F Poitrasson

L Cruz Vieira

P Seyler

G Márcia

Santos Pinheiro

D Santos Mul

values. . . . . . . .

Iron isotopes compositions of various terrestrial and extraterrestrial high-

temperature samples against the IRMM-014 reference standard. The grey line represents the mean ffi 57 Fe value for chondrites = -0.015 ± 0.009 2SE % [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]. Data are from : Craddock and Dauphas (2011a); [START_REF] Needham | An Fe isotope study of ordinary chondrites[END_REF]; Poitrasson et al. (2005); [START_REF] Schoenberg | Modes of planetary-scale Fe isotope fractionation[END_REF]; Wang et al. (2013Wang et al. ( , 2014) ) for chondrites; [START_REF] Craddock | Abyssal peridotites reveal the nearchondritic Fe isotopic composition of the Earth[END_REF]; [START_REF] Huang | Iron and magnesium isotopic compositions of peridotite xenoliths from Eastern China[END_REF]; [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF]; Weyer and Ionov (2007); Williams et al. (2005); [START_REF] Zhao | Iron isotope variations in spinel peridotite xenoliths from North China Craton: Implications for mantle metasomatism[END_REF][START_REF] Zhao | Iron isotope evidence for multistage melt-peridotite interactions in the lithospheric mantle of eastern China[END_REF][START_REF] Zhao | Effects of melt percolation on iron isotopic variation in peridotites from Yangyuan, North China Craton[END_REF] for peridotites; Gleeson et al. (2020); Konter et al. (2016); [START_REF] Nebel | Iron isotopic evidence for convective resurfacing of recycled arc-front mantle beneath back-arc basins[END_REF]Nebel et al. ( , 2018Nebel et al. ( , 2019)); Sossi et al. (2016); Teng et al. (2013) for the basalts ; Poitrasson et al. (2004); [START_REF] Schoenberg | Modes of planetary-scale Fe isotope fractionation[END_REF]; Wang et al. (2012); Weyer et al. (2005) for the Moon; Poitrasson et al. (2004); Sossi et al. (2016); Wang et al. (2012) for Mars ; Poitrasson et al. (2004); [START_REF] Schoenberg | Modes of planetary-scale Fe isotope fractionation[END_REF]; Wang et al. (2012); Weyer et al. (2005) for angrites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Compilation of the iron isotopes compositions of igneous crustal rocks ver-

sus SiO 2 (in wt.%) as an index of differentiation (Du et al., 2017;Foden et al., 2015;He et al., 2017;Heimann et al., 2008;Li et al., 2020;Poitrasson and Freydier, 2005;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012;Williams et al., 2018;Xia et al., 2017;Zambardi et al., 2014). . . . . After Poitrasson (2017) and [START_REF] Lameyre | Roches et Minéraux[END_REF]. . . . . . . . . . . . . . . . . . 1.5 Silicon isotopes compositions of the major natural reservoirs (terrestrial and extraterrestrial) against the NBS-28 reference standard. The grey line represents the ffi 30 Si value of bulk silicate earth (BSE) of 0.29 ± 0.07 % and the grey dashed line represents the global ocean ffi 30 Si signature of 1.1 ± 0.3 % . Modified from [START_REF] Zambardi | Recherche de marqueurs de processus de formation des planètes à travers les isotopes stables de masse moyenne[END_REF] and Poitrasson (2017) and references therein). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Compilation of the silicon isotopes compositions of crustal igneous rocks versus SiO 2 (in wt.%) as an index of differentiation (Poitrasson and Zambardi, 2015;Savage et al., 2011Savage et al., , 2012;;Zambardi et al., 2014) . . . . . . . .

Schematic representation of the self-consistent iterative resolution of Kohn-

Sham equations (modified from [START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF]). . . . . . . . . . . . . . . . represents a scenario with a shifted grid from the origin leading to only 3 inequivalent k-points (Martin, 2004). . . . . . . . . . . . . . . . . . . . . . 2.4 Schema of the difference between a wave-function fully calculated (black plain line) and one calculated using a pseudopotential (dotted line). Beyond the r cut-of f , the wave-functions, as well as the external potentials [START_REF] Sholl | Density Functional Theory: A Practical Introduction[END_REF], where valence and conduction bands are clearly distinguishable. The band gap is located around 5 eV. Figures (b) (b) at T =1280 K. The linear fit between 10 3 lnβ 57 Fe/ 54 Fe and the iron force constant is : f (x) = 0.0026x and is in total agreement with the equation of Dauphas et al. (2014). The linear fit between 10 3 lnβ 30 Si/ 29 Si and the silicon force constant is: f (x) = 0.0062x. . . . . . . . . . . . . . . . . . . . sharing an edge or a corner with the central Fe or Si polyhedron (small symbols). For comparison, large circles correspond to the mean force constant evolution with the Fe molar concentration in the mineral M sites (upper axis). For example, in panel a, fayalite (red large circle) contains only iron and no magnesium in its M1 and M2 sites. Therefore, the corresponding iron concentration in M sites is 100 mol% Fe, and the mean Fe force constant (F = 99.6 N/m) is an average of the Fe force constants of the four Fe atoms in M1 sites (88.1, 88.3, 87.3 and 87.3 N/m) and of the four Fe atoms in M2 sites (109.8, 109.8, 112.9 and 112.9 N/m). . . . . . . . . . . xiii 3.6 Comparison of the 10 3 lnβ 57 Fe/ 54 Fe (a) and 10 3 lnβ 30 Si/ 29 Si (b) calculated in the present study at T = 1280 K, with previous data computed from NRIXS measurements (Dauphas et al., 2012;Dauphas et al., 2014), Mössbauer spectroscopy (Polyakov and Mineev, 2000), experimental petrology (Sossi and Foden, 2017), and DFT calculations (Huang et al., 2014, Qin et al. 2016). The uncertainties are ≤ 0.05 % for NRIXS-derived data (Dauphas et al., 2014(Dauphas et al., , 2012;;Polyakov et al. 2007, Mineev et al. 2007) and ≤ 0.10 % for Mössbauer-derived data (Polyakov and Mineev, 2000). . . .

V P P ef f ,

Models of δ 57

Fe evolution during magmatic differentiation: the black line represents model 1 where a bulk fractionation factor α f orsterite-melt is used with forsterite representing all Fe 2+ -bearing minerals, and the red line represents model 2 where the differences between the relevant Fe 2+ -bearing minerals are taken into account (i.e., α enstatite-melt , α augite-melt ). Models were obtained using the MCS software (Bohrson and Spera, 2007) with starting andesite composition (SiO 2 = 58.23, TiO 2 = 0.94, Al 2 O 3 = 16.94, Cr 2 O 3 = 0.03, FeO = 7.31, MgO = 3.99, CaO = 7.08, Na 2 O = 3.47, K 2 O=1.57, P 2 0 5 = 0.24, H 2 O = 0.20) with an FMQ buffer (Dauphas et al., 2014). The circles in the background correspond to δ 57 Fe from the literature (Du et al., 2017;Foden et al., 2015;He et al., 2017;Poitrasson and Freydier, 2005;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012;Teng et al., 2013;Williams et al., 2018;Xia et al., 2017;Zambardi et al., 2014) FeO (c), MgO (d), CaO (e) and Na 2 0 (f) content of the melts. . 145 4.11 Relationship between the average Si-cation distance and the Si force constant.146 4.12 Models of δ 57 Fe (a; blue line) and δ 30 Si (b; dark blue cross) evolution during magmatic differentiation : Models were obtained using the MCS software (Bohrson and Spera, 2007) = 0.20) with an FM buffer Dauphas et al. (2014). The grey circles in the background correspond to δ 57 Fe (a) and δ 30 Si (b) from the literature (Du et al., 2017;Foden et al., 2015;He et al., 2017;Poitrasson and Freydier, 2005;Poitrasson and Zambardi, 2015;Savage et al., 2011Savage et al., , 2012;;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012;Teng et al., 2013;Williams et al., 2018;Xia et al., 2017;Zambardi et al., 2014). The fractionation factor ∆ 57 Fe min-melt (∆ 57 Fe enstatite-trachyte ; ∆ 57 Fe diopside-trachyte ;

Al 2 O 3 (b),
∆ 57 Fe ulvospinel-trachyte ; ∆ 57 Fe magnetite-trachyte and ∆ 30 Si min-melt (∆ 30 Si enstatite-trachyte ;

∆ 30 Si diopside-trachyte ; ∆ 30 Si anorthite-trachyte ; ∆ 30 Si albite-trachyte ; ∆ 30 Si quartz-trachyte ) used in these models are displayed in Table 4.7. The linear trends in b) represent the linear fit of bulk rock analyses obtained by Savage et al. (2011) in black and Zambardi et al. (2014) in green from natural magmatic rocks analysis as well as the linear trend of our fractional crystallization modelled value (in red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.1 Geological maps of the South East of Kerguelen Archipelago constituted of Gallieni Peninsula and Mont Ross (A) modified from [START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF] and Jeanne d'Arc and Ronarch' Peninsula (B) modified from [START_REF] Leyrit | Kerguelen : Cartographie et magmatologie des presqu'iles Jeanne d'Arc et Ronarc[END_REF] reference and the San Carlos Olivine. The line represents the mass-dependent fractionation line with a slope of 1.438 ± 0.047, that is indistinguishable from previously measured isotopic mass-dependent fractionation trends in natural and references samples of 1.466 ± 0.013 (Malinovsky et al., 2003), 1.475 ± 0.022 (Craddock and Dauphas, 2011), and from theoretical prediction at equilibrium (1.475). . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Iron isotope compositions of three USGS references glasses (BHVO-2, BIR-1 and BCR-2) determined by fs-La-MC-ICP-MS in this study compared with data from fs-La-MC-ICP-MS of (1) Oeser et al. (2014) and rock powders analyzed in other studies : [START_REF] Bennett | Iron isotope fractionation in a buoyant hydrothermal plume, 5S Mid-Atlantic Ridge[END_REF] Craddock and Dauphas (2011), (3) Millet et al. (2012) ,( 4) Nebel et al. (2015) ,(5) Peters et al. (2019) ,(6) Dideriksen et al. (2006),(7) Weyer and Ionov (2007) ,(8) Wombacher et al. (2009) ,(9) Schuessler et al. (2009) ,(10) Williams and Bizimis (2014) List of Tables 1.1 Most common iron isotopes elemental and argide isobars [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]. The numbers in parentheses are the resolving power required for mass analysis. The resolving power can be calculated by dividing the mass difference between the isobar and the Fe isotopes of interest ∆m and the mass of the Fe isotopes of interest (∆m/m). The majority of mass spectrometers are not able to resolve elemental isobars, so for Fe, the 54 Cr and 58 N i are monitored using 53 Cr and 60 N i respectively. . . . . . . . . . . . . (with x = 10 6 /T 2 ) as well as fractionation factor (1000lnα 30 Si/ 28 Si) between minerals and quartz under the form y 2 + y 3 (with y = 10 2 /T). . . . 4.6 Comparison of the iron average force constants of the melts (in N/m) using either the "snapshot" method or the "VAF" method as well as the silicon force constants of the melts using the "snapshot" method. . . . . . . . . . . xxii 4.7 Polynomial fits of iron and silicon fractionation factors between various minerals and melts. Polynomial fits use equation of type C 1 x 2 + C 2 x for ∆ 57 Fe/ 54 Fe and C 1 x 3 + C 2 x 2 + C 3 x for ∆ 30 Si/ 28 Si with x = 10 6 /T 2 for T between 0 and 1500 K. The iron and silicon β-factor of minerals were taken from Rabin et al. (2021) for all minerals except quartz, albite and anorthite that were taken from Qin et al. (2016)). . . . . . . . . . . . . . . 4.8 Comparison between ∆ melt-mineral 30 Si/ 28 Si calculated in this study and the ones estimated in [START_REF] Deng | An oceanic subduction origin for Archaean granitoids revealed by silicon isotopes[END_REF] at 1123 K. . . . . . . . . . . . . . . . Leyrit 1992, Weis et al. 1993 andthis study), as well as Sr, Nd, Pb isotopic ratios [START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF]) and GPS coordinates of the Kerguelen lavas. . . . . . . . . . . . . . . . . . . . . . .

Major element compositions (from

Whole-rock Iron isotope measurements from LMS, UMS and MRS relative

to the IRMM-014 standard data as well as the corrected basalt iron isotope signature of the initial parental meltsδ 57 Fe prim (see Sossi et al. 2016).

∆ 57 Fe raw-corr represents the difference between the measured δ 57 More details avalaible : here . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Calculated and experimental lattice parameters (in Å) for aegirine [START_REF] Redhammer | Singlecrystal X-ray diffraction and temperature dependent 57Fe Mössbauer spectroscopy on the hedenbergite-aegirine (Ca,Na)(Fe2+,Fe3+)Si2O6 solid solution[END_REF] and forsterite [START_REF] Böstrom | Single-crystal X-ray diffraction studies of synthetic Ni-Mg olivine solid solutions[END_REF]

General introduction (English version)

Iron (Fe) and Silicon (Si) are among the most abundant element in the Solar System.

Subsequent analytical developments involving high mass resolution Multiple Collector -Inductively Coupled Plasma -Mass Spectrometry (MC-ICP-MS) and ion-exchange chromatography enabled scientists to measure Fe and Si fractionation in high-temperature domain with precision < 0.05 and 0.1% respectively (e.g., Craddock and Dauphas, 2011b;Poitrasson and Freydier, 2005;[START_REF] Zambardi | Precise Determination of Silicon Isotopes in Silicate Rock Reference Materials by MC-ICP-MS[END_REF]. The investigation of the Fe isotopic composition of the different Earth reservoirs revealed significant isotopic fractionation between the mantle and the crust : both continental and oceanic crust exhibit a non-chondritic Fe isotopes composition, heavier than the mantle composition (e.g., Foden et al., 2015;Gleeson et al., 2020;[START_REF] Nebel | Iron isotopic evidence for convective resurfacing of recycled arc-front mantle beneath back-arc basins[END_REF]Nebel et al., , 2018Nebel et al., , 2019;;Poitrasson and Freydier, 2005;Sossi et al., 2016). On the contrary, Si displays a homogeneous isotopic composition in the mantle and the basaltic crust (e.g., Savage et al., 2010Savage et al., , 2011)). However, Si isotopes measurements exhibited significant variations in meteorites and terrestrial planets (e.g., [START_REF] Fitoussi | Si isotope systematics of meteorites and terrestrial peridotites: implications for Mg/Si fractionation in the solar nebula and for Si in the Earth's core[END_REF]Georg et al., 2007;[START_REF] Pringle | Silicon isotopes in angrites and volatile loss in planetesimals[END_REF][START_REF] Savage | The silicon isotope composition of the upper continental crust[END_REF] as well as iron isotopes (e.g., Poitrasson et al., 2004Poitrasson et al., , 2005)). Consequently, Fe and Si stable isotopes system quickly appeared as promising tool for a better understanding of the Earth formation and differentiation as well as extraterrestrial planetesimal formation. After a great interest in the Earth core and mantle Fe and Si isotopes compositions (e.g., Georg et al., 2007;[START_REF] Liu | Iron isotopic fractionation between silicate mantle and metallic core at high pressure[END_REF]Savage et al., 2010;[START_REF] Shahar | An assessment of iron isotope fractionation during core formation[END_REF]Shahar et al., 2011[START_REF] Shahar | Sulfur-controlled iron isotope fractionation experiments of core formation in planetary bodies[END_REF]Williams and Bizimis, 2014;Williams et al., 2004;Ziegler et al., 2010), it became obvious to constrain more precisely the composition of the Earth crust. A significant isotopic variability was observed in the crust of approximatively 0.6 % and 0.3% for Fe and Si isotopic system respectively. The differentiated igneous rocks constituting the crust displays indeed heavier Fe and Si isotopes signatures relative to the basalts (e.g., Foden et al., 2015;Gajos et al., 2016;Poitrasson and Freydier, 2005;Poitrasson and Zambardi, 2015;Savage et al., 2011;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012;Williams et al., 2018;Zambardi et al., 2014). The processes involved in the heavier isotopic compositions of the basaltic crust relative to the mantle as well as the enrichment in heavy isotopes during magmatic differentiation are still debated in the literature. Several processes were proposed along the years to explain the peculiar signature of the crust such as: partial melting (e.g., Dauphas et al., 2014;Nebel et al., 2019;Sossi et al., 2016;Weyer and Ionov, 2007;Williams and Bizimis, 2014), fractional crystallization (e.g., Qin et al., 2016;Sossi et al., 2012), fluid exsolution (e.g., Heimann et al., 2008;Poitrasson and Freydier, 2005) or diffusion-driven kinetic fractionation (e.g., Gajos et al., 2016;Lundstrom, 2009;Zambardi et al., 2014). The incomplete comprehension of the mechanisms involved in Fe and Si isotopes fractionation at equilibrium and out of equilibrium (i.e.

kinetic fractionation) prevents the scientific community, for now, to determine with precision the process or processes controlling Fe and Si isotopes fractionation during magmatic differentiation.

Consequently, the aim of this thesis was to deepen our knowledge on the mechanisms controlling equilibrium Fe and Si isotopes fractionation in silicate minerals and melts.

This study is based on a multidimensional approach combining first-principles calculation of Fe and Si isotopic fractionation between minerals and melts, conventional MC-ICP-MS Fe isotopes measurements on natural volcanic rocks, and preliminary in-situ Fe isotopes measurements on olivines using a femtosecond laser (fs-LA) coupled to a MC-ICP-MS.

This first chapter introduces the fundamental concept of isotopes and isotopic fractionation followed by a state of the art on iron and silicon isotopes fractionation in the context of magmatic differentiation.

The second chapter focuses on the theoretical and numerical methodology of the firstprinciple calculation performed on minerals as well as the first-principles molecular dynamics (FPMD) methods used for the modelling of silicates melts. Therefore, the concept of Density Functional Theory (DFT) and the associated numerical calculation techniques are detailed in this chapter.

The third chapter is based on first-principles calculations of the iron and silicon isotopic properties of various silicate minerals relevant to the upper-mantle and oceanic crust. We present, in this chapter, the parameters that influence iron and silicon isotopes within Fe 2+ -bearing minerals. This study has been submitted for publication to Geochimica and Cosmochimica Acta (accepted with minor revisions). The authors are: Ségolène Rabin, Marc Blanchard, Carlos Pinilla, Franck Poitrasson and Michel Grégoire.

The chapter 4 is based on first-principles molecular dynamics calculations in order to determine Fe and Si isotopic properties of silicate melts. Several melts compositions were modelled, with composition in major elements close to natural melt, in order to assess the impact of alkaline, SiO 2 and FeO contents on the Fe and Si isotopic signature of the melts.

Firstly, iron and silicon isotopic composition of silicate melts as well as the parameters inducing fractionation are discussed. Secondly, equilibrium Fe and Si fractionation factor between various iron-bearing minerals and silicate melts are determined and the implications of partial melting and fractional crystallization on the isotopic signature of the natural igneous rocks are discussed. This study has not been submitted for publication yet. The authors are: Ségolène Rabin, Marc Blanchard, Carlos Pinilla, Franck Poitrasson and Michel Grégoire.

The chapter 5 is related to the study of iron isotopes compositions and redox conditions of three ocean island basalt (OIB) magmatic series that differ by their alkaline contents, from Kerguelen Archipelago. Ion-exchange chromatography associated to MC-ICP-MS measurements were performed on a total of 23 lavas. The methodology used to measure iron isotopes compositions of bulk rocks is described in this chapter. The processes involved in iron isotopes fractionation during magmatic differentiation is discussed using the Fe isotopes compositions of Kerguelen lavas as well as the equilibrium Fe fractionation factor between minerals and melt calculated in chapter 4. The possible upper-mantle heterogeneity beneath Kerguelen Archipelago from the scope of Fe isotopes and Fe 3+ /Fe tot content is also discussed. This study has not been submitted for publication yet.

Introduction générale (French version)

Le fer (Fer) et le silicium (Si) font partie des éléments les plus abondants du système solaire. Les progrès analytiques, en termes de chimie et d'analyse par spectrométrie de masse MC-ICP-MS, ont permis de mesurer les compositions isotopiques du fer et du silicium d'échantillons provenant de domaines de haute-température, avec des précisions < 0,05 % et 0,1 % , respectivement (e.g., Craddock and Dauphas, 2011b;Poitrasson and Freydier, 2005;[START_REF] Zambardi | Precise Determination of Silicon Isotopes in Silicate Rock Reference Materials by MC-ICP-MS[END_REF]. L'étude de la composition isotopique en fer des différents réservoirs terrestres a mis en avant un fractionnement significatif entre le manteau et la croûte : la croûte continentale ainsi que la croûte océanique ont une composition achondritique, enrichie en isotopes lourds par rapport au manteau(e.g., Foden et al., 2015;Gleeson et al., 2020;[START_REF] Nebel | Iron isotopic evidence for convective resurfacing of recycled arc-front mantle beneath back-arc basins[END_REF]Nebel et al., , 2018Nebel et al., , 2019;;Poitrasson and Freydier, 2005;Sossi et al., 2016). Le silicium, quant à lui, a une composition homogène au sein du manteau et de la croûte basaltique (e.g., Savage et al., 2010Savage et al., , 2011)). Cependant, des variations isotopiques (en Fe et Si) ont été mesurées dans les météorites et les planètes telluriques (e.g., [START_REF] Fitoussi | Si isotope systematics of meteorites and terrestrial peridotites: implications for Mg/Si fractionation in the solar nebula and for Si in the Earth's core[END_REF]Georg et al., 2007;Poitrasson et al., 2004Poitrasson et al., , 2005;;[START_REF] Pringle | Silicon isotopes in angrites and volatile loss in planetesimals[END_REF][START_REF] Savage | The silicon isotope composition of the upper continental crust[END_REF]. En conséquence, les systèmes isotopiques du fer et du silicium sont apparus comme des outils prometteurs pouvant aider à la compréhension de la formation de la Terre, et des autres planètes du système solaire. Un grand intérêt fut d'abord porté à la composition isotopique du noyau ainsi que du manteau terrestre (e.g., Georg et al., 2007;[START_REF] Liu | Iron isotopic fractionation between silicate mantle and metallic core at high pressure[END_REF]Savage et al., 2010;[START_REF] Shahar | An assessment of iron isotope fractionation during core formation[END_REF]Shahar et al., 2011[START_REF] Shahar | Sulfur-controlled iron isotope fractionation experiments of core formation in planetary bodies[END_REF]Williams and Bizimis, 2014;Williams et al., 2004;Ziegler et al., 2010), puis il est apparu essentiel de caractériser également la composition de la croûte terrestre. Une variabilité de l'ordre de 0,6 à 0,3 % est observée pour les isotopes du fer et du silicium au sein de la croûte. Les roches magmatiques les plus différenciées ont en effet, des signatures isotopiques plus lourdes que celle des basaltes (e.g., Foden et al., 2015;Gajos et al., 2016;Poitrasson and Freydier, 2005;Poitrasson and Zambardi, 2015;Savage et al., 2011;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012;Williams et al., 2018;Zambardi et al., 2014). Les processus générant ces signatures isotopiques plus lourdes au sein de la croûte terrestre sont encore débattus. Dans la littérature, différents processus ont été proposés afin d'expliquer ces signatures, comme la fusion partielle (e.g., Dauphas et al., 2014;Nebel et al., 2019;Sossi et al., 2016;Weyer and Ionov, 2007;Williams and Bizimis, 2014), la cristallisation fractionnée (e.g., Qin et al., 2016;Sossi et al., 2012), l'exsolution de fluides (e.g., Heimann et al., 2008;Poitrasson and Freydier, 2005) ou bien encore le fractionnement cinétique causé par la diffusion (e.g., Gajos et al., 2016;Lundstrom, 2009;Zambardi et al., 2014)

AY + BY <=> AY + BY (1.1)
Isotopic exchange between two phases results in the isotopic enrichment of one phase over the other. This phenomenon called isotope fractionation can occur out of equilibrium or at equilibrium. Non-equilibrium isotope fractionation is a kinetic fractionation related to the contrast of the diffusion rate, linked to the difference of masses, of the isotopologues.

Kinetic effets can therefore easily be described by classical mechanics whereas equilibrium fractionation requires a quantum description of the system (e.g., Blanchard et al., 2017). Equilibrium isotope fractionation is indeed related to the discrete variation of free energy of the systems. It is important to keep in mind that both kinetic and equilibrium fractionations are dependent of the mass however, mass-indenpendent fractionation also exists in nature (see e.g. Blanchard et al. (2017) for details on the theory related to mass-independent fractionation). In the following part, we will focus exclusively on mass-dependent equilibrium isotope fractionation as it represents the central part of this thesis.

Gibbs free energy and equilibrium constant

The Gibbs free energy per atom during an isotopic exchange at thermodynamic equilibrium can be expressed as :

∆G = -k B T lnK eq (1.2)
where k B is the Boltzmann constant, T the temperature and K eq is the equilibrium constant of the isotopic exchange reaction.

Equilibrium isotope exchange has no influence on the molar volume (Schauble, 2004).

The exchange can therefore be considered with a constant volume and the variation of free energy ∆G is equal to the Helmoltz free energy variation ∆F . The variation of free energy ∆F can be expressed in term of products and reactants :

∆F = F (AY ) + F (BY ) -F (AY ) -F (BY ) (1.
3)

The equilibrium constant K eq can be defined by the free energy variation associated to the isotopic exchange :

K eq = e (-∆F/k B T ) (1.4)
Equation 1.4 underlines the importance of temperature (T) on isotope fractionation : for a given ∆F , a high temperature will result in a small isotopic fractionation with a K eq close to 1.

Partition function

In statistical physics, the free energy F is correlated to the function of partition Q :

F = -k B T lnQ (1.5)
where Q can be defined by the general expression :

Q = n e (-En/k B T ) (1.6)
E n represents the energy of the considered state n.

As a consequence, the equilibrium constant K eq (1.4) can be expressed as a function of Q of the reactants and products:

K eq = Q products Q reactants = Q AY × Q BY Q AY × Q BY (1.7)

The fractionation factor α

The isotopic enrichment of one phase over the other is characterised by the fractionation factor α that is defined as the ratio of the classical (cl) and quantum (qm) equilibrium constant [START_REF] Urey | The thermodynamic properties of isotopic substances[END_REF] :

α AY -BY = K qm eq K cl eq
(1.8)

Based on equation 1.7, the fractionation factor α AY -BY can be expressed as :

α AY -BY = Q qm AY × Q qm BY Q qm AY × Q qm BY × Q cl AY × Q cl BY Q cl AY × Q cl BY (1.9)
However, the ratio of the classical partition function of species AY and AY' depends exclusively of the masses m and m of isotopes Y and Y' :

Q cl AY Q cl AY = ( m m ) 3/2 (1.10)
Consequently, the classical mechanics does not predict equilibrium isotope fractionation as the equilibrium constant K cl eq is equal to 1 :

K cl eq = Q cl AY × Q cl BY Q cl AY × Q cl BY = 1 (1.11)
Equilibrium isotope fractionation originates exclusively from quantum mechanical effects.

The factor β

From equations 1.9 and 1.11, it is possible to define the reduced partition function ratio, also called β-factor, for phase A and B :

β AY = Q qm AY Q qm AY (1.12) β BY = Q qm BY Q qm BY (1.13)
The β-factor is the central parameter for numerical modelling as the β-factor allows us to determine the enrichment in heavy isotopes for each phase separatly. Once the β-factor of phase A and B calculated, the fractionation factor α AY -BY can be determined :

α AY -BY = β AY β BY (1.14)
The isotope fractionation is relatively small resulting to α and β-factor close to 1. It is therefore convenient to express theses factors in terms of their logarithms (when x is close to 0):

ln(1 + x) ≈ x (1.15)
By convention, lnα and lnβ are expressed in permil (% ) : 10 3 lnα AY -BY and 10 3 lnβ AY .

Calculation of the β-factor

As underlined part 1.3.4, only quantum partition function Q qm have to be determined in order to assess the expression of the α and β-factor. Determination of the quantum partition function is based on the fact that the degree of liberty associated to the movements in a solid are vibrations. The most common methods for the calculation of isotope fractionation is to use the harmonic approximation for the interaction potential between two atoms. In this case, the interaction potential between the two masses of the oscillator has a quadratic shape. This harmonic approximation is only valid if the atoms are in their equilibrium positions. The vibrational energy of an harmonic oscillator is expressed as : 1.16) where is the reduced Plank constant, ω q,i is the vibrational frequency of the crystal (phonons) as a function of the q wavevector in the reciprocal space 1 and for the vibrational mode i.. In the case of a periodic system constituted of harmonic oscillators, the number 1 These terms (reciprocal space, Brillouin zone, wavevector) will be explicited in chapter 2

E n,q,i = n + 1 2 ω q,i (
of vibrational modes (i) is three times the number of atoms (3N) and the vibrational partition function Q qm vib is : (1.17) where N q corresponds to the number of q wavevectors used to sample the Brillouin Zone.

Q qm vib =   {q} 3N i=1 e (-ω q,i /2k B T ) 1 -e (-ω q,i /k B T )   1/Nq
When the calculation is performed in the center of the Brillouin Zone (Γ), the 3 first modes represent the translation modes that can be ignored as their frequencies are theoretically null.

Following the equations 1.10 and 1.17, the equation of the β-factor is :

β AX = m m 3/2 ×   {q} 3N i=1 e (-ω q,i /2k B T ) 1 -e (-ω q,i /k B T ) 1 -e (-ω q,i /k B T ) e (-ω q,i /2k B T )   1/Nq (1.18)
The theorem of Teller-Redlich (Bigeleisen and Mayer, 1947) states that at high temperature the quantum effects become negligible and therefore the fractionation factor tends to 0. The high-temperature rules [START_REF] Kieffer | Thermodynamics and lattice vibrations of minerals: 5. Applications to phase equilibria, isotopic fractionation, and high-pressure thermodynamic properties[END_REF] for crystal systems is :

m m 3/2 ×   {q} 3N i=1 ω q,i ω q,i   1/Nq = 1 (1.19)
Using this rule, it is possible to write the β-factor equation at high-temperature (Bigeleisen and Mayer, 1947):

β AX =   {q} 3N i=1 ω q,i ω q,i e (-ω q,i /2k B T ) 1 -e (-ω q,i /k B T ) 1 -e (-ω q,i /k B T ) e (-ω q,i /2k B T )   1/Nq (1.20)
For conveniency, the β -f actor is expressed as 10 3 lnβ 57 Fe (in % ).

In the case of melts modelling, we also used a second method to calculate the βfactor. This method, less conventional, is based on the kinetic energy of the atoms and is developed in chapter 4.

Determination of experimental isotope fractionation factor

The isotope composition of a phase A can be determined experimentally by measuring the isotopic ratio using a mass spectrometer. The isotopic ratio R(Y /Y ) AY is the ratio of the concentration N of the two isotopes Y and Y' :

R(Y /Y ) AY = N AY Y N AY Y (1.21)
In order to be able to compare the measured values internationally, the isotopic ratio of a certified standard is also measured and the deviation of the phase A from the standard values is calculated. The isotopic composition of phase A is expressed as :

δ AY Y = R(Y /Y ) AY R(Y /Y ) standard -1 × 1000 (1.22)
where δ is the deviation expressed in % . The certified international standard for iron and silicon isotopes systems are IRMM-014 and NBS-28 (NIST RM-8546) respectively.

The fractionation factor between two phases A and B, called ∆ AY -BY , is obtain by subtracting the δ-values of each phase :

∆ AY -BY = δ AY Y -δ BY Y (1.23)
The fractionation factor α defined in equation 1.14 can also be expressed in terms of δ values :

α AY -BY = 1 + δ AY Y 1 + δ BY Y (1.24)
As the measured δ-values/1000 are close to 0, it is possible to approximate a relationship between ∆ AY -BY and α AY -BY : 1.25) This relationship (1.24) allows us to compare easily theoretical and experimental isotope fractionation results.

α AY -BY -1 ≈ δ AY Y -δ BY Y = ∆ AY -BY (
It is important to notice that in the case of iron isotopes system, fractionation factor can also be determined by other experimental methods such as Nuclear Resonant Inelastic X-ray Scattering (NRIXS; (e.g., Dauphas et al., 2014)) and Mössbauer spectroscopy (e.g., Polyakov and Mineev, 2000). Both methods use similar notation as in first-principles calculations (β-factor, α fractionation factor).

Iron isotopes systematics

Iron geochemistry

Iron is of particular interest for geochemist due to the fact that it is a ubiquitous element with complex chemical behaviour. Iron is the most abundant redox-active metal in the entire Solar System and the fourth most abundant in the Earth's crust. Iron is moderately volatile and a refractory element. Following the Goldschmidt's classification, iron is a siderophile element. However, at high sulphide level iron is chalcophile and at high oxygen levels, iron becomes lithophile. The aptitude of iron to bind with both sulfur and oxygen explains the wide variety of minerals (silicates, oxides/hydroxydes, carbonates, sulphides) that contain iron on Earth. The most interesting characteristic of iron is however its sensitivity to redox changes. Iron has three oxidation states, metallic iron (Fe 0 ), ferrous iron (Fe 2+ ) and ferric iron (Fe 3+ ). The electronic configuration of F e 0 is 3d 6 4s 2 whereas under oxidizing conditions Fe 2+ and Fe 3+ lose the two electrons of the outer 4s

shell and display electronic configuration of 3d 6 and 3d 5 , respectively. Consequently, iron contributes significantly to the electron transfer in numerous reactions in Earth.

In magmatic systems, iron is a major element in the magma and is stoichiometrically incorporated into mantle and crust minerals (e.g. olivines, pyroxenes, oxides such as magnetite). Iron abundance decreases with magmatic differentiation (increasing SiO 2 content; [START_REF] Bowen | The evolution of the igneous rocks[END_REF]), but remains present in amounts that have significant effects on melts (Mysen and Richet, 2019b) and mineral compositions. Iron, and more precisely the Fe 3+ /Fe tot ratio, is also the most common proxy of oxygen fugacity (f O 2 ) of the upper-mantle and basalts. At high temperature, the changes in the relative abundances of ferrous and ferric iron can be simply defined by the reaction :

2 Fe 2+ + 1/2 O 2 -→ 2Fe 3+ + O 2-
where O 2-represents the structural O in oxides and silicates.

The f O 2 is commonly calculated from O 2-buffered mineral reactions. The most common oxygen fugacity buffers are, in increasing order of oxidation, the Iron-Wurstite (IW), the Fayalite-Magnetite-Quartz (FMQ) and the Magnetite-Hematite (MH) buffer.

Iron stable isotopes

Iron has four natural stable isotopes 54 Fe, 56 Fe, 57 Fe, 58 Fe with respective mean abundances of 5. 80, 91.72, 2.20 and 0.28 % and atomic masses of 53.9396105, 55.9349375, 56.9353940 and 57.9332756 a.m.u (atomic mass unit). In the literature, iron isotopic compositions are given alternatively as δ 56 Fe or δ 57 Fe. Measurement of 57 Fe is more chal-lenging because of its low abundance compared to 56 Fe. However, the mass difference between 57 Fe and 54 Fe is greater than between 56 Fe and 54 Fe, as a consequence a larger range of fractionation (by ≈ 1.5 times) can be observed (Schauble, 2004).

Measurements of Fe isotopes There are two major challenges for Fe isotopes accurate analysis using Multi-Collector Inductively Coupled Plasma Mass Spectrometry (MC-ICP-MS) : 1) Resolving Fe isotopes from the different elemental and molecular isobaric interferences (Table 1.1), 2) correcting for instrumental bias (matrix effects) in order to preserve the natural mass-dependent isotopic fractionation. Instrumental bias can be corrected using Ni isotopes spike associated to standard bracketing methods (e.g., Weyer et al., 2005) or daily regression method between ln 57 Fe/ 54 Fe and ln 61 Fe/ 60 Fe (Poitrasson and Freydier, 2005) assuming that Ni and Fe behave similarly to the matrix-induced massbias fluctuation. However, the use of Ni as a mass-bias monitor during analytical session is only valid if the 2 interference-free Ni isotopes can be collected simultaneously in the collector array [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]. Details of the mass-bias correction can be found in Poitrasson and Freydier (2005). Furthermore, ion-exchange chromatography is necessary before the introduction in the MC-ICP-MS in order to remove the isobars (i.e. [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]. The numbers in parentheses are the resolving power required for mass analysis.

The resolving power can be calculated by dividing the mass difference between the isobar and the Fe isotopes of interest ∆m and the mass of the Fe isotopes of interest (∆m/m). The majority of mass spectrometers are not able to resolve elemental isobars, so for Fe, the 54 Cr and 58 N i are monitored using 53 Cr and 60 N i respectively.

Parameters controlling iron isotopes fractionation

As underlined part 1.4.1, iron is a complex element and the parameters influencing iron isotope fractionation are many. The main parameter influencing Fe isotope fractionation, as for all elements, is temperature (Part 1.3.2). Other parameters controlling iron isotopes fractionation were investigated by both theoretical (e.g., Blanchard et al., 2009;[START_REF] Rustad | Iron isotope fractionation in the Earth's lower mantle[END_REF][START_REF] Yang | Iron isotopic fractionation in mineral phases from Earth's lower mantle: Did terrestrial magma ocean crystallization fractionate iron isotopes?[END_REF]) and experimental (e.g., Dauphas et al., 2014;Roskosz et al., 2015;Shahar et al., 2008;Sossi and O'Neill, 2017) studies. By now, four bonding-characteristics emerged as major parameters driving the iron isotopic signature of a phase of interest :

1. The nature of the chemical bond. A material with covalent-like bonds tends to concentrate more heavy Fe isotopes than material where iron forms ionic-like bonds.

2. The bond length. A material with short Fe-O (or S) bonds length, considered also as a strong bonds, concentrates heavy Fe iron isotopes, on the contrary of long Fe-O bonds length, that are considered as weaker bonds. A linear relationship between the 10 3 lnβ 57 Fe and the Fe-O bond length in silicate minerals is indeed observed in the literature (e.g., Sossi and O'Neill, 2017).

3. The coordination number. A material with four-fold coordinated iron will display a heavier iron isotopes signature than a material containing exclusively six-fold coordinated iron. This relationship is directly related to the stiffness of the Fe-O (or S) bonds: Fe that has a low coordination number in a material tends to have shorter and stronger bonds than Fe that has a high coordination number.

4. The valence. Ferric iron, displaying stronger bonds, tends to favour heavy isotopes ( 57 Fe) compared to ferrous iron. Minerals containing ferric iron, such as magnetite, display higher 10 3 lnβ 57 Fe than minerals containing mainly ferrous iron such as olivine. In addition, the material of interest will display heavier isotopic composition under high-redox conditions than under low redox conditions. Dauphas et al. (2014) demonstrated indeed this relationship for silicates glasses, with melts quenched under oxidising condition (f O 2 buffer : QFM +2.5) displaying 10 3 lnβ 57 Fe higher than the ones quenched under more reduced conditions.

Applications As iron is an ubiquitous element, the analysis of iron isotopes were carried on various major geological reservoirs, and display major variations especially at low temperature with ≈ 5 % fractionation range for Banded Iron Formations (BIFs) and cherts (Fig. 1.1). Iron isotopes are indeed a powerful mean for tracing processes associated with mass transfer and/or redox change. Iron isotope variations are studied in multiple environments for a better understanding of (i) the mechanical and chemical weathering of the Earth's surface (e.g., [START_REF] Bergquist | Iron isotopes in the Amazon River system: Weathering and transport signatures[END_REF]Fantle and DePaolo, 2004), (ii) the parameters influencing Fe mobility in soils (e.g., [START_REF] Fantle | Iron isotopic fractionation during continental weathering[END_REF], (iii) the dissolved and suspended Fe fluxes in rivers (e.g., [START_REF] Chen | Iron isotopes in the Seine River (France): Natural versus anthropogenic sources[END_REF]Poitrasson et al., 2014), (iv) the (bio)chemical cycle of iron (e.g., [START_REF] Beard | Iron Isotope Biosignatures[END_REF]Radic et al., 2011), (v) the Fe budget of the ocean (e.g., [START_REF] Beard | Iron isotope constraints on Fe cycling and mass balance in oxygenated Earth oceans[END_REF][START_REF] Conway | The cycling of iron, zinc and cadmium in the North East Pacific Ocean -Insights from stable isotopes[END_REF][START_REF] Zhu | Secular Variation of Iron Isotopes in North Atlantic Deep Water[END_REF] and rivers (e.g., [START_REF] Bergquist | Iron isotopes in the Amazon River system: Weathering and transport signatures[END_REF]Poitrasson et al., 2014) 2004;[START_REF] Frost | Preservation of Fe isotope heterogeneities during diagenesis and metamorphism of banded iron formation[END_REF][START_REF] Heimann | O isotope compositions of banded iron formation carbonates demonstrate a major role for dissimilatory iron reduction in ˜2.5Ga marine environments[END_REF]Planavsky et al., 2009;Steinhoefel et al., 2009;Tsikos et al., 2010) and (vii) ancient climate changes (e.g., Craddock and Dauphas, 2011a). Iron isotopes fractionation were also investigated at high temperature in order to better constrain processes such as : (i) the formation of planets and planetary cores (e.g., [START_REF] Hezel | A nebula setting as the origin for bulk chondrule Fe isotope variations in CV chondrites[END_REF][START_REF] Mullane | Nebular and asteroidal modification of the iron isotope composition of chondritic components[END_REF][START_REF] Schoenberg | Modes of planetary-scale Fe isotope fractionation[END_REF], (ii) formation of differentiated crust (e.g., Foden et al., 2015;Poitrasson and Freydier, 2005;Sossi et al., 2012;Telus et al., 2012), (iii) partial melting (e.g., Gleeson et al., 2020;Nebel et al., 2019;Sossi et al., 2016;Weyer and Ionov, 2007;Williams and Bizimis, 2014) (iv) metasomatism of the Earth mantle (e.g., [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF]Sun et al., 2020;[START_REF] Zhao | Iron isotope variations in spinel peridotite xenoliths from North China Craton: Implications for mantle metasomatism[END_REF], (v) hydrothermal alteration of basalts (e.g., [START_REF] Bennett | Iron isotope fractionation in a buoyant hydrothermal plume, 5S Mid-Atlantic Ridge[END_REF][START_REF] Li | Fe Isotopic Compositions of Modern Seafloor Hydrothermal Systems and Their Influence Factors[END_REF]Rouxel et al., 2003[START_REF] Rouxel | Subsurface processes at the lucky strike hydrothermal field, Mid-Atlantic ridge: evidence from sulfur, selenium, and iron isotopes 1 1Associate editor[END_REF]) and (vi) magmatic ores formations (e.g., [START_REF] Graham | Tracing Cu and Fe from source to porphyry: in situ determination of Cu and Fe isotope ratios in sulfides from the Grasberg Cu-Au deposit[END_REF][START_REF] Levasseur | The global variation in the iron isotope composition of marine hydrogenetic ferromanganese deposits: implications for seawater chemistry?[END_REF][START_REF] Markl | Iron isotope fractionation during hydrothermal ore deposition and alteration[END_REF].

Iron isotopes fractionation at high temperature

The improvement of iron isotopes analysis, in term of accuracy and precision, through time (see [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF] for details) promoted the use of iron isotopes in hightemperature field. Iron isotope fractionation at high temperature is indeed small, in the order of 1 % or less, but significant (Fig 1 .2). This section will focus on Fe isotopes fractionation occurring in the Earth's mantle and crust and the processes associated. The effect of metasomatism on the iron isotopes signatures of mantle rocks, the influence of partial melting on peridotites and basalts Fe isotopes signatures as well as the evolution of Fe isotopes compositions of the lavas during magmatic differentiation will therefore be discussed as it is at the center of this phD thesis.

The Earth mantle and metasomatism

Investigation of iron isotopes compositions of the Earth mantle was carried on a variety of mantle rocks including lherzolites, harzurgites, dunites, wehrlites, pyroxenites and various metasomatised peridotites (e.g., [START_REF] Craddock | Abyssal peridotites reveal the nearchondritic Fe isotopic composition of the Earth[END_REF][START_REF] Huang | Iron and magnesium isotopic compositions of peridotite xenoliths from Eastern China[END_REF]Macris et al., 2015;[START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF]Weyer and Ionov, 2007;Williams and Bizimis, 2014;Williams et al., 2005;[START_REF] Zhao | Iron isotope variations in spinel peridotite xenoliths from North China Craton: Implications for mantle metasomatism[END_REF][START_REF] Zhao | Iron isotope evidence for multistage melt-peridotite interactions in the lithospheric mantle of eastern China[END_REF][START_REF] Zhao | Effects of melt percolation on iron isotopic variation in peridotites from Yangyuan, North China Craton[END_REF]. Mantle rocks display the largest δ 57 Fe range at high temperature with values varying from -0.82 to 0.22 % (Fig. 1.2). However, most of the peridotites fall in a more limited δ 57 Fe range between -0.16 and 0.2% [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF], indistinguishable from that of the chondrites -0.015 ± 0.009 2SE (n = 146; [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]). Whereas most of the mantle rocks are indistinguishable in their δ 57 Fe signature (Fig. 1.2), wehrlites display heavier iron isotopes composition of about 0.156 ± 0.038 2SE (Weyer and Ionov, 2007;[START_REF] Zhao | Iron isotope evidence for multistage melt-peridotite interactions in the lithospheric mantle of eastern China[END_REF]. However, the studied wehrlites were affected by melt percolation resulting in the generation of secondary clinopyroxene [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]. Pyroxenites tend also to be enriched in heavy iron isotopes (Williams and Bizimis, 2014) however there is a great variability of pyroxenite δ 57 Fe in function of the localities they come from (e.g., [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF]Williams and Bizimis, 2014).

The most negative δ 57 Fe in the mantle were observed in several lherzolites and harzburgites that suffered from intermediate to high level of metasomatism [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF][START_REF] Zhao | Iron isotope variations in spinel peridotite xenoliths from North China Craton: Implications for mantle metasomatism[END_REF]. Mantle metasomatism can simply be defined as "compositional changes in the mantle wall-rock due to interaction with mantle fluids" (O'Reilly and Griffin, 2013). Mantle metasomatism is expected indeed to have strong effects on both chemical and isotopic composition of the mantle creating disequilibrium within the peridotites (Dauphas et al., 2017). This disequilibrium results in chemical reactions or chemical/isotopic diffusion of distinct elements between the melt and the mantle minerals. Isotopic disequilibrium between minerals, such as Fe diffusion, can be preserved if the xenoliths are brought to the surface at short timescales after the metasomatism event, limiting the diffusive inter-mineral re-equilibration. The metasomatic overprinting of the initial Fe isotopes compositions of peridotites is variable and highly depends on the mantle source that generated the melt, the chemical and isotopic composition of the melt and the physico-chemical conditions of the partial melting that drive the mineral reaction [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]. Recent in-situ analysis on olivine using fs-LA-MC-ICP-MS demon- strated that chemical Fe-Mg diffusion results in significant iron isotope fractionation > 1 % in δ 57 Fe (e.g., Collinet et al., 2017;Oeser et al., 2015Oeser et al., , 2018)). The light isotopes diffuse faster than the heavy isotopes, consequently it results in an enrichment in light isotopes in the direction of the diffusion. In the case of melt (or a fluid) percolation in the mantle, the mantle rocks would be enriched in light isotopes. Therefore, the negative δ 57 Fe measured in the metasomatised peridotites may be interpreted as being caused by melt (or a fluid) percolation through the mantle generating Fe isotopes diffusion-driven fractionation (e.g., [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF][START_REF] Zhao | Iron isotope variations in spinel peridotite xenoliths from North China Craton: Implications for mantle metasomatism[END_REF].

Oceanic crust and partial melting

Iron isotopes compositions of basaltic rocks were investigated in various studies (e.g., Gleeson et al., 2020;Konter et al., 2016;[START_REF] Nebel | Iron isotopic evidence for convective resurfacing of recycled arc-front mantle beneath back-arc basins[END_REF]Nebel et al., , 2018Nebel et al., , 2019;;Sossi et al., 2016;Sun et al., 2020;Teng et al., 2013). The majority of these studies focus on the composition of the oceanic crust by analysing mid-ocean ridge basalt (MORB) and ocean island basalt (OIB). Most basaltic rocks, independently of their geodynamical settings and origin, display heavier Fe isotopes composition than chondrites and mantle compositions (Fig. 1.2). Komatiites, that mainly erupted during Archean times and originated from high degree of melting, are the only basaltic rocks displaying δ 57 Fe of 0.0015 ± 0.014 % similar to the mantle value [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]. The similar iron isotopic composition of Komatiites and mantle rocks seems to indicate that Komatiites sampled the underlying mantle at the time of the formation. The iron isotope composition of MORB is generally homogeneous and the δ 57 Fe is 0.100 ± 0.02 % . Only enriched-MORB (E-MORB) type tends to display Fe heavier isotopes composition (Fig. 1.2 and Sossi et al. (2016)). The Fe isotope signatures of OIB tend to be slightly heavier than MORB (Fig. 1.2) and are more heterogenous even among samples from the same localities. The highest δ 57 Fe values of basalt have been measured in Samoan OIB with a values of 0.45 ± 0.05% (Konter et al., 2016) and in Galapagos Archipelago MORB with a value of 0.347±0.08% (Gleeson et al., 2020). Two processes were inferred to be responsible for the general heavier Fe isotopes signature of basalts compared to the peridotites: 1) the variable degree of differentiation of the melt (Sossi et al., 2016) and 2) the degree of mantle partial melting (e.g., Dauphas et al., 2014;Foden et al., 2015;Gleeson et al., 2020;Nebel et al., 2019;Sossi et al., 2016;Weyer and Ionov, 2007;Williams and Bizimis, 2014) associated or not to isotopic heterogeneity of the mantle.

The variable degree of differentiation of the melt was investigated by Sossi et al. (2016), as most basalts do not reflect the composition of the initial parental magma due to early olivine crystallisation. Consequently, basalts δ 57 Fe signature cannot be interpreted as a true signature of the mantle source as the heaviest signature observed can be caused by incorporation of light Fe isotopes in olivines. For this reason, Sossi et al. (2016) developed a correction for early olivine fractionation for basalt displaying MgO concentration < 8

Wt% in order to recover for the initial Fe isotopes signatures of the parental melt. By applying this correction to MORB (Sossi et al., 2016), and OIB (Gleeson et al., 2020;Nebel et al., 2019), some of the heaviest δ 57 Fe basalt signature remained and partial melting processes were consequently explored in order to explain the heavy Fe isotope composition of the parental magma.

The impact of the degree of mantle partial melting on the variable δ 57 Fe of basalts was investigated in various studies (e.g., Dauphas et al., 2014;Foden et al., 2015;Gleeson et al., 2020;Nebel et al., 2019;Sossi et al., 2016;Weyer and Ionov, 2007;Williams and Bizimis, 2014). Partial melting appears indeed as a favourable mean of fractionating significantly iron isotopes because of the difference of incompatibility between Fe 2+ and Fe 3+ . Ferric iron is a moderately incompatible element with a K ≈ 0.4 whereas ferrous iron is compatible with a K ≈ 1 (Davis et al., 2013;Mallmann and O'Neill, 2009). Therefore, Fe 3+ will partition in the melt at relatively low degree of partial melting. Significant iron isotopes fractionation is consequently expected as ferric iron form stronger bonds and concentrates 57 Fe (Part 1.4.2). If most studies agree on the fact that partial melting has an impact on the Fe isotope signature of the basalts, its significance is still debated. It is still unclear whether partial melting can only explain partly the Fe isotope signature of the MORB (e.g., Dauphas et al., 2014) or if one (e.g., Sossi et al., 2016) or multiple stages of partial melting (e.g., Weyer and Ionov, 2007) are necessary to recreate the entire range of MORB signatures. The major difference between the different models elaborated in the literature are the fractionation factor between Fe 3+ and Fe 2+ -minerals and melt. The iron isotopes composition of olivine and pyroxene are considered indeed as indistinguishable by some study (e.g., Beard and Johnson, 2004;Dauphas et al., 2014) whereas some display significant fractionation between them (e.g., Sossi et al., 2016;Williams and Bizimis, 2014): this has its importance as during partial melting clinopyroxene represents the major mineral phase that melts [START_REF] Hirschmann | Calculation of Peridotite Partial Melting from Thermodynamic Models of Minerals and Melts. II. Isobaric Variations in Melts near the Solidus and owing to Variable Source Composition[END_REF]. Moreover, the iron isotopes compositions of melts are also poorly constrained as the only available values are from synthetic glasses proxies analysed by NRIXS by Dauphas et al. (2014) and Prissel et al. (2018) for terrestrial and lunar glasses, respectively.

Concerning the heavy Fe isotopes signatures in both MORB and OIB, recent partial melting Monte-Carlo modelling from Gleeson et al. (2020) demonstrated that redox changes cannot be responsible for such iron isotopes signature in basalts and implies that an heavy-isotopes component (i.e. pyroxenite) is necessary in the mantle source material.

The latter hypothesis supports the fact that Fe could be a viable proxy for the determination of mantle heterogeneities as proposed by Williams and Bizimis (2014), Konter et al. (2016) and Nebel et al. (2019). The variability of δ 57 Fe among the different OIB also supports the hypothesis that the upper-mantle has a heterogenous Fe isotope composition. Ocean island basalt may indeed reflect more the variable Fe isotopes compositions of the different mantle regions than MORB as they sample deeper mantle sources and the plume regions are typically more heterogeneous in their chemical and isotopic composition [START_REF] Hofmann | Sampling Mantle Heterogeneity through Oceanic Basalts: Isotopes and Trace Elements[END_REF].

Magmatic differentiation

While the Fe isotopes compositions of basalts from different geodynamical settings vary moderately from Earth's mantle composition, the Fe isotopes compositions of highly differentiated igneous rocks are considerably heavier, with δ 57 Fe up to ≈ 0.9% (Fig 1 .3).

The figure (1.3) compiles the δ 57 Fe composition of various igneous and volcanic magmatic series from distinct tectonic settings and localities. A general trend emerges from figure 1.3 with a general increase in δ 57 Fe with increasing degree of differentiation (increasing SiO 2 content). Above 60 Wt% of SiO 2 , the mean δ 57 Fe for both plutonic and volcanic rocks is ≈ 0.3% (n=360; [START_REF] Johnson | Iron Geochemistry: An Isotopic Perspective[END_REF]). In the case of volcanic series such as Kilauea Iki (Teng et al., 2008) and Red Hill (Sossi et al., 2012), the increase in δ 57 Fe is observable at lower SiO 2 content. The processes involved in the enrichment in heavy iron isotopes of the most evolved igneous rocks are still debated. Different processes were proposed such as :

1. Fluid exsolution. The heavy δ 57 Fe of granites and rhyolites measured by Poitrasson and Freydier (2005) and Heimann et al. (2008) were assessed to be due to exsolution of a fluid from highly evolved melts during late magmatic stages. An aqueous fluid rich in chlorine can indeed be exsolved from the evolved magmas resulting in the mobilization of elements such as Zn, Cu, Mo or Fe. These fluids enriched in light Fe isotopes would drive the Fe isotopes compositions of the residual magma to heavier values. However, subsequent studies displayed arguments against the potential effect of fluid exsolution on the δ 57 Fe of evolved rocks : i) The A-type granite, that displays heavy Fe isotopes compositions, are among the most anhydrous (Sossi et al., 2012;Telus et al., 2012), ii) The study of Zn and Li isotopes, both elements being very mobile in chlorine-rich fluids, in evolved magmatic rocks display no significant isotopes fractionation (Schuessler et al., 2009;Telus et al., 2012), except for some pegmatites (Telus et al., 2012). Therefore, fluid exsolution is not expected to play a major role on Fe isotopes fractionation during magmatic differentiation.

2. Diffusion-driven processes. Lundstrom (2009), Zambardi et al. (2014) and Gajos et al. (2016) proposed thermal diffusion as the one of the processes driving the Fe isotopes composition to heavier value in the evolved igneous rocks based on the fact that thermal gradient could control Fe isotope fractionation in magmas [START_REF] Huang | Chemical and isotopic fractionation of wet andesite in a temperature gradient: Experiments and models suggesting a new mechanism of magma differentiation[END_REF]Richter et al., 2003). However, Telus et al. (2012) measured Zn isotopes as well as other isotopic systems supposed to behave similarly as Fe in case of thermal diffusion, and no correlations were observed. Another scenario proposed by [START_REF] Zhao | Effects of melt percolation on iron isotopic variation in peridotites from Yangyuan, North China Craton[END_REF] to explain the A-type granites δ 57 Fe, is based on diffusion between a silica-rich and a Fe-rich immiscible melt. The potential effect of diffusion between minerals and melt were also proposed as great δ 57 Fe variations were shown in olivines (e.g., Collinet et al., 2017;Oeser et al., 2015Oeser et al., , 2018;;Sio et al., 2013;Teng et al., 2008) and illmenite (Tian et al., 2020). [START_REF] Wu | Diffusion-driven magnesium and iron isotope fractionation at a gabbro-granite boundary[END_REF] also evidenced significant iron isotopes fractionation in an adjacent gabbro and granite profile in Dabie Orogen,

showing that large isotopic fractionation can be produced in igneous rocks by kinetic diffusion at least on a decimeter scale. However, if Fe-Mg inter-diffusion plays a role on the isotopic signature of some magmatic series studied (e.g., Teng et al., 2008;[START_REF] Wu | Diffusion-driven magnesium and iron isotope fractionation at a gabbro-granite boundary[END_REF], it is unlikely to play a role on large granitic intrusion.

3. Partial melting. The effect of partial melting of the lower crust was also investigated in order to understand the heavy Fe isotopes compositions of evolved igneous rocks (e.g., Foden et al., 2015;Xia et al., 2017;[START_REF] Xu | Iron isotope fractionation during crustal anatexis: Constraints from migmatites from the Dabie orogen, Central China[END_REF]. Partial melting of the lower crust appears indeed as a viable mechanism to re-create the high δ 57 Fe signatures of the S-type granites generated under reduced conditions (Foden et al., 2015), of the migmatites from Dabie Orogen [START_REF] Xu | Iron isotope fractionation during crustal anatexis: Constraints from migmatites from the Dabie orogen, Central China[END_REF] and of Hair rhyolite (Xia et al., 2017).

4. Fractional crystallization. Fractional crystallization (FC) is considered to be the main driver behind the chemical differentiation of magmatic rocks. Many studies invoked that fractionation crystallization could also be responsible of isotopic fractionation during magmatic differentiation (e.g., Schuessler et al., 2009;Sossi et al., 2012;Williams et al., 2018). Removal of isotopically light Fe 2+ -rich minerals (olivine, orthopyroxene, clinopyroxene, illmenite, ulvospinel) from the magma during FC should indeed drive the melt to heavier isotopic compositions. Some studies (Shahar et al., 2008;Zambardi et al., 2014) et al. (2015), going a step further by predicting variations in δ 57 Fe in lavas with lower SiO 2 content, as observed in different magmatic suites (e.g., Sossi et al., 2012;Teng et al., 2008). As for partial melting modelling (Part 1.4.3.2), the choice of equilibrium fractionation factors between mineral and melt used in FC modelling is essential to precisely determine if FC could explain entirely or partly the iron isotopes evolution during magmatic differentiation.

The evolution of the δ 57 Fe during FC were also inferred to be coupled with f O 2 evolution of the melt (Sossi et al., 2012). The two magmatic suites of Red Hill volcanoes studied by Sossi et al. (2012), clearly demonstrated two types of δ 57 Fe variations for a system open and closed in oxygen.

In a system open to oxygen, the δ 57 Fe as well as the concentration of Fe 3+ increase in the last stage of differentiation due to the removal of Fe 2+ generated by olivines and pyroxenes cristallisation in the early stage of magmatic differentiation. In the case of a system closed to oxygen, the concentration of Fe 3+ is finite. Therefore, the δ 57 Fe will increase, as in an open system to oxygen, until the crystallisation of Fe 3+ -bearing minerals (e.g. magnetite)

that will trigger the δ 57 Fe to lower values in the most evolved lavas (Sossi et al., 2012).

Figure 1.3: Compilation of the iron isotopes compositions of igneous crustal rocks versus SiO 2 (in wt.%) as an index of differentiation (Du et al., 2017;Foden et al., 2015;He et al., 2017;Heimann et al., 2008;Li et al., 2020;Poitrasson and Freydier, 2005;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012;Williams et al., 2018;Xia et al., 2017;Zambardi et al., 2014).

Mineral-Mineral and Mineral-Melt fractionation factors

Our understanding of iron isotope fractionation in high-temperature geological processes is limited by our knowledge of equilibrium isotopic fractionation factors between minerals and melt as underlined part 1.4.3.2 and 1.4.3.3 for partial melting and fractional crystallization. Iron isotopes fractionation factors are derived from experimental works, from first-principles calculations (Chapter 2) or from using spectroscopic measurements such as Mössbauer and NRIXS. Inter-mineral iron isotopes fractionation among mantle and oceanic crust minerals were investigated in various studies (e.g., [START_REF] Dauphas | Iron isotope, major and trace element characterization of early Archean supracrustal rocks from SW Greenland: Protolith identification and metamorphic overprint[END_REF]Dauphas et al., , 2014;;Polyakov and Mineev, 2000;Roskosz et al., 2015;Shahar et al., 2008;Sossi and O'Neill, 2017;Williams and Bizimis, 2014;[START_REF] Yang | Iron isotopic fractionation in mineral phases from Earth's lower mantle: Did terrestrial magma ocean crystallization fractionate iron isotopes?[END_REF][START_REF] Ye | Calibrating equilibrium Fe isotope fractionation factors between magnetite, garnet, amphibole, and biotite[END_REF], especially Fe isotopic fractionation between Fe 2+ and Fe 3+ minerals. The Fe β-factors of minerals relevant to the mantle and oceanic crust formation available in the literature are displayed in Table 6.2 and 1.3. The first major study on determination of silicate minerals and oxides 10 3 lnβ 57 Fe was carried on by Polyakov and Mineev (2000) using Mössbauer

Spectroscopy. Since then, many of the silicates β-factor (e.g. diopside and aegirine) were not reassessed. Moreover, first-principles calculations were only performed on Fe-bearing sulfides and oxides (Blanchard et al., 2009) or high-pressure Fe-bearing silicates (i.e Ferropericlase, Ferroperovskite, Bridgmanite ; [START_REF] Rustad | Iron isotope fractionation in the Earth's lower mantle[END_REF], [START_REF] Yang | Iron isotopic fractionation in mineral phases from Earth's lower mantle: Did terrestrial magma ocean crystallization fractionate iron isotopes?[END_REF] preventing any comparison. If the β-factors calculated by Polyakov and Mineev (2000) were revised later by their authors in the case of magnetite and pyrite β-factor (Blanchard et al., 2012;Polyakov et al., 2007), it remains the first study underlying the significance of fractionation between Fe 2+ and Fe 3+ minerals. This latter observation was confirmed by other studies using NRIXS (Dauphas et al., 2014) and experimental measurements (Roskosz et al., 2015;Shahar et al., 2008;Sossi and O'Neill, 2017), with fractionation on the order of 0.2-0.4% . However, fractionation between the Fe 2+ -bearing minerals is still debated in the literature. Some studies considered Fe isotope fractionation between olivine and pyroxene as negligable (e.g., Beard and Johnson, 2004;Dauphas et al., 2014) whereas Sossi and O'Neill (2017) measured significant fractionation between Fe 2+ minerals in accordance with measured values on natural samples (e.g., Williams and Bizimis, 2014) and Mössbauer results of Polyakov and Mineev (2000). 1.5 Silicon isotopes systematics

Silicon geochemistry

Silicon is one of the most abundant element on Earth. This is a lithophile element, and therefore has a great affinity for oxygen. Silicon is monovalent and has an oxydation state of +4 in Earth crust and upper mantle. Silicon is also of particular interest in the study of Earth's mantle and crust as they are composed in majority (75%; [START_REF] Ding | Silicon Isotope Geochemistry[END_REF]) by silicate minerals. The building block of these minerals is the silicon tetrahedron, that is a combination of four oxygen atoms bonded to one silicon atom (SiO 4 ). In silicate minerals, the silica tetrahedron are arranged and related to each other in a variety of ways from single units (i.e. nesosilicates), to more complex frameworks (Fig. 1.4) depending on the polymerisation of (SiO 4 ) . In silicate melts and glasses at low pressure, silicon is also four-fold coordinated and is considered as the major network-former (Mysen and Richet (2019a) and reference therein).

Silicon stable isotopes

Silicon has three natural stable isotopes 28 Si, 29 Si, 30 Si with respective mean abundances of 92.22, 4.69 and 3.09 % and atomic masses of 27.97692649, 28.97649468 and 29.97377018 a.m.u (atomic mass unit).

Measurements If silicon isotopes measurements started in the middle of 20 th century using gass-mass spectrometer, the methods commonly used nowadays are performed on MC-ICP-MS. The correction of mass-bias fractionation is performed by Mg isotopes measurements on dynamical modes (e.g., Savage et al., 2011;[START_REF] Zambardi | Precise Determination of Silicon Isotopes in Silicate Rock Reference Materials by MC-ICP-MS[END_REF].

Isobaric interference are observed for the three masses of silicon during MC-ICP-MS analysis: 14 N 2 on 28 Si, 14 N 1 2 H on 29 Si and 14 N 16 O on 29 Si. The method of measurements developed by [START_REF] Zambardi | Precise Determination of Silicon Isotopes in Silicate Rock Reference Materials by MC-ICP-MS[END_REF] enable silicates material analyses with a precision < 0.1% , allowing the observation of Si isotopes fractionation at high-temperature. As for iron, ion-exchange chromatography is necessary before the introduction of the samples in the MC-ICP-MS (see [START_REF] Zambardi | Precise Determination of Silicon Isotopes in Silicate Rock Reference Materials by MC-ICP-MS[END_REF] for details).

Parameter controlling Si fractionation The major parameters that drive Si isotopes fractionation are relatively similar to the ones exposed for iron : temperature, Si-O bond-lengths (related to volume of SiO 4 tetrahedra; Qin et al. (2016)) as well as the nature of the bonds. In the case of silicon, the redox effect on the bonds is not considered as Si is monovalent. Moreover, there is no effect of the coordination number on the Si force constant as the Si-O coordination number in silicate minerals and melts is always Oxygen and silicon atoms are represented by large spheres and smaller red dots, respectively. After Poitrasson (2017) and [START_REF] Lameyre | Roches et Minéraux[END_REF].

four. Other parameters that could control Si isotopes fractionation were also proposed in the literature :

1. Polymerization degree The theoretical study of Grant (1954) suggested that polymerization of (SiO 4 ) could be a key parameter to explain silicon stable isotope fractionation in minerals. Poitrasson and Zambardi (2015) supported this hypothesis in demonstrating a positive linear correlation between δ 57 Fe of bulk lunar and terrestrial rocks and the degree of polymerisation (NBO/T; chapter 4). However, this relationship was not observed, at the rock scale, by Savage et al. (2011) and the degree of polymerisation did not appeared as the main parameter driving silicon isotope fractionation at the mineral scale by Méheut and Schauble (2014).

2. Electronegativity of the surrounding cations. The electronegativity of the cations bonded to Si via O atoms in minerals, such as Mg and Al, was suggested to explain the difference in Si isotope fractionation between minerals that display similar bond-lengths and polymerization degree: The lower the electronegativity of the cation bonded to Si is, the larger is the enrichment in light Si isotopes (Méheut and Schauble, 2014). However, the effect of cations other than Mg and Al has not been considered so far.

Applications Silicon is one of the principle actors of geological and biochemical cycles and Si isotopes display a significant range of composition at low temperature of about ≈ 9% (Fig. 1.5). Consequently, Si isotopes appeared as a valuable tracer in order to understand multiple near-surface processes such as the soil formation (e.g., Pogge von Strandmann et al., 2012), weathering mechanisms along with biological activity (e.g., [START_REF] Cockerton | Stable-isotope (H, O, and Si) evidence for seasonal variations in hydrology and Si cycling from modern waters in the Nile Basin: Implications for interpreting the quaternary record[END_REF] as well as the global silicon cycling of the atmosphere (loess; [START_REF] Savage | The silicon isotope composition of the upper continental crust[END_REF]). Si isotopes fractionations were also investigated in the ocean to constrain the diatom productivity for a better understanding of the carbon cycle in past oceans (e.g., [START_REF] Hendry | The relationship between silicon isotope fractionation in sponges and silicic acid concentration: Modern and core-top studies of biogenic opal[END_REF] and also in the rivers to assess the global Si output in the oceans (e.g., [START_REF] Georg | Silicon fluxes and isotope composition of direct groundwater discharge into the Bay of Bengal and the effect on the global ocean silicon isotope budget[END_REF]. At high temperature, Si isotopes were investigated in various igneous rocks more or less differentiated (e.g., Gajos et al., 2016;Savage et al., 2011Savage et al., , 2012;;Zambardi et al., 2014). Metal-silicate Si isotopes fractionations were also deeply investigated in order to better understand the Earth's core formation (e.g., Poitrasson and Zambardi, 2015;[START_REF] Pringle | Silicon isotopes in angrites and volatile loss in planetesimals[END_REF][START_REF] Savage | High temperature silicon isotope geochemistry[END_REF][START_REF] Zambardi | Silicon isotope variations in the inner solar system: Implications for planetary formation, differentiation and composition[END_REF]Ziegler et al., 2010). et al. (2011, 2012) as well as Zambardi et al. (2014) and Poitrasson and Zambardi (2015).

Moon

As for iron isotopes system, a general Si isotopes trend can be observed during magmatic differentiation with an increasing δ 30 Si with increasing SiO 2 content (Fig. bardi (2015) suggested that the heavy Si isotopes signatures of the most evolved magmas could result from an ensemble of parameters such as melt polymerization, nature of the cations surrounding Si and thermal diffusion. Qin et al. (2016) model, using their firstprinciples isotopes fractionation factors between minerals and melt (Part. 1.5.3.2), showed that fractional crystallization could reproduce the δ 30 Si trend measured in the different studies. Alike for iron isotopes system, the lack of precise (calculated or measured) equilibrium fractionation factors between minerals and melts prevents scientists to conclude on the processes involved in Si isotope fractionation during magmatic differentiation.

Mineral-Mineral and Mineral-Melt fractionation factors

Only few theoretical studies were performed in order to access the Si isotopes compositions of the high-temperature silicates minerals (Huang et al., 2014;Méheut and Schauble, 2014;Méheut et al., 2009;Qin et al., 2016). The effect of different cationic environments in olivine and pyroxene on Si β-factors was not explored by now (cf. part 1.5.2). Besides theoretical studies, silicon isotopes measurements on mineral-separated of pegmatite and granite samples from the Cornubian batholith localites were also performed by Savage et al. (2012).

The silicate melt Si β-factors were neither calculated nor measured. However, Qin et al. (2016) approximated Si isotope compositions of different melts using the weighted sum of the Si β-factors of the minerals present in their normative mineral compositions. 1.4: Silicon β-factor (1000lnβ 30 Si/ 28 Si) of minerals relevant to the Earth uppermantle and crust from ab-initio studies under the form Ax 3 +Ax 2 +Cx (with x = 10 6 /T 2 ) as well as fractionation factor (1000lnα 30 Si/ 28 Si) between minerals and quartz under the form y 2 + y 3 (with y = 10 2 /T).

In chapter 1, the theory of equilibrium isotope fractionation (Bigeleisen and Mayer, 1947;[START_REF] Urey | The thermodynamic properties of isotopic substances[END_REF] was presented based on theoretical works (Bigeleisen and Mayer, 1947;[START_REF] Richet | A Review of Hydrogen, Carbon, Nitrogen, Oxygen, Sulphur, and Chlorine Stable Isotope Fractionation Among Gaseous Molecules[END_REF] and review papers (Blanchard et al., 2017;Schauble, 2004). This chapter introduces the theoretical and numerical methods used in this study in order to perform first-principles calculation on minerals (Chapter 3) as well as first-principles molecular dynamics (FPMD) on melts (Chapter 4). More thorough description of the quantum physical principles presented below can be found in various textbooks (e.g., [START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF][START_REF] Sholl | Density Functional Theory: A Practical Introduction[END_REF] and PhD thesis (e.g., [START_REF] Ducher | Fractionnement isotopique du zinc a l equilibre par calcul ab initio[END_REF][START_REF] Dupuis | Calculs réalistes du fractionnement isotopique du silicium et du lithium pour des équilibres impliquant des phases liquides[END_REF].

Theoretical analysis of the electronic structure

Electronic structure theory aims to describe accurately real systems found in nature.

Stable structures of solids are classified in term of the electronic ground state. The main electronic ground state properties are cohesive energy, equilibrium crystal structure, phase transitions between structures, magnetic order and susceptibility, nuclear vibrations and motions (adiabatic approximation). The electronic ground state can be defined as the lowest-energy configuration of the electrons. In order to determine this ground state electronic structure, Born-Oppenheimer approximation, that consists in dividing nuclei and electrons in separate mathematical problems, is used. This approximation is based on the fact that one electron is 1800 times lighter than each proton or neutron of the atomic nucleus. This means that electrons are much faster than the nuclei, implying that they will be more rapidly affected by a change than the nuclei. The equation that describes the change of energy of a material as we move its atoms is the time-independent nonrelativistic Schrödinger equation (2.1). In this section, we will describe the Schrödinger equation and one of the "ab-initio" method that permits to solve this equation, called Density Functional Theory (DFT).

The Schrödinger equation

Generalities

The general notation of the time-independent non-relativistic Schrödinger equation is :

Ĥψ = Eψ (2.1)
where Ĥ is the Hamiltonian operator and ψ is a set of solutions (or eigenstates) of the

Hamiltonian. E, is the associated eigenvalues to ψ that satisfies the equation.

The definition of the Hamiltonian in the Schrödinger equation depends on the physical system studied. In the case, like in this thesis, where multiple electrons interact with multiple fixed nuclei, the Hamiltonian is defined as:

Ĥ = [ T + Vext + Vee + E II ] (2.2)
where T represents the kinetic energy of each electron and is expressed as :

T = i - 1 2 ∇ 2 i (2.3)
Vext is the interaction energy between each electron and the collection of atomic nuclei :

Vext = i,I V I (|r i -R I |) (2.4)
where r i and R I are respectively the electronic and atomic coordinates and V I represents the interatomic Coulombic interactions.

Vee defines the electron-electron interaction :

Vee = 1 2 i =I 1 |r i -R i | (2.5)
The final term E II represents the classical nuclei-nuclei interaction.

For the Hamiltonian chosen here, ψ (2.1) represents the electronic wave function that depends on the spatial coordinates of each of the N electrons, ψ = ψ(r 1 ,. . . ,r n ) where the spin is assumed to be included in the coordinate r i . The Schrödinger equation is therefore a function of 3N coordinates. E is the ground state energy of the electrons, which is independent of time.

It is also possible to approximate ψ as a product of invidual wave function, ψ = ψ 1 (r), ψ 2 (r),. . . ,ψ N (r). This approximation of the full wave function into individual one-electron wave function is known as the Hartree product. It is however important to notice that the number of electrons is much higher than the number of nuclei in a system and if we take a simple crystal such as Fayalite (Fe 8 Si 4 O 16 in the primitive cell), the full wave-function is a 1176-dimensional function (3 dimensions for each 392 electrons) which requires an unreasonable large amount of time to be solved.

Electron density

A noteworthy point on Schrödinger equation is that the wavefunction for any set of coordinates cannot be directly observed. The real quantity of physical interest is the probability that a set of N electrons (in any order) have coordinates r 1 , . . . , r N . The closest quantity describing this, is the density of electrons at a particular position in space, noted n(r), and expressed in terms of the individual electron wave functions as,

n(r) = 2 i ψ * i (r)ψ i (r) (2.6)
The term inside the summation represents the probability that an electron in individual wave function ψ(r) is located at position r. As each electron wave function can be occupied by two separate electrons provided they have different spins as described by the Pauli exclusion principle, a factor 2 is applied.

The main point is that electron density n(r) is a function of only three coordinates and contains great information physically observable from the full wave function solution to the Schrödinger equation, which is a function of 3N (2.1.1). This is why electron density plays a central role on electronic structure theory and is the foundation of the Density Functional Theory (DFT) described in part 2.1.2.

Density Functional Theory

Hohenberg and Kohn theorems

The entire density functional theory relies on two mathematical theorems demonstrated by Hohenberg and Kohn (1964[START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF] and derived equations of [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF] described in part 2.1.2.2.

Theorem 1: The ground state of any interacting many particle system with a given fixed inter-particle interaction is a unique functional of the electron density n(r).

Thus, the potential V ext (r i ), in a system of interacting particles, is determined by the ground state particle density n 0 (r). It implies that all the properties of the system can be fully determined given only the ground state density.

Theorem 2: The electron density that minimizes the energy of the overall functional is the true electron density corresponding to the full solution of the Schrödinger equation.

This theorem means that there is a universal functional for the energy E[n] in terms of electron density n(r), for any potential V ext (r) and the functional E[n] alone is sufficient to determine the ground state energy. As only the density is needed to obtain the ground state energy, the system of 3N variables (2.1.1) can be replaced by a system of 3 spatial variables. Electron density is a variable and a solution of the system. In order to obtain the ground state electron density an iterative method is necessary (2.1.2.3) As a consequence, energy functional can be expressed as follow :

E HK [n(r)] = T[n(r)] + E Hartree [n(r)] + V ext (r)n(r)dr + E II (2.7) 2.1.2.

Kohn and Sham ansatz

Hohenberg and Kohn theorems are essentials, and reduce the number of variables, however they don't give the form of the density functional. The ansatz (approach) of [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF] simplifies the resolution of the Hamiltonian by replacing the many-body problem by an auxiliary independent-particle problem. Indeed, this Ansatz "assumes that the ground state density of the original interacting system is equal to that of some chosen non-interacting system" [START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF]. The energy functional can then be described as independent-particle equations for the non-interacting system with all the complex manybody problem incorporated in a functional called the exchange-correlation functional of the density. The ground state energy and density can then be calculated with accuracy by solving this equation, where the only approximations are limited to the exchangecorrelation functional.

The auxiliary independent-particle system can be described by an auxiliary Hamiltonian (using Hartree atomic units):

Ĥσ aux = - 1 2 ∇ 2 + V σ (r) (2.8)
Where V σ is an effective local potential acting on an electron of spin (σ) at point r.

Unlike the previous equations, the system is described by single-electron wave functions and depends only on three spatial variables. These equations are independent and can be solved numerically.

The Kohn-Sham reformulation of the Hohenberg-Kohn expression for the ground state energy functional has the form:

E KS = T s [n] + E Hartree [n] + V ext (r)n(r)dr + E II + E xc [n] (2.9)
Where V ext (r) is the external potential due to the nuclei and any other fields (independent of spin), E II is the interaction between the nuclei (2.1.1). E Hartree is the classical Coulomb interaction energy of the electron density n(r) interacting with itself.

E Hartree = 1 2 d 3 r d 3 r n(r)n(r') |r -r'| (2.10)
The density of the auxiliary system is the sums of squares of the orbitals (ψ) for each spin.

n(r) = σ n(r, σ) = σ N σ i=1 |ψ σ i (r)| 2 (2.11)
T s is the independent-particle kinetic energy given by :

T s = 1 2 σ N σ i=1 d 3 r |∇ψ σ i (r)| 2 (2.12)
Finally, E xc is the exchange-correlation functional where all the many-body problems are grouped and can be expressed in term of Hohenberg-Kohn functional (F HK ):

E xc [n] = F HK [n] -(T s [n] + E Hartree [n]) (2.13)
If the exact E xc functional were known, the exact ground state energy of a system could be calculated by solving the equation of Kohn-Sham. A good approximate form of the E xc [n] makes the Kohn-Sham a practical approach to calculate the ground state properties of a many-body system (2.1.1).

The self-consistent coupled Kohn-Sham equations

Kohn-Sham equations simplify considerably the resolution of the Schrödinger equation.

However, the resolution of such equations appears to be "circular". Indeed, to solve Kohn-Sham equations the Hartree potential needs to be defined and to define it the electron density is mandatory. But to determine the electron density, the single-electron wave functions are needed and to know these wave functions, the Kohn-Sham equations must be solved. In order to overcome this circular path, an iterative method is used to solve Kohn and Sham equations that is described in figure 2.1 . This led to a solution of the Kohn-Sham equation that is self-consistent. This is the method the most widely used today for electronic structure calculations.

Exchange -correlation functional

The crucial quantity in the Kohn-Sham approach is the exchange-correlation energy which is a functional of electron density, called E xc [n]. By explicitly separating out the long-range Hartree term (E Hartree ) and the particle-independent kinetic effect (T s ), the remaining

E xc [n]
can be approximated as a local functional of the density. This part is devoted to the approximations of E xc [n] in DFT. There is an important number of functionals that can be used depending on the system, but we will introduce here only the two major types of approximations: the local density approximation (LDA) and the Generalized-Gradient Approximation (GGA).

Initial guess

Calculate effective potential Solve Kohn-Sham equation

Calculate electron density

Self-consistent ?

Output quantities

Energy, forces, stresses, eigenvalues... n (r r), n (r r) The local density approximation (LDA) In the LDA, the exchange-correlation energy is "an integral over all space with the exchange-correlation energy density at each point assumed to be the same as in a homogeneous electron gas with that density [START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF]. This means that the electron density is constant at all points in space; that is

V eff (r r) = V ext (r r) + V Hartree [n] + V xc [ n , n ] [ [ 1 2 - Δ 2 i i i i i =1 N σ σ σ + V eff (r r) ψ σ (r r) = ε σ ψ σ (r r) n σ (r r) = ∑ ∑ | ψ σ (r r)| 2
n(r) = constant. E LDA XC [n] = n(r) XC [n]dr (2.14)
where XC is the exchange and correlation energy per electron.

The LDA gives accurate results specially for solids close to homogeneous gas, like a nearly-free electron metal or solids where covalent bonds dominate.

The Generalized-gradient approximation (GGA) The generalized-gradient approximation goes a step beyond the local approximation in considering not only the electron density as the LDA, but also the magnitude of the gradient of the density. The general expression of the GGA functional has a form of

E GGA XC [n] = n(r) GGA XC (n(r), |∇n(r)|)dr (2.15)
There are many ways to construct GGA functional from the general equation (2.15) in order to preserve the desired properties of the system. This is why there is a large number of distinct GGA functionals. Two GGA functionals are widely used: GGA-BLYP from [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF] and GGA-PBE from Perdew et al. (1996). GGA-PBE functional has the advantage of being unique and has no adjustable parameters.

The main advantage of GGA functionals is that they lead to lower exchange energy than the LDA. This results in the reduction of binding energy correcting the LDA overbinding tendency. This is why GGA tends to improve agreement with experiment compared to LDA. However, it is commonly observed that GGA functionals overestimate of about 1 to 2% the lattice parameters compared to the experimental values (e.g. Ducher et al. ( 2016)).

DFT calculation of solids characteristics

Description of a crystal : structure optimization

The cell containing the minimum number of atoms that can be used to fully define the crystal structure is called primitive cell. A cell is defined by the direct lattice parameters a 1 ,a 2 ,a 3 . The ground-state energy is calculated on fixed atomic positions. It is therefore essential to optimize the cell volume and atomic positions to have the equilibrium structure. Indeed, a stable structure at a given pressure and temperature determines all the other properties of the crystal. In DFT, the calculation of the total energy E is done at T = 0 K. The volume of the cell (Ω) is the most convenient quantity that can be tuned to control the system theoretically. The main goal of structure optimization is to determine a theoretical equilibrium volume (Ω 0 ) and equilibrium atomic positions where the total energy E is minimum or P = 0. In order to find the ground state energy, the Newton method can be used. This method consists in finding the new position R n+1 at the step n + 1, knowing R n at step n and the Jacobian matrix J(∇ R E 0,e ), that is the Hessian matrix H(E 0,e ) of E 0,e . R n+1 positions are then expressed as

R n+1 = R n -[H(E 0,e )] -1 ∇ R E 0,e (2.16)
However, the calculation of the Hessian matrix at each iteration necessitates an important calculation time. This is why, a quasi-Newton method called BFGS [START_REF] Broyden | The Convergence of Single-Rank Quasi-Newton Methods[END_REF][START_REF] Fletcher | A new approach to variable metric algorithms[END_REF][START_REF] Goldfarb | A Family of Variable-Metric Methods Derived by Variational Means[END_REF][START_REF] Shanno | Conditioning of Quasi-Newton Methods for Function Minimization[END_REF] is widely used. Indeed, this method allow us not to calculate the Hessian matrix. In practice, the Kohn-Sham equations are solved at each BFGS iteration until the equilibrium positions are found.

Bloch theorem and plane waves

For a periodic system, a fundamental property known as the Bloch theorem must be satisfied to solve Schrödinger equation. Indeed, Felix Bloch [START_REF] Bloch | Über die Quantenmechanik der Elektronen in Kristallgittern[END_REF] elaborated the concept of electron bands in crystals. The Bloch theorem is based on the fact that monoelectronic wavefunction can be represented as the sum of multiple plane waves. The Bloch theorem is expressed as

φ k (r) = exp(ik • r)u k (r) (2.17)
The function exp(ik • r) is called plane waves, this is why many DFT calculations are called "plane-wave calculations" and k is a reciprocal wave-vector (2.2.3). As the Schrödinger equation is defined for each k separately: "each state can be labeled by the wavevector k and the eigenvalues and eigenvectors for each k are independent unless they differ by a reciprocal lattice vector. In the limit of large cell volume, the k-points become a dense continuum and the eigenvalues become continuous bands" [START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF]. This band theory for independent electrons, is also the foundation of crystals classification (Wilson and Fowler, 1931), in term of insulators, semiconductors and metals (2.2.6).

The Bloch theorem can also be developed in Fourier series:

u k,i (r) = 1 √ Ω G c k,i (G)e jG.r (2.18)
The sum in this equation is in theory infinite. However, in practice, it is necessary to limit the summation at the certain point G c of the G vector, called the energy cut-off.

This energy cut-off is determined in order to the ground state energy to only vary a little (depending on the convergence criteria) if the G c is increased. For this reason, the energy cut-off needs to be carefully tested before any DFT calculation.

Reciprocal lattice parameters and the Brillouin zone

The idea of reciprocal space is central to the plane-wave DFT calculations as underlined in the previous section (2.2.2). Multiple mathematical problems encounter in DFT calculation are indeed much more convenient to solve using the k-space (permitting the use of plane waves). It is therefore important to define some features of the reciprocal space.

The reciprocal vectors k can be defined, in three dimensions, by geometric arguments b

1 , b 2 , b 3 : b 1 = 2π a 2 × a 3 a 1 • (a 2 × a 3 ) b 2 = 2π a 3 × a 1 a 2 • (a 3 × a 1 ) b 3 = 2π a 1 × a 2 a 3 • (a 1 × a 2 ) (2.19)
It is important to notice that the length of the reciprocal lattice vectors are inversely related to the real space lattice vectors: larger lattice vectors in real space correspond to shorter lattice vectors in reciprocal space.

The volume of the primitive cell in the reciprocal space Ω BZ has the form for any dimension d,

Ω BZ = (2π) d Ω cell (2.20)
This represents the volume of the first Brillouin Zone (BZ), that is usually called Brillouin Zone. This volume is defined by "the planes that are the perpendicular bisectors of the vectors for the origin to the reciprocal lattice points" [START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF]. The center of the BZ, k=0, is called the gamma (Γ).

The electron density can be then calculated as an integral over the possible values of k in the Brillouin Zone of the form

n(r) = 1 V BZ V BZ Ne i |ψ k,i (r)| 2 dk (2.21)
However, such integral (2.21) takes up long calculation times, especially when the number of k points is large. [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] found a solution that is widely implemented in DFT calculation packages such as Quantum Espresso (2.2.8) and VASP (2.3.6). This method, specifies the number of k points in each direction of the reciprocal space, as a uniformly spaced grid along the BZ. The number of k-points noted M x M x M, is dependent of the length of the reciprocal cell. For a cubic cell, the value M will be similar for each direction but for a monoclinic system the number of k-points M along the longer lattice vector (in the reciprocal space) will be larger than for the two other lattice vectors. A larger number of k-points will considerably slower the calculation, therefore it is crucial to optimize their number to perform DFT calculations. If the total energy (or the property of interest) varies considerably while the number of k-points is increased means that the calculation result is not well-converged. This is why k-point density is a parameter to test before calculating any equilibrium structure.

Crystal symmetry can be used also to reduce the number of k-points since all the independent information can be found in the Irreducible Brillouin Zone (IBZ) which is a zone of the BZ reduced by point group symmetry of the primitive cell. Looking at figure

2.3 a and b, it is clear that symmetry leads to 6 inequivalent k-points in the IBZ against 16 k-point if no symmetry were found. The use of a shifted grid from the gamma point (Fig. 2.3 c) is even more efficient because this leads to the same grid density but with only 3 inequivalent points.

Pseudopotentials

Valence electrons have a major impact on the crystal properties as they are the only participating to the interatomic bonding. Therefore, during bonds perturbation, wave function of the valence electrons is much more affected than the one of the "core" electron that are almost unaltered. This is why in order to simplify the DFT calculation, only the valence electrons should be considered in the resolution of Kohn and Sham equations.

Indeed, when the core electrons are considered a large number of k-points is needed to shifted grid from the origin leading to only 3 inequivalent k-points [START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF].

reproduce the oscillations of the valence electron wave functions in the core region due to the Pauli exclusion principle. In addition to the large amount of k-points needed, a large energy cut-off is necessary, because the one-electron wave functions have to be orthogonal, which slows down as well the calculation. If the detailed description of the core electrons is avoided, less plane waves are necessary to construct the one-electron orbitals. This is basically the aim of the pseudo-potential. Pseudo-potentials are established in order to replace the external potential (V ext ) of the nuclei and core electrons that are considered as "frozen". The pseudopotential external potential V P P ef f must be equal to the external potential above a cut-off radius r cut-of f (Fig. 2.4).

One of the most widely used pseudopotential type in DFT is the Ultra-Soft Pseudo-Potential (USPP) based on work of [START_REF] Vanderbilt | Soft self-consistent pseudopotentials in a generalized eigenvalue formalism[END_REF]. The USPP have the advantage of requiring lower cut-off energies than other approaches. However, the main disadvantage of USPP for each atom is that a number of empirical parameters needs to be specified, and the degree of softness for each element varies. Another frozen core approach that can avoid some of the disadvantages of USPP is the projector augmented-wave (PAW) method based on work from [START_REF] Blöchl | Projector augmented-wave method[END_REF] and Kresse and Joubert (1999). PAW method is based on the fact that the calculation is done on the full all-electron wavefunction. As described above, full wavefunction varies rapidly near the nuclei, therefore all integrals are considered to be a combination of integrals of smooth wavefunctions (represented by plane waves) extending through space plus the localised part. The smooth wavefunctions can be described by a linear transformation operator T :

|ψ n = T | ψn (2.22)
An augmentation sphere |r -R a | < r a c around each atom is created where r c is the cut-off radius, a the atom index and R a is the atom position.

T can then be expressed as

T = 1 + a T a (2.23)
where T a only acts in the augmentation sphere. Partial waves |φ a i in the sphere are chosen, and smooth partial waves can be obtained using T.

|φ a i = T | φa i = (1 + T a )| φa i (2.24)
As T a only acts in the sphere, it follows that, as for the UPSS,

|φ a i = | φa i for r > r a c (2.25)
Outside the sphere, the smooth wavefunctions can be then expanded using the partial waves basis.

Whatever the pseudopotentials used, USPP or PAW, both are giving close results to the all-electron calculations.

r c ψ k,i PP ψ k,i v eff PP Z/r r Figure 2
.4: Schema of the difference between a wave-function fully calculated (black plain line) and one calculated using a pseudopotential (dotted line). Beyond the r cut-of f , the wave-functions, as well as the external potentials V P P ef f , are identical.

Magnetism

Magnetic systems can be described as systems in which the ground-state symmetry is broken with non-zero spin and/or orbitals moments of electrons. The magnetic ground state is caused by the electron-electron interactions and is by definition a many-body problem.

In diamagnetic materials, each electronic state is populated by a pair of electrons, one in spin up another in spin down (Fig. 2.5 a). Magnetic systems can be simply defined by the existence of unpaired electrons. The ordering of the unpaired electrons leads to different magnetic configurations with electron magnetic moments per atom = 0 (e.g. ferromagnetic state, antiferromagnetic state). A ferromagnetic state is characterized by all electron spins pointing exclusively in the same direction (Fig. 2.5 b) whereas an antiferromagnetic state is characterized by electron spins alternating in direction on adjacent atoms (Fig. 2.5 c). The average of electron magnetic moments per atom is defined as the magnetic moment of the system, called m. Ferromagnetic systems display a non-zero magnetic moment, unlike antiferromagnetic system where m = 0.

Including electron spin slows down the DFT calculations, however it is essential for 3d elements such as Fe where the spin effect is important. Indeed, the ground state energy as well as the equilibrium lattice parameters can be significantly modified by the addition of electron spin. Including electron spin in the calculation produces lower ground state energy and larger lattice parameters than in no-spin calculation. Whenever it was necessary, we investigated the magnetic ordering in order to find the ground-state configuration.

Electronic density of states

One of the most important quantities used to describe an electronic structure is the electronic density of states (DOS) per energy unit (E) that can be expressed as :

ρ(E) = Ω cell (2π) d BZ dk δ( i,k -E) (2.26)
where i,k represents the energy of an electron and ρ(e) is the number of electron (independent particle in the case of independent-particle states) per energy unit. As explain in part 2.2.2, the basic of plane-waves DFT calculations is to express the electron density in functions of the form exp(ik.r). Electrons associated with plane waves have energy E = (hk) 2 /2m. Therefore, once a DFT calculation is performed, it is possible to obtain the electronic DOS by integrating the resulting electron density in k-space (eq. 2.26). From the electronic DOS it is then possible to determine the nature of the system studied: insulator, semi-conductor or metal. In order to do this, it is necessary to introduce the Fermi energy E f which is the energy of the highest occupied electronic state. For metals, the electronic density of state is positive at the Fermi level. Metals are therefore conductive materials. Whereas a semi-conductor, will display a small gap of electronic states at E f , also called band gap, separating the valence bands and the conduction bands (Fig. 2.6). This means that if an electric field is applied to this system, this material will lead to small (less than metals) electron conduction. If the band gap between valence and conduction bands is wide (> 3 eV), then the system is called an insulator and the conduction is impossible.

Vibrational frequencies

Once the atomic equilibrium positions are calculated by DFT, it is possible to determine the vibrational frequencies using perturbation approaches. These approaches are based on small external perturbation of the system with an energy smaller than E 0 of the system.

This perturbation affects the atomic positions, and therefore their forces and energies.

The vibrational states that characterise a crystal are called phonons. Different methods can be used to obtain these vibrational frequencies such as the DFPT (Density Functional Perturbation Theory) or "frozen phonons".

Those methods are both based on Born-Oppenheimer approximation (2.1), so any dependence of electronic energies on the velocities of the nuclei is neglected. Therefore, the equation of motion of the nuclei are determined by the total energy E(R) of the system of electrons and nuclei, with R the position of nuclei as parameter.

Density Functional Perturbation Theory

The vibrational modes at frequency ω are described by displacements u I expressed as

u I (t) = R I (t) -R 0 I (t) = u I e iωt (2.27)
where R 0 I (t) is the equilibrium nuclei positions, R I (t) the new nuclei positions. Within the harmonic approximation, the response to the external perturbation that induces atomic displacements corresponds to the second order derivatives of energy:

C I;γ;J;δ = ∂ 2 E ∂R I,γ ∂R J,δ
(2.28)

The matrix C I;γ;J;δ , defined above, is the interatomic force constant matrix, or the so-called dynamical matrix.

DFPT method consists in the direct calculation of second-order derivatives of energy.

The vibrational frequencies are finally obtained by diagonalising the dynamical matrix:

det 1 √ M I M J C I;γ;J;δ (q) -ω 2 (q) = 0 (2.29)
q here is a reciprocal-space wave vector of the perturbation, and C I;γ;J;δ (q) is the Fourier transform of C I;γ;J;δ .

In a crystalline system, the vibrational frequencies are calculated from a grid of qpoints in the reciprocal-space. When the system is large in real space, it is possible to calculate the vibrational frequencies only at Γ (q = 0 0 0) because the dispersion effect can be neglected at Γ when the Brillouin Zone is small. When the calculation is performed at Γ, there must be three acoustic modes with ω ≈ 0 cm -1 .

Frozen Phonon

This approach is based on the fact that the perturbation is frozen in with a finite amplitude in the system. Then, Fourier transform of force constants at q are calculated from finite differences of forces induced on all the atoms of the supercells by the perturbation.

The interatomic force constant matrix, in this approach, is defined as:

C I;γ;J;δ = ∆F I,γ ∆R J,δ (2.30)
In the following chapters, the force constant is called <F> and is calculated from the dynamical matrix by averaging the force constants for an atom X in three spatial dimensions x, y, z.

Quantum Espresso

Use of Quantum Espresso Codes

Mineral β-factors (Chapter 3) were calculated using Quantum Espresso (QE) package.

These codes are based on DFT, plane-waves and pseudopotentials. Self-consistent-field (scf) calculations and structure optimisations, are performed with the code pw.x while dynamical matrices are obtained using the phonon code ph.x. The functionals used for the minerals are GGA-PBE (Perdew et al., 1996) and the pseudo-potentials used are USPP from the GBRV library (Garrity et al., 2014).

Calculation optimization

The duration of calculations is one of the major limitations of the quantum mechanics field. There are some parameters as described above that will influence the calculation time such as the E cuf -of f , the k-point grid, the r cuf -of f of the pseudopotentials. . . Once these internal parameters are tuned in order to get converged results, the only way to improve the calculation duration is the choice of core-setting on which the calculation is run. Indeed, DFT codes are parallelized in order to perform efficiently calculation on multiple core system with different programing techniques such as Open-MP, MPI or hybrid Open-MP/MPI systems used by QE. It is then important to understand the configuration of the supercalculator used (e.g. CALMIP) in term of threads and cores to run the calculation on the most efficient number of CPU (central processing units) cores. Moreover, the use of GPU (graphics processing units) has been encouraged by CALMIP computing center these past few years for codes, such as pw.x in QE, that have been optimized for it (by programming in CUDA or OpenAcc) in order to reduce the calculation time. We performed many tests on GPU configuration with QE and calculation time can be up to 2 times shorter than when the calculation is run on CPU.

This allowed us to perform calculations with bigger cells, or with more complex chemical compositions without being discouraged by the calculation duration.

Ab-initio molecular dynamics

Ergodicity principle

The liquids are by definition disordered systems, where there is no periodicity. The size of the system is also much bigger than the simulation box. However, it is possible to describe a global liquid with an ensemble of microstates, which is an ensemble of atomic configuration in the system with a certain probability associated. Ab-initio molecular dynamics can produce the ensemble of microstates (λ) of a system, and therefore reproduce a liquid evolution through time. However, in order to consider that the liquid is accurately represented, the ergodicity principle needs to be satisfied. This implies that the system explored all the possible states of the system (with the same probability as in the liquid)

during the simulation time. This is why, molecular dynamics (MD) simulations need to be long enough to have an equilibrated system, according to the ergodicity principle.

In mathematics terms, ergodicity can be expressed as,

A λ = λ a(i)e -βEλ dλ λ e -βEλ dλ = lim t→∞ 1 t t 0 t 0 +t A dt = A temp (2.31)
where λ represents the configurations ensemble of the trajectory, A is an observable at macroscopic scale, and a(i) is the function associated to A.

Born -Oppenheimer Molecular Dynamics

The Born-Oppenheimer Molecular Dynamics (BOMD) principle is based on the resolution of the Kohn-Sham equations in order to find the ground-state energy of a system by separating nuclei and electrons motions into two distinct mathematical problems. The large difference in relative masses between electrons and nuclei allows us indeed to approximately separate the wavefunction as a product of nuclear and electronic terms. The Lagrangian for this approach can be written as:

L = 1 2 3N i=1 m i v 2 i -E[φ(r 1 , r 2 , ..., r 3N )] (2.32)
where the first term corresponds the kinetic energy of the atomic nuclei based on classical motion equation and the second term φ(r 1 , r 2 , ..., r 3N ) represents the full Kohn-Sham oneelectron wave functions for the electronic ground state of the system.

Verlet algorithm

The integration of atomic positions is based on the Verlet algorithm [START_REF] Verlet | Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules[END_REF]. This algorithm, widely used in molecular dynamics, simplifies the complicated motion equations by calculating the atomic positions at t+∆t using Taylor expansion between t+∆t and t-∆t.

The position r n-1 is expressed as

r n-1 = r n -v n ∆t + 1 2 F n m ∆t 2 -O(∆t 3 ) (2.33)
And r n+1 is expressed as

r n+1 = r n + v n ∆t + 1 2 F n m ∆t 2 -O(∆t 3 ) (2.34)
where v n is the velocity of the atoms, the ratio F n /m corresponds to the acceleration, and

O(∆t 3 ) represents the local error.
Therefore, the Verlet algorithm is defined as the sum of the two expression (2.33; 2.34):

r n+1 = 2r n -r n-1 + F n m ∆t 2 + O(∆t4) (2.35)
For sufficiently small ∆t, this algorithm is a good approximation to the true trajectory.

The velocity of the atoms v n at t can be defined as a mean velocity between t + ∆t and t -∆t by subtracting equation (2.34) from (2.33) :

v n = r n+1 -r n-1 2∆t + O(∆t 2 ) (2.36)

Thermodynamic particle ensemble

Different thermodynamic ensembles exist to numerically reproduce the experimental conditions. The first ensemble is the micro-canonical ensemble, called NVE, that consists of an isolated system where the number of particle N, the volume V and the total energy E is conserved. However, this system is not representative of experimental systems where there is energy exchange with the surrounding environment. For this reason, the two other ensembles are composed, in addition to the studied system, of an infinite-sized reservoir that exchanges with the system to control dynamic parameter such as the temperature T (). The canonical ensemble, called NVT, is a system at temperature equilibrium and the isobaric ensemble, called NPT, corresponds to a system controlled in pressure P and temperature. The grand canonical ensemble is at thermal and chemical equilibrium (µVT).

Depending on the system that needs to be modelled and the parameters that have to be controlled, a thermodynamic ensemble is chosen. For our melt models, a first simulation under NPT ensemble is performed in order to obtain an equilibrium volume for the target pressure. Then, a calculation in NVT ensemble is performed in order to keep the volume at equilibrium and at fixed temperature as we want to study isotopic fractionation at finite temperature. The NVT simulations produce the trajectories used to calculate βfactors (Chapter 4).

Thermostats

Born-Oppenheimer molecular-dynamics (BOMD) approach that is based on Newtonian equations of motion conserves the total energy. Trajectories are then generated in the NVE microcanonical ensemble. This means that the T is not controlled. However, under experimental conditions the atoms are able to exchange heat with their surroundings. In this situation, the atoms exist in a canonical ensemble, where N, V and T are constant (2.3.4). In order, to reproduce a canonical ensemble a thermostat is used, that consists of adding a heat bath. Here, the Nosé-Hoover thermostat Nosé (1984) is used which introduces an additional degree of freedom to the system to create a "heat bath particle".

The Nosé-Hoover thermostat can be expressed as an extended Lagrangian Nosé (1984) :

L = 1 2 3N i=1 m i s 2 v 2 i -U (r 1 , r 2 , ..., r 3N ) + Q 2 ds dt 2 -gk B T ln s (2.37)
where m i and v i are respectively the mass and the velocity of the atom associated with the i th coordinate and U (r 1 , r 2 , ..., r 3N ) represents the total potential energy. The term Q is assimilated to a mass associated to the thermostat, g corresponds to the degree of liberty associated to the thermostat and k B is the Boltzmann constant.

The equations of motion from this extended lagragian were written by Hoover (1985) :

dr i dt = v i , (2.38
)

dv i dt = - 1 m i ∂U (r 1 , ..., r 3N ) ∂r i - ζ m I v i , (2.39) dζ dt = 1 Q i=1 3N m i v 2 i -3N k B T , (2.40) d ln s dt = ζ. (2.41)
where, the second term of equation ( 2.39) represents the 'friction' term, that will increase or decrease the velocity depending on the sign of ζ. Equation (2.40) controls the magnitude and the sign of ζ and can be rewriten as

dζ dt = 3N k B Q [T M D -T ], (2.42)
ζ is adjusted in order to control the instantaneous temperature of the simulation to stay close from the target temperature T. Q is the parameter that defines how feedback between the temperature difference T M D -T is applied to ζ. If the temperature is too low, ζ will be adjusted by eq (2.42), leading to an adjustment of the atom velocities (2.39) that increases their average kinetic energy and vice versa.

VASP

In this thesis, the ab-initio molecular dynamics simulations were performed using Vienne Ab-initio Simulation Package (VASP) that is a DFT plane-wave code (Kresse and Furthmüller, 1996). The choice of VASP compared to QE as for the minerals has been made

according to the duration of the calculation. Indeed, the simulations have a large number of atoms (N > 100) with up to 7 different atomic species including magnetisation of Fe, which implies that the calculations are heavy and as mentioned part 2.3.1 the simulations need to be long enough to satisfy the ergodicity of the system. VASP appears to be much faster than the best optimised calculation on QE for our simulations (2.2.8.2).

Physical properties investigated

In addition to volume (V), density (ρ), temperature (T) and pressure (P), the atomic Mean Square Displacement (MSD) and the Radial Distribution Function (RDF) represent major properties of the melts and therefore, are detailed below. The aim of our FPMD calculations is to determine the isotopic properties of melts however, the different methods of β-factor calculation used in this study won't be developed here as the methods are detailed in Chapter 1 and 4.

Mean Square Displacement

The first characteristic to check on the MD trajectory is the atomic Mean Square Displacement as it measures the deviation of the atomic positions with respect to a reference position over time t.

The MSD can be expressed as:

M SD ≡ 1 N i=1 N |x i (t) -x i (0)| 2 (2.43)
where N is the number of atoms averaged, x i (0) vector is the reference position of the i -th atoms and x i (t) is the positive vector of the i-th atom at time t.

MSD is therefore a key parameter that will determine if the trajectory represents a liquid or a solid phase. Indeed, depending on the nature of the atoms, the size of the systems, the pressure or temperature, the liquid can solidify and the motion of atoms becomes then equal to 0. If the MSD keeps increasing with time, then this means that there is motion and the system is still liquid.

Radial distribution function

The structure of disordered materials can be conveniently described by the radial distribution function expressed as:

g(r) = ρ(r) 4πr 2 ρ (2.44)
where ρ(r) is the average density of atoms found in a thin shell at a radius r from an atom X in the material, and ρ is the average density of the entire material.

In our case, RDFs were essential to determine the coordination number of the 6 cations of our melt models (i.e. the number of oxygen atoms surrounding each cation) in order to investigate the evolution of the melt structures (Chapter 4).

Abstract

In order to elucidate the processes involved in iron and silicon isotopes partitioning during magmatic differentiation, it is essential to know the precise value of equilibrium fractionation factors between the main minerals present in the evolving silicic melts. In this study, we performed first-principles calculations based on the density functional theory to determine the equilibrium iron and silicon isotopes fractionation factors between eleven relevant silicate or oxide minerals in the context of magmatic differentiation, namely: aegirine, hedenbergite, augite, diospide, enstatite, fayalite, hortonolite, Fe-rich and Fe-free forsterites, magnetite and ulvospinel. Results show that Fe 2+ -bearing silicate minerals display significant differences in iron isotope fractionation factors that cannot be neglected, even at high temperature (1000 • C). Various physical and chemical parameters control the iron isotopic fractionation of silicate minerals. However, the main parameter, after temperature and the iron oxidation state, is the nature and number of iron second neighbors

Introduction

Silicon and iron represent, respectively, the second and fourth most abundant elements on Earth. Isotopic ratios of these elements are perfect candidates to become useful tracers of a variety of geological processes because of their ubiquitous nature. Despite the large mass difference between Fe isotopes ( 54 Fe, 56 Fe, 57 Fe, 58 Fe) and Si isotopes ( 28 Si, 29Si, 30 Si), the magnitude of their high-temperature isotope fractionations is limited (e.g., Dauphas et al., 2012;Méheut et al., 2009;Savage et al., 2010;Teng et al., 2013;Williams et al., 2004;Zambardi et al., 2014). Iron is interesting because of its multiple oxidation states (Fe 0 , Fe 2+ and Fe 3+ ) that make it a perfect element to trace the redox conditions in igneous rocks such as the evolution of oxygen fugacity (f O 2 ) in the mantle associated with depth [START_REF] Mccammon | A redox profile of the Slave mantle and oxygen fugacity control in the cratonic mantle[END_REF][START_REF] Woodland | Variation in oxygen fugacity with depth in the upper mantle beneath the Kaapvaal craton, Southern Africa[END_REF][START_REF] Yaxley | An oxygen fugacity profile through the Siberian Craton -Fe K-edge XANES determinations of Fe 3+ / Fe in garnets in peridotite xenoliths from the Udachnaya East kimberlite[END_REF] or in the lavas during magmatic differentiation. In nature, silicon has two valence states (Si 0 , Si 4+ ) (Poitrasson, 2017). Metallic silicon (Si 0 ) can be found in metallic inclusions in meteorites, and most likely in the metallic cores of terrestrial planets (e.g., Georg et al., 2007;[START_REF] Pack | Silicon in iron meteorite metal[END_REF][START_REF] Wai | Silicon concentrations in the metal of iron meteorites[END_REF]. In the magmatic differentiation context, Si can be considered as only tetravalent. Silicon concentration increases during magmatic differentiation whereas Fe concentration decreases. Therefore, silicon isotopes are more likely reflecting cumulative (Zambardi et al., 2014). Considering their contrasted behavior during magmatic differentiation, Si and Fe isotopes represent complementary approaches to constrain the mechanisms of isotopic fractionation involved in magmatic differentiation. Both isotopic systems were extensively studied in that context, either independently (Dauphas et al., 2014;Foden et al., 2015;Li et al., 2020;Lundstrom, 2009;Nebel et al., 2015;Poitrasson and Freydier, 2005;Poitrasson and Zambardi, 2015;Savage et al., 2011;Schoenberg et al., 2009;Schuessler et al., 2009;Shahar et al., 2008;Sossi and O'Neill, 2017;Sossi et al., 2012;Teng et al., 2008;Weyer et al., 2005;Williams et al., 2005Williams et al., , 2009Williams et al., , 2018) ) or together (Gajos et al., 2016;Zambardi et al., 2014) (Zambardi et al., 2014;Gajos et al., 2016). Nonetheless, the processes leading to heavy Fe and Si isotopic compositions in the most evolved lavas during magmatic differentiation are still debated. Fractional crystallization (e.g., Du et al., 2017;Foden et al., 2015;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012), partial melting (Williams et al., 2009), fluid exsolution processes (Poitrasson and Freydier, 2005), thermal diffusion (e.g., Gajos et al., 2016;Lundstrom, 2009;Zambardi et al., 2014) and sulfide saturation in the magma (Williams et al., 2018) were suggested to explain the Fe systematics observed among igneous rocks. As Shahar et al. (2008) pointed out, "In order to understand the meaning of small but significant differences in Fe isotope ratios among minerals representing differentiated planetary materials, the mechanisms of high-T Fe isotope fractionation must be quantified". Different techniques can be used to determine isotope fractionation factors in silicate minerals, including: 1) conventional analysis of mineral separates involving mineralization followed by iron or silicon separation by anion exchange chromatography and MC-ICP-MS analyses, for both natural or synthetic samples (Savage et al., 2011;Shahar et al., 2008;Sossi and O'Neill, 2017;Williams et al., 2005); 2) In-situ laser ablation coupled to MC-ICP-MS techniques using nanosecond or femtosecond laser on natural samples (Collinet et al., 2017;Oeser et al., 2014Oeser et al., , 2015Oeser et al., , 2018;;Sio et al., 2018Sio et al., , 2013;;Teng et al., 2013). 3) For Fe isotopes, the vibrational properties of minerals can be determined by nuclear resonance inelastic X-ray scattering (NRIXS) from which reduced partition function ratios, called β-factors, are derived (Dauphas et al., 2012(Dauphas et al., , 2014;;Roskosz et al., 2015); 4) Also for iron, β-factors can be calculated from second-order Doppler (SOD) shift derived from Mössbauer spectroscopic measurements [START_REF] Polyakov | Equilibrium iron isotope fractionation at core-mantle boundary conditions[END_REF]Polyakov and Mineev, 2000;[START_REF] Polyakov | New data on equilibrium iron isotope fractionation among sulfides: Constraints on mechanisms of sulfide formation in hydrothermal and igneous systems[END_REF]Polyakov et al., 2007); 5) Theoretical fractionation factors can also be determined from electronic structure calculations usually based on the density functional theory (Blanchard et al., 2009;Huang et al., 2014;Méheut and Schauble, 2014;Méheut et al., 2009;[START_REF] Wang | Effect of Ca content on equilibrium Ca isotope fractionation between orthopyroxene and clinopyroxene[END_REF]. Concerning iron, only few fractionation factors are available for silicate minerals. Because olivine and most of pyroxenes are Fe 2+ -bearing minerals, the Fe isotope fractionation between them is often considered as negligible (Dauphas et al., 2014;Roskosz et al., 2015) or even inexistent (Beard and Johnson, 2004). However, some studies on natural samples highlight small but distinct isotopic signatures for olivine and coexisting clinopyroxene and orthopyroxene (Williams et al., 2005;[START_REF] Zhu | Mass fractionation processes of transition metal isotopes[END_REF]. This effect was also proposed earlier from Mössbauer data by Polyakov and Mineev (2000). Besides temperature, iron oxidation state is described as one of the main parameters influencing the iron isotope fractionation with the coordination number of iron (Dauphas et al., 2014;Polyakov and Mineev, 2000;Sossi and O'Neill, 2017). Indeed, Fe 3+ -rich phases have stiffer bonds and therefore tend to concentrate heavy 57 F e isotopes. This is the reason why most studies focused on the common Fe 3+ -bearing oxide of magmas, i.e., magnetite, in order to constrain the isotopic evolution during magmatic differentiation (Dauphas et al., 2012;Polyakov and Mineev, 2000;Polyakov et al., 2007;Shahar et al., 2008;Sossi and O'Neill, 2017). Our study provides the first ab-initio values of inter-mineral fractionation factors for iron isotopes in silicate minerals. Regarding Si isotopes, only few theoretical studies were performed on high-temperature silicate minerals (Grant, 1954;Huang et al., 2014;Méheut and Schauble, 2014;Qin et al., 2016). Grant (1954) suggested that polymerization was the key factor influencing Si isotopes among the differentiation path with enrichment of heavy isotopes with increasing polymerization. The study of Poitrasson and Zambardi (2015) on lunar and terrestrial rocks supported Grant's hypothesis in displaying a clear relationship between increasing polymerization degree and increasing silicon isotope composition of lunar rocks. However, at the mineral scale, Méheut and Schauble (2014) displayed large isotopic fractionations between minerals having the same structure and silicon polymerization degree (kaolinite, lizardite). Silicon-Oxygen bond lengths (Huang et al., 2014) as well as the electronegativity of the cations Mg and Al (Méheut and Schauble, 2014) were also proposed to explain the difference in Si isotope signatures between minerals, but the effects of other cations have not been considered so far. Here, the influence of Ca, Fe, and Na on Si

Methodology

Calculation of equilibrium isotope fractionation factors

The equilibrium isotope fractionation factor α(a,b,Y) of an element Y between two phases a and b is defined as the ratio of isotope ratios and can also be written as the ratio of the reduced partition functions, called β-factors:

α (a,b,Y ) = β (a,Y ) β (b,Y ) (3.1)
Where β (a,Y ) can be seen as the equilibrium fractionation factor of Y isotopes between the phase a and a perfect gas of Y atoms. The reduced partition function ratio of each mineral was calculated from their harmonic vibrational frequencies (Bigeleisen and Mayer, 1947) using the following equation:

β a,Y =   {q} 3N i=1 ν q,i ν q,i e (-ν q,i /2k B T ) 1 -e (-ν q,i /k B T ) 1 -e (-ν q,i /k B T ) e (-ν q,i /2k B T )   1/Nq×N (3.2)
where, ν q,i and ν q,i are the vibrational frequencies of the heavy and light isotopologues of the phase a, identified by a wavevector q and a branch index i from 1 to 3N at ; N at is the number of atoms in the crystal unit cell, is the Planck constant, k B is the Boltzmann constant, and T the temperature (in Kelvin). The second product in Eq.3.2 is performed on a sufficiently large grip of N q q-vectors in the Brillouin zone. In Eq. 3.2, the three translational modes at the center of the Brillouin zone (Γ) with ν 0,i = 0, are not considered. Equation 3.2 accounts for the fact that there should be no isotopic fractionation at the high-temperature limit, called the "rule of the high-temperature product" or "Redlich-Teller rule" (Bigeleisen and Mayer, 1947).

Isotope fractionation is often reported in permil (% ) and thus, we will adopt the usual notation: 10 3 lnα(a, b, Y ) = 10 3 lnβ(a, Y ) -10 3 lnβ(b, Y ). In laboratory studies, 10 3 lnα(a, b, Y ) can be approximated (if alpha is not too different from 1) by the fractionation factor ∆ a-b = δ aδ b , where δ a and δ b represent the isotopic compositions of phases a and bin isotopic equilibrium, obtained as follows for iron isotopes:

δ 57 F e( ) =
57 F e/ 54 F e sample 57 F e/ 54 F e standard -1 × 1000 (3.3)

Modeling approach

The reduced partition function ratios of each mineral were calculated from Eq 3.2, using the phonon frequencies, ν q,i , computed from first-principles methods based on density functional theory (DFT). Initial atomic positions and lattice parameters taken from experimentally determined structural models were relaxed first until the residual forces on atoms are less than 10 -4 Ry/au, and the residual pressures smaller than 0.03 kbar. For each mineral, the dynamical matrix was calculated within density functional perturbation theory (DFPT) and then diagonalized for the two atomic masses of interest (54 and 57 for Fe / 28 and 30 for Si) in order to obtain the vibrational frequencies for both isotopologues.

Structural relaxations as well as vibrational frequency calculations were performed using PWscf and PHonon codes included in the Quantum Espresso suite of codes (Giannozzi et al., 2009). All calculations were performed using the Perdew-Burke-Ernzerhof (PBE) parametrized generalized gradient approximation (GGA) functional (Perdew et al., 1996).

The ionic cores were described by ultrasoft pseudopotentials (USPP) from the GBRV library (Garrity et al., 2014). The wave-functions cutoff was taken equal to 40 Ry for all minerals, such as the total energies are converged within 1 mRy/atom. A regular grid in k-space, shifted from Γ point, according to the Monkhorst-Pack scheme [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF]) is used for Brillouin zone sampling. The size of the converged k-point grid for all the studied minerals is displayed in Table 3.1. For all minerals, vibrational frequencies were calculated along one shifted q-point except for magnetite vibrational frequencies that were calculated along a 2 × 2 × 2 shifted q-point grid. For the systems investigated here, the reduced partition function ratio converges rapidly with respect to the number of q-points. Shifting the q-point grid relative to the center of the Brillouin zone (Γ point) avoids us to deal with the long-range effects that should be considered otherwise. Such long-range effects give rise to the LO-TO frequency splitting and thus affect the reduced partition function ratios.

As mentioned above, experimental structures were taken as starting configurations.

However, in the investigated silicate minerals, there are two sites available for iron (M 1 , M 2 ) that are shared by other cations such as Mg, Ca or Na and experimental models could only give us the occupation proportion of iron in each site. Thus, we performed self-consistent field (scf) calculations on each structure for all non-equivalent iron atomic positions in M 1 and M 2 sites, with various magnetic moments, in order to find the configuration with the lowest energy to perform the calculation of the phonon frequencies (Fig. 1 in Appendix A). Calculations for all minerals were spin-polarized and magnetic moments were free to relax. Iron electronic charge density of the optimized structures were determined using the Bader Charge Analysis code [START_REF] Henkelman | A fast and robust algorithm for Bader decomposition of charge density[END_REF]. [START_REF] Bader | The zero-flux surface and the topological and quantum definitions of an atom in a molecule[END_REF] developed a method based on zero flux surfaces to divide crystal/molecule in atoms.

By doing so, the charge enclosed in the Bader volume approximates the total electronic charge of the atom.

Results

Structural properties

In terms of structure, olivine is a nesosilicate that consists of isolated silicon tetrahedra connected by interstitial cations, with the general formula The clinopyroxene and olivine unit cells contain four formula units, and the orthoenstatite and ulvospinel unit cells contain eight formula units (Table 3 suggest that 72% of iron will preferentially be incorporated in M 1 sites over M 2 sites at 700 • C. This calculation is consistent with our modeled hortonolite that has 75% of the iron in M 1 sites. It also agrees with the study of [START_REF] Brown | High-Temperature Crystal Chemistry of Hortonolite[END_REF], indicating that iron is mainly in M 1 sites in hortonolite. However, we decided to perform phonon calculation on both forsterite models in order to assess the impact of iron positions on the 57 Fe/ 54 Fe β-factors as this element can be equally distributed in M 1 and M 2 sites at high temperature in forsterite of the upper mantle (e.g., Blanchard et al., 2017;[START_REF] Mccarty | Transition metal cation site preferences in forsterite (Mg2SiO4) determined from paramagnetically shifted NMR resonances[END_REF]. Regarding ulvospinel, the positions of iron in the octahedral and tetrahedral sites are based on [START_REF] Gatta | Static positional disorder in ulvöspinel: A single-crystal neutron diffraction study[END_REF] in order to result in an asymmetric charge distribution which, preserve the non-cancelling cation-cation repulsions and the observed out-of-center distortion of the central Ti 4+ [START_REF] Kunz | Out-of-center distortions around octahedrally coordinated d0 transition metals[END_REF]. As ulvospinel is antiferromagnetic [START_REF] Gatta | Static positional disorder in ulvöspinel: A single-crystal neutron diffraction study[END_REF], only one spin configuration was compatible with an antiferromagnetic structure that consists of all the Fe T in spin up configuration and all the Fe M in spin down configuration. Magnetite is known to be ferrimagnetic, and the spin configurations were taken from [START_REF] Wright | Charge ordered structure of magnetite (formula presented) below the Verwey transition[END_REF], where the Fe T is in spin down configuration and all the Fe M in spin up configuration.

[M 2 ][M 1 ](T)
For the end-members of the solid solutions (i.e., aegirine, hedenbergite, fayalite) and for diopside where all iron positions are equivalent, only the magnetic order was investigated. Fayalite is known to be antiferromagnetic. The spin configuration is taken from [START_REF] Cococcioni | Structural, electronic, and magnetic properties of Fe2SiO4 fayalite: Comparison of LDA and GGA results[END_REF] where the magnetic ground state corresponds to a similar Fe M2 magnetization to that of the closest Fe M1 site and the antiferromagnetic ordering occurs between corner-sharing octahedra (Fig. 1 in Appendix A). As there is no information about the spin configuration of the antiferromagnetic hortonolite, we decided to take the same spin configuration as for fayalite (Fig. 1 Calculated lattice parameters for the most stable model of each mineral agree with the experimental values within 1 -2% (Table 3.1). The Fe-Fe interchain distance in hedenbergite (6.534 Å) is larger than in aegirine (6.352 Å) as described in [START_REF] Baum | Magnetic properties of the clinopyroxenes aegirine and hedenbergite: A magnetic susceptibility study on single crystals[END_REF] that displays an Fe-Fe interchain distance 4.7% larger in hedenbergite than in aegirine. The distortion of Fe 2+ octahedral sites in augite and enstatite is observed in the modeled structures in accordance with experiments [START_REF] Cameron | Structural and chemical variations in pyroxenes[END_REF][START_REF] Cameron | High-Temperature crystal chemistry of acmite, diopside, hedenbergite, jadeite, spodumene, and Ureyite[END_REF].

Reduced partition function ratios of iron and silicon

Temperature dependence of the Si and Fe reduced partition function ratios of the silicate and oxide minerals are displayed in Fig 3 .1. The parameters of the corresponding polynomial fits are given in Table 3.2. A linear correlation between β-factors and their corresponding average force constants (F) is observed for iron and silicon (Fig. 3.2), where F corresponds to the average of the force constants for an atom X in three orthogonal directions. This parameter F is calculated for each crystallographic site where atom X is located and averaged in order to obtain an average force constant for each mineral. This linear relationship allows us to discuss isotopic fractionations in terms of both β-factors or force constants. Following the general discussion on the uncertainties related to the harmonic approximations and the PBE functionals performed by

Discussion

Parameters controlling Fe isotope fractionation in minerals

As mentioned, the reduced partition function ratios calculated from equation 3.2 correlate linearly with the corresponding average force contants, (Fig. 3.2). This observation is in agreement with the following relationship (Bigeleisen and Mayer, 1947):

1000lnβ = 1000 1 m - 1 m 2 8k 2 B < F > T 2 (3.4)
where m and m are atomic masses of the light and heavy isotopes respectively. Compared to the frequency-based equation 3.2, the force constant-based equation 3.4 implies some approximations (i.e. hν/k B T ≤ ≈ 2 and harmonic approximation). The divergence between the two equations increases when masses and temperature decrease (Blanchard et al., 2017). In the case of iron, the constant of proportionality of the approximate equa- at T =1280 K. The linear fit between 10 3 lnβ 57 Fe/ 54 Fe and the iron force constant is : f (x) = 0.0026x and is in total agreement with the equation of Dauphas et al. (2014).

The linear fit between 10 3 lnβ 30 Si/ 29 Si and the silicon force constant is: f (x) = 0.0062x.

Iron electronic charge density

One of the main characteristics of iron is its redox sensitivity. As described by Polyakov and Mineev (2000) and subsequent studies (Blanchard et al., 2009;Dauphas et al., 2012;Polyakov et al., 2007;Roskosz et al., 2015;Shahar et al., 2008;Sossi and O'Neill, 2017), Fe 3+ -bearing minerals, such as magnetite, display a heavier isotopic signature compared to Fe 2+ -bearing minerals. As mentioned in the method section, the valence of iron is approximated by the Bader charge [START_REF] Bader | The zero-flux surface and the topological and quantum definitions of an atom in a molecule[END_REF]. Except for aegirine that is a Fe 3+bearing mineral and magnetite that contains both Fe 3+ and Fe 2+ , all minerals are only bearing Fe 2+ . Here, we observe that all minerals display a mean valence number of about 1.4 (Fig. 3.4b). Aegirine shows a valence of 1.8 and magnetite shows an intermediate valence of 1.5. Here again, the results suggest that the iron oxidation state strongly affects the iron force constant and therefore the isotopic fractionation. Aegirine as well as magnetite, the trivalent Fe-bearing minerals, display significantly higher force constants than all other minerals that contain only divalent iron. However, within Fe 2+ -bearing minerals, there is no clear correlation between the Bader charge and the force constant.

For nearly the same value of the Bader charge, the iron force constant varies over more than 50 N/m.

Iron concentration

Dependency of the reduced partition function ratio of an element on its concentration in a mineral has been underlined for several stable isotope systems. For example, [START_REF] Wang | Effect of Ca content on equilibrium Ca isotope fractionation between orthopyroxene and clinopyroxene[END_REF] showed that the Ca concentration in pyroxenes affects the equilibrium Ca isotope fractionation down to very low concentrations (x = 1/48, corresponding to the molar ratios of Ca x Fe y Mg 1-x-y SiO 3 ). These authors also highlighted that Fe concentration, to a lesser extent, has an impact on Ca isotope fractionation with addition of Fe in orthopyroxene leading to a lower lnβ 44 Ca/ 40 Ca. In the present study, we can also see an effect of iron concentration in the minerals on their iron β-factors for a given mineral solid-solution (Fig 4c). In the olivine solid-solution, the iron β-factor decreases with increasing iron content.

Fayalite has the lowest iron force constant, forsterite the highest while hortonolite shows an intermediate value. The same trend is observed for the three Fe 2+ -bearing clinopyroxenes, namely augite, hedenbergite and diopside. Note that because of the presence of trivalent iron, aegirine is outside this clinopyroxene trend. We also observe that the orthopyroxene enstatite has a slightly higher force constant than the clinopyroxene diopside while the iron content is the same.

Iron second neighbors

Coordination number as well as iron oxidation state can explain major fractionation effects between Fe-bearing minerals. However, the scattered correlations displayed in previous sections can be explained by the iron local environment in the silicates. Indeed, for Fe 2+bearing minerals displaying the same coordination number, the only parameters that lead to different fractionation are the nature and the number of the surrounding cations (i.e. neighboring sites will affect the force constant of the central iron, whether located in M 1 or M 2 sites. The more iron is present among these cationic neighbors, the weaker is the iron force constant (fig. 3.5a). This is especially noticeable when looking at the force constant evolution in M 1 sites. The decrease is less apparent for M 2 sites, and this could be explained by the fact that there are only 2 neighbors sharing an edge in M 2 sites against 4 in the M 1 sites. This would mean that the closest cations from the central iron affect more the iron force constant. So, the force constant is not only related to the nature of the second neighbors (or first cationic neighbors) but also to their number and crystallographic arrangement. The effect of the local cationic environment leads to an average correlation between reduced partition function ratios and iron concentration in the minerals (Fig. 3.5a, large symbols). However, the effect of the local environment cannot only be reduced solely in term of concentration of iron in the minerals. As described above in part 3.5. 1.4, Wang et al. (2017) showed that Ca concentration in pyroxenes has no effect on Ca fractionation factor for small Ca concentration below x = 1/48. From our observations, this concentration limit could correspond to the point where the local environment of Ca does not change anymore. Our conclusion also agrees with Roskosz et al. (2015) that underlined that the iron force constant of Cr-bearing spinel is much lower (244 ± 22 N/m) than the Cr-free spinel (296 ± 18 N/m) and explained it as the effect of Cr substitution by Al in the octahedral sites. Moreover, the impact of second neighbors have been observed by Ducher et al. (2016) on zinc isotopes in the case of zincite and franklinite that display different lnβ 66 Zn/ 64 Zn despite the fact that both minerals have the same Zn-O bond lengths and coordination number (4-fold). The only difference is the nature of the second neighbors, iron for franklinite and zinc for zincite. This observation could also explain the difference of iron force constant between dacite and rhyolite glasses with similar Fe 3+ /Fe tot in Dauphas et al. (2014): substitution of Ca for K in dacite or for Na in rhyolite as second neighbor of iron atoms could at least partly explain this variation in β-factors.

Parameters controlling Si isotope fractionation in minerals

In contrast to iron, silicon is not a redox sensitive element in silicic melts and the concentration of silicon is constant in the different silicate groups studied. As a result, the evolution of the silicon force constant in the olivine solid-solution or in the pyroxene group cannot be addressed by a change in silicon concentration. In order to examine the variations in Si β-factors (Fig. 3.1b), we focus here on bond lengths and the nature of the second neighbors.

Si-O bond length

The relationship between silicon force constant and the average Si-O bond length is inexistent in the olivines and pyroxenes investigated here (Fig 3b). In silicon tetrahedra, the Si-O bond lengths show negligible fluctuations around 1.67 Å, while the Si force constant ranges between 608 and 700 N/m. This implies that a parameter other than the Si-O bond length controls the Si isotope fractionation in both olivine and pyroxene groups. This is supported by the theoretical study of Huang et al. (2014) that found similar average Si-O bond length for olivine, clinopyroxene and orthopyroxene of about 1.64 Å. This is also supported by Méheut and Schauble (2014) and Qin et al. (2016) who observed a welldefined correlation between Si-O bond length and Si β-factor for most silicates except for forsterite, enstatite, and clinopyroxenes. In addition, the various clinopyroxenes display the same silicon polymerization but distinct force constants, upholding that polymerization of SiO 4- 4 is not the key factor causing silicon isotope fractionation in silicate minerals (Méheut and Schauble, 2014;Qin et al., 2016). Huang et al. (2014) highlighted the fact that Si isotopes can fractionate between minerals with similar coordination number if their Si-O bond lengths are different. Here, we go one step further in showing that Si isotopes can fractionate between minerals where silicon has the same coordination number and very similar average bond lengths.

Silicon second neighbors

As described for Fe isotopes, it seems that the nature of the silicon second neighbors is the major parameter affecting its force constant among olivines and pyroxenes. In the olivine solid-solution, silicon is, by definition, in an isolated oxygen tetrahedron sharing 3 edges and 6 corners with iron or magnesium octahedra. Here, we identified, as we did for iron, the nature of each cation sharing an edge or corner with the central silicon. Results Méheut and Schauble (2014) discussed that electronegativity of the cations may have an impact on Si isotopes. Depending on the electronegativity difference between Si and neighboring cations, the distribution of the electronic density is modified, the bridging oxygen receive more or less electrons affecting in turn the Si-O bond strength. In their phyllosilicate study, Méheut and Schauble (2014) found that the presence of cations with lower electronegativity (Ca, Na compared to Mg) leads to lighter silicon isotopic signature. This conclusion is true for the geometrical arrangement of cations in the phyllosil-icate structure, but as the same authors already suggested it, this conclusion does not hold for olivine. In phyllosilicates, the presence of cations with lower electronegativity brings more electrons on the bridging oxygen. The Si-O apical bond becomes stronger but the equatorial bonds counteract this effect by weakening, which leads to an overall lowering of the force constant (see Fig. 9 of Méheut and Schauble (2014)). In olivine, because of the more isotropic distribution of the cations around Si atom, there is no such counterbalancing effect. Replacing Mg atoms (χ M g = 1.31 under Pauling's definition) by the more electronegative Fe atoms (χ F e = 1.83) decreases the electronic charge of oxygen atoms (i.e. Bader charge of oxygen atoms decreases from 1.83 in forsterite to 1.66 in fayalite) and therefore decreases the Si force constant. In olivine solid-solution, this electronegativity effect is also enhanced by structural relaxations that lead to Si-Fe distances in fayalite 1.5% longer than Si-Mg distances in forsterite. These longer distances in fayalite contribute to the decrease of Si force constant. Our results on olivine agree with the measurements done by Savage et al. (2011) on natural samples. The Si isotope fractionations between plagioclase and olivine of various compositions indeed suggest that olivine Si β-factor decreases with increasing iron content.

Comparison with previous data

Fe isotopes

We compare our ab-initio iron isotope fractionation β-factors and iron force constants (part 3.4.2) with previous values reported in the literature. The available experimental and theoretical β-factors are displayed in Fig. 3.6. One must note that the uncertainties of NRIXS and Mössbauer results for the various minerals are respectively in the order of ≤ 0.05 % and 0.10 % . And applying the general 5% of relative uncertainty on DFT results would lead to an error bar of maximum ± 0.03 % . The general trend of the silicate βfactors is similar, whatever the techniques used, displaying that, at equilibrium, aegirine is with magnetite the most enriched in 57 Fe followed by ulvospinel, enstatite and forsterite.

Slightly below are diopside, hortonolite, augite, hedenbergite and fayalite. Considering the uncertainties, our theoretical β-factors are lower than the β-factors determined by NRIXS measurement for forsterite (Dauphas et al., 2014) and the approximate theoretical calculations for fayalite from Sossi and O'Neill (2017). In constrast, our β-factors are higher for enstatite and lower for aegirine and magnetite, than those determined from Mössbauer measurements (Polyakov and Mineev, 2000). All other values agree within uncertainties (Fig. 3.6a). The discrepancies between theoretical and Mössbauer β-factor values might be due to the slight chemistry difference between the minerals or/and to the quality of the Mössbauer data used. Indeed, the first value for magnetite β-factor in Polyakov and Mineev (2000) using Mössbauer SOD shift from De [START_REF] De Grave | Mössbauer study of the high-temperature phase of Co-substituted magnetites, CoxFe3-xO4. I. x0.04[END_REF], appeared to be overestimated compared to subsequent magnetite β-factors measurements (Dauphas et al., 2012;[START_REF] Mineev | Equilibrium iron isotope fractionation factors for magnetite from Mössbauer spectroscopy and inelastic nuclear resonant X-ray scattering data[END_REF]Polyakov et al., 2007;Sossi and O'Neill, 2017).

Discrepancies between Mössbauer-derived and DFT-based β-factors have also been observed for pyrite (Blanchard et al., 2009(Blanchard et al., , 2012;;Polyakov and Mineev, 2000). [START_REF] Polyakov | Iron and Sulfur Isotope Factors of Pyrite: Data from Experimental Mössbauer Spectroscopy and Heat Capacity[END_REF] showed that the Mössbauer temperature (θ) used in Polyakov and Mineev (2000) were erroneous by recording a new Mössbauer spectra of pyrite. Pyrite β-factor calculated from this revised Mössbauer data is in good agreement with that obtained by DFT calculations (Blanchard et al., 2009) and by NRIXS measurements [START_REF] Polyakov | Fe β-Factors for Sulfides from NRIXS Synchrotron Experiments[END_REF]. With this in mind, one cannot exclude that aegirine and enstatite β-factor calculated in Polyakov and Mineev (2000) using De Grave and Van Alboom (1991) data could be erroneous for the same reason. A general advice is to avoid combining β-factors obtained from different techniques, whenever possible. Due to systematic errors, relative values (i.e. isotopic fractionation between two phases) within a same technique are expected to be more accurate than differences taken from two distinct techniques. Looking at the olivine solid-solution, one could have doubts about the reliability of the 68 N/m difference between the forsterite iron force constant calculated from NRIXS measurement by Dauphas et al. (2014) and the fayalite one approximated from theoretical calculation using Born-Landé equation [START_REF] Young | Spinel-olivine magnesium isotope thermometry in the mantle and implications for the Mg isotopic composition of Earth[END_REF] by Sossi and O'Neill (2017). However, our DFT approach, where the two end-members have been treated in a consistent way, provides a significant difference too (47 N/m) and thus supports the existence of a Fe isotope fractionation at equilibrium between forsterite and fayalite.

The calculated iron force constant of magnetite (F = 207 N/m) is in agreement with the NRIXS-derived iron force constant of Polyakov et al. (2007) of 228 ± 15 N/m later revised downwards by [START_REF] Mineev | Equilibrium iron isotope fractionation factors for magnetite from Mössbauer spectroscopy and inelastic nuclear resonant X-ray scattering data[END_REF] to 198 ± 15 N/m. The calculated ∆ 57 Fe Mag-Fay = +0.44× 10 6 /T 2 overlaps with the ∆ 57 Fe Mag-Fay = +0.40 × 10 6 /T 2 determined by Sossi and O'Neill (2017). The ∆ 57 Fe Mag-Fay = +0.30 × 10 6 /T 2 calibrated experimentally using the three-isotope method by Shahar et al. (2008) is lower than our calculated ∆ 57 Fe Mag-Fay . However, looking at the only fully equilibrated experiments 'QFM 8' of Shahar et al. (2008) the estimated ∆ 57 Fe Mag-Fay is about × 10 6 /T 2 (Sossi and O'Neill, 2017). Ulvospinel is poorly studied for iron isotopes, however some spinels were studied such as Al-rich spinels (Polyakov and Mineev, 2000) and Ti-free spinels (Roskosz et al., 2015). Ulvospinel in our study displays an average iron force constant of about 151.8 N/m when Polyakov and Mineev (2000) reported a force constant of 106 N/m for their Fe 2+ -spinel (Mg 0.9 Fe 0.1 Al 2 O 4 ), and Roskosz et al. (2015) found a much higher value for their Fe 2+ -spinel of about 196 N/m. Beyond the limitations related to the use of different techniques as discussed above, the difference in term of cationic compositions of the three types of spinel studied may be the reason of such apparent iron force constant differences between these Fe 2+ -minerals.

Iron isotope fractionation factor between olivines and pyroxenes were also determined on natural samples, especially from mantle xenoliths, in various studies (e.g., Beard and Johnson, 2004;Schoenberg et al., 2009;Weyer and Ionov, 2007;Williams et al., 2004).

At relevant mantle temperature (1100K), our calculated ∆ 57 Fe/ 54 Fe diopside-forsterite = -0.03 is identical, within 2 standard deviation uncertainties, to the mean ∆ 57 Fe/ 54 Fe cpx-ol of about 0.04 ± 0.08% measured by Weyer and Ionov (2007) in lherzolite. Schoenberg et al. (2009) measured a ∆ 57 Fe/ 54 Fe augite-ol = 0.10 ± 0.05 % that is close to the calculated ∆ 57 Fe/ 54 Fe augite-fayalite = 0.14 % at 923 K. Moreover, the ∆ 57 Fe/ 54 Fe cpx-ol from Williams et al. ( 2004) of about 0.09 ± 0.07 % obtained on natural samples by mineral separation and solution analysis falls in the range of the calculated ∆ 57 Fe/ 54 Fe diopside-fayalite = 0.15 % and ∆ 57 Fe/ 54 Fe diopside-hortonolite = 0.01 % at T ≈ 1273 K (minerals reflecting the most likely composition of natural samples). Beard and Johnson (2004) determined ∆ 57 Fe/ 54 Fe cpx-ol between -0.06 ± 0.14 % and 0.28 ± 0.14 % and ∆ 57 Fe/ 54 Fe opx-ol between -0.18 ± 0.14 % and 0.28 ± 0.14 % at magmatic temperature (between 1108 and 1373K) while [START_REF] Zhu | Mass fractionation processes of transition metal isotopes[END_REF] measured a mean ∆ 57 Fe/ 54 Fe cpx-ol = 0.28 ± 0.12 % and a ∆ 57 Fe/ 54 Fe opx-ol = 0.32 ± 0.12% . Our calculated values ( ∆ 57 Fe/ 54 Fe diopside-forsterite ≈ -0.03 % and ∆ 57 Fe/ 54 Fe enstatite-forsterite = 0.01 % at the same temperature range) fall within the wide fractionation range measured by Beard and Johnson (2004). These observations as well as the relationship between the δ 57 Fe and Fe concentration of clinopyroxenes pointed out in Beard and Johnson (2004), may suggest that most of the measured iron isotope fractionation factors in the various studies reflect the equilibrium fractionation. However, some iron fractionation factors measured by Beard and Johnson (2004), Schoenberg et al. (2009) and [START_REF] Zhu | Mass fractionation processes of transition metal isotopes[END_REF], are slightly higher than the ones calculated in this study. This slight difference between the measured and calculated iron isotope fractionation factors could be explained by the compositional difference of the mineral phases but could also reflect an isotopic disequilibrium. Previous studies displayed indeed the importance of kinetic effects on iron isotope fractionation in natural samples (e.g., Collinet et al., 2017;Dauphas et al., 2017;Oeser et al., 2015Oeser et al., , 2018;;Sio et al., 2013;Teng et al., 2011).

In summary, the fractionation factors calculated here is a stand-alone set that can be used for a majority of magmatic series, due to its large variety of mineral compositions. This will avoid introducing additional uncertainties related to the use of β-factors derived from different methods (Dauphas et al., 2012;Dauphas et al., 2014), Mössbauer spectroscopy (Polyakov and Mineev, 2000), experimental petrology (Sossi and Foden, 2017), and DFT calculations (Huang et al., 2014, Qin et al. 2016). The uncertainties are ≤ 0.05 % for NRIXS-derived data (Dauphas et al., 2014(Dauphas et al., , 2012;;Polyakov et al. 2007, Mineev et al. 2007) and ≤ 0.10 % for Mössbauer-derived data (Polyakov and Mineev, 2000).

Si isotopes

Silicon isotope fractionation between silicates and metals in the context of core formation has been extensively studied in the past years [START_REF] Bibliography | Silicon isotopes in lunar rocks: Implications for the Moon's formation and the early history of the Earth[END_REF]Georg et al., 2007;[START_REF] Javoy | First-principles investigation of equilibrium isotopic fractionation of O-and Si-isotopes between refractory solids and gases in the solar nebula[END_REF]Poitrasson, 2017;Poitrasson and Zambardi, 2015;[START_REF] Pringle | Silicon isotopes in angrites and volatile loss in planetesimals[END_REF]Shahar et al., 2011;Ziegler et al., 2010). However, only few studies concern silicon fractionation between silicates themselves in the context of magmatic differentiation (Huang et al., 2014;Méheut and Schauble, 2014;Méheut et al., 2009;Qin et al., 2016). Among those by Qin et al. (2016). Secondly, our calculations were performed at zero-pressure and zero-temperature like in Qin et al. (2016), but using the GGA-PBE functional instead of the LDA one. We thus found equilibrium volumes about 7% larger than in the study of Qin et al. (2016) for the three minerals. This observation that explains our lower β-factors, is consistent with the general knowledge of the functional effect on the bond length description. LDA functional is known to underestimate by 2-3 % the experimental volumes (e.g., Qin et al., 2016) whereas GGA functionals tend to overestimate volumes (Table 3.1). Thus, the computational parameters likely explain this systematic difference on β-factors, but do not affect the inter-mineral fractionations. This emphasizes again the importance of using a consistent data set to compute inter-mineral isotopic fractionations. Lastly, in addition to these temperature and functional effects, one must note that the pyroxenes in the present study have some atoms of iron in their M sites whereas the pyroxenes in the previous studies are iron-free. Following our observations, the nature of second atomic neighbors likely impacts the Si β-factors. For instance, in the olivine solid-solution, having iron atoms as silicon second neighbors lowers the Si force constant, and as a consequence the Si β-factor.

Geochemical implications on magmatic differentiation

In many studies, the Fe isotope fractionation factor between the melt and olivine (∆ melt-olivine ) is considered to represent ∆ melt-Fe 2+ silicates (e.g., Dauphas et al., 2014;Sossi et al., 2012).

In a core-mantle differentiation context or in less differentiated planetary bodies than Earth, pyroxenes have mainly the composition of enstatite and olivines are forsterite-like.

In these contexts, calculations agree that there is no significant Fe isotope fractionation between silicate minerals. This inference is supported by experimental determination of isotopic partitioning (Prissel et al., 2018) and interplanetary comparisons [START_REF] Poitrasson | A reassessment of the iron isotope composition of the Moon and its implications for the accretion and differentiation of terrestrial planets[END_REF]. However, considering the upper crust formation, our results evidence significant iron isotope fractionation between Fe 2+ -bearing minerals, even with similar Fe-O bond lengths. The use of our set of isotopic fractionation factors for modeling the fractional crystallization occurring during magmatic differentiation can help to assess the role of Fe 2+ -bearing silicates on the iron heavy signature of the granites and evolved volcanic lavas compared to basalt. Here, simple models are performed based on Rayleigh equation:

R = R 0 f α bulk -1 (3.5)
where, R is the 57 Fe/ 54 Fe isotopic ratio and R 0 is the initial ratio, f is the fraction of remaining melt and α bulk represent the bulk fractionation factor between residual melt and crystallizing minerals computed according to Eq. 3.6 as described below.

We used the Magma Chamber Simulator (MCS; [START_REF] Bohrson | Thermodynamic model for energy-constrained open-system evolution of crustal magma bodies undergoing simultaneous recharge, assimilation and crystallization: The magma chamber simulator[END_REF]) based on MELTS software [START_REF] Gualda | Rhyolite-MELTS: a Modified Calibration of MELTS Optimized for Silica-rich, Fluid-bearing Magmatic Systems[END_REF] in order to model fractional crystallization of an andesitic melt as in Dauphas et al. (2014) that simulates the formation of peraluminous A-type granite with QFM (Quartz-Fayalite-Magnetite) buffer. In the MCS, composition and proportion of the minerals removed from the melt are described every 5 • C step, during a temperature drop from 1165 • C to 885 • C. From this information, the bulk fractionation factor is calculated by taking the calculated mineral-melt fractionation factor (α 1 , . . . , α n ) weighted by their proportions (X 1 , . . . ,X n ) :

α bulk = α 1 × X 1 + ... + α n × X n (
F = F 0 + 41 1 + e 34-SiO 2 2 (3.7)
The iron force constant value of the initial melt F 0 (199 N/m) is similar to the forsterite one (197 N/m) according to Dauphas et al. (2014). In this study, the initial value is replaced by the DFT force constant of forsterite (147 N/m) in order to correct for any method bias between our first-principles calculations and the NRIXS-derived data.

Two different models were performed in order to see the impact of Fe 2+ -bearing silicates on iron isotope partioning during magmatic differentiation: i) The first model applies a general α f orsterite-melt to represent the fractionation between the melt and all

Fe 2+ -bearing silicates as previously done in Dauphas et al. (2014) and Sossi et al. (2012).

ii) The second model uses distinct fractionation factors for orthopyroxene and clinopyroxene α enstatite-melt and α augite-melt . In order to be representative of the spinel composition along the magnetite-ülvospinel solid-solution, the α spinel-melt is considered as a mixture of α lvospinel-melt and α magnetite-melt in models 1 and 2. In order to account for ilmenite crystallization at the last stage of magma differentiation (SiO 2 > 75 Wt%), the ilmenite β-factor was calculated from the ilmenite Fe force constant of 137 N/m determined by Sossi and O'Neill (2017).

Despite the mentioned approximation for the melt β-factor, the modeled iron isotopic evolution (Fig. 3.7), is consistent with the values of natural samples taken from the literature (Foden et al., 2015;Poitrasson and Freydier, 2005;Schuessler et al., 2009;Sossi et al., 2012;Teng et al., 2008;Williams et al., 2018;Zambardi et al., 2014) with a small increase of δ 57 F e from 0.1 to 0.15 % up to 68% SiO 2 followed by a sharp increase up to 0.71 % for model 1, and up to 0.98 % for model 2 at 76 % SiO (Dauphas et al., 2014). The circles in the background correspond to δ 57 Fe from the literature (Du et al., 2017;Foden et al., 2015;He et al., 2017;Poitrasson and Freydier, 2005;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012;Teng et al., 2013;Williams et al., 2018;Xia et al., 2017;Zambardi et al., 2014).

Conclusion

Equilibrium iron and silicon isotope fractionation factors were calculated from first-principles methods in silicate and oxide minerals covering a wide range of minerals formed during magmatic differentiation. Significant isotopic fractionations are observed between the studied minerals even at high temperature. Our results allow us to conclude that Fe and Si isotope fractionations among Fe 2+ -bearing minerals are not negligible in magmatic contexts. Among the parameters investigated, the results show that oxidation state and coordination number explain to a certain extent isotopic fractionation. Accordingly, Fe 3+bearing minerals and minerals that have four-fold coordinated iron display higher force constants than Fe 2+ -bearing minerals and minerals that have six-fold coordinated iron.

However, this study allows to go a step further in showing that after temperature, the main parameter influencing the β-factors of minerals containing atomic species displaying similar coordination number and charge is the local crystal chemical environment. It can be described as the nature and number of the cations bound to the atom of interest.

This theoretical study of some of the main magmatic minerals provides a self-consistent set of fractionation factors to better assess the processes involved in isotopic fractionation during crust formation. As such, it could help to better understand the bulk Si and Fe isotopic compositions of igneous crustal rocks. Simple fractional crystallization models using our DFT-derived β-factors enabled us to underline the significant impact of the Fe 2+bearing minerals on iron isotope fractionation during magmatic differentiation. However, in order to precisely quantify the impact of fractional crystallization on the Fe isotopic evolution of melts, it is essential to have the theoretical β-factors for different types of melt (e.g with various Fe 3+ /Fe tot ratios and with different cationics environment).

Acknowledgements

CHAPTER 4

Iron and silicon isotope fractionation in silicate melts using first-principles molecular dynamics.

Ségolène RABIN, Marc BLANCHARD, Carlos PINILLA, Franck POITRASSON, Michel isotopes signature. Finally, we calculated that olivine at equilibrium with a basaltic melt could display an iron isotope composition down to -0.14 % for δ 57 Fe. Hence, the lower isotopic compositions (δ 57 Fe < -0.14 % ) observed in natural olivines are most likely due to diffusion-driven kinetic fractionation.

Introduction

Understanding the detailed structural properties of iron-bearing silicate melts as well as the behavior of the constituting cations remains one of the great challenges in geosciences and in the glass-making industry. Many experimental and theoretical studies on melts from simple composition to natural-like melts have been carried out showing that cations could be either network former or network modifier depending on the nature of the cations and their coordination number that are dependent also of pressure, temperature and redox state of the melt [START_REF] Angeli | Insight into sodium silicate glass structural organization by multinuclear NMR combined with first-principles calculations[END_REF][START_REF] Bajgain | Structure and density of basaltic melts at mantle conditions from first-principles simulations[END_REF][START_REF] Bauchy | Atomic scale foundation of temperature-dependent bonding constraints in network glasses and liquids[END_REF][START_REF] Caracas | Melt-crystal density crossover in a deep magma ocean[END_REF][START_REF] Gerlach | Self-diffusion of iron in alkali -magnesia -silica glass melts[END_REF]Ghosh andKarki, 2018, 2020;Guillot and Sator, 2007a,b;[START_REF] Jackson | Multi-spectroscopic study of Fe(II) in silicate glasses: Implications for the coordination environment of Fe(II) in silicate melts[END_REF][START_REF] Karki | First-principles molecular dynamics simulations of silicate melts: Structural and dynamical properties[END_REF][START_REF] Karki | First-principles computation of mantle materials in crystalline and amorphous phases[END_REF][START_REF] Kim | Effect of iron content on the structure and disorder of iron-bearing sodium silicate glasses: A high-resolution 29Si and 17O solid-state NMR study[END_REF][START_REF] Le Losq | The role of Al3+ on rheology and structural changes in sodium silicate and aluminosilicate glasses and melts[END_REF][START_REF] Mcmillan | Structural studies of silicate glasses and melts-applications and limitations of Raman spectroscopy[END_REF][START_REF] Oestrike | High-resolution 23Na, 27Al and 29Si NMR spectroscopy of framework Aluminosilicate glasses[END_REF][START_REF] Rossano | Environment of ferrous iron in CaFeSi2O6 glass; contributions of EXAFS and molecular dynamics[END_REF]Solomatova andCaracas, 2019, 2021;[START_REF] Stein | Molecular dynamics simulations of liquids and glasses in the system NaAlSiO4-SiO2: Physical properties and transport mechanisms[END_REF]. Silicon in melt is mostly four-fold coordinated and acts as the main network former of the melt structure. Despite the fact that silicon could be found as Si0 in metallic inclusions of meteorites and planetary cores (Poitrasson, 2017), silicon is considered as only tetravalent in the upper mantle and crust of the Earth.

Hence, silicon is not redox sensitive in terrestrial melts whereas iron is. Iron in melts has been assessed to behave differently in the melt depending on its redox state (Fe 2+ or Fe 3+ ). At low pressure, ferric iron is most likely to be four-fold coordinated and acts as a network former, as Al 3+ , whereas ferrous iron is five to six-fold coordinated and acts as a network modifier (see [START_REF] Mysen | The structural behavior of ferric and ferrous iron in aluminosilicate glass near meta-aluminosilicate joins[END_REF]; Mysen and Richet (2019) and references therein).

However, the role of ferrous and ferric iron in melts is still debated as some studies demonstrated that both could be four to six-fold coordinated and therefore they could both act as network modifier [START_REF] Ghosh | Effects of valence and spin of Fe in MgSiO3 melts: Structural insights from first-principles molecular dynamics simulations[END_REF]. Recent study of [START_REF] Ghosh | Effects of valence and spin of Fe in MgSiO3 melts: Structural insights from first-principles molecular dynamics simulations[END_REF] displayed indeed that iron redox state has limited impact on the general melt properties.

In contrast, the iron spin state has more influence on the structural properties of the melt. Some uncertainties on the behavior of cations in the melt remain and, one of the major unknown so far is the isotopic properties of melts, including those of silicon ( 28 Si, 29 Si, 30 Si) and iron ( 54 Fe, 56 Fe, 57 Fe, 58 Fe). Analytical developments and improvements over the past decades enabled scientists to investigate non-traditional stable isotope fractionation at high-temperature. Several studies carried out on natural and experimental samples have revealed measurable (> 0.05 % ) silicon and iron isotopic fractionations in igneous systems: i) Iron and silicon display heavy isotope enrichment during magmatic differentiation, although the trends are different (e.g., Dauphas et al., 2014;Foden et al., 2015;Gajos et al., 2016;Li et al., 2020;Nebel et al., 2015;Poitrasson and Freydier, 2005;Savage et al., 2011;Schoenberg et al., 2009;Shahar et al., 2008;Sossi et al., 2012;[START_REF] Weyer | Coupled lithium-and iron isotope fractionation during magmatic differentiation[END_REF]Weyer et al., 2005;Williams et al., 2005Williams et al., , 2018;;Zambardi et al., 2014),

ii) Oceanic crust (MORB and OIB) displays heavier iron isotope signature than some mantle peridotites (e.g., Nebel et al., 2018;Sossi et al., 2016;Williams and Bizimis, 2014;Williams et al., 2009), iii) There is significant iron isotopes fractionation, associated to Fe-Mg chemical zoning, within single crystals of olivine (Collinet et al., 2017;Oeser et al., 2015Oeser et al., , 2018;;Prissel et al., 2018;Sio et al., 2018;[START_REF] Sio | Thermal and crystallization histories of magmatic bodies by Monte Carlo inversion of Mg-Fe isotopic profiles in olivine[END_REF]Sio et al., 2013;Teng et al., 2011). The potential processes involved in the documented isotopic variations, such as partial melting (e.g., [START_REF] Deng | An oceanic subduction origin for Archaean granitoids revealed by silicon isotopes[END_REF]Sossi et al., 2016;Weyer and Ionov, 2007;Williams et al., 2009), fractional crystallization (e.g., Du et al., 2017;Foden et al., 2015;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012), diffusion-driven kinetic fractionation (Lundstrom, 2009;Zambardi et al., 2014) or even fluid exsolution during late stage of differentiation (Heimann et al., 2008;Poitrasson and Freydier, 2005;Telus et al., 2012), were then investigated. However, interpretations of these observed isotopic varia- 2015)). Grant (1954) and Poitrasson and Zambardi (2015) proposed polymerization degree as a key factor influencing the Si isotope fractionation during magmatic differentiation. However, at the mineral scale, this relationship does not seem to be the sole parameter involved (Méheut and Schauble, 2014;Qin et al., 2016;Rabin et al., 2021). The mechanisms listed above could play as well an important role on silicon and iron isotope fractionation in melts.

To investigate the mechanisms involved in melt isotope fractionation, we have calculated consistent Si and Fe β-factors for various melt compositions using first-principles molecular dynamics (FPMD). In this study, five silicic melt compositions were investigated in order to assess the effect of increasing sodium, iron and silicon contents on the Si and Fe isotope composition of melts at 1 GPa. The pressure effect on isotope fractionation is also explored through a FPMD basalt simulation at 12 GPa. Our calculated Si and Fe β-factors for melts were combined to the set of Fe and Si β-factors of silicate and oxide minerals calculated by Rabin et al. (2021) and Qin et al. (2016) to obtain a full set of equilibrium fractionation factors between minerals and melts of various compositions.

These data enable us to discuss the impact of equilibrium fractionation, through different processes such as fractional crystallization and partial melting, on Fe and Si isotopes systematics in igneous environment.

Methodology

Modelling approach

The structure properties as well as the reduced partition function ratios (β-factors) of each melt were obtained from first-principles molecular dynamics simulations (FPMD)

based on density functional theory (DFT). All calculations were performed using the Perdew-Burke-Ernzerhof (PBE) parametrized generalized gradient approximation (GGA)

functional (Perdew et al., 1996) implemented in the Vienna Ab-initio Simulation Package (VASP; Kresse and Furthmüller (1996)). The interaction between the valence and core electrons were modelled using the projector-augmented wave (PAW) method (Kresse and Joubert, 1999). Plane-wave cutoff of 500 meV and Brillouin-zone sampling at Gamma are used. The simulation cell of each system is constrained to a cube (Table 4.1). The design of the different melt structures in this study was conducted in 3 steps: i) A primary melt structure including 4 compounds (MgO, CaO, Al 2 O 3 , SiO 2 ) was generated using classical MD simulations, for few nanoseconds, based on DL-POLY code [START_REF] Smith | DL_POLY: Application to molecular simulation[END_REF] and using empirical potentials proposed by [START_REF] Matsui | A Transferable Interatomic Potential Model for Crystals and Melts in the System CaO-MgO-Al2O3-SiO2[END_REF]. ii) Substitution of some Mg atoms by Fe, some Ca atoms by Na or K was then performed in order to reach the desired melt compositions. iii) The structure thereby created was used as starting composition for the FPMD simulations. The structure was first equilibrated at high temperature (well above the melting point, > 6000 K) using NVT canonical ensemble (fixed number of atoms N, constant volume V and constant temperature T) and then cooled down to a desired lower temperature (4000 K). Once the system was at the desired temperature, the isothermal-isobaric ensemble NPT (fixed number of atoms N, constant pressure P and constant temperature T) with variable-cell dynamics was used, for few picoseconds (ps)

to optimize the cell parameters of each different melt composition, in order to approximate the experimental volume of the liquid under consideration (Table 4.1). The volume of the melt is indeed a key parameter as the liquid structure is primarily a function of volume, rather than pressure [START_REF] Karki | First-principles molecular dynamics simulations of silicate melts: Structural and dynamical properties[END_REF]. Then, NVT canonical ensemble were used to explore the effect of the various liquid compositions, as well as the pressure effect, on iron and silicon isotopic properties. Both NPT and NVT ensembles used Nosé thermostats (Nosé, 1984). Each system contains at least 100 atoms in order to have a simulation cell large enough to calculate equilibrium properties [START_REF] Karki | First-principles molecular dynamics simulations of silicate melts: Structural and dynamical properties[END_REF]. To confirm that 100-atoms simulation cells were sufficient to describe the short-and medium-range properties of the silicate melts, a calculation on a 183-atoms simulation cell was also performed. The duration of the FPMD NVT simulation is also a critical parameter in order to satisfy the ergodicity principles (i.e. the system explores all the possible states of the system, with the same probability as in a liquid, during the simulation time). Therefore, the simula-nb atoms V (Å 3 ) Density ρ (g/cm 3 ) Pressure (GPa) Temperature (K) tions were carried on for approximately 20 picoseconds in order to acquire representative ensemble at the desired temperature [START_REF] Bajgain | Structure and density of basaltic melts at mantle conditions from first-principles simulations[END_REF]. Analysis of the mean-square displacements (MSD) as a function of time (Allen and Tildesley, 1989) were performed for each temperature-volume-composition condition, in order to confirm that the system stays in the liquid state (Appendix B, Fig. 3 to 9).

The mean pressure applied during the simulations is chosen to be around 1 GPa (corresponding to a crustal depth of ≈ 35 km), in order to reproduce the pressure conditions of deep magmatic chambers such as in Mont Pelée [START_REF] Pichavant | Physical conditions, structure, and dynamics of a zoned magma chamber: Mount Pelée (Martinique, Lesser Antilles Arc)[END_REF] or Iceland [START_REF] Kelley | Pressures of crystallization of Icelandic Magmas[END_REF]. Moreover, partial melting happens at pressures between 1 and 3 GPa (e.g., [START_REF] Bertka | Anhydrous partial melting of an iron-rich mantle I: subsolidus phase assemblages and partial melting phase relations at 10 to 30 kbar[END_REF]. The exact mean pressures of each melt are displayed in Table 4.1 and more detailed analysis of pressure evolution during the MD simulations are available in Appendix B, figure 10.

The melts simulated here, are composed of 7 to 8 chemical elements (Na, Ca, K, Fe, Mg, Al, Si, O) in order to reproduce faithfully natural melt compositions. Three calc-alkaline melts (iron-rich (IR) basalt, iron-depleted (ID) basalt and trachyte), and two peralkaline melts (basanite, phonolite) were studied in order to assess the isotopic evolution of the melt through differentiation (increasing SiO 2 content) as well as the effect of alkaline content in the melts on iron and silicon isotopic fractionation. The melts compositions are displayed in Table 4.2. Ghosh and Karki (2020) showed that iron spin state configuration is a major parameter that influences greatly (> 15 %) the structural parameters of the melt. In our study, iron atoms are set up in high spin-state in each melt because it is known that iron is to be predominantly in high-spin states in glasses and melts (Mysen and Richet (2019) and references therein). The Hubbard U term was not added to the DFT calculation as previous studies showed that there are no significant changes in terms of Fe local environment and melts properties (pressure, temperature, spin and valence state) between GCA and GCA+U calculations in liquid systems where Fe is in high-spin state, especially at high temperature (e.g., [START_REF] Caracas | Melt-crystal density crossover in a deep magma ocean[END_REF]Ghosh andKarki, 2018, 2020;[START_REF] Solomatova | Pressure-Induced Coordination Changes in a Pyrolitic Silicate Melt From Ab Initio Molecular Dynamics Simulations[END_REF]. All iron atoms were set up as Fe 2+ , in order to reproduce realistic Fe 3+ /Fe tot composition in the melts as MORB displays

Fe 3+ /Fe tot of about 0.09 (Bézos et al., 2021) and OIB of about 0.2 (Cottrell and Kelley, 2014). This allowed us also to observe structural parameter evolution unrelated to change of iron spin polarization and iron oxidation state.

In addition to melt simulations, first-principles calculations are performed on two silicate minerals (Appendix B, Table 7), using the same PBE-GGA functional and PAW pseudo-potentials as described above.

Calculation of equilibrium isotope fractionation factors

The equilibrium isotope fractionation factor α_(a,b,Y) of an element Y between two phases a and b is defined as :

α (a,b,Y ) = β (a,Y ) β (b,Y ) (4.1)
where β(a,Y) ,called β-factor, is the reduced partition function ratio between phase a and a perfect gas of Y atoms. The usual notation of isotope fractionation, in permil (% ), is written 10 3 lnβ (a,Y ) . The isotope fractionation factor is therefore expressed as

10 3 lnα (a,b,Y ) = 10 3 lnβ (a,Y ) -10 3 lnβ (b,Y ) ≈ ∆ a-b = δ a -δ b .
The notations δ a and δ b , used in laboratory studies, represent the isotopic compositions of phase a and b in isotopic equilibrium and are expressed as, in the case of iron:

δ 57 F e( ) =
57 F e/ 54 F e sample 57 F e/ 54 F e standard -1 × 1000 (4.

2)

The reduced partition function ratios of the melts were calculated in this study using two different methods referred to as the "snapshot" method and the "VAF" method.

The "snapshot" method

The molecular dynamics trajectories are constituted of up to 40 000 time-steps (∆ t = 0.5 fs). In order to obtain the β-factor of each melt, a number of snapshots are extracted from the trajectory, at regular time interval (1 ps interval), large enough to consider the snapshots as independent from each other. Then, each snapshot structure is relaxed to their equilibrium structure at T = 0 K and a finite pressure of 1 GPa, until the residual forces on atoms are less than 10 -4 Ry/au, and the residual pressures smaller than 0.002 GPa. Hence, the equilibrated structure does not correspond anymore strictly to a melt but rather to a glass. The "frozen phonon" approach is then used to determine the dynamical matrix for each snapshot: 1) Phonopy code [START_REF] Togo | First principles phonon calculations in materials science[END_REF] is used to generate the 6N (N = number of atoms in the simulation cell) configurations corresponding to each atomic displacement in the three dimensions (3N in the positive axis, 3N in the negative axis as it is a glass without symmetry). 2) self-consistent field (scf) calculations are then performed on each configuration using VASP. 3) Post-treatment is performed with Phonopy to obtain the dynamical matrix that is subsequently diagonalized for the two atomic masses of interest in order to obtain the vibrational frequencies for both isotopologues. The reduced partition function ratio is finally calculated from these vibrational frequencies (Bigeleisen and Mayer, 1947):

β a,Y = 3Nat i=1 ν i ν i e (-ν i /2k B T ) 1 -e (-ν i /k B T ) 1 -e (-ν i /k B T ) e (-ν i /2k B T ) 1/N (4.3)
where, ν i and ν i are the vibrational frequencies of the two isotopologues (the prime symbol refers to the light isotope of the element Y), identified by a branch index i from 1 to 3N at ;

Nat is the number of atoms in the simulation cell containing N atoms of the element Y; h is the Planck constant, k B is the Boltzmann constant, and T the temperature (in Kelvin). Equation 4.3 takes into account that there should be no isotopic fractionation at the high-temperature limit (i.e. the "Redlich-Teller" rule, [START_REF] Redlich | Eine allgemeine beziehung zwischen den schwingungsfrequenzen isotoper molekeln[END_REF]).

In order to obtain accurate and precise Si and Fe β-factors for each melt, "frozen phonon" calculations were performed on 6 to 10 snapshots per FPMD simulation. The snapshot β-factors are then averaged in order to obtain a mean β-factor for each melt and the uncertainties associated correspond to the standard error of the mean β-factor. In this method, the interatomic force constant (F) can be calculated from the dynamical matrix by averaging the force constants of the atom of interest (Fe or Si) in the three spatial dimensions x,y,z. The number of snapshots is large enough as shown by the cumulative mean of the β-factor and Fe force constant (Appendix B, fig. 11).

The same methodology based on the vibrational frequencies was used to calculate the β-factors of the two minerals, except that only 3N configurations were generated by Phonopy, because of crystal symmetry.

The "VAF" method

The "snapshot" method described above is a tried-and-tested method, especially for solvated species [START_REF] Ducher | Equilibrium isotopic fractionation between aqueous Zn and minerals from first-principles calculations[END_REF][START_REF] Dupuis | Fractionation of silicon isotopes in liquids: The importance of configurational disorder[END_REF][START_REF] Kowalski | Prediction of equilibrium Li isotope fractionation between minerals and aqueous solutions at high P and T: An efficient ab initio approach[END_REF][START_REF] Pinilla | Equilibrium magnesium isotope fractionation between aqueous Mg2+ and carbonate minerals: Insights from path integral molecular dynamics[END_REF] but the resultant β-factor does not strictly correspond to a liquid but rather to a glass as the atomic positions are relaxed at T = 0 K prior to the calculation of the β-factor. Another approach that allows to keep the temperature effect and the liquid properties (i.e. configurational disorder as well as possible anharmonic effects), is a method based on the kinetic energy of the atoms during the simulations following the thermodynamic perturbation (Landau and Lifshitz, 1980) where the β-factor is expressed as:

lnβ AY = m -m m K RT - 3 2 (4.4)
where K is the kinetic energy of the element of interest and can be calculated from the partial vibrational density of state (VDOS) noted g(e): Here, the VDOS, g(e), is obtained from the Fourier transform of the velocity autocorrelation function (VAF) thermodynamically averaged over all atoms [START_REF] Meyer | Vibrational density of states of silicon nanoparticles[END_REF]:

g(e) = +∞ -∞ dt N i=1 V i (t)|V i (0) N i=1 V i (0)|V i (0) e i2πνt (4.7)
where V i is the velocity of the i-th atom.

In this method, the interatomic force constant (F) can also be retrieved from the VDOS, using the following equation (e.g., Dauphas et al., 2012) :

< F >= m 2 +∞ 0
g(e)e 2 de (4.8)

In this study, the "VAF" method is applied on 10 000 time-steps trajectories (5 ps), enabling us to obtain 2 to 4 β-factors for each melt that are then averaged. Uncertainties are estimated as the 2-standard error of this calculated mean. The "VAF" method was only applied on iron isotopes system as iron can be a network modifier, more likely to be affected by the transition liquid-glass (i.e. Fe-O coordination number, Fe-O distance) compared to Si that is a network former.

The main limitation of using the VDOS to calculate the β-factor of a liquid is the fact that the VDOS display a non-zero value at zero frequency due to diffusion effects.

In principle, this diffusive contribution should be removed before calculating the β-factor.

Such corrections in complex silicate melts would be difficult to achieve since it would require precise knowledge of the molar volume contribution of each species in the melt.

Furthermore, [START_REF] Pinilla | Firstprinciples investigation of equilibrium iron isotope fractionation in fe1x s x alloys at earth's core formation conditions[END_REF] showed that these fluidicity effects were negligible in the calculation of equilibrium fractionation factor of iron in liquid Fex-1Sx alloys. Thereby, such effects were neglected here

Results

Structural melt properties

The partial radial distribution functions (RDF) of the different atomic species pairs present in the silicate melts were analyzed in order to determine the average interatomic distances as well as the atomic coordination (Appendix B, fig. 12 to 18). The atomic coordination (C) for a given species A with respect to another species B can be expressed as:

C AB = 4πρ B r min 0 r 2 g AB (r) dr (4.9)
where g AB (r) is the partial radial distribution function of the A-B atomic pairs and C AB represents the number of atoms of species B which lie within a spherical region of radius r min centered around the atom of species A.

The average cation-oxygen distances and coordination numbers are displayed in Table 4.3. The results are generally consistent with previous molecular dynamics results performed at high temperature (e.g., [START_REF] Bajgain | Structure and density of basaltic melts at mantle conditions from first-principles simulations[END_REF][START_REF] Caracas | Melt-crystal density crossover in a deep magma ocean[END_REF]Ghosh andKarki, 2018, 2020;Guillot and Sator, 2007a;[START_REF] Solomatova | Buoyancy and Structure of Volatile-Rich Silicate Melts[END_REF]. The 183atoms simulation of the basalt_IR displays similar cation-oxygen coordination numbers (difference < 1.8 %) and cation-oxygen average bond lengths (difference < 1%) as the 100-atoms simulation (Table 4.3). This allows us to confirm that 100-atoms simulations are large enough to discuss about the short-and medium-range structural properties and therefore equilibrium isotopic properties. At low pressure, the calculated Si-O distance is about 1.63 Å and the mean coordination number is about 4.1 for all melts studied (Table 4.3) showing that Si-O pairs are not sensitive to composition as observed in Guillot and Sator (2007a). The proportion of five-fold and six-fold Si-O coordination increase in the simulated liquids at high pressure, leading to a mean Si-O CN of 4.64. This pressure effect on Si-O pairs, even if not observed by experimental studies, was also observed in the molecular dynamics study of [START_REF] Bajgain | Structure and density of basaltic melts at mantle conditions from first-principles simulations[END_REF]. In the case of aluminum, the Al-O distance is about 1.76-1.81 Åfor all melts corresponding to a coordination number between 4.57 and 4.75 at low pressure. Aluminum coordination number distribution (Fig.

4.1)

shows that about 50% of the aluminum is 4-fold coordinated and behaves as network former in the different melts as expected (Mysen and Richet, 2019). The pressure effect on Al-O pairs is greater than the effect observed on silicon, with an average coordination number of about 5.45. The population of 6-fold coordinated aluminum has indeed tripled compared to the basalt_IR under low-pressure condition (Fig. Guillot and Sator (2007a). the number of Non-Bridging Oxygen atoms, and T stands for tetrahedral representing the network formers (Si, Al in this study). These NBO/T ratios (Table 4.2) show that the more evolved melts such as trachyte and phonolite (NBO/T = 0.28) are more polymerized than the basaltic melts (NBO/T=1).

Ca-O Na-O K-O Fe-O Mg-O Al-O Si-O CN d(Å) CN d(Å) CN d(Å) CN d(Å) CN d(Å) CN d(Å) CN d(Å) Basalt_IR 6.
The structural homogeneity of melts can be defined according to glass specialists using a simple parameter characterizing the degree of occupation of the interstitial site C=M/T [START_REF] Kuryaeva | Degree of polymerization of aluminosilicate glasses and melts[END_REF], where M represents the network-modifiers, and T the network formers.

A melt structure is considered homogeneous when C < 0.27. The melt homogeneity is mostly driven by the total concentration of alkali and alkali-earth metals oxides [START_REF] Kuryaeva | Degree of polymerization of aluminosilicate glasses and melts[END_REF]. In this study, melts are heterogeneous, displaying C values ranging between 0.45 for phonolite and 0.67 for the calc-alkaline basalt_IR and basalt_ID. This high degree of occupation of interstices is indeed favorable to individual structural groupings in the melts (i.e. microheterogeneous structure). The heterogeneity of aluminosilicate melts, caused by high content of sodium cations, leads to changes in the properties that are not accounted for in the degree of polymerization calculation. This explains why there is no apparent relationship between NBO/T and the Fe-O coordination number and poor correlation with Si-O coordination number.

Reduced partition function ratios of iron and silicon.

The theoretical reduced partition function ratios (β-factors) have been calculated for iron and silicon. Their temperature dependences are displayed in figure 2 as well as their corresponding polynomial fits in Table 4.5. As observed for minerals (Blanchard et al., 2017;Rabin et al., 2021), there is a linear correlation between melt β-factor and their associated average interatomic force constant (Fig. 4.3). As described in part 4.3.2, two methods were used to obtain the Fe β-factors and force constants: the "snapshot" and "VAF" methods. Using the "snapshot method", the force constant <F> is calculated from the dynamical matrix by averaging the force constants of the atom of interest (Fe or Si) in the three spatial dimensions x,y,z. For all melts, both methods give similar results for F within uncertainties (Table 4.6). At 1000 K, the 10 3 lnβ 57 Fe/ 54 Fe for all melts span in a range of ≈ 1.1 % and the 10 3 lnβ 30 Si/ 28 Si are within a range of ≈ 0.5 % . The variation of Fe isotope compositions within melts are greater than the one observed in silicate minerals whereas the variation of Si isotope compositions is divided by two between melts and minerals (Rabin et al., 2021). Phonolite, the most evolved peralkaline melt displays the highest iron β-factors followed by the calc-alkaline iron-rich basalt (Basalt_IR) whereas the calc-alkaline iron-poor basalt (Basalt_ID) displays the lowest iron β-factors. On the other hand, the highest Si β-factor is found in trachyte, followed closely by phonolite. The linear fit for silicon is f (x) = 0.0046x for melts and minerals.

Fe force constant

Si force constant « Snapshot » method « VAF » method « Snapshot » method 4.6: Comparison of the iron average force constants of the melts (in N/m) using either the "snapshot" method or the "VAF" method as well as the silicon force constants of the melts using the "snapshot" method.

Discussion

4.5.1 Iron and silicon environment in the silicate melts.

Our results on the iron and silicon structural properties allow us to discuss the iron and silicon environment in the different melts studied. As mentioned in part 4.4.1, silicon structural properties are independent of the melt composition whereas iron is sensitive to the compositional change of the melt.

The nature of the network former in the melt is one of the parameters investigated that have an impact on iron atoms bonding-properties. We observe a linear relationship between Fe-O coordination number and the Al/(Si+Al) ratio in melts as described in previous studies [START_REF] Mysen | The structural behavior of Al3+ in peralkaline melts and glasses in the system Na2O-Al2O3-SiO2[END_REF]. Phonolite and basanite display the highest Fe-O coordination numbers that correspond to the highest Al/(Si+Al) ratio. Peralkaline melts are indeed, by definition, depleted in silicon. The increasing aluminum content as network former over silicon results in an increase of the coordination number of all the network modifiers (Table 4.3) including a part of aluminum acting as network modifier (Fig 4.1).

The iron effective charge in the studied melts shows little variation (2.28-2.37) that cannot be correlated to systematic changes in coordination number or Fe-O distances (Table 4.2 and 4.3). [START_REF] Ghosh | Effects of valence and spin of Fe in MgSiO3 melts: Structural insights from first-principles molecular dynamics simulations[END_REF] showed indeed that ferric and ferrous iron can have the same coordination number in melts and that variations of iron effective charge in melts cause no significant change of the structural properties of iron.

The structural heterogeneity of the melts, defined in part 4.4.1 as related to the abundance of network modifiers, indicates the potential presence of individual structural groupings that could also induce segregation of cations in the melts. Therefore, it is interesting to look at the nature and distribution of the network modifiers cations around iron and silicon (Table 4.4). Iron tends to favor iron as its closest neighbor in melts when iron concentration is relatively high (basalt_IR, basanite; Table 4.4). The distribution of iron atoms as a next nearest neighbor (NNN) of another iron can be determined by dividing the number of Fe NNN by the total number of its NNN's (i.e. CN Fe-Fe /(CN Fe-X , with X = Fe, Ca, Na, Si, Mg, and Al, Table 4.4, Rossano et al. (2000)). In the case of a random distribution of cations in the melt structure, this ratio is only dependent of the melt stoichiometry. For the basaltic melt (basalt_IR), a random cation distribution model predicts a value of 0.1, while our FPMD simulation indicates a value of 0.14. Concerning the basanite melt, a random distribution of iron in the melt corresponds to a value of 0.12. FPMD simulation of basanite gives a ratio of 0.14. These results suggest a slight segregation of iron in the basanite and basalt_IR melts. These results match with a random distribution of iron in the melts with a potential slight segregation in the basaltic melt. Concerning silicon, unlike iron there is no systematic preferences in term of the nature of second neighbors. The next nearest neighbor of silicon is alternatively silicon itself, aluminum, potassium or even calcium.

4.5.2 "Snapshot" vs "VAF" method.

The Fe reduced partition function ratios were calculated using two methods: the "VAF" method and the "snapshot" method. In the "snapshot" method the β-factors are extrapolated to magmatic temperatures using Eq. (3) using the vibrational frequencies calculated at 0 K and P = 1 GPa. On the other hand, β-factors determined by the "VAF" method are extrapolated from the MD run at 4000 K and P 1 GPa using Eq. ( 4). Results

show that the "VAF" and "snapshot" methods give similar force constants (Table 6) and therefore β-factors since both quantities are linearly correlated (Fig. 2). This is consistent with the fact that the structures obtained with the "snapshot" method are really similar to the ones from the "VAF" trajectories in terms of bond lengths, coordination number and nature of Fe neighbors. The calculated uncertainties are twice higher with the "VAF" method than using the "snapshot" methods but uncertainties could be certainly reduced by increasing significantly the duration of the simulation. Concerning phonolite, the discrepancy of force constant between the "VAF" and "snapshot" methods associated to extremely high uncertainties for both methods prevent us to use these results. It is clear that phonolite tends to have heavier Fe isotopic values than the other melts. However, the Fe force constant cannot be clearly quantified. This may be caused by the model itself of the phonolite melt that contains only one Fe atom over the 102-atoms system which induces a significant statistical noise. However, with that exception, the similar Fe β-factors obtained by the "VAF" and the "snapshot" methods allow to conclude that both methods are relevant to determine the isotopic properties of a melt as previously observed by [START_REF] Pinilla | Firstprinciples investigation of equilibrium iron isotope fractionation in fe1x s x alloys at earth's core formation conditions[END_REF] in the case of Fe-S alloys at high pressure and temperature conditions. In the further discussion, the only results that will be considered are the ones obtained from the "snapshot" method for clarity and consistency with Si isotopes results.

In many experimental studies, glasses are the only proxy for melts. When the melt quenching is fast enough to avoid cation reorganization, then the glass structure associated is most likely to be close to the melt one as observed with the snapshot method. A comparative study of MD simulation and experimental glasses shows that cation-oxygen distances are slightly larger in glasses than in melts even if the structure properties are very similar [START_REF] Rossano | Environment of ferrous iron in CaFeSi2O6 glass; contributions of EXAFS and molecular dynamics[END_REF]. [START_REF] Ghosh | Effects of valence and spin of Fe in MgSiO3 melts: Structural insights from first-principles molecular dynamics simulations[END_REF] also pointed out that the effects of temperature on the Fe-O coordination is not negligible, with a silicate glass at 300 K displaying Fe-O coordination of 4.5 whereas melt at 3000 K have a Fe-O coordination of 3.7. In the case of isotopic variation between melts and glasses, this question has not really been assessed. As stated in the methods section, the "VAF" method takes into account all features of the melt system whereas the "snapshot" method involves a relaxation of atomic positions at zero temperature. Both methods give similar β-factors.

However, it is noteworthy that this computational structural relaxation does not mimic exactly the quench of a glass that is inevitably slower. Many studies observed that the coordination number of an element is a parameter that will affect its isotopic signature in a material (e.g., Méheut et al., 2009;Polyakov and Mineev, 2000;Rabin et al., 2021;Sossi and O'Neill, 2017). Therefore, it is possible that silicate glasses are not an ideal proxy for melts, as glasses display significantly higher coordination numbers and therefore will underestimate the melts β-factor.

4.5.3 Iron isotope fractionation in melts.

Parameters controlling Fe isotope fractionation

In silicate minerals, the main parameters involved in iron isotope fractionation, besides temperature, are the oxidation state of iron, i.e. Fe 3+ /Fe tot (Dauphas et al., 2014;Shahar et al., 2008), the Fe-O bond length, the Fe-O coordination number (Sossi and O'Neill, 2017) and the number and nature of iron second-neighbors (Macris et al., 2015;Rabin et al., 2021). All these parameters are also expected to have an impact on melt iron isotope composition. However, the importance of a parameter over another can change due to the structural differences between melts and minerals. Furthermore, the range of Fe β-factors within those melts is greater than what was observed in silicate minerals (Fig. 2 in (Rabin et al., 2021)).

The first important observation from these FPMD simulations is that the variations in Fe-O coordination number (from 3.7 to 4.7) does not seem to be correlated to the iron force constants (Fig. However, the variation of Fe force constant in melts with similar Fe 2+ content definitely means that iron isotope fractionation within silicate melts cannot be uniquely explained in term of iron redox state.

In addition to the lack of relationship between Fe force constant and SiO 2 content, there seems to be no correlation between iron force constant and the melt polymerization parameter NBO/T (Fig. 4.7a). The only parameter that seems to greatly influence the melt iron isotope fractionation along magmatic differentiation for our investigated melt compositions is the nature and average number of the cationic neighbors around iron atoms, at least for some elements. There is a strong relationship between the Fe-Fe and Fe-Mg distances and iron force constants: a decrease of these distances leads to higher iron force constant (Fig. 4.8). This explains why basalt_IR and basanite display higher force constant than the basalt_ID and trachyte (Table 4.6). Moreover, Fe segregation could play a role on the higher Fe force constant calculated for the basaltic melt as we determined part 4.5.1 that iron in the basaltic melt tends to slightly segregate compared to iron in the basanite that is randomly distributed. However, there is no clear relationship between iron/magnesium concentration in the melts and the iron force constant (Fig. 4.11: Relationship between the average Si-cation distance and the Si force constant.

(Wt %) (Wt %) (Wt %) (Wt %) (Wt %) (Wt %) Basanite Trachyte Basalt-H Basalt Basanite Trachyte Basalt-H Basalt Basanite Trachyte Basalt-H Basalt a) a) b) c) d) e) f)
<F> (N/m) Si-Al distance (Å) Phonolite Basanite Trachyte Basalt-ID Basalt 2.9 3.0 3.1 3.2 <F> (N/m) Si-Si distance (Å) Phonolite Basanite Trachyte Basalt-ID Basalt Basanite Basalt-ID Basalt Phonolite Basanite Trachyte Basalt-ID Basalt Phonolite Basanite Trachyte Basalt-ID Basalt Phonolite Basanite Trachyte Basalt-ID a) b) ) d ) c ) f ) e

Iron fractionation factors between minerals and melts.

One of the major issues in determining the processes involved in iron isotope fractionation during magmatic differentiation is the lack of equilibrium iron fractionation factors between minerals present at magmatic temperatures (i.e. pyroxene, olivine, oxides...) and their associated melts. In this study, we combine the iron β-factors of various minerals present in magmatic setting, obtained from first-principles calculation (Rabin et al., 2021) with the present iron β-factors of the melts in order to obtain a full set of equilibrium fractionation factors between minerals and melts (∆ 57 Fe mineral-melt ). The minerals β-factors in Rabin et al. (2021) were calculated using ultra-soft pseudo-potentials Garrity et al. ( 2014) with the Quantum Espresso (QE) codes (Giannozzi et al., 2009). Here, we employed VASP code with PAW pseudopotentials, which could lead to systematics bias between the different calculated β-factors. For this reason, we also calculated iron β-factors of two distinct minerals (aegirine and forsterite) already studied by Rabin et al. (2021) in order to estimate this bias. First, we checked that results from both ab-initio calculation methods display the same linear correlation between 10 3 lnβ 57 Fe/ 54 Fe and their associated force constants (y = 0.0019 x, at T = 1500 K, Fig. 4.2). Second, aegirine and forsterite display iron force constant of 244 N/m and 162 N/m respectively using VASP against 216 N/m and 146 N/m using QE. There is thus a difference of about +10% between the VASP-determined force constants and those calculated by QE which is assumed to be present for all other minerals iron β-factors. Therefore, a systematic correction of +10% is applied to minerals iron β-factors previously obtained with QE (Rabin et al., 2021) before calculating the mineral-melt fractionation factors. The entire set of equilibrium fractionation factors between minerals and melts is displayed in Table 4.7. It is noticeable that most of the minerals display lighter iron isotope composition than the different melts at equilibrium (Appendix B,fig. 19 fig. 22). Only aegirine and magnetite display heavier iron isotopic signature than basalt_ID, basanite and trachyte melts. As expected, fayalite, with its low iron force constant (F = 100 N/m), is the mineral that shows the greatest iron isotope fractionation with the different melts, followed by aegirine, hedenbergite and augite (Appendix B, Comparison between theoretical and experimental data. Dauphas et al. (2014) were the first to provide experimental iron β-factors of silicate terrestrial glasses of various compositions, as proxy of melts, using Nuclear Resonant Inelastic X-ray Scattering (NRIXS). In figure 6, iron force constant obtained by NRIXS (Dauphas et al., 2014) are compiled with our calculated ones. The range of iron force constants of their Fe 2+ -basaltic glasses (from 197 ± 8 N/m up to 292 ± 18 N/m; IW+3 Si with x = 10 6 /T 2 for T between 0 and 1500 K. The iron and silicon β-factor of minerals were taken from Rabin et al. (2021) for all minerals except quartz, albite and anorthite that were taken from Qin et al. (2016)).

buffer) are similar to our calculated iron force constants for Fe 2+ -basaltic melts at 1 GPa (between 171 ± 8 N/m up to 243 ± 17 N/m), although their compositions in major elements are different. For the most evolved lavas, our trachyte iron force constant is consistent with that of Dauphas et al. (2014). They found no major increase in the iron force constant until late stage of magmatic differentiation (≈ 70 wt% of SiO 2 ) for the silicasaturated series. No measurement on peralkaline glasses were performed which prevents us to compare our phonolite and basanite iron β-factor. Later, Prissel et al. (2018) determined iron force constants between 189 ± 9 and 203 ± 11 N/m for their synthetic lunar glasses using NRIXS that is undistinguishable from Dauphas et al. ( 2014 Other studies mainly focused on inter-mineral iron isotope fractionations (Dauphas et al., 2012;Macris et al., 2015;Polyakov and Mineev, 2000;Roskosz et al., 2015;Shahar et al., 2008;Sossi and O'Neill, 2017) as measuring iron isotope composition of glasses in equilibrium with minerals is a great analytical challenge on natural and experimental samples. This is why there is only few data on iron isotope fractionation factor between silicate melts and minerals. However, ∆ 57 Fe mineral-melt were assessed empirically in many studies performed on natural samples in order to model fractional crystallization and other related processes (e.g., Sossi et al., 2012Sossi et al., , 2016;;Williams et al., 2018). Williams et al. (2018) determined a ∆ 57 Fe melt-cpx of 0.15 % , based on the observed difference in mean Fe isotope composition between oceanic basalts and fertile mantle peridotites (Williams and Bizimis, 2014) with no indication of the temperature dependence. Assuming a temperature of 1000 K, our calculated ∆ 57 Fe Basalt I D-Augite and ∆ 57 Fe Basalt I D-Hedenbergite of about 0.12 % as well as ∆ 57 Fe Basalt_ID-Diopside = 0.08 % are only slightly lower than the empirically determined one. However, looking at iron-rich basalt, our calculated ∆ 57 Fe Basalt I R-Augite and ∆ 57 Fe Basalt I R-Hedenbergite of about 0.42 % as well as our ∆ 57 Fe Basalt I D-Diopside = 0.38 % appear significantly larger than the 0.15 % value proposed by Williams et al. (2018). Depending on the iron content in the melt ∆ 57 Fe cpx-melt can vary between 0.08 % up to 0.42 % at 1000 K. This is why strong cautions need to be applied when using ∆ 57 Fe melt-mineral to model iron isotope composition of natural samples. It is important to know the exact chemistry of the bulk rock and of the minerals (e.g., Rabin et al., 2021), along with the melt f O 2 (see Fig. 4.6 and Dauphas et al. (2014); Prissel et al. (2018)). The ∆ 57 Fe melt-cpx = 0.25 × 10 6 /T 2 used by Sossi et al. (2012) to model fractional crystallization at Red hill volcano also fits in the calculated range. Our calculated ∆ 57 Fe forsterite-basalt = -0.15 % at 1500 K agrees with the ∆ 57 Fe ol-melt of -0.15 % observed by Teng et al. (2008) at magmatic temperature.

4.5.4 Silicon isotope fractionation in melts. The parameters proposed to control silicon isotope fractionation in silicate minerals are Si-O bond lengths, Si-O polyhedron volume (Qin et al., 2016), Si-O coordination number (Méheut and Schauble, 2014;Méheut et al., 2009;Qin et al., 2016), the nature of silicon second neighbors (Rabin et al., 2021), the electronegativity of the surrounding cations (Méheut and Schauble, 2014), and the polymerization degree (Grant, 1954;Poitrasson and Zambardi, 2015).

Looking at the melts investigated here, we find that the range of Si isotope β-factors in melts is relatively similar to that observed in silicate minerals, though to the higher end (Fig. 4.3b), except for fayalite that displays a significantly lower β-factor. The average Si-O bond length and coordination number are similar in all the melts studied (Table 4.3).

This means that neither Si-O bond length nor Si-O coordination number are responsible for the Si β-factor variation observed in these melts (Fig. 4.4b and Fig. 4.5b). This lack of relationship between those parameters and the silicon isotope fractionation were also observed in the silicate minerals investigated in Rabin et al. (2021) that all display a coordination of 4, and an average Si-O bond length of 1.65 Å.

The melt polymerization has been proposed to be the main factor causing silicon isotope fractionation in natural lavas study (Poitrasson and Zambardi, 2015). However, polymerization of SiO 4- 4 tetrahedra did not appear to be the sole parameter influencing Si isotope fractionation in silicate minerals Meheut2009, Meheu2014, Qin2016, RABIN2021a.

In this study, there seems to be a correlation between NBO/T and Si force constant from basalt to trachyte, considering the associated uncertainties (Fig 4.7b). However, Phonolite has a Si force constant of 700 ± 15 N/m associated with a NBO/T ratio of 0.28 whereas basalt_ID has a similar Si force constant (695 ± 13 N/m) but a NBO/T ratio of 1 (Fig. 4.7b). As underlined part 4.4.1, variable melts composition, generating structural heterogeneities, can blur the potential correlation with the polymerization parameter NBO/T. This scatter is well observed among terrestrial igneous rocks, as opposed to lunar ones with a simpler petrogenetic history, notably in terms of protolith that were likely less varied on the Moon Poitrasson and Zambardi (2015). This suggests that in melts displaying structural heterogeneities, the polymerization degree is not the main parameter influencing silicon isotope composition.

A limited increase in Si force constants with increasing SiO 2 content, a proxy for magmatic differentiation, can also be inferred (Fig. 4.10). However, there is no relationship between Si force constant and Si-Si distance (Fig. 4.11) or coordination number. The variation of SiO 2 content (from 49.1 to 63.2 wt%) in our simulated melts is relatively reduced. To evidence any strong relationship between Si force constant and SiO 2 content a melt with > 70 W-wt% silica content would need to be modeled. Silicon force constants are really similar for basalt_ID, basanite and phonolite, with values of 695 ± 13, 687 ± 8 and 700 ± 15 N/m respectively. This means that increasing alkaline and aluminum content in the melts, that seems to play a role on iron isotope fractionation, has no effect on silicon fractionation (Fig. 4.10). This also demonstrates that electronegativity differences of the cations constituting the melt (χ N a = 0.93; χ M g = 1.31; χ Al = 1.63) does not affect the silicon isotope composition of the melts. Silicon is indeed a network former in the melts and silicon structural properties as well as the Si force constant are much less sensitive to melt composition than iron. The only significant variation of Si force constant is between basalt_IR and trachyte with values of 669 ± 11 N/m and 716 ± 10 N/m, respectively. Besides the polymerization degree (Fig. 4.7b), it seems that the major difference between these two melts are the Si-Fe average distance, of about 3.1 Åfor basalt_IR and 2.4 Å for trachyte (Table 4.4). The shortest average Si-Fe distance corresponds to the highest Si force constant in the melts (Fig. 4.11).

To conclude, the small but significant silicon isotope fractionation in the melts seems to be mainly driven by the presence of iron as silicon second neighbors. Although we cannot exclude a potential effect of the polymerization degree, at this stage. The nature of the melt, whether peralkaline or calco-alkaline, is not influencing the silicon isotope signature. Further studies with more extended SiO 2 content would need to be performed in order to assess whether SiO 2 concentration is affecting or not the Si isotope composition of the melts.

Silicon fractionation factors between minerals and melts.

The set of mineral silicon β-factors from Rabin et al. (2021) using QE code were taken to calculate most of the ∆ mineral-melt 30 Si/ 28 Si. As observed for the iron isotope system (4.3.2), the linear correlation between the 10 3 lnβ 30 Si/ 28 Si and their associated force constant obtained from QE and VASP DFT codes are identical (y = 0.0102x, Fig. 4.2b).

Aegirine and fayalite Si β-factors and Si force constants obtained using the two different code packages display also similar results with F = 662 N/m and 692 N/m respectively using VASP and F = 648 N/m and F = 686 N/m using QE codes. The change of DFT packages and pseudo-potentials only affect the Si β-factors by about ≈ 1.3 %, unlike iron β-factor that was much more affected (≈ 10%). In order to be consistent with the correction applied to Fe β-factors, a correction of + 1.3 % were also applied on the mineral Si β-factors from Rabin et al. (2021) before the calculation of ∆ mineral-melt 30 Si/ 28 Si. In order, to obtain ∆ albite-melt 30 Si/ 28 Si, ∆ anorthite-melt 30 Si/ 28 Si and ∆ quartz-melt 30 Si/ 28 Si, silicon β-factors from Qin et al. (2016) were used. The forsterite Si β-factor determined by Qin et al. (2016) is 5% heavier compared to the Si β-factor calculated by Rabin et al. (2021) and Huang et al. (2014) respectively. In order to limit the bias due to difference in the calculation methods, a correction of -5% is applied on the Si β-factor of Qin et al. (2016) prior to calculate the ∆ mineral-melt 30 Si/ 28 Si. Except for albite, anorthite, quartz and fayalite the other silicate minerals display lighter silicon isotopic signature relative to the different melts (Appendix B, fig. 19 fig. 22). As for iron, the entire set of equilibrium silicon isotope fractionation factors between minerals and melts is displayed in Table 4.7. 4.5.4.3 Comparison between theoretical and experimental data.

As for iron isotopes, most studies focus on inter-mineral silicon isotope fractionation (Méheut and Schauble, 2014;Méheut et al., 2009;Qin et al., 2016;Savage et al., 2011) and there are only two studies that display ∆ melt-mineral 30 Si/ 28 Si. Qin et al. (2016) and [START_REF] Deng | An oceanic subduction origin for Archaean granitoids revealed by silicon isotopes[END_REF], used the silicon β-factors of silicate minerals calculated by Qin et al. (2016) in order to assess Si fractionation factors between silicate minerals and different melt compositions. These calculations were performed assuming that silicate melts have Si isotope fractionation factors equal to the weighted sum of those for their normative mineral compositions. Our set of Si isotope fractionation factors as well as the one calculated by [START_REF] Deng | An oceanic subduction origin for Archaean granitoids revealed by silicon isotopes[END_REF] at 1123 K are displayed in Table 4.8. Both studies display a similar evolution of the Si fractionation factor with ∆ melt-diopside 30 Si/ 28 Si > ∆ melt-orthoenstatite 30 Si/ 28 > ∆ melt-olivine 30 Si/ 28 Si > ∆ melt-anorthite 30 Si/ 28 Si > ∆ melt-albite 30 Si/ 28 Si. The silicon fractionation factors calculated by Qin et al. (2016) and [START_REF] Deng | An oceanic subduction origin for Archaean granitoids revealed by silicon isotopes[END_REF] are however systematically higher than our calculated ∆ melt-mineral 30 Si/ 28 Si, by 0.1-0.2 % for the trachytic melt and by 0.4-0.5 % for basaltic (IR) melt. The discrepancies between our calculated silicon fractionation factor between minerals and trachytic melt and the one estimated from Qin et al. (2016) lies within our estimated calculations uncertainties of ≈ 0.3 % at 1123 K. Concerning the basaltic melt, the ∆ melt-mineral 30 Si/ 28 Si of Qin et al. (2016) is significantly higher than our calculated ∆ melt-mineral 30 Si/ 28 Si within uncertainties. As a result, Qin et al. (2016) would suggest that olivine and enstatite have a lighter silicon isotope composition than the basaltic melts and that the silicon fractionation factor between the feldspars and the melt is insignificant which is not what is observed in our study (Table 4.8). The overestimation of the Si β-factor of the melt [START_REF] Deng | An oceanic subduction origin for Archaean granitoids revealed by silicon isotopes[END_REF]Qin et al., 2016) is probably due to the high content of feldspars in the composition of the magmatic rocks, as a consequence estimated 10 3 lnβ melt is mainly driven by anorthite and albite Si β-factors. From our results, it appears that the melt Si β-factor cannot be assessed by using the weighted sum of the mineral composition of the melt and their associated Si β-factor, especially for the more mafic ones. This study Deng. et

Applications

Magmatic differentiation

Iron isotopes Iron isotope fractionation has been widely studied on natural samples in the context of magmatic differentiation. Most of the magmatic series display heavy iron isotope compositions in the most evolved lavas (e.g., Sossi et al., 2012;Teng et al., 2008;Williams et al., 2018), especially with SiO 2 > 70 wt% (e.g., Poitrasson and Freydier, 2005;Schuessler et al., 2009;Zambardi et al., 2014). One of the candidates to explain such isotopic signature is fractional crystallization. However, the impact of fractional crystallization on iron isotope fractionation is still debated. Many studies considered that there is no significant iron isotopic fractionation between Fe 2+ -minerals and the associated melt and that magnetite, an Fe 3+ oxide, is the main mineral phase driving the iron isotope compositions in the lavas. Hence, the evolved lavas should display lighter iron isotopic compositions (Shahar et al., 2008). Therefore, fractional crystallization has been discarded

as potential candidate to explain the observed heavy iron isotope signature at the last stage of magmatic differentiation (Zambardi et al., 2014). However, Dauphas et al. (2014) displayed the first experimental β-factors for glasses (part 4.5.3.3) and demonstrated that the heavy isotope signature of evolved melts could compensate for the magnetite crystallization during magmatic differentiation and that fractional crystallization could lead to the heavy δ 57 Fe observed. Besides, Polyakov and Mineev (2000) and Rabin et al. (2021) showed that isotopic fractionations between Fe 2+ -bearing minerals are significant.

Taking this into account in fractional crystallization models could reproduce the isotopic trend observed in natural lavas (Rabin et al. 2021). Here, the model produced in Rabin et al. (2021) was modified by replacing the approximated ∆ 57 Fe min-melt (using melt βfactor of Dauphas et al. (2014)), by the ones calculated in this study (Table 4.7, Appendix B -Table 8). Our model shows a smooth increase of δ 57 Fe from 58 wt.% SiO2 to 74 wt% SiO 2 followed by a sharp increase up to 0.65% (Fig. 4.12a). If most of the Fe isotope signatures of the evolved lavas and granites (e.g., Foden et al., 2015;Poitrasson and Freydier, 2005;Telus et al., 2012) can be explained by our fractional crystallization model, some of the heaviest iron isotopic signatures observed in some granites (δ 57 Fe = + 0.9% ; e.g. He et al. (2017); Xia et al. (2017)) are however not reproduced by the model. Despite the fact that this is the most constrained fractional crystallization model so far, the lack of calculated β-factor of rhyolitic melt could under-estimate the δ 57 Fe of the most evolved melt as rhyolitic melt should have a higher β-factor than trachyte (Dauphas et al., 2014).

Overall, from this new model, it is possible to conclude that fractional crystallization can explain iron isotope evolution during magmatic differentiation for δ 57 Fe values at least up to 0.65% .

Silicon isotopes Silicon isotopes show progressive and systematic enrichment in heavy isotopes during magmatic differentiation (Savage et al., 2011;Zambardi et al., 2014) going from δ 30 Si = -0.30 ± 0.05 % for SiO 2 = 47 wt% up to -0.10 ± 0.07 % for SiO 2 = 76 wt% in terrestrial volcanic rocks. Savage et al. (2011) proposed fractional crystallization as the main process involved in silicon isotope fractionation during magmatic differentiation. However, modelling of the effect of fractional crystallization on δ 30 Si along magmatic differentiation performed by Savage et al. (2011) and Zambardi et al. (2014), diverge from the measured silicon isotopic trends. This discrepancy was mainly due to the lack of precise fractionation factor between silicate minerals and melt at this time.

Many studies (Méheut and Schauble, 2014;Méheut et al., 2009;Qin et al., 2016;Rabin et al., 2021) underlined that silicon isotope fractionation between silicate minerals, even at high temperature, are significant. The use of approximated ∆ 30 Si min-mineral allowed Qin et al. (2016) to show, through different models, that fractional crystallization is a viable mechanism to explain heavy silicon isotope signatures in the most evolved lavas.

Here, we are modelling the effect of fractional crystallization on silicon isotopes using our ∆ 30 Si min-mineral in order to evaluate this hypothesis (Electronic appendix -Table E2). We used the same starting composition in the MCS software as in the model of fractional crystallization for iron isotopes. In this model, silicon isotope evolution is expressed as follow:

δ 30 Si(melt) j+1 = δ 30 Si(melt) j + i X i × 10 3 lnα i (T ) (4.10)
where X i is the modal proportion of minerals crystallized at each 5 • C step. δ 30 Si melt is the initial δ 30 Si of the melt. In this case δ 30 Si melt = -0.22 % for SiO 2 = 58 wt%.

Our model is in good agreement with the measured silicon isotopic trend in terrestrial lavas (Fig. 4.12). The linear trend of our model (y = 0.0061 x -0.588) is close to the linear trend measured of the Hekla volcanic suite (y = 0.0059 x -0.591; Savage et al. (2011) and Zambardi et al. (2014)). Our new set of fractionation factors between silicate minerals and melts allow us to better constrain the fractional crystallization model for the first time, and to show that fractional crystallization is most likely responsible for the observed heavy silicon isotope signature in the most evolved lavas.

Partial melting effect on iron isotopes

Previous studies have suggested that mantle-derived melts may show heavier iron isotope signatures compared to their presumed mantle peridotite source rocks (e.g., Dauphas et al., 2014;Nebel et al., 2018;Sossi et al., 2016;Weyer and Ionov, 2007;Williams and Bizimis, 2014;Williams et al., 2004Williams et al., , 2009)). The changes in Fe 3+ /Fe tot between crystal and liquid due to partial melting has been argued as a key parameter to explain this.

The moderately incompatible behavior of Fe tried to assess whether partial melting of mantle peridotite, through single or numerous stages, could create the heavier isotopic signature of MORB (0.7 % < δ 57 Fe > 0.22 % ; Sossi et al. (2016)). However, the role of partial melting on the iron isotope signature of MORB remains debated (e.g., Dauphas et al., 2014;Sossi et al., 2016;Weyer and Ionov, 2007;Williams and Bizimis, 2014;Williams et al., 2009). Many parameters in these partial melting models are well constrained such as the Fe 3+ /Fe tot of MORB (Fe 3+ /Fe tot = 0.09 ± 0.02; (Bézos et al., 2021)) and the temperature at depth during mantle melting. One of the main critical parameters is the fractionation factor between the melt and the minerals (olivine, orthopyroxene). Recent study of Dauphas et al. (2014) displays that partial melting is only responsible for one third (0.023 % ) of the isotopic fractionation between MORB and mantle peridotite. Sossi et al. (2016) proposed a correction of Fe isotope signature of MORB for the effect of olivine early fractionation allowing them to assess whether mantle heterogeneity could be the source of the heavy iron isotope signature of MORB. From this correction, it appears that normal MORB displays a mean δ 57 Fe of 0.10 ± 0.01 % . Sossi et al. (2016) demonstrated then that for an upper mantle δ 57 Fe = 0.05 ± 0.01 % , a Fe 3+ /Fe tot = 0.037 and a degree of partial melting comprised between 0.1 and 0.2, the MORB iron isotope composition can be reproduced. As mentioned above, the choice of ∆ 57 Fe min-melt is critical in partial melting models. Sossi et al. (2016) applied a ∆ 57 Fe min-melt = ∆ 57 Fe ol-melt = -0.4 × 10 6 /T 2 as olivine is the major iron carrier in peridotites before orthopyroxene. At 1500 K, the ∆ 57 Fe ol-melt is about -0.17 % in their models, which is slightly higher, but similar within uncertainties, than our calculated ∆ 57 Fe forsterite-basalt = ∆ 57 Fe orthoenstatite-basalt = -0.15 % . Our new set of equilibrium isotope fractionation factor between minerals and melts allows us to confirm that normal MORB iron isotopic signature can be explained by 10 to 20 % partial melting of the peridotite mantle as demonstrated by Sossi et al. (2016).

Iron isotope intracrystalline diffusion

Chemical zoning is commonly observed in phenocrysts (i.e. olivine, pyroxene) of volcanic rocks. This has been interpreted as the results of crystal growth in an evolving melt during magmatic differentiation or/and diffusion-controlled chemical exchange due to the contrast (disequilibrium) between crystal and melt composition (e.g., Teng et al., 2011). In the past few years, studies demonstrated that iron chemical zoning in olivine is mirrored by iron isotope zoning (Collinet et al., 2017;Oeser et al., 2014Oeser et al., , 2015Oeser et al., , 2018;;Sio et al., 2018Sio et al., , 2013;;Teng et al., 2011). Iron isotopes profiles across olivine display significant iron isotopes variations (> 1 % ; Oeser et al. (2015); Oeser et al. (2018) that exceeds potential equilibrium isotope fractionation. Iron isotopes profiles can therefore be used to evidence for the diffusion origin of the chemical zoning in olivine, as the light isotopes ( 54 Fe) diffuse faster than the heavy isotopes ( 57 Fe). Quantifying kinetic isotope effects on minerals and melts from first-principles calculations is beyond the scope of this study. We would require a large simulation cell for the melt with an important number of atoms (N) in order to assess with precision diffusion coefficients for the different elements in our systems, as diffusivity follows the inverse of N 1/3 dependence Karki (2010). However, from our set of iron equilibrium fractionation factors between silicate melts and minerals, we can assess the magnitude of iron isotope fractionation during olivine crystallization over diffusiondriven kinetic isotope fractionation. The net isotope fractionation measured across zoned olivine crystals depends on the rate of crystal growth relative to the rate of chemical diffusion [START_REF] Sio | Thermal and crystallization histories of magmatic bodies by Monte Carlo inversion of Mg-Fe isotopic profiles in olivine[END_REF]Sio et al., 2013). A fast crystal growth relative to diffusion process would generate olivine zoning driven by crystal growth associated with magmatic differentiation. Some authors (Dauphas et al., 2014;Prissel et al., 2018;Sio et al., 2018), implied that in this case only little isotopic fractionation between mineral and melt is expected as they consider ∆ 57 Fe olivine-melt to be insignificant at magmatic temperature. Alternatively, when crystal growth is slow relative to diffusion, then crystal zoning is induced by diffusive re-equilibration with the evolving melts and higher isotopic variations within the olivine are expected. The lack of fractionation between forsterite and melt was also observed in our study in the case of a basaltic melt with low iron content, basalt_ID, with a ∆ 57 Fe forsterite-basalt-ID = -0.01 % at 1500 K. Our study shows however that the equilibrium fractionation factors between olivine and a basalt with higher iron content (13.1 wt%) are significant with a ∆ 57 Fe forsterite-basalt = -0.15 % and a ∆ 57 Fe fayalite-basalt = -0.25 % at 1500 K. Therefore, at equilibrium, assuming that iron isotope fractionation is exclusively driven by crystal growth process, an olivine may display according to our calculation δ 57 Fe as low as -0.14 % , considering a melt value of 0.09% (BIR1g, Craddock and Dauphas (2011); Oeser et al. (2014)). Iron isotopes composition lighter than -0.14 % , as observed in in-situ isotopic profiles with olivine rims δ 57 Fe as low as -2.1 % (Oeser et al., 2015), can only be explained by diffusion-driven fractionation (Collinet et al., 2017;Oeser et al., 2015;Prissel et al., 2018;[START_REF] Sio | Thermal and crystallization histories of magmatic bodies by Monte Carlo inversion of Mg-Fe isotopic profiles in olivine[END_REF].

Our new equilibrium ∆ 57 Fe olivine-basalt allows us to propose that the low δ 57 Fe signatures observed in olivines can partly be explain by equilibrium fractionation. However, our results also confirm that kinetic diffusion-driven fractionation has a major influence on the iron isotope chemical profiles of olivine as implied by other studies Collinet et al.

Conclusion

This study provides, for the first time, equilibrium Fe and Si isotope β-factors of various melts, using first-principles molecular dynamics. We first demonstrated, through comparison of the "snapshot" and "VAF" methods, that both methods are appropriate to study isotope fractionation of melts. Iron, that acts mainly as a network-modifier in the melts, displays significant isotope fractionation among the different compositions of melt. The Fe-O mean coordination number appears not playing a significant role on the observed Fe β-factor variations in melts. The main parameters driving the iron isotope composition in melts is the Fe-O bond length and the chemical environment around iron (i.e. iron second neighbors); increase of iron contents drives indeed Fe β-factor to heavier values.

These results show that significant Fe isotope fractionation is expected without changes of iron redox. On the other hand, silicon isotope fractionation appears to be less sensitive to compositional changes in melts, and displays limited isotope fractionation among the different melts. This is most likely due to its network-former role in the melt. Since structural parameters such as Si-O bond length and coordination number remain constant, they do not play a major role on the Si β-factors variation observed in these melts. The silicon isotope compositional variations were also expected to be related to the polymerization degree (NBO/T; Grant (1954); Poitrasson and Zambardi (2015)). However, this relationship may be blurred by the high alkaline content of the melts. Silicon β-factors appear to be also driven by its local environment, notably the presence of iron as silicon second neighbors.

This Fe and Si β-factors of melts, allowed us to calculate a consistent set of equilibrium fractionation factors between various minerals and melts. This represents a major step forward in the understanding and quantifying of the processes involved in iron and silicon isotope fractionation in magmatic settings. In this perspective, we used our new ∆ 57 Fe min-melt and ∆ 30 Si min-melt in several models from the literature in order to assess the impact of the addition of more constrained fractionation factors on them. Our results support fractional crystallization as a possible driving process involved in both Si and Fe isotope evolution during magmatic differentiation for the melts chemical range investigated. Our calculated ∆ 57 Fe olivine-melt allows us to confirm the model of Sossi2016 showing that generation of MORB with δ 57 Fe = 0.1 % can be explained by partial melting of peridotites peridotites (with a degree of partial melting comprised between 10 and 20%).

We also showed that there is significant iron isotope fractionation between Fe 2+ -bearing minerals and melts. Olivine crystallization at equilibrium could show a ∆ 57 Fe olivine-basalt range from 0 to -0.21 % . Our new set of iron fractionation factors can therefore help to quantify the equilibrium fractionation versus the diffusion-driven kinetic fractionation in future iron isotope diffusion models.

Chapters 3 and 4 are based on first-principles calculations. The following chapter is instead based on the analysis of natural lavas from Kerguelen Archipelago. Ion-exchange chromatography was performed in order to access the Fe isotopes composition of three different magmatic series. These magmatic series differ in term of alkaline content and silica-saturation. This study allows us to discuss the origin of such different magmatic series from the same geodynamical context by using the theoretical Fe isotopes fractionation factors determined in the previous chapters.

Abstract

Processes generating iron isotope fractionation during magmatic differentiation are investigated through three volcanic series from the Kerguelen Archipelago. This allows us to compare the evolution of iron isotope fractionation in silica-saturated and silicaundersaturated volcanic series, with increasing alkalinity, from the same geodynamical setting. The three series reflect the temporal evolution of magmatism in the Kerguelen Archipelago after the formation of the Kerguelen Plateau; the silica-saturated alkaline series was emplaced in the Late Oligocene -Lower Miocene followed by the silicaundersaturated alkaline series in the Upper Miocene, whereas the silica-saturated alkaline series from Mont Ross is Pleistocene in age (< 1 My). Iron isotope and Fe 3+ /Fe tot measurements, as well as petrological studies, partial melting and fractional crystallization models have been performed on 23 lavas. Our results display different δ 57 Fe and Fe 3+ /Fe tot for each magmatic series implying a heterogenous mantle source with various oxygen fugacity. Moreover, iron isotopic signatures differ for each series along their differentiation paths. The Lower Miocene silica-saturated (LMS) series displays no δ 57 Fe increase during differentiation with a δ 57 Fe of about 0.120 ± 0.064 % associated to elevated and constant Fe 3+ /Fe tot = 0.46. Partial melting model implies the necessity of metasomatized peridotites in the Kerguelen Plume mantle source, with a δ 57 Fe mantle of -0.20 ± 0.035 % , to generate the primitive LMS basalt, δ 57 Fe prim ≈ -0.00 % . The low δ 57 Fe of the LMS trachytes could reflect altered basaltic crust assimilation during magma residence or reflect the presence of another magmatic series with similar age but with lower δ 57 Fe mantle source. The Upper Miocene silica undersaturated magmatic series (UMS) displays a hint of an increase in δ 57 Fe during differentiation from 0.163 ± 0.064 % up to 0.308 ± 0.53 % for the most evolved phonolites mirrored by Fe 3+ /Fe tot ratio going from 0.35 to 0.6. This can be explained by a system open to oxygen where the iron isotope fractionation is mainly driven by fractional crystallization of pyroxenes. The δ 57 Fe prim = 0.236 % and Fe 3+ /Fe tot = 0.35 of the UMS can only be reproduced by the introduction of an isotopically-heavy mantle component, such as pyroxenites, to the metasomatized peridotites in the hotspot source, with a δ 57 Fe mantle = 0.09 ± 0.059 % . The Mont Ross series (MRS) displays a slight increase of δ 57 Fe along the differentiation path up to 0.266 ± 0.062% and it is reflected by the Fe 3+ /Fe tot evolution from 0.21 to 0.45. As for the UMS, the MRS δ 57 Fe evolution along the magmatic differentiation path could be reproduced by a fractional crystallization model. The δ 57 Fe prim ≈ 0.1 % and Fe 3+ /Fe tot ≈ 0.2 of the MRS can also be generated by a two-component mantellic source with a δ 57 Fe mantle = -0.06 ± 0.035 % .

Iron isotopes are therefore the witnesses of the complex magmatic evolution of Kerguelen Plume volcanism. We suggest in this study that stable iron isotopes associated to Fe 3+ /Fe tot ratio could be an efficient tracer for lithological heterogeneities in the mantle source. Our fractional crystallization models can also reproduce the heavy isotopic signatures of the most evolved lavas.

Introduction

The development in multicollector inductively coupled plasma mass spectrometry (MC-ICP-MS) techniques has improved the precision and accuracy of the measurements of stable metal isotope compositions. Of particular interest is iron, the fourth most abundant element in the crust and also the most abundant redox-variable element on Earth.

Multiple valence states of iron exist on Earth: Fe 0 , Fe 2+ , Fe 3+ . There are four iron stable isotopes, 54 Fe, 56 Fe, 57 Fe, 58 Fe with respective mean abundances of 5.80, 91.72, 2.20 and 0.28 %. Due to its multiple oxidation states, processes involving redox equilibrium are effective in fractionating iron isotopes. In terrestrial reservoirs, the oxygen fugacity under which igneous rocks equilibrated can be indicated by the relative activities of Fe 2+ and Fe 3+ species in silicate melts and minerals (e.g., [START_REF] Canil | Distribution of ferric iron in some upper-mantle assemblages[END_REF][START_REF] Carmichael | The redox states of basic and silicic magmas: a reflection of their source regions?[END_REF], and the associated isotopic fractionation between them (Dauphas et al., 2014;Rabin et al., 2021a,b;Sossi and O'Neill, 2017;Williams et al., 2004). Equilibrium stable isotope fractionation theory predicts that the heavier isotopes are incorporated preferentially in bonds where Fe 3+ predominates over Fe 2+ (Dauphas et al., 2014;Polyakov and Mineev, 2000;Polyakov et al., 2007;Sossi and O'Neill, 2017;Williams et al., 2004). Rabin et al. 2021a, also demonstrated that equilibrium iron isotope fractionations among Fe 2+bearing minerals are not negligible in magmatic contexts. Iron isotope are then sensitive tracers to constrain mechanisms of magmatic evolution (Dauphas et al., 2014;Du et al., 2017;Peters et al., 2019;Poitrasson and Freydier, 2005;[START_REF] Schuessler | Oxidation state of iron in hydrous phono-tephritic melts[END_REF]Sossi et al., 2012;Teng et al., 2008;Zambardi et al., 2014) in addition to variations in mantle oxygen fugacity (e.g., Dauphas et al., 2014;Williams et al., 2009). Recent studies have demonstrated that there is significant iron isotope variation during magmatic differentiation among a range of magmatic series, regardless of their chemical affinity (e.g., [START_REF] Schuessler | Oxidation state of iron in hydrous phono-tephritic melts[END_REF]Sossi et al., 2012;Zambardi et al., 2014). Invariably the more differentiated lavas, such as rhyolites, display heavier isotopic composition than basaltic rocks from a given magmatic suite. However, the fundamental processes involved in fractionating iron isotopes in igneous rocks are still debated since fluid exsolution, fractional crystallization and/or thermodiffusion have been successively proposed (Du et al., 2017;Heimann et al., 2008;Lundstrom, 2009;Poitrasson and Freydier, 2005;Sossi et al., 2012;Telus et al., 2012;Teng et al., 2008;Williams et al., 2018;[START_REF] Wu | Mineral composition control on inter-mineral iron isotopic fractionation in granitoids[END_REF]. For the advocates of the fractional crystallization hypothesis, changing mineral assemblage (e.g., Foden et al., 2015), the degree to which the magma is open to oxygen exchange (e.g., Sossi et al., 2012) and/or melt composition (e.g., Dauphas et al., 2014) have been invoked to account for the progressive enrichment in heavy iron isotopes in evolved igneous rocks. However, the relative importance of each of these controls is difficult to disentangle. Therefore, the goal of this study is to better understand these mechanisms by mainly using a new set of iron fractionation factors between minerals and melts determined by ab-initio calculation (Rabin et al., 2021b).

Iron isotopes were also investigated as a proxy for identifying lithological heterogeneity of the mantle source region of diverse oceanic basalts (e.g., Gleeson et al., 2020;Konter et al., 2016;Nebel et al., 2018;Williams and Bizimis, 2014). The variations of δ 57 Fe appeared to provide additional information on mantle lithology heterogeneity relative to the traditional radiogenic isotope ratio (e.g. Sr, Nd, Pb) for three reasons: 1) Iron is more representative of mantle components by mass given it is a major element that is stoichiometrically incorporated into mantle minerals ; 2) Significant differences exists in iron isotopic fractionation during melting of pyroxene-rich mantle material compared to peridotite (Dauphas et al., 2014;Rabin et al., 2021a;Sossi and O'Neill, 2017;Williams et al., 2005) and 3) mantle pyroxenites display significantly heavier δ 57 Fe signature relative to other peridotites (Nebel et al., 2018;[START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF]Williams and Bizimis, 2014;Williams et al., 2005). The investigation of iron isotope variations of the Kerguelen Archipelago basalts and basanites will thus allow us to deepen the knowledge on the presumed heterogeneity of the Kerguelen Mantle Plume (e.g., [START_REF] Borisova | Lead isotope signatures of Kerguelen plume-derived olivine-hosted melt inclusions: Constraints on the ocean island basalt petrogenesis[END_REF][START_REF] Doucet | Constraining the components of the Kerguelen mantle plume: A Hf-Pb-Sr-Nd isotopic study of picrites and high-MgO basalts from the Kerguelen Archipelago[END_REF][START_REF] Doucet | Primitive neon and helium isotopic compositions of high-MgO basalts from the Kerguelen Archipelago, Indian Ocean[END_REF][START_REF] Frey | Flood basalt from mont tourmente in the Central Kerguelen Archipelago: The change from transitional to alkalic basalt at ≈ 25 Ma[END_REF].

The Kerguelen Archipelago is one of the largest oceanic island systems on Earth with an area of 5800 km 2 . The peculiarity of the magmatism in the Kerguelen archipelago is its chemical evolution through time linked to the geodynamics of the Kerguelen plume and its location near the South East Indian Ridge (SEIR) 40 My ago. Early activity was typified by magmas that have tholeiitic-transitional compositional affinities, with magmatism becoming progressively more alkaline with time. A very interesting characteristic of the alkaline magmatic activity is the occurrence of the two main types of alkaline series: silica-saturated alkaline series and silica-undersaturated alkaline series, both consisting of volcanic and plutonic rocks (e.g., [START_REF] Gautier | Petrology and geochemistry of the Kerguelen Archipelago basalts (South Indian Ocean): evolution of the mantle sources from ridge to intraplate position[END_REF]. The aim of the present study is to focus, for the first time, on the iron isotope evolution of lavas from both silica-undersaturated and silica-saturated alkaline volcanic series from the same area. We discuss the processes involved in fractionating iron isotopes during magmatic differentiation and the impact of partial melting and mantle source heterogeneities on the basaltic iron isotopic signatures. With this information, we comment on the nature and origin of the intraplate magmatism of the Kerguelen Archipelago.

Geological setting

The Kerguelen Archipelago, located in the Antarctic oceanic domain, is the largest subaerial exposure of the Kerguelen Plateau; the second largest oceanic plateau (25 × 106 km 3 ) on Earth after the Ontong Java plateau [START_REF] Coffin | Scratching the surface: Estimating dimensions of large igneous provinces[END_REF]. Most of the surface exposure (80-85%) of the archipelago consists of flood basalts characterised by temporal trends from older tholeiitic-transitional magmas to younger alkaline magmas (e.g., [START_REF] Frey | Origin and evolution of a submarine large igneous province: The Kerguelen Plateau and Broken Ridge, southern Indian Ocean[END_REF][START_REF] Gautier | Petrology and geochemistry of the Kerguelen Archipelago basalts (South Indian Ocean): evolution of the mantle sources from ridge to intraplate position[END_REF][START_REF] Nicolaysen | Vredefort shatter cones revisited[END_REF][START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF]. The oldest transitional flood basalt in Kerguelen Archipelago erupted at 30 My [START_REF] Frey | Flood basalt from mont tourmente in the Central Kerguelen Archipelago: The change from transitional to alkalic basalt at ≈ 25 Ma[END_REF][START_REF] Nicolaysen | 40Ar/39Ar geochronology of flood basalts from the Kerguelen Archipelago, southern Indian Ocean: Implications for Cenozoic eruption rates of the Kerguelen plume[END_REF]. After the tholeiitic