. 7).

Given that activity is a product of mole fraction and activity coefficient (a = X*ɣ) where X is mole fraction and ɣ is activity coefficient, equation 7 may be expanded to:

𝑒 -Δ𝐺 0 RT = X 𝐶𝑟3+ 𝐶ℎ𝑟 * ɣ Cr3+ 𝐶ℎ𝑟 X 𝐶𝑟2+ 𝑙𝑖𝑞 * ɣ 𝐶𝑟2+ 𝑙𝑖𝑞 * 𝑓 𝑂 2 - 1 4
(eqn. 8).

Since the mole fraction of total chromium in the liquid is the sum of the mole fractions of Cr 2+

and Cr 3+ in the liquid:

X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 = X 𝐶𝑟 2+ 𝑙𝑖𝑞 + X 𝐶𝑟 3+
𝑙𝑖𝑞 , (eqn. 9), where X 𝐶𝑟 2+ 𝑙𝑖𝑞 and X 𝐶𝑟 3+

𝑙𝑖𝑞 are molar fractions of 𝐶𝑟 2+ and 𝐶𝑟 3+ in the liquid.

At the same time, given that the activity of Cr 3+ is the same in the liquid and chromite:

X 𝐶𝑟 3+ 𝑙𝑖𝑞 * ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 = X 𝐶𝑟 3+ 𝐶ℎ𝑟 * ɣ 𝐶𝑟 3+ 𝐶ℎ𝑟 (eqn. 10),
Combining rearranged forms of (eqn. 9) and (eqn. 10), it may be shown that:

X 𝐶𝑟 2+ 𝑙𝑖𝑞 = X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 - X 𝐶𝑟 3+ 𝐶ℎ𝑟 * ɣ 𝐶𝑟 3+ 𝐶ℎ𝑟 ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 (eqn. 11a), X 𝐶𝑟 2+ 𝑙𝑖𝑞 = X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 * ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 -X 𝐶𝑟 3+ 𝐶ℎ𝑟 * ɣ 𝐶𝑟 3+ 𝐶ℎ𝑟 ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 (eqn. 11b).
From (eqn. 8) and (eqn. 11b) the following equation may be obtained:

𝑒 -Δ𝐺 0 RT = X 𝐶𝑟 3+ 𝐶ℎ𝑟 * ɣ 𝐶𝑟 3+ 𝐶ℎ𝑟 * 𝑓 𝑂 2 - 1 4 * ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 ( X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 * ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 -X 𝐶𝑟 3+ 𝐶ℎ𝑟 * ɣ 𝐶𝑟 3+ 𝐶ℎ𝑟 ) * ɣ 𝐶𝑟 2+ 𝑙𝑖𝑞 (eqn. 12),
Rearranging equation 12 it may be shown that:

𝑒 -Δ𝐺 0 RT *ɣ 𝐶𝑟 2+ 𝑙𝑖𝑞 * X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 X 𝐶𝑟 3+ 𝐶ℎ𝑟 * ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 = 𝑒 -Δ𝐺 0 RT * ɣ 𝐶𝑟 2+ 𝑙𝑖𝑞 * ɣ 𝐶𝑟 3+ 𝐶ℎ𝑟 + ɣ 𝐶𝑟 3+ 𝐶ℎ𝑟 * 𝑓 𝑂 2 - 1 4 * ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 (eqn. 13),
Given that X liq Cr tot and X Chr Cr3+ are both measurable quantities, equation 13 is rearranged such that the ratio of these terms is alone on the left hand side the equation. For simplicity, in the following we will assume ideality of the chromite solid solution (i.e. ɣ 𝐶𝑟 3+ 𝐶ℎ𝑟 has been set to 1). This is not strictly the case (e.g. [START_REF] Sack | Chromian spinel as petrogenetic indicators: thermodynamics and petrological applications[END_REF], but as long as

ɣ 𝐶𝑟 3+
𝐶ℎ𝑟 is constant, our approach remains valid. In this case, equation 14b becomes:

X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 X 𝐶𝑟 3+ 𝐶ℎ𝑟 = 1 ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 (1 + 𝑓 𝑂 2 - 1 4 * 𝑒 Δ𝐺 0 RT * ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 ɣ 𝐶𝑟 2+ 𝑙𝑖𝑞 ) (eqn. 15).
This thermodynamically derived equation thus predicts how the partition coefficient of Cr between liquid and chromite (left hand side of equation 15) should vary as a function of T, fO2 and composition. However, the activity coefficient terms of this equation may themselves be a function of temperature and will almost certainly be a function of melt composition. We thus need to consider in a little more detail how each term of equation 15 will vary in order to derive an equation that can be fitted to experimental data with a minimum number of variable parameters.

For example, 1 ɣ 𝐶𝑟 3+ 𝑙𝑖𝑞 is the limit of where optical basicity (⋋) was calculated based on the work of [START_REF] Duffy | A common optical basicity scale for oxide and fluoride glasses[END_REF][START_REF] Duffy | A review of optical basicity and its applications to oxidic systems[END_REF]: and predicted values has a slope of 0.9425 (rather than 1 in the ideal case) and an intercept of 0.0001 (rather than 0 in the ideal case). The correlation coefficient R 2 is 0.94. Overall, we thus conclude that the model has a level of agreement that is excellent.

⋋ = ∑ ⋋𝑜𝑥 *

Discussion

Figure 7 depicts chromium content in basaltic glasses in equilibrium with chromites as a function of oxygen fugacity and temperature (this study and literature data). This figure clearly illustrates that chromium content in the chromite-saturated melts increases when oxygen fugacity decreases. In line with previous interpretations, we assign this behavior to a combination of a higher Cr 2+ fraction in the liquid at reducing conditions [START_REF] Berry | A XANES determination of the oxidation state of chromium in silicate glasses[END_REF][START_REF] Berry | The effect of composition on Cr2+/Cr3+ in silicate melts[END_REF]) and a higher solubility of Cr 2+ compared to Cr 3+ . Higher solubility of Cr 2+ can be explained by its larger ionic radius and lower charge.

It is clear from Figure 7 that increasing temperature also favors high concentrations of Cr at chromite/magnesiochromite saturation. Our new data are thus in perfect accordance with the previous literature data on chromite (Fe-bearing) solubility. Our point for chromite-MORB system at 1440 C and FMQ buffer is very close to the point of chromite dissolution in basalt 401 at FMQ and 1450 C of [START_REF] Murck | The effects of temperature, oxygen fugacity and melt composition on the behaviour of chromium in basic and ultrabasic melts[END_REF].

Moreover, at FMQ+2, the Cr concentrations are higher for Fe-free system with higher optical basicity (0.62) compared to the Fe-bearing system with lower optical basicity (0.58).

This is in contrast to FMQ-2, where the solubility of all four systems is quite close to each other. Thus, at high oxygen fugacity the influence of the optical basicity is very important, but at lower oxygen fugacity this effect becomes negligible. At high fO2 the values of In addition, [START_REF] Roeder | Crystallization of chromite and chromium solubility in basaltic melts[END_REF] noted that higher iron content in the liquid corresponds to lower concentration of chromium in the liquid. They explained this assuming that Cr 2+ and Cr 3+ hold the same structural positions in the melt as Fe 2+ and Fe 3+ , respectively.

In this study, magnesiochromite was formed in the iron-free (Cr2O3-haplobasalt) system according to the reaction Cr2O3solid + MgOliq = MgCr2O4solid. The equilibrium melt formed in this system has the highest chromium concentration of the mafic melts studied. This result may have important implications for natural systems as discussed below.

To estimate silicate liquid composition effect on solubility of chromite, optical basicity was calculated for range of compositions from picritic to rhyolitic from GeoRoc (http://georoc.mpch-mainz.gwdg.de/georoc/) and PetDB (http://www.petdb.org) databases.

The distribution coefficients that the function has a minimum, due to the fact that there is competition between the effect of melt composition on the Cr content at high fO2 and the highest Cr 2+ /Cr 3+ ratios (equation 19b).

However, overall, solubility typically increases from felsic to ultramafic melts.

To study the temperature effect, the distribution coefficients (starting MORB 3786/3). It can be seen that solubility decreases asymptotically with an increase of log fO2. Thus, our model reproduces the oxygen fugacity effect found in all experiments (Fig. 7).

Comparing the effects of individual parameters, we find that the range of log Considering temperature and fO2 in a little more detail, these two parameters were considered independently in the discussion above. However, geological systems tend to evolve along paths that are parallel to solid buffers such as FMQ. For this reason, we have also predicted the variation of

X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 X 𝐶𝑟 2 𝑂 3 𝐶ℎ𝑟
along IW (iron-wustite buffer), FMQ-2, FMQ, FMQ+2 mineral buffers in the range of temperatures from 900 to 1500 C for MORB 3786/3 with optical basicity 0.588. fractions corresponding to that of the starting MORB 3786/3 glass). These assumptions both introduce some uncertainty, but combined uncertainty is estimated to be <±20%. At high oxygen fugacity (Fig. 12) an increase in optical basicity (i.e. more mafic liquids) increases the concentration of Cr in chromite saturated liquids. For example, at a log fO2 = -2, Cr concentration increases from 250 to 750 ppm. Interestingly, at very low oxygen fugacity the effect of optical basicity is opposite. For example, at log fO2 of -12.0, increasing optical basicity from 0.495 to 0.636 (from rhyolite to picrite), Cr concentrations drop from 3250 ppm to 1750 ppm. At log fO2 = -8.4 (corresponding to FMQ buffer at 1200 C) a modest increase from 600 to 1000 ppm is observed during increase of optical basicity along whole range of melt compositions from rhyolite to picrite. Thus, picritic melts saturated with chromite are enriched in Cr at FMQ buffer and 1200 C, whereas at extremely high oxygen fugacity at the same temperature of 1200 C, rhyolitic melts would have much lower Cr contents. Therefore, mixing or hybridization of typical basaltic melt with SiO2 -rich melts/rocks would result in massive precipitation of chromite from the mafic liquid at FMQ and 1200 C, that is in excellent accordance with classic model of [START_REF] Irvine | Crystallization sequences in the Muskox intrusion and other layered intrusions-II. Origin of chromitite layers and similar deposits of other magmatic ores[END_REF] in implications mostly for the layered intrusions.

Values of

In summary, compared to the existing models, our model provides dependence of the chromite/melt partitioning on the most important parameters controlling the partitioning: temperature, oxygen fugacity and compositional parameter, which is optical basicity. In contrast, Murck and Campbell's eq.17 ignores the temperature parameter. This is a problem of the model, which does not consider the most important temperature parameter controlling chemistry and crystallization field of chrome spinel. Other parameter, which may be quantitatively estimated and compared to the estimations made by our models, is XCr2O3 in chrome spinel. Our model compared to the models of [START_REF] Poustovetov | The distribution of Cr between basaltic melt and chromian spinel as an oxygen geobarometer[END_REF] and [START_REF] Nikolaev | SPINMELT-2.0: Simulation of spinelmelt equilibrium in basaltic systems under pressures up to 15 kbar: I. model formulation, calibration, and tests[END_REF] gives better results when calculating XCr2O3 in chromite (Fig. 13a,b). It may be seen that the accordance of the calculated XCr2O3 to the experimental ones at co-existing equilibrium chromite is much better with our model. Additionally, our model expands the compositional field of the model application to felsic and intermediate melts never investigated experimentally.

Implications

If a mantle-derived magma is initially reduced, but starts to degas and become more oxidized, the chromite would be crystallized. Figure 10 demonstrates that stability of chromite may be caused by the mafic magma oxidation in the initially water-rich system. Indeed, the chromitite formation in the oceanic mantle-crust transition zone due to serpentinite dehydration, reaction with basaltic melt and assimilation (Borisova et al., 2012;2020a,b) corresponds to such natural examples and settings, where initially reduced and hydrous magmas (enriched in CH4 and H2) would degas and progressively oxidize. The massive chromite crystallization is responsible for the chromitite formation at the mantle-crust transition zone. Future model of chromitite formation will be developed in details; it is beyond of our stydy. Additionally, our results show that iron poor/Mg-rich melt systems are favorable for high chromite dissolution, at high temperature, at reducing conditions, and in equilibrium with Cr-rich chromites, all parameters that increase Cr content in silicate liquid. Given that serpentinized mantle is enriched in Mg and that magnesiochromite is the Mg-Cr-rich chromite mineral frequently presents in the serpentinized rocks (e.g. Borisova at al. 2012), assimilation of serpentinized lithosphere by mafic/ultramafic melts would be a viable way to reach high concentrations of Cr in the melt. Fractional crystallization of olivine from these chromium-rich melts would decrease magnesium index (Mg/(Mg+Fe)) of the residual melts, the iron concentration in the melts would be increased accordingly. This effect, combined with the cooling and degassing, may be an efficient mechanism to trigger massive chromite crystallization from hybrid mafic/ultramafic melts. Our model also expands the field of the chromite-saturated melts, mostly of mafic and ultramafic compositions, to intermediate and felsic melts and predicts that the melt crystallization is a viable mechanism to trigger massive chromite crystallization.

Indeed, our experiments demonstrate Cr concentration ranges from 225 to 580 ppm at fO2 above FMQ in experimental felsic to intermediate melts formed upon the hydrated peridotite melting, in particular at the mantle-crust transition zone, and reaction with the basaltic melts (Borisova et al., 2020a,b). These Cr concentrations are in excellent agreement with those predicted by the thermodynamic model constructed in our work (Fig. 12). Therefore, the proposed thermodynamic model may be applied to model formation of the oceanic mantle-crust transition zone.

Conclusions

1. New data on solubility of chromite and magnesiochromite have been obtained for basaltic and haplobasaltic (λ = 0.58 to 0.60) systems at high temperature (1440 C) and oxygen fugacity in the range from FMQ-2 to FMQ+2.

2. Magnesiochromite is the equilibrium phase in haplobasatic melt with high Cr contents (3500 -6800 ppm). This shows high solubility of magnesiochromite in haplobasaltic melt, that has an important implication for serpentinized mantle assimilation by basaltic magmas.

3. A predictive model of chromite and magnesiochromite solubility in silicate melts has been developed. This model is applicable for the explanation of massive chromite crystallization from hybrid mafic/ultramafic melts, happening in the mantle-crust transition zone according to model of disseminated, stratiform and nodular chromitite formation by Borisova et al. (2012), [START_REF] Zagrtdenov | Anatomy of a chromitite dyke in the mantle/crust transition zone of the Oman ophiolite[END_REF] and [START_REF] Rospabé | Multi-scale development of a stratiform chromite ore body at the base of the dunitic mantle-crust transition zone (Maqsad diapir, Oman ophiolite): the role of repeated melt and fluid influxes[END_REF].

  𝑎 + 𝑏 * ⋋ +𝑐/𝑇) * (1 + 𝑓 𝑜2 -1 4 * 𝑒𝑥𝑝(𝑑 + 𝑘/𝑇 + 𝑔 * ⋋)) (eqn. 19b),

  Figure 6 illustrates X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞

  the observed compositional dependence must be related to that term. At lower fO2, two terms of equation 19b. Given that the constants b and g are opposite in sign this explains why a reduction of fO2 leads to attenuation of the compositional effect.

  this range of optical basicity at fixed temperature of 1200 C and an oxygen fugacity corresponding to the FMQ buffer according to the eqn. 19b. Log function of optical basicity in Figure 8. It can be seen

Figure 10 .

 10 . 19b for a range of temperature from 900 to 1500 C at fixed log fO2 = -8.4 and optical basicity = 0.588 (corresponding to the optical basicity of the initial basalt in our experiments: MORB 3786/3). As shown in Figure9, temperature strongly increases the solubility of chromite in the silicate melt.Finally, the dependence of Cr distribution coefficient The coefficients were calculated according to eqn. 19b for range of log fO2 from -13.0 to -0.1 at fixed temperature of 1200°C for basalt with optical basicity = 0.588

  change of optical basicity corresponding to a variation from picrite to rhyolite (-3.2 to -2.9, Figure8) represent change of temperature of less than 140C or a change of oxygen fugacity of 6.3 logarithmic units, the latter range being large because of the fact that values of log high fO2 when there is little or no Cr 2+ . When taken together, Figures 8 to 10 indicate that temperature has the most significant influence on solubility of chromite, followed by oxygen fugacity at low fO2 then melt composition and oxygen fugacity at high fO2.

  . 19b were then multiplied by the molar fraction of Cr in typical chromite (50 wt% of Cr2O3: 𝑋 𝐶𝑟 2 𝑂 3 𝐶ℎ𝑟 =0.308) to obtain the mole fraction of the chromium in the liquid, that was then recalculated to concentration in ppm. The logarithm of Cr concentrations in the silicate liquid along IW, FMQ-2, FMQ, FMQ+2 mineral buffers versus temperature is illustrated in Figure 11. The lines are close to each other at low temperatures, which implies a weak influence of oxygen fugacity, but lines diverge at high temperature. For instance, the difference of Cr concentrations in the silicate melt in equilibrium with chromite derived at 900 C at different fO2 is insignificant. At 1200 C, near basaltic liquidus, the difference between Cr concentrations at FMQ-2 (940 ppm Cr) and FMQ (610 ppm Cr) is near 300 ppm. However, at high temperature 1500 C this difference between FMQ-2 (7680 ppm Cr) and FMQ (3750 ppm Cr) buffers is almost 4000 ppm. The simultaneous influence of oxygen fugacity and melt composition on chromite solubility at a fixed temperature 1200 C is illustrated in Figure 12. These values were calculated in the same way as for Figure 11 (i.e. assuming 𝑋 𝐶𝑟 2 𝑂 3 𝐶ℎ𝑟 = 0.308 and a sum of mole

4.

  At moderate to high oxygen fugacities (for example, >FMQ-1 at 1200 C), chromite solubility increases from felsic to ultramafic melts, but in general terms, temperature is the most important variable. Our model may thus predict physico-chemical conditions of massive crystallization of chromitites. The Cr concentrations in felsic to intermediate melts saturated in chromite and formed upon the hydrated peridotite melting and reaction with basaltic melts(Borisova et al., 2020a,b) are in excellent agreement with those predicted by the thermodynamic model constructed in our work. The formation and involvement of intermediate and felsic melts with high SiO2 contents and low chromite solubility has a direct impact on the formation of the terrestrial chromitebearing mantle-crust transition zone represented by chromitites and dunites.

Figure 1 .

 1 Figure 1. Chromium concentrations in natural basaltic and boninitic melts according to the PetDB database (http://www.petdb.org).

Figure 2 .

 2 Figure 2. Residual chromite (Chr) in the basaltic (MORB) glass in the experiment E19. Spongy texture in chromite may be clearly recognized. The image was obtained using an optical microscope in reflected light.

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Chromium content in the quenched basaltic and haplobasaltic melts in equilibrium with chromite (chromite or magnesiochromite) as a function of oxygen fugacity at the run 1440 °C temperature. Error bars (one standard deviation, characterizing homogeneity of the sample) are smaller than a symbol. Note that the chromite solubility is not strictly linear function of Cr contents on this range of oxygen fugacity.

Figure 6 .

 6 Figure 6. Fitting of

Figure 9 .

 9 Figure 9. Influence of temperature to the chromium distribution coefficient according to eqn. 19b. Section is presented for log fO2 = -8.4 and optical basicity (O.B.) = 0.588, corresponding to composition of initial basalt of our experiments (MORB 3786/3).

Figure 10 .FMQFigure 11 .

 1011 Figure 10. Dependence of Cr distribution coefficient on oxygen fugacity according to eqn. 19b. Section at 1200 °C for basalt with optical basicity = 0.588 (starting MORB 3786/3). Dashed line depicts FMQ buffer at 1200 °C.

Figure 12 .

 12 Figure 12. Contour plot of the estimated chromium concentrations in dry silicate melts (in ppm) at chromite saturation as function of optical basicity and logarithm of oxygen fugacity at fixed temperature 1200 C according to the model (eqn. 19b). Typical ranges of optical basicities for the main silicate melts (rhyolite, andesite, dacite, boninite, basalt and picrite) are marked in brackets at the right side of the plot.

Figure 13

 13 Figure 13(b). Comparison of our model with the model of Nikolaev et al. (2018). We demonstrate that new model predicts experimental XCr2O3 better than model of Nikolaev et al. (2018).

  ox is the optical basicity of each oxide (taken from Table2in[START_REF] Duffy | A review of optical basicity and its applications to oxidic systems[END_REF]), Cox is concentration of the oxide in the glass in wt%. No is number of oxygens in the formula of oxide. Mox is molar weight of the oxide.

	𝐶𝑜𝑥 * 𝑁𝑜 * 𝑀𝑜𝑥 𝑀𝑜𝑥 * ∑ 𝐶𝑜𝑥 * 𝑁𝑜	,	(eqn. 20),
	where ⋋ The model was fitted using the experimental data of Murck & Campbell (1986), Barnes
	(1986), Roeder & Reynolds (1991), Forsythe & Fisk (1994), Poustovetov & Roeder (2000),
	Voigt et al. (2016) and those of this study. For this fitting, we have used experimental data for
	basaltic melts obtained at 1 bar and the following parameters: temperatures ranging from 1153
	°C to 1450 °C; logfO2 ranging from -12.8 to -0.7; and λ from 0.56 to 0.61 (see Electronic
	Appendix). Coefficients for equation 19b were found using the "curve_fit" function of
	PYTHON (see Appendix 1). The final coefficients for equation 19b are: a (-7.01 ± 2.10), b
	(13.72 ± 2.73), c (-12405 ± 1253), d (24.46 ± 3.13), k (-24395 ± 2037), g (-23.59 ± 4.20).

  Chromium contents in the mafic and ultramafic quenched melts in equilibrium with chromite and magnesiochromite as function of oxygen fugacity at different temperatures based on this study and literature data. The color dot lines are interpolation of the experimental data obtained at a given temperature at variable oxygen fugacities and at a given mineral buffer (FMQ). Logarithm of Cr distribution coefficient between silicate liquid and chromite in equilibrium with the liquid (expressed as ratio of chromium molar fraction in liquid to Cr2O3 molar fraction in the chromite) versus optical basicity of the liquid at oxygen fugacity corresponding to FMQ mineral buffer and temperature 1200 C. Range of different effusive rocks from databases GeoRoc (http://georoc.mpch-mainz.gwdg.de/georoc/) and PetDB (http://www.petdb.org) is taken as examples of silicate melts.

		-2.85
		10000 12000 -2.90	Picrite Oceanic glasses PetDB Basalt Boninite Dacite	Poustovetov&Roeder 1200 C Poustovetov&Roeder 1250 C Forsythe&Fisk 1150 C Forsythe&Fisk 1164 C Forsythe&Fisk 1200 C Forsythe&Fisk 1260 C Forsythe&Fisk 1300 C
		-2.95	Andesite Rhyolite	This study Chr-MORB This study Chr-Di63An37 This study Cr2O3-MORB
				This study Cr2O3-Di63An37
				Murck & Campbell QFM
		8000 -3.00		Murck & Campbell 1250 C Murck & Campbell 1400 C
				Roeder&Reynolds 1400 C
				Roeder&Reynolds 1300 C
		-3.05
	ppm	6000
	Cr,	-3.10
		4000
		-3.15
	X 𝐶𝑟 𝑡𝑜𝑡 𝑙𝑖𝑞 X 𝐶𝑟 2 𝑂 3 𝐶ℎ𝑟 -11 measured based on available experiments (abscissa, correlation coefficient is R 2 = 0.94). calculated from the model (eqn. 19b) (ordinate) vs. the values -13 -9 -7 -5 -3 -1 1 log f O2 Figure 7. 0 2000 0.49 0.51 0.53 0.55 0.57 0.59 0.61 0.63 Figure 8. -3.20 Optical basicity

Table 1 . Starting materials compositions

 1 Digits in brackets correspond to standard deviation of number of analyzes. c Emission-ICP, relative error <2% for all oxides.

		Na2O	SiO2	Al2O3	K2O	CaO	MnO	FeO	MgO	TiO2	P2O5	Cr2O3	NiO	Total Cr(ppm)
	S2 chromite (31) a			26.45(42) b			0.21(04) 15.87(62) 14.81(34) 0.10(02)		41.15(0.27) 0.16(05) 98.54
	Cr2O3 (19)			0.14(1)					0.28(28)	1.45(3)		97.92(1.19)		99.79
	MORB 3786/3 c	3.04	50.31	15.31	0.31	10.29	0.17	8.91	8.21	1.45	0.18	0.04		98.22	275
	Di63An37 (15)		50.52(35) 15.22(25)		22.95(31)			11.30(19)					99.99
	Notes:
	a Number of analyzes.

b

Table 2 . Composition of crystals after experiments and chromium distribution coefficients between crystals and glass (averages from multiple analyses)

 2 Analyses of central part of crystals (c) or rims (r) which are in equilibrium with melt. Number in brackets is a number of analyses. b Logarithm of chromium distribution coefficient, ratio [concentration of Cr2O3 in crystal in wt% / concentration of Cr2O3 in melt in wt% ].

	Run	Rim or Center a	System	Buffer	Phase	Al2O3	SiO2	MnO	FeO	MgO	TiO2	Cr2O3	V2O3	Total Log Kd b
	E23	r (5)		FMQ-2	Chr c	14.71 (1.09) d	D.L.	0.29 (4)	12.04 (19)	14.47 (16)	0.39 (2)	55.42 (1.03)	0.32 (3)	97.66	1.78
	E19	r (5)	Chr-MORB	FMQ	Chr	16.24 (1.68)	0.09 (5)	0.24 (4)	16.82 (12)	13.07 (33)	0.41 (1)	51.53 (2.17)	0.15 (2)	98.55	2.09
	E27	r (14)		FMQ+2	Chr	16.35 (3.21)	D.L.	0.27 (3)	16.49 (33)	15.05 (91)	0.36 (4)	49.67 (4.25)	D.L.	98.18	2.23
	E24 E28	r (5) r (10)	Chr-DiAn	FMQ-2 FMQ+2	Chr Chr	11.45 (1.65) 10.26 (1.57)	D.L. D.L.	D.L. D.L.	1.73 (7) 1.06 (5)	20.6 (21) 21.26 (23)	D.L. D.L.	63.78 (1.84) 65.43 (2.08)	D.L. D.L.	97.56 98.01	1.86 2.14
	E25	c (2)		FMQ-2	Chr	9.37 (1.97)	D.L.	0.27 (5)	13.27 (25)	13.06 (32)	0.36 (0)	61.22 (1.98)	0.19 (1)	97.75	1.81
	E25	r (5)		FMQ-2	Chr	11.65 (0.49)	D.L.	0.30 (03)	13.25 (12)	13.47 (23)	0.41 (1)	58.67 (0.51)	0.27 (2)	98.04	1.80
	E31 E31	c (12) r (17)	Cr2O3-MORB	FMQ FMQ	Cr2O3 Chr	D.L. 4.15 (3.26)	D.L. D.L.	D.L. 0.28 (4)	0.64 (28) 16.52 (38)	0.18 (16) 10.47 (1.14)	1.53 (3) 0.45 (5)	97.48 (0.9) 66.64 (4.59)	0.11 (1) 0.1 (3)	99.94 98.61	2.34 2.17
	E29	c (5)		FMQ+2	Cr2O3	0.22 (3)	0.14 (11)	D.L.	1.08 (75)	0.30 (27)	1.55 (58)	95.32 (2.69)	D.L.	98.60	2.52
	E29	r (15)		FMQ+2	Chr	5.1 (3.07)	D.L.	0.28 (5)	18.56 (2.48)	10.92 (1.37)	0.34 (5)	63.22 (7.14)	D.L.	98.42	2.35
	E26	c (3)		FMQ-2	MgChr	4.77 (7)	D.L.	D.L.	1.28(4)	20.14 (7)	D.L.	71.78 (7)	D.L.	97.97	1.86
	E26	r (5)		FMQ-2	MgChr	7.72 (72)	D.L.	D.L.	1.25(3)	20.48 (2)	D.L.	68.16 (92)	D.L.	97.61	1.83
	E33 E33	c (19) r (12)	Cr2O3-DiAn	FMQ FMQ	Cr2O3 MgChr	0.14 (1) 4.29 (2.68)	D.L. D.L.	D.L. D.L.	D.L. D.L.	0.28 (28) 20.68 (46)	1.45 (3) 0.14 (11)	97.92 (1.19) 73.32 (3.93)	D.L. D.L.	99.79 98.43	2.17 2.05
	E30	c (14)		FMQ+2	Cr2O3	0.13 (13)	D.L.	D.L.	0.21(6)	0.31 (34)	1.11 (53)	98.13 (1.39)	0.12 (1)	100.00	2.28
	E30	r (15)		FMQ+2	MgChr	3.11 (2.05)	D.L.	D.L.	2.72(33)	19.31 (48)	0.15 (11)	73.35 (3.5)	D.L.	98.64	2.16

a c Chrchromite and MORBstarting MORB glass. d Digits in brackets correspond to standard deviation of number of analyses. e D.L. -value below detection limit. 'Phase'means the final phase observed in the experimental samples.

a Digits in brackets correspond to standard deviation from average value for the performed analyses.

b D.L. -value below detection limit.