Nicki C Siersch
email: nsiersch@gmail.com

Giacomo Criniti

Alexander Kurnosov

Konstantin Glazyrin

Daniele Antonangeli

Thermal equation of state of Fe3O4 magnetite up to 16 GPa and 1100 K

Keywords: magnetite, thermal equation of state, X-ray diffraction, high-pressure, hightemperature

Fe 3 O 4 magnetite is an important mineral commonly found in various geological settings, including the planet Mars, whose thermo-elastic properties at high pressure and temperature are still poorly constrained. We performed X-ray diffraction measurements on natural magnetite using resistive-heated diamond anvil cells up to 16 GPa and 1100 K. We fitted a thermal equation of state (EoS) to the collected data resulting in K 0 = 182(1) GPa, K' 0 = 4, 𝜃D = 660 K, 𝛾 = 1.8(1), and q = 2.7. Moreover, it was possible to explore the structural evolution of magnetite in detail using single-crystal measurements. Over the studied pressure and temperature range, we found no evidence of a transformation from an inverse to a normal spinel structure. The EoS parameters obtained in this study will allow to be implemented into currently available databases for self-consistent thermodynamic modeling. In particular, the obtained results are used to model and compare the sound wave velocities of a magnetite-bearing and magnetite-free Martian upper mantle assemblage. We observe that the incorporation of 2 magnetite reduces the sound wave velocities, however, the magnitude of the effect is below the current seismic detection limit of the InSight mission on Mars at the low abundance of magnetite expected in the Martian mantle.

Introduction

Magnetite (α-Fe 3 O 4 ) is an important mineral commonly found in igneous, sedimentary and metamorphic rocks of the Earth and in meteorites, which also finds industrial applications, going from being a traditional recording medium to the emerging field of spin electronics.

Magnetite was also recently suggested to be potentially stable in the most oxidized regions of the upper mantle of Mars [START_REF] Xu | Low Velocity Zones in the Martian Upper Mantle Highlighted by Sound Velocity Measurements[END_REF]. Magnetite crystallizes at ambient pressure in a cubic structure above the Verwey temperature of T V ~ 120 K [START_REF] Verwey | Electronic Conduction of Magnetite (Fe3O4) and its Transition Point at Low Temperatures[END_REF], in the space group Fd3 ̅ m with Z = 8. It has an inverse spinel structure with the tetrahedrally-coordinated sites being fully occupied by Fe 3+ , whereas the remaining Fe 2+ and Fe 3+ are randomly distributed between the octahedrally-coordinated sites ([Fe 3+ ] T [Fe 2+ Fe 3+ ] O O 4 ). This is in contrast to a normal spinel structure where the tetrahedrally-coordinated sites are occupied by the divalent cation and the octahedrally-coordinated sites are filled with the trivalent cation. With increasing pressure, magnetite was found to undergo a gradual phase transformation to the so-called h-Fe 3 O 4 [START_REF] Mao | Isothermal compression of magnetite to 320 KB[END_REF][START_REF] Huang | Rapid Determination of Fe3O4 Phase Diagram by Synchrotron Radiation[END_REF][START_REF] Fei | In situ strucutre determination of the high-pressure phase of Fe3O4[END_REF][START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF][START_REF] Lazor | High-pressure study of stability of magnetite by thermodynamic analysis and synchrotron X-ray diffraction[END_REF] or, following a newer nomenclature, β-Fe 3 O 4 [START_REF] Khandarkhaeva | Structural Diversity of Magnetite and Products of its Decomposition at Extreme Conditions[END_REF]. [START_REF] Mao | Isothermal compression of magnetite to 320 KB[END_REF] were the first to report the transformation from magnetite to β-Fe 3 O 4 above 25 GPa as observed by powder X-ray diffraction in a diamond anvil cell (DAC). The transformation was described to be sluggish, especially at ambient temperature, with the completion of the phase transformation being faster at 600 K. These findings were confirmed by the study of [START_REF] Huang | Rapid Determination of Fe3O4 Phase Diagram by Synchrotron Radiation[END_REF], which reported a large two-phase hysteresis upon compression and decompression. The structure of β-Fe 3 O 4 was initially suggested to be monoclinic [START_REF] Mao | Isothermal compression of magnetite to 320 KB[END_REF]), but subsequent Rietveld refinements performed on powdered magnetite in a resistively-heated DAC proposed β-Fe 3 O 4 to have the orthorhombic CaMn 2 O 4 -type structure in the space group Pbcm [START_REF] Fei | In situ strucutre determination of the high-pressure phase of Fe3O4[END_REF]. A more recent study by [START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF] confirmed β-Fe 3 O 4 to be orthorhombic, but having the CaTi 2 O 4 -type structure in the space group Cmcm. This structure was further confirmed by [START_REF] Lazor | High-pressure study of stability of magnetite by thermodynamic analysis and synchrotron X-ray diffraction[END_REF] who also performed Rietveld refinements on polycrystalline magnetite and a recent study by [START_REF] Khandarkhaeva | Structural Diversity of Magnetite and Products of its Decomposition at Extreme Conditions[END_REF], who performed single-crystal Xray diffraction experiments in laser-heated DACs. The latter also reported two new Fe 3 O 4 polymorphs, namely γ-Fe 3 O 4 having the Yb 3 S 4 -type structure in space group Pnma stable at 64-73 GPa upon heating at 1500-2000 K, and δ-Fe 3 O 4 having the Th 3 P 4 -type structure in space group I4 ̅ 2d stable at 78(1) GPa and 4800(300) K [START_REF] Khandarkhaeva | Structural Diversity of Magnetite and Products of its Decomposition at Extreme Conditions[END_REF]).

In the stability field of α-Fe 3 O 4 at pressures below the structural transformation into β-Fe 3 O 4 , the material retains the cubic structure. However, within this pressure range, the electronic state of magnetite is quite controversial. Several studies using Mössbauer spectroscopy and powder X-ray diffraction introduced the notion of a possible coordination crossover in the magnetite structure, with the electronic charge density shifting from the octahedral to tetrahedral sites, hence inducing an inverse to normal spinel transition [START_REF] Pasternak | Pressureinduced coordination crossover in magnetite; the breakdown of the Verwey-Mott localization hypothesis[END_REF][START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF]). Above 7 GPa, magnetite is suggested to first transform from an inverse spinel to an intermediate state, and then to a normal spinel above 17 GPa at 300 K [START_REF] Pasternak | Pressureinduced coordination crossover in magnetite; the breakdown of the Verwey-Mott localization hypothesis[END_REF][START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF]). The same process was observed with decreasing temperature [START_REF] Pasternak | Pressureinduced coordination crossover in magnetite; the breakdown of the Verwey-Mott localization hypothesis[END_REF]. A more recent investigation by single-crystal XRD and Mössbauer spectroscopy [START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF], however, could not confirm these observations and the proposed scenario, at least up to a pressure of 25 GPa at 300 K. An additional magnetic transition was proposed by [START_REF] Ding | Novel pressure-induced magnetic transition in magnetite (Fe 3 O 4 )[END_REF] who, based on Fe K-edge X-ray magnetic circular dichroism measurements, suggested a high-spin to intermediate-spin transition of the Fe 2+ ions between 12 and 16 GPa that, noteworthy, is incompatible with an inverse to normal spinel transition as described by [START_REF] Pasternak | Pressureinduced coordination crossover in magnetite; the breakdown of the Verwey-Mott localization hypothesis[END_REF] and [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF]. [START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF] found the volume change associated to such a transition too small to be resolved with single-crystal XRD, and further found no evidence for a spin transition in their Mössbauer spectroscopy data.

Additional investigations are needed to confirm the existence and clarify the nature of these pressure-and/or temperature-induced modifications affecting either electronic or magnetic degrees of freedom, also as a function of different pressure media and the effects of nonhydrostatic conditions.

Recently, experimental studies confirmed the presence of magnetite next to majoritic garnet in Earth's mantle assemblages [START_REF] Tao | Experimental investigation of Fe3+-rich majoritic garnet and its effect on majorite geobarometer[END_REF]) and in equilibrium with pyroxene and olivine in upper mantle assemblages representative for Mars [START_REF] Xu | Low Velocity Zones in the Martian Upper Mantle Highlighted by Sound Velocity Measurements[END_REF]) making it even more relevant to investigate the thermo-elastic properties of magnetite. Until now, the elastic properties of magnetite have been determined only at room temperature using a large number of different techniques including powder X-ray diffraction (e.g. [START_REF] Mao | Isothermal compression of magnetite to 320 KB[END_REF]Haavik et al. 2000 and references therein;[START_REF] Lazor | High-pressure study of stability of magnetite by thermodynamic analysis and synchrotron X-ray diffraction[END_REF][START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF]), single-crystal X-ray diffraction (e.g. [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Nakagiri | Nakagiri et al[END_REF][START_REF] Kuriki | High-Pressure Structural Analysis of Fe3O4[END_REF][START_REF] Reichmann | High-pressure elasticity of a natural magnetite crystal[END_REF][START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]) and ultrasonic interferometry (e.g. [START_REF] Reichmann | High-pressure elasticity of a natural magnetite crystal[END_REF]. The bulk modulus K 0 obtained in the above-mentioned studies fall in a large range between 141 and 222 GPa, with its pressure derivative K' 0 ranging between 3.6 and 7.5.

Once limiting to studies performed after the year 2000, predominantly conducted using singlecrystal XRD [START_REF] Reichmann | High-pressure elasticity of a natural magnetite crystal[END_REF][START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]) and powder XRD [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF], results are more comparable, with K 0 between 180 and 189 GPa and K' 0 in the range 3.6 to 5.2. Until now, only one study [START_REF] Lazor | High-pressure study of stability of magnetite by thermodynamic analysis and synchrotron X-ray diffraction[END_REF]) attempted to determine the compressibility of magnetite at high temperatures, without reporting the actual data. The thermo-elastic properties of magnetite, however, are critically needed, for example to be able to model density and wave velocities through a magnetite-bearing phase assemblage, e.g. representing the Martian upper mantle [START_REF] Xu | Low Velocity Zones in the Martian Upper Mantle Highlighted by Sound Velocity Measurements[END_REF].

The aim of this study is two-fold: 1) to determine the thermal equation of state of Fe 3 O 4 magnetite up to 16 GPa and 1100 K using X-ray diffraction combined with resistive-heated diamond anvil cells and 2) to investigate the structural evolution of magnetite in order to elucidate any structural changes leading towards a phase transformation at high pressure and temperature.

Methods

A natural magnetite sample grown on a mica schist bedrock sourced from Chester, Vermont, USA has been used as starting material for all powder and single-crystal measurements at high pressure and high temperature. The electron microprobe, operated at 15 keV and 10 nA with a focused beam, was calibrated using diopside as standard for Mg, Si and Ca, albite for Na, orthoclase for Al, Fe 2 O 3 for Fe, Cr 2 O 3 for Cr, and used to obtain the chemical composition of the natural magnetite. Analyses on 60 individual spots on a one-sided polished magnetite sample embedded in epoxy resulted in the chemical composition reported in Table 1, confirming the high purity of this natural magnetite sample. This natural sample is purer (99.5%) in comparison to the natural magnetite sample investigated by [START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF] having a purity of 99.2%. The sample was crushed dry in an agate mortar for 30 min until a fine-grained homogeneous powder was obtained.

High pressure and high temperature experiments were prepared and performed at the Extreme Conditions Beamline P02.2 of PETRA III (Hamburg, Germany) [START_REF] Liermann | The Extreme Conditions Beamline P02.2 and the Extreme Conditions Science Infrastructure at PETRA III[END_REF]. Two Mao-type symmetric DACs were prepared and dedicated to XRD measurements at highpressure and ambient temperature on a single-crystal (run hereafter referred to as DAC_300Ksc) and on a powdered sample (DAC_300Kpwd), respectively. DAC_300Ksc was equipped with two Boehler-Almax diamonds [START_REF] Boehler | New anvil designs in diamond-cells[END_REF] having a culet diameter of 400 µm. A 200 µm thick Re gasket was indented to 73 µm and a hole of 200 µm diameter was laser-drilled in the center of the indentation acting as sample chamber. A singlecrystal of around 40x60x20 µm 3 was placed in the center of the culet next to a small ruby sphere acting as a fluorescence pressure gauge [START_REF] Dewaele | Equations of state of six metals above 94 GPa[END_REF]). DAC_300Kpwd was prepared with standard diamonds of 400 µm culet diameter. The Re gasket was prepared in the same way as described above. The powdered pelleted sample was placed inside the sample chamber together with a ruby sphere. Both DACs were gas-loaded with neon acting as quasi-hydrostatic pressure transmitting medium [START_REF] Liermann | The Extreme Conditions Beamline P02.2 and the Extreme Conditions Science Infrastructure at PETRA III[END_REF] as it was reported to provide conditions as hydrostatic as helium gas up to at least 19 GPa at ambient temperature [START_REF] Scheidl | Extending the single-crystal quartz pressure gauge up to hydrostatic pressure of 19 GPa[END_REF].

During all experiments the pressure was increased remotely using a gas-driven membrane.

Three resistively-heated DACs (DAC_600K, DAC_900K and DAC_1100K) were prepared with one standard (upstream) and one Boehler-Almax diamond (downstream) each, employing diamonds of 300 µm culet for DAC_600K and DAC_900K and of 400 µm culet for DAC_1100K. A couple of 0.5 mm thick graphite heaters connected by two Mo electrodes built inside the DACs allowed generating stable high temperatures (designed after Lawrence Livermore 4-pin cells). Heaters from layered graphite surrounding the Re gasket are forming a homogenous heating field at the diamond tips and temperatures up to 1100 K inside the sample chamber. Re gaskets were pre-indented to a thickness of ~40-70 µm and sample chambers of 130-150 µm in diameter were drilled. The temperature was monitored using two individual R-type thermocouples per DAC, positioned as close to the sample chamber as possible. The average of the two readings was assumed as the sample's temperature and the standard deviation was taken as uncertainty (Table 2). DAC_600K was loaded with a single-crystal magnetite sample and SrB 4 O 7 :Sm 2+ acting as pressure gauge throughout the experiment [START_REF] Rashchenko | Revised calibration of the Sm:SrB4O7 pressure sensor using the Sm-doped yttrium-aluminum garnet primary pressure scale[END_REF]. DAC_900K and DAC_1100K were loaded with a powdered and single-crystal magnetite sample, respectively, together with gold nano-powder for pressure determination [START_REF] Dorogokupets | Equations of state of MgO, Au, Pt, NaCl-B1, and NaCl-B2: Internally consistent high-temperature pressure scales[END_REF]. For DAC_1100K fitting of 1D patterns was preferred at high pressure and temperature above 900 K because the quality of the single-crystal data was not ideal anymore. All three DACs were loaded with silicon oil as pressure transmitting medium, which provides a good compromise between hydrostaticity and risk of diamond failure events. Our experiment at 600 K confirmed the efficiency of silicon oil as quasi-hydrostatic pressure medium at high temperatures. The membrane-driven resistivelyheated DACs were placed into a vacuum chamber, to avoid oxidation at high temperatures of graphite heaters, cell components and diamonds.

Reference values for the unit cell volumes of magnetite (both powder and single-crystal) and gold nano-powder, were measured at ambient conditions prior to the compression experiments, as well as the fluorescence lines of ruby and SrB 4 O 7 :Sm 2+ . In this procedure, both the magnetite samples and the pressure standards were measured being placed inside the sample chamber of a DAC without any pressure transmitting medium. By employing an X-ray beam having a focused spot size of 8(H) x 3(V) µm 2 full width at half maximum (FWHM) at 25.6 keV, diffraction patterns of magnetite and gold nano-powder were recorded using a Perkin-Elmer XRD 1621 flat panel detector [START_REF] Liermann | The Extreme Conditions Beamline P02.2 and the Extreme Conditions Science Infrastructure at PETRA III[END_REF]. Polycrystalline CeO 2 and a single-crystal of natural enstatite (Mg 1.93 Fe 0.06 Si 1.93 Al 0.06 O 6 , Pbca; a=18.2391(3) Å; b=8.8117(2) Å; c=5.18320(10) Å) were used to calibrate the sample to detector distance and the detector parameters for powder and single-crystal diffraction. Single-crystal diffraction data at high pressure were collected upon continuous rotation of samples around a vertical axis (𝜔) between ±30° in steps of 0.5°. The acquisition time of each frame was 0.5 s/frame and a Pt absorber of 50 µm was used to avoid saturating the detector. Diffraction data used for precise lattice parameter determination from integrated 1D profiles were collected upon rotation of the DAC between ±20° in 𝜔 with an exposure time of 20 s and using a Pt absorber of 50 µm.

For the high-temperature runs, the samples were pre-compressed to a low starting pressure, within the silicon oil field of quasi-hydrostaticity. Then the temperature was increased to the target value and left to stabilize for half an hour. XRD patterns were collected upon compression every 1 to 2 GPa at a constant temperature up to ~14 GPa. For DAC_1100K, XRD patterns were collected every 100 K starting from 600 K while increasing to the target temperature of 1100 K. The collected diffraction data were integrated using DIOPTAS [START_REF] Prescher | DIOPTAS: a program for reduction of twodimensional X-ray diffraction data and data exploration[END_REF] and analyzed using full profile LeBail refinements with the GSAS software package in the EXPGUI interface [START_REF] Toby | EXPGUI, a graphical user interface for GSAS[END_REF][START_REF] Larson | General structure analysis system (GSAS)[END_REF]. Processing of single-crystal diffraction data, which included orientation determination, indexing, intensity integration, correction for Lorentz and polarization factors, and empirical absorption correction based on spherical harmonics (ABSPACK), was carried out using CrysAlisPro 1.0.43 (Rigaku, Oxford Diffraction).

Structural refinements were performed for data collected in DAC_300Ksc, DAC_600K and for the first four points of DAC_1100K using the ShelXle software [START_REF] Hübschle | ShelXle: A Qt graphical user interface for SHELXL[END_REF][START_REF] Sheldrick | Crystal structure refinement with SHELXL[END_REF]. Isotropic displacement parameters were used to reduce the number of refined parameters due to the limited number of reflections at high pressure and temperature in the DAC. Between 26 and 38 unique reflections could be used, depending on pressure and temperature to refine 5 individual parameters. Details of the structural refinements are reported in the deposited CIF and in Table S1.

Results and Discussion

P-V-T equation of state of Fe3O4 magnetite

The variation with pressure of the unit-cell volumes of magnetite at ambient temperature collected on a single-crystal and a powdered sample in two separate runs, normalized with respect to their room pressure values, shows a smooth decrease up 15.7 GPa, the highest pressure reached in this study (Figure 1a). Plotting the normalized stress F E versus Eulerian finite strain f E (Figure S1) shows a horizontal line over the entire pressure range. Therefore, a second-order Birch-Murnaghan equation of state (BM2 EoS) can be applied to fit the combined P-V data, normalized to the corresponding V 0 , as implemented in the EoSfit7 GUI software [START_REF] Angel | EosFit7c and a Fortran module ( library ) for equation of state calculations[END_REF][START_REF] Gonzalez-Platas | EosFit7-GUI: A new graphical user interface for equation of state calculations, analyses and teaching[END_REF], resulting in an isothermal bulk modulus K T0 = 184(1) GPa with its pressure derivative K' T0 fixed to 4. Fit to a third-order BM EoS (BM3 EoS)

gives comparable results, with K T0 = 181(3) GPa and K' T0 = 4.6(5). Fitting the two datasets individually with BM2 EoS results in values of K T0 of 183(1) GPa and 184(1) for powder and single-crystal data, respectively, thus, within mutual uncertainties. The unit-cell volumes reported by several studies [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Nakagiri | Nakagiri et al[END_REF][START_REF] Kuriki | High-Pressure Structural Analysis of Fe3O4[END_REF][START_REF] Reichmann | High-pressure elasticity of a natural magnetite crystal[END_REF][START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF][START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF] are in very good agreement with the P-V data collected in this study (Figure 1a). We note that all these studies, except for [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF], were performed on synthetic or natural single-crystals and using quasi-hydrostatic pressure transmitting media such as a 4:1 methanol:ethanol mix for measurements below 4.5 GPa [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Nakagiri | Nakagiri et al[END_REF]), 16:3:1 methanol:ethanol:water below 9 GPa [START_REF] Reichmann | High-pressure elasticity of a natural magnetite crystal[END_REF]Gatta et al. 2007) or neon up to 21 GPa [START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]. The P-V data reported by [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF] is the only study performed on powdered magnetite that is in agreement with our study, most likely because their data have been collected using helium as pressure transmitting medium, so under hydrostatic conditions. Four further studies by [START_REF] Mao | Isothermal compression of magnetite to 320 KB[END_REF], [START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF], [START_REF] Kuriki | High-Pressure Structural Analysis of Fe3O4[END_REF] and [START_REF] Lazor | High-pressure study of stability of magnetite by thermodynamic analysis and synchrotron X-ray diffraction[END_REF] show a stiffer compression behavior, especially at pressures above ~10 GPa (Figure 1a). Three out of the four X-ray diffraction studies have been performed on powders in DACs, with the exception of [START_REF] Kuriki | High-Pressure Structural Analysis of Fe3O4[END_REF], who performed measurements on a single-crystal. Employed pressure-transmitting media, namely N 2 [START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF], NaCl [START_REF] Mao | Isothermal compression of magnetite to 320 KB[END_REF][START_REF] Lazor | High-pressure study of stability of magnetite by thermodynamic analysis and synchrotron X-ray diffraction[END_REF]) and a methanol:ethanol mixture [START_REF] Kuriki | High-Pressure Structural Analysis of Fe3O4[END_REF]) are known to have low hydrostatic limits of 3-10.5 GPa (N 2 and NaCl, e.g. [START_REF] Angel | Effective hydrostatic limits of pressure media for high-pressure crystallographic studies[END_REF][START_REF] Klotz | Hydrostatic limits of 11 pressure transmitting media[END_REF]) and <4 GPa (methanol:ethanol, e.g. [START_REF] Piermarini | Hydrostatic limits in liquids and solids to 100 kbar[END_REF][START_REF] Otto | Nonhydrostatic compression of elastically anisotropic polycrystals. II. Direct compression and plastic deformation[END_REF]. The stiffer compressibility reported in these studies might therefore arise from non-hydrostatic stresses developed upon compression, in particular at the highest pressures of the data collection. The observed behavior could also be explained by the effect of deviatoric stress resulting in 'pressure overestimation' when, although the sample volume is measured correctly, the corresponding pressure is systematically biased, e.g. due to a different stress on a pressure sensor [START_REF] Glazyrin | Compression of a multiphase mantle assemblage: Effects of undesirable stress and stress annealing on the iron spin state crossover in ferropericlase[END_REF].

The unit-cell volumes of magnetite collected in this study along four isotherms, at 300 K, 600(50) K, 869(57) K and 1068(72) K, are reported in Table 2 and illustrated in Figure 1b. The expected increase of the unit-cell volume with increasing temperature is well visible. The single-crystal and powder data (represented by open and closed circles, respectively) collected up to 1070 K (DAC_1100K) every 100 K starting from 600 K are in good agreement with the volumes collected in the runs DAC_600K and DAC_900K within uncertainties. The entire P-V-T dataset was fitted by a BM2 EoS combined with a Mie-Grüneisen-Debye (MGD) model for the thermal pressure, using the EoSfit7 GUI software [START_REF] Angel | EosFit7c and a Fortran module ( library ) for equation of state calculations[END_REF][START_REF] Gonzalez-Platas | EosFit7-GUI: A new graphical user interface for equation of state calculations, analyses and teaching[END_REF]) to obtain V 0 and K T0 and the high-temperature parameters, namely the Debye temperature 𝜃 D , the Grüneisen parameter 𝛾 and its logarithmic volume derivative q. We took advantage of the q-compromise Mie-Grüneisen-Debye approach implemented in EoSfit7 GUI [START_REF] Angel | Commentary on "Constraints on the Equations of State of stiff anisotropic minerals: rutile, and the implications for rutile elastic barometry[END_REF]) in order to reduce the number of fitting parameters and the otherwise large correlation of q with 𝛾 in the relatively limited pressure and temperature range targeted in this study. Still, refining K 0 , 𝜃D, and 𝛾 was yielding larger correlations and thus uncertainties, especially for 𝜃D and γ (𝜃D = 1397(865) K with V 0 = 44.604(7) cm 3 /mol, K 0 = 182(1) GPa and 𝛾 = 2.3(8)). The value of 𝜃D was therefore fixed to 660 K [START_REF] Kouvel | Specific Heat of a Magnetite Crystal at Liquid Helium Temperatures[END_REF]) which is also in agreement with the Debye temperature calculated for an Fe oxide having a mean atomic mass of ~29 [START_REF] Stixrude | Thermodynamics of mantle minerals -II. Phase equilibria[END_REF]. The final fitting results are V 0 = 44.603(7) cm 3 /mol, K 0 = 182(1) GPa and 𝛾 = 1.8(1) (solid lines in Figure 1b). In order to test the influence of q on 𝛾, the fitting procedure was repeated with q fixed to 2.7, the value proposed by Stixrude andLithgow-Bertelloni (2005, 2011) for the two spinels reported in their database, namely the spinels MgAl 2 O 4 and FeAl 2 O 4 . The resulting fitting parameters were exactly the same as the results obtained from the q-compromise approach reported above.

Structural evolution of Fe3O4 magnetite at high pressure and high temperature

Single-crystal data collected along the two isotherms at 300 K and 600 K and four additional points obtained while increasing the temperature in the DAC to 1100 K have been used to refine the structure of magnetite at high pressure and temperature. Figure 2a shows the evolution of the polyhedral volumes, namely the octahedral and tetrahedral volumes, with increasing pressure at room temperature. Both data sets at 300 K were fit to a BM2 EOS resulting in bulk moduli K 0,tet and K 0,oct of 210(13) and 169(4) GPa, respectively. The compressibility of the tetrahedra is therefore smaller compared to compressibility of the octahedra. The here-measured compressibilities are in agreement with the values expected according to a model based on ionic potential (IP=valence state/ionic radius) proposed by [START_REF] Bruschini | The elasticity of MgAl 2 O 4 -MnAl 2 O 4 spinels by Brillouin scattering and an empirical approach for bulk modulus prediction[END_REF], namely K 0,tet and K 0,oct of 230 and 180 GPa, respectively. Our measurements thus support the validity of this relatively simple model for predicting the bulk moduli for spinel compositions. Figure 2a also compares the evolution of polyhedral volumes with pressure to literature data obtained from single-crystal [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF] or powder XRD [START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF][START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF]. The data collected in this study are in good agreement with the data reported by [START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]. The compressibility data by [START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF] shows slightly softer tetrahedral and stiffer octahedral volumes, respectively.

Single-crystal data by [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF] are in very good agreement with the tetrahedral volumes reported in this study, whereas a stronger decrease with pressure for the octahedral volumes were observed. The compressibility of the tetrahedral site measured by [START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF] is in good agreement until ~ 16 GPa, but then becomes stiffer. The octahedral volumes already start deviating to a stiffer behavior above ~10 GPa. This directly reflects the stiffer behavior of the unit-cell volume above 10 GPa (Figure 1a), likely due to non-hydrostatic stresses in their DAC experiment [START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF]. The pronounced decrease and increase for octahedral and tetrahedral volumes starting from ~7 GPa reported by [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF], which was ascribed to the onset of an inverse to normal spinel transition, is not supported by the present data set. However, as the pronounced changes reported by [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF] are exactly opposite for tetrahedral and octahedral volumes inside the magnetite structure, the mean compressibility of the unit-cell volume is not influenced, resulting in the good agreement with the unit-cell volumes reported in this study (Figure 1a).

Figure 2b shows the polyhedral volumes versus pressure along the two isotherms measured in this study. As expected, the polyhedral volumes are slightly larger at high temperatures (Figure 2b), while temperature-induced softening is not clearly visible. No abrupt change in polyhedral volumes can be observed that could be linked to a phase transformation or a distinct structural change in magnetite. Furthermore, we performed a test on our structural refinement data allowing the site occupancy factor of Fe in octahedral coordination to be refined. Results show no evidence of a transformation to maghemite (Fe 3+ 0.67 ☐ 0.33 )Fe 3+ 2 O 4 upon heating, arguing against the possible oxidation of our samples during the high-temperature experiments.

The evolution of the oxygen coordinate u, controlling the individual volumes of tetrahedra and octahedra, was used by [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF] to support their hypothesis of the onset of inverse to normal spinel transition. In Figure 3, we report the oxygen coordinate u versus pressure at 300 K, and the high-temperature single-crystal data collected in this study. The data points show an almost constant trend, with only a slight increase with pressure. We also note that, within uncertainties, the values of u are very similar for ambient temperature as well as for the high temperature datasets. This provides evidence that no normal spinel component appears upon heating in our samples. Our data are in good agreement with values reported in literature, which however generally have a larger scatter and larger uncertainties [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]. The oxygen coordinates reported by [START_REF] Nakagiri | Nakagiri et al[END_REF] and [START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF] are smaller than the values obtained in this study and show a little decrease with pressure. [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF] report oxygen coordinates that first decrease slightly up to pressures of ~5 GPa, then show a strong increase with a maximum around ~15 GPa, followed by a decrease up to 20 GPa. This behavior follows the same pressure evolution previously discussed for the tetrahedral volumes, and is not confirmed by the single-crystal data collected in this study nor by any other study on magnetite available in literature [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Nakagiri | Nakagiri et al[END_REF][START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF][START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]). Since the change in polyhedral volumes depend on a single coordinate of the O atoms inside the crystal structure of magnetite, data collected on a powder sample are often more difficult to analyze and more prone to misinterpretation. [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF] could have experienced problems due to data quality, resulting in biased values produced by Rietveld refinements on their powdered magnetite sample. In particular, [START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF] suggested that the peculiar trend reported from a powder sample analysis by [START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF] could have originated from an insufficient number of diffracting crystallites potentially a with strong grain size hysteresis as well as from texture evolution. Our single-crystal results are consistent with the single-crystal structural refinements shown in [START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF], and we consider that they resolve an important controversy.

Implications

Magnetite is commonly found as a stable phase in many geological contexts, however, until now, its thermal equation of state (P-V-T relations) was not experimentally determined. The mixed iron valence at the octahedral site makes magnetite also a challenge for calculations. As such the here-obtained thermal EoS does not only greatly enhance current knowledge of the thermo-elastic properties of magnetite, but it also adds the thermal EoS of a Fe 3+ -bearing mineral to the available databases for self-consistent thermodynamic modeling (e.g., [START_REF] Stixrude | Thermodynamics of mantle minerals -II. Phase equilibria[END_REF][START_REF] Holland | New thermodynamic models and calculated phase equilibria in NCFMAS for basic and ultrabasic compositions through the transition zone into the uppermost lower mantle[END_REF]) and provides a critical benchmark to ab initio approaches (e.g. [START_REF] Dorogokupets | Thermodynamic properties of rock-forming oxides, α-Al 2 O 3 , Cr 2 O 3 , α-Fe 2 O 3 , and Fe 3 O 4 at high temperatures and pressures[END_REF].

Recently magnetite was found to be a potentially important constituent of a phase assemblages representative of the Martian upper mantle [START_REF] Xu | Low Velocity Zones in the Martian Upper Mantle Highlighted by Sound Velocity Measurements[END_REF]. Although present only in a small amount, this experimental observation prompts for a reconsideration of the importance of magnetite for our understanding of the Martian mantle and asks for including magnetite in modeling the sound velocities and densities of a bulk assemblage representative of Mars.

Noteworthy, the effects of magnetite, or of Fe 3+ -bearing minerals more in general, were not considered in any of the a priori velocity models used as guidance to the interpretation of the Mars seismic records [START_REF] Smrekar | Pre-mission InSights on the Interior of Mars[END_REF][START_REF] Giardini | The seismicity of Mars[END_REF][START_REF] Stähler | Seismic detection of the martian core[END_REF]) due to the absence of Fe 3+ in end-member phases in the [START_REF] Stixrude | Thermodynamics of mantle minerals -II. Phase equilibria[END_REF] database.

The data collected in our study enables a more sophisticated modeling of Mars' interior at least concerning the bulk sound velocity 𝑣 Φ = √ 𝐾 𝜌 ⁄ , where K corresponds to the bulk modulus and ρ to the density, respectively. Indeed, it allows evaluating the contribution of magnetite to the seismic signal of the Martian upper mantle.

As a first example, we can consider the phase proportions proposed in [START_REF] Xu | Low Velocity Zones in the Martian Upper Mantle Highlighted by Sound Velocity Measurements[END_REF], namely 50 vol% orthopyroxene, 44 vol% olivine and 6 vol% of magnetite, representative of a Martian mantle mineralogy at a depth of ~250 km, versus a magnetite-free bulk rock containing 53 vol% orthopyroxene and 47 vol% olivine. The orthopyroxene was modeled to consist of 80 % enstatite (MgSiO 3 ) and 20 % ferrosilite (FeSiO 3 ) end-members and the olivine of 77.5 % forsterite (Mg 2 SiO 4 ) and 22.5 % fayalite (Fe 2 SiO 4 ) end-members, respectively [START_REF] Xu | Low Velocity Zones in the Martian Upper Mantle Highlighted by Sound Velocity Measurements[END_REF].

The change in bulk velocity was calculated using the finite strain approach with the Debye-Mie-Grüneisen model and the elastic properties for orthopyroxene and olivine reported by Stixrude andLithgow-Bertelloni (2005, 2011) Figure 4 shows the bulk velocity of both phase assemblages versus depth. An incorporation of 6 vol% magnetite in the phase assemblage lowers the bulk velocity by ~12 m/s corresponding to ~0.21 % compared to the magnetite-free bulk assemblage. Not surprisingly, magnetite being the minor phase in the assemblages and having 𝑣 Φ very close to 𝑣 Φ of the modelled Martian assemblage (Figure S2), the effect is quite small and is well below the current seismic detection limit of the InSight mission on Mars. Nonetheless, our study improves the methodological approach, fully contributing to the general understanding and providing a deeper insight with respect to the previously uncharacterized effect of an Fe 3+ -rich mineral on the bulk velocities of a Martian upper mantle. When considering the large diversity of possible chemical and mineralogical compositions of rocky exoplanets (e.g. Putirka and Rarick 2019), any modeling omitting Fe 3+ -rich minerals could result in an oversimplification, particularly in the case of planetary environments having formed in relatively oxidizing conditions. In the case of Mars, Fe 3+ -bearing phases such as magnetite are very important for mineralogical models since they bear a signature for the redox state of Mars' mantle involving complex thermodynamic equilibrium, even when their effects of seismic velocity are minor.

Tables Table 1: Major element composition of the natural magnetite investigated in this study reported in oxides (wt%) and atoms per formula unit (apfu) X-ray diffraction [START_REF] Mao | Isothermal compression of magnetite to 320 KB[END_REF][START_REF] Nakagiri | Nakagiri et al[END_REF][START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF][START_REF] Kuriki | High-Pressure Structural Analysis of Fe3O4[END_REF][START_REF] Lazor | High-pressure study of stability of magnetite by thermodynamic analysis and synchrotron X-ray diffraction[END_REF][START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF]this study) are shown as open symbols and singlecrystal data [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Reichmann | High-pressure elasticity of a natural magnetite crystal[END_REF][START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]this study) [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF][START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF][START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]). b) Polyhedral volumes measured in this study at high pressure and high temperature. Data collected upon heating follows the same color code as reported in Figure 1b. pressure obtained using data collected in this study (both room temperature and hightemperature) and reported in the literature [START_REF] Finger | High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels[END_REF][START_REF] Nakagiri | Nakagiri et al[END_REF][START_REF] Haavik | Equation of state of magnetite and its high-pressure modification: Thermodynamics of the Fe-O system at high pressure[END_REF][START_REF] Gatta | Effect of non-hydrostatic conditions on the elastic behaviour of magnetite: An in situ single-crystal X-ray diffraction study[END_REF][START_REF] Rozenberg | Structural characterization of temperature-and pressure-induced inverse normal spinel transformation in magnetite[END_REF][START_REF] Glazyrin | Effect of high pressure on the crystal structure and electronic properties of magnetite below 25 GPa[END_REF]). The dashed line is a

  along a representative areotherm for Mars established through 3-D thermal evolution models (case 12 by Plesa et al. (2018)). This specific areotherm assumes a relatively thin average crustal thickness for Mars of 45 km, in agreement with recent findings by Knapmeyer-Endrun et al. (2021) who argue for a crustal thickness less than 40 km below the InSight lander. Generalization to different temperature profiles is straightforward.

Figure 1 :

 1 Figures

  are plotted as closed symbols. b) High-pressure and high-temperature volume of Fe 3 O 4 magnetite obtained in this study. High-temperature data collected upon heating to 1100 K are shown as color-coded circles for increasing temperature: red = 574(50) K, rose = 687(50) K, purple = 772(57) K, blue = 874(68) K, light blue = 977(63) K, green = 1075(65) K. Lines across experimental points are the outcome of fitting to a BM2 EoS combined with a Mie-Grüneisen-Debye (MGD) model for the thermal pressure (see text for details).

Figure 2a Figure 2b Figure 2 :

 2a2b2 Figure 2a

Figure 3 Figure 3 :

 33 Figure 3

Table S1 :

 S1 Single-crystal unit-cell volumes of magnetite measured at different pressures and temperatures, oxygen coordinate u, R int and R1 and the number of unique reflections obtained in the structural refinements.

	1.77(1) 601(50) 591.86(8) 0.2550(2)	2.40	1.83	37
	2.47(1) 601(50) 589.45(8) 0.2549(2)	2.99	1.77	37
	3.73(1) 603(50) 585.58(8) 0.2548(2)	2.07	1.62	38
	4.09(1) 604(50) 583.50(8) 0.2549(2)	2.41	1.56	37
	5.62(8) 604(50) 580.39(8) 0.2548(2)	1.60	1.44	37
						No. Unique
	P (GPa) 6.90(16) 602(50) 576.45(10) 0.2549(2) T (K) V (Å 3 ) u (-)	R int (%) R1 (%) 1.98 1.94	36
						reflections
	7.54(1) 601(50) 575.08(10) 0.2547(2)	2.14	2.16	39
			DAC_300Ksc
	8.85(8) 600(50) 570.82(10) 0.2550(2)	3.07	2.46	34
	0.0001(1)	298	592.26(6) 0.2549(3)	3.59	2.44	28
	9.38(4) 600(50) 569.97(14) 0.2550(3)	2.79	2.60	34
	0.77(1)	298	590.04(6) 0.2545(2)	3.14	2.28	29
	11.43(8) 598(50) 564.50(16) 0.2551(3)	2.76	2.22	35
	1.65(1)	298	587.07(8) 0.2548(3)	3.61	2.63	30
	12.49(12) 597(50) 561.52(18) 0.2547(4)	4.54	2.77	33
	3.00(3)	298	583.82(8) 0.2546(3)	2.59	2.70	30
	3.97(8)	298	579.41(8) 0.2547(3)	2.76	2.70	30
			DAC_1100K
	5.69(6)	298	574.58(8) 0.2549(3)	3.19	2.43	28
	1.75(1) 574(50) 592.47(6) 0.2547(3)	4.13	3.85	29
	6.69(14)	298	572.20(6) 0.2547(3)	4.20	2.29	27
	2.29(1) 687(50) 592.32(6) 0.2547(3)	3.80	3.59	33
	7.66(9)	298	570.05(8) 0.2547(2)	2.38	2.41	27
	2.84(1) 772(57) 592.17(6) 0.2546(3)	2.52	3.56	36
	8.71(9)	298	567.60(8) 0.2550(2)	2.71	1.93	26
	3.40(1) 874(68) 593.04(6) 0.2548(2)	3.81	1.89
	9.89(1)	298	563.42(8) 0.2548(2)	3.34	2.38	27
	10.99(6)	298	561.01(6) 0.2549(2)	4.66	2.62	27
	12.02(6)	298	559.68(6) 0.25493(2)	2.67	1.79	27
	13.21(1)	298	555.82(6) 0.2548(2)	2.43	2.70	29
	14.40(1)	298	553.65(8) 0.2549(2)	2.58	2.27	27
	15.74(3)	298	550.27(5) 0.2552(4)	3.18	2.49	28
			DAC_600K
	1.11(1)	298	589.77(8) 0.2548(3)	1.82	1.80	38

Acknowledgements

We thank the technical editor, Giovanni B. Andreozzi and one anonymous reviewer for their constructive comments and G. Diego Gatta for handling the revision process of this manuscript.

We thank the Sorbonne University Mineral Collection for the provision of the samples. The authors wish to thank Imène Estève for her help with sample analysis by SEM at the Institut de Minéralogie de Physique des Matériaux et de Cosmochimie (IMPMC, Paris) and Nicolas Rividi for his help during microprobe analysis at the Centre Camparis, Sorbonne Université (Paris, France). Tainá Maciel, Ludovic Delbes, Benoît Baptiste and Eglantine Boulard are acknowledged for preliminary measurements at the IMPMC X-ray diffraction platform. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation Programme (Grant agreement 724690). This work was supported by the French Space Agency (CNES), focused on SEIS instrument of the InSight mission. The Scanning Electron Microscope (SEM) facility at IMPMC is supported by Région Ile de France grant SESAME 2006 N°I-07-593/R, INSU-CNRS, Institute de Physique (INP)-CNRS, University Pierre et Marie Curie-Paris 6, and by the French National Research Agency (ANR) grant ANR-07-BLAN-0124-01. This research was carried out at the P02.2 Extreme Conditions Beamline at DESY, a member of the Helmholtz Association (HGF). The research leading to this result has been supported by the project CALIPSOplus under the Grant

Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020. This paper is InSight contribution 261.

Table 2: Unit-cell volumes of magnetite measured at listed pressures and temperatures. For the high-temperature datasets, the average of the two thermocouple readings was assumed as the sample's temperature and the standard deviation were taken as uncertainties. The pressure uncertainties were determined by the half difference between the measurements of the pressure standards taken before and after each XRD measurement. guide to the eye showing the evolution of u with pressure at ambient temperature observed in this study. for Mars [START_REF] Plesa | The Thermal State and Interior Structure of Mars[END_REF] for olivine (Fo77.5, green), magnetite (blue) and orthopyroxene (En80, orange). These bulk velocities are compared to the two phase assemblages 1) containing 50 % orthopyroxene, 44 % olivine and 6 % magnetite (black) and 2) a magnetite-free bulk assemblage with 53 % orthopyroxene and 47 % olivine (red) as shown in Figure 4.

Supplementary Information


Marcos Serrano

Andres Lucero

Pourang Irani

Anne Roudaut

Andrés Lucero

Stranger Screens: Exploring the Application Themes for Interactive Freeform Devices

Keywords:

HAL is

INTRODUCTION

The pervasive rectangular screen, which has dominated the display industry for decades is slowly giving room to a future in which devices and their displays may take any arbitrary shape. For example, it is now possible to manufacture displays having any 2D shape [START_REF] Folium | Folium Optics[END_REF][START_REF] Sharp | Sharp Free-Form Display[END_REF], and circular displays are already available on many smartwatches. Devices with such non-traditional displays are commonly referred to as Freeform devices [START_REF] Serrano | Mold-It: Understanding how Physical Shapes affect Interaction with Handheld Freeform Devices[END_REF][START_REF] Serrano | Investigating Text Legibility on Non-Rectangular Displays[END_REF][START_REF] Serrano | Visual Composition of Graphical Elements on Non-Rectangular Displays[END_REF], as they can assume any non-planar and non-rectilinear shape. With this fundamental shift https://doi.org/10.1145/3491101.3519707 in display form factors, the looming question that arises for interaction designers includes how to integrate and take advantage of the non-rectilinearity of such interactive devices that may soon become common. Furthermore, opening the space of possible topologies for displays creates a tremendous need for understanding the interplay between shapes, interactions, applications and contexts.

Freeform devices take their roots in Shape-Changing (SCI) [START_REF] Alexander | Grand Challenges in Shape-Changing Interface Research[END_REF] and Organic User (OUI) Interfaces [START_REF] Holman | Organic User Interfaces: Designing Computers in Any Way, Shape, or Form[END_REF] but constitute a specific subcategory defined by their non-rectilinear nature. To date, characteristics of freeform devices and their supersets, including SCIs or OUIs, have been primarily defined by taxonomies driven by technological innovations [START_REF] Roudaut | Morphees: Toward High "Shape Resolution" in Self-actuated Flexible Mobile Devices[END_REF][START_REF] Sturdee | Analysis and Classification of Shape-Changing Interfaces for Design and Application-based Research[END_REF]. Previous explorations of the design space of such devices heavily focus on shape transformation, for example how a change in shape may relate to a user's emotions [START_REF] Strohmeier | An Evaluation of Shape Changes for Conveying Emotions[END_REF], input gestures [START_REF] Lahey | PaperPhone: Understanding the Use of Bend Gestures in Mobile Devices with Flexible Electronic Paper Displays[END_REF] or interaction metaphors [START_REF] Majken | Sketching Shape-changing Interfaces: Exploring Vocabulary, Metaphors Use, and Affordances[END_REF]. In contrast, little is known on how the non-rectilinearity of the display itself drives thematic applications on such devices.

In this paper, we address this gap by specifically examining freeform devices, or 'Stranger Screens', in the context of applications they benefit. This work takes a complementary approach to recent developments in freeform device research [START_REF] Serrano | Investigating Text Legibility on Non-Rectangular Displays[END_REF][START_REF] Serrano | Visual Composition of Graphical Elements on Non-Rectangular Displays[END_REF][START_REF] Simon | Finding Information on Non-Rectangular Interfaces[END_REF], that has involved limited perspectives beyond those of the research community. In contrast to past literature on SCIs, our work moves away from looking at the transformation of the devices and rather investigates the state of the device at a given time. In particular, we know little about how such displays may integrate into everyday lives through meaningful applications.

We adopt the established speculative design techniques used in HCI [START_REF] Dunne | Speculative Everything: Design, Fiction, and Social Dreaming[END_REF] to explore, in an unconstrained manner, how freeform devices could become part of our daily living. We carried this through an exploratory one-week long workshop where participants produced, in an unrestricted manner (to either technology or scenario), novel freeform devices. The goal was to imagine possible futures. The analysis of their creations unveiled three design axes behind the concept of freeform devices: Introspection, Community and Magic exploration. We believe that these concepts go beyond what current applications have proposed for typical usage of freeform devices and pave way for new usage directions.

'Stranger Screens' is an exploration of the potential impact that freeform devices can have on our everyday digital interactions. In this vein, our contribution consists of an understanding of inspirational application themes related to freeform devices.

Freeform Interfaces

The term 'Freeform device" was used by Serrano et al. [START_REF] Serrano | Investigating Text Legibility on Non-Rectangular Displays[END_REF][START_REF] Serrano | Visual Composition of Graphical Elements on Non-Rectangular Displays[END_REF]to refer to devices having a non-rectangular display. By extension, in this paper we consider a freeform device any device not having a rectangular shape, in contrast to current laptops or smartphones for instance. Research on Freeform devices has mostly focused on the design of non-rectangular content according to static shape features [START_REF] Serrano | Investigating Text Legibility on Non-Rectangular Displays[END_REF][START_REF] Serrano | Visual Composition of Graphical Elements on Non-Rectangular Displays[END_REF][START_REF] Simon | Finding Information on Non-Rectangular Interfaces[END_REF], or on how to dynamically organize its content according to the available display surface [START_REF] Niyazov | Dynamic Decals: Pervasive Freeform Interfaces Using Constrained Deformable Graphical Elements[END_REF].

Freeform devices take their roots in Organic User (OUIs) and Shape-Changing (SCI) Interfaces. Researchers have produced a number of explorations on the design space of shape-changing interfaces or organic user interfaces. The common element among these explorations is their focus on the shape transformation [START_REF] Majken | Shape-changing Interfaces: A Review of the Design Space and Open Research Questions[END_REF]. Researchers have also investigated the best design approaches for ideating shape-changing devices. Rasmussen et al. [START_REF] Kwak | The Design Space of Shape-changing Interfaces: A Repertory Grid Study[END_REF][START_REF] Majken | Sketching Shape-changing Interfaces: Exploring Vocabulary, Metaphors Use, and Affordances[END_REF] investigated the use of sketches, Fuchs et al. [START_REF] Fuchs | Foldwatch: Using Origami-inspired Paper Prototypes to Explore the Extension of Output Space in Smartwatches[END_REF] employed origami paper prototypes, Everitt et al. [START_REF] Everitt | ShapeCanvas: An Exploration of Shape-Changing Content Generation by Members of the Public[END_REF] conducted a deployment of a prototype in a public environment, while Sturdee et al. [START_REF] Sturdee | A Public Ideation of Shape-Changing Applications[END_REF] first employed brainstorm sessions within a public engagement study, and later proposed an approach combining low-fidelity prototypes, high fidelity video footage, with end-user diagrams and scenario sketching [START_REF] Sturdee | Visual Methods for the Design of Shape-Changing Interfaces[END_REF].

However, previous work did not answer the question "how such shapes are influenced by specific applications", which is the goal of this work.

Design Futuring Approaches

In this paper we adopt a research method inspired by recent design futuring approaches [START_REF] Kozubaev | Expanding Modes of Reflection in Design Futuring[END_REF], such as design fiction [START_REF] Coulton | Design fiction as world building[END_REF] and speculative design [START_REF] Dunne | Speculative Everything: Design, Fiction, and Social Dreaming[END_REF]. The common goal of these approaches is to use design methods to imagine alternative futures. These approaches are well suited to explore emerging technological concepts and their future adoption [START_REF] Lindley | Implications for Adoption[END_REF]: as such, design futuring has been applied to brain computer interaction [START_REF] Wong | When BCIs Have APIs: Design Fictions of Everyday Brain-Computer Interface Adoption[END_REF], interaction with robots [START_REF] Auger | Living with Robots: A Speculative Design Approach[END_REF] or animalcomputer interaction [START_REF] Hirskyj | On the Internet, Nobody Knows You're a Dog... Unless You're Another Dog[END_REF] among others. To our knowledge, such an approach has not been applied to the topic of freeform interfaces. The closest exploration we could find was the work by Rasmussen et al. [START_REF] Kirkegård | Envisioning Future Challenges and Possibilities for Shape-Changing Interfaces through Speculative Scenarios[END_REF] who present three speculative scenarios imagining the future of living with SCIs, again from a shape transformation perspective.

WORKSHOP: EXPLORING APPLICATIONS AND FORM FACTORS

To explore the topic of interactive freeform interfaces we faced the challenges involved with the nascent and emerging nature of such technologies. We adopted a Research through Design1 (RtD) approach [START_REF] Frayling | Research in Art and Design[END_REF][START_REF] Koskinen | Design Things: Models, Scenarios, Prototypes[END_REF][START_REF] Zimmerman | Research through Design as a Method for Interaction Design Research in HCI[END_REF], where the act of designing is in itself a confrontation of various forms of knowledge, both formalized and experiential, which brings about new knowledge. This knowledge can for instance be generated by designing an artifact, by the artifact itself, and by evaluations of use, and later be generalized as design recommendations, theories or frameworks. We organized a one-week exploratory workshop to identify application ideas inspired by freeform devices.

Method

The workshop involved 12 students (10F/2M, 21 years old on average) from a Master course on design at the University of Toulouse (France). Students had no previous knowledge of HCI. Students were supervised by two senior external designers (i.e. not teaching in their course). We decided to carry out a long duration workshop as this gives the participants time to really expand on their ideas. The workshop spanned over five full days, from Monday to Friday.

Monday: Introduction.

The first day consisted of a set of seminars given by different individuals related to the perspectives on Stranger Screens: three researchers in HCI working on nonrectangular interfaces [START_REF] Serrano | Visual Composition of Graphical Elements on Non-Rectangular Displays[END_REF], spatial immersive analytics [START_REF] Ens | Spatial Analytic Interfaces: Spatial User Interfaces for In Situ Visual Analytics[END_REF] and authoring tools for designers [START_REF] Maudet | Beyond Grids: Interactive Graphical Substrates to Structure Digital Layout[END_REF]; a collector of old screens and displays; as well as the two designers that supervised the rest of the workshop. Each of the presenters showed slides and video examples related to their topic.

Tuesday-Thursday: Design Rounds.

From Tuesday to Friday, the students worked in groups on their projects. The 12 students could organize into groups as they wanted: 8 of them formed 4 groups of 2 students, while the other 4 students worked individually.

The schedule was similar each day: each group met the supervisors once in the morning, and had to present their daily advances to everyone else by the end of the day. All the ideas came from the participants, and were refined through iterations with the two supervising senior designers (graphic and UX designers). During these three days, all participants shared the same space, a large prototyping room (i.e. with a few scattered chairs and tables), that they could rearrange if needed. The room included a fabrication workbench with multiple craft tools. They could also use videoprojectors as well as their personal laptops (all of them had a version of Adobe Creative Suite).

Friday: Idea Sharing.

The last day of the workshop consisted in a group presentation of the resulting design artifacts. For each project, the participants presented the final prototype, as well as contextual illustrations, mind maps with the main themes, and the intermediate prototypes and drawings (which were hung up on the wall behind the final prototype).

Data Collection and Analysis

For each project, we collected a written essay by the participants describing their project in their own words; a set of pictures of the final prototype; as well as pictures of the design process (see Figure 1). In total, we collected and analyzed 212 photos. Two researchers carried a thematic analysis of the projects using both pictures and text, refined by using affinity diagrams [START_REF] Lucero | Using Affinity Diagrams to Evaluate Interactive Prototypes[END_REF], to find the main themes that emerged from the design process. This followed a process of developing from lower level codes to higher level themes. A first coder proceeded to create initial codes that were refined with the other coder, before coding all the projects and proceeding to the grouping by themes.

Results

Our analysis revealed three main themes, that we describe hereafter and illustrate with the different projects.

Theme 1: Introspection (Intimacy).

We tagged the first theme, Introspection, on three of the presented projects: Flux, Game Shower and Monocle. These projects share elements of intimacy in terms of being directly related with or coming in close contact with one's body.

Flux is a spherical object that displays the evolution of the menstrual cycle through a colorful wave (Figure 2 -left). Flux creators said the following regarding their idea: "Menstruation is invisible in our society and the project FLUX, through an offbeat object, intends to make them visible and normal. " Softly caressing the sphere with a hand allows to reveal precise information, thus increasing a woman's awareness of their menstrual cycle. This design contrasts with menstrual tracking phone apps, that tend to use stereotypical feminine attributes, such as using the color pink or flowers, and that unveil concerns about accidentally disclosing the cycle [START_REF] Daniel | Examining Menstrual Tracking to Inform the Design of Personal Informatics Tools[END_REF].

Game Shower is an interactive device for showers in public swimming pools (Figure 2 -right). A game is projected onto the water of the shower where users pass through, thus effectively turning an ordinary shower into a shower display that covers one's body. This idea is related to the interactive water surface of the AquaTop display [START_REF] Koike | AquaTop Display: Interactive Water Surface for Viewing and Manipulating Information in a Bathroom[END_REF], that uses the water surface of the bathroom to project information.

Monocle is a portable video-guide intended for museums that one straps onto their forehead. Its creators describe Monocle as such: "The curved form of the device, like a hand mirror, will appear to the visitor like a second face that initiates a discussion, like a speech bubble." This project extends other see-through displays [START_REF] David Hincapié-Ramos | cAR: Contact Augmented Reality with Transparent-Display Mobile Devices[END_REF] by focusing on the overall shape of the display and the handle.

Overall, these projects employ the freeform shape to get the device in close contact with one's body: the spherical shape of Flux can be caressed, the water in Game Shower covers one's skin, and the curved form of Monocle can be grasped like a hand mirror.

Theme 2:

Community. We tagged the second theme, Community, on four of the presented projects: Pass the Sound, Internal Tissue, Focus, and Nautical Mile. The common thread among these ideas is a multi-user and caring-for-others aspect, be it another person, an animal, or the environment.

Pass the Sound is a project that deals with group music listening practices (Figure 3 -left). It consists of a set of screens that can be separated, or collated to collect something [START_REF] Goguey | Pick-Cells: A Physically Reconfigurable Cell-composed Touchscreen[END_REF]. Its creators describe their idea like this: "[Pass the Sound] is a neutral support that allows people to have common access to a music library. It's a way to listen to music together, in a more equitable way. "

Focus is a device that allows an animal to be brought back to the owner in case the animal is lost and found by a stranger. This is how its creators describe the idea: "A QR code contains the coordinates of the owner in order to be able to contact if the animal is found. "

Finally, Nautical Mile is a professional exploration instrument for students (Figure 3 -right). "At the end of the game, students use sonar to keep memories of the discussion when meeting with the person. " 3.3.3 Theme 3: Magic Exploration. We identified the third and final theme, Magic Exploration, based on these three projects: the aforementioned Flux, Imaginarium and Monocle. These three ideas share the notion of augmenting a person's senses and abilities, allowing them to see and access information that would otherwise not be available to them.

First, the aforementioned Flux project uses the metaphor of a crystal ball [START_REF] Benko | Sphere: Multitouch Interactions on a Spherical Display[END_REF] to tell the future of the menstrual cycle, or in this case, revealing information that is normally hidden. Its creators describe Flux thusly, "by bringing the crystal ball closer to the digital screen, it is a question of associating the technology with a form of magic. "

Imaginarium is a parabolic screen for kids in hospitals (Figure 4 -left). As per the authors, "the shape has been chosen to offer an immersive journey through different universes, like a window or an escape from the hospital. "

Finally, the aforementioned Monocle (Figure 4 -right) is a device that gives its user some sort of superpowers. Their authors say, "it enriches and enhances the reality. "

Summary

This exploratory workshop revealed some inspirational themes that emerge from the concept of Stranger Screens. In addition, it suggests a number of possible domains, applications, shapes, ways to interact, and multi-user scenarios, among other things.

DISCUSSION AND FUTURE WORK

In this section we discuss our results in light of our goal which was to explore application themes inspired by freeform displays. We particularly discuss the differences we found with previous work that were more technologically driven and highlight directions for future work.

Our study unveiled three inspirational themes imagined by nonspecialist designers for freeform devices: introspection, community and magic exploration. It is interesting to note that these themes differ considerably from those mentioned by prior work on freeform devices [START_REF] Serrano | Investigating Text Legibility on Non-Rectangular Displays[END_REF][START_REF] Serrano | Visual Composition of Graphical Elements on Non-Rectangular Displays[END_REF]. The main application scenarios of these previous work were in-vehicle displays, pocket devices, non-rectangular furniture such as cooktops and road signs. In contrast, non-specialist designers placed more emphasis on the Why (i.e. what is the design/usage implication of freeform shapes), rather than on the Where (i.e. where can a freeform shape fit in our environment).

In the future, we plan to explore the freeform devices imagined by non-specialist designers. However, since these devices have a variety of form factors, we will need to adopt approaches that could better fit the scale and characteristics of these ideas, such as bodystorming or play acting [START_REF] Lucero | The dialogue-labs method: process, space and materials as structuring elements to spark dialogue in co-design events[END_REF]. We plan to develop design probes corresponding to the proposed ideas of handheld devices. We will carry both controlled studies on the dexterity and discoverability of such devices, as well as longitudinal studies similar to [START_REF] Everitt | ShapeCanvas: An Exploration of Shape-Changing Content Generation by Members of the Public[END_REF] so that we can investigate the practicality of freeform devices.

Figure 1 :

 1 Figure 1: Workshop participants used a variety of ideation tools, such as sketching, computer-aided design, crafting and video projection.

Figure 2 :

 2 Figure 2: Some projects shared elements of intimacy (Introspection theme), as in Flux and Game Shower.

Figure 3 :

 3 Figure 3: A common thread among some ideas is a multiuser and caring-for-others aspect (Community theme), as in Pass the Sound (left) and Nautical Mile (center and right).

Figure 4 :

 4 Figure 4: Some ideas share the notion of augmenting a person's senses and abilities (Magic Exploration theme), as in Imaginarium (left) and Monocle (right).

https://www.interaction-design.org/literature/book/the-encyclopedia-of-humancomputer-interaction-2nd-ed/research-through-design

ACKNOWLEDGMENTS

This work was supported by the ANR JCJC PERFIN grant (ANR-18-CE33-0009). We thank Anthony Masure and all the students of the Masters program on Transdisciplinary design, cultures and territories from the University of Toulouse.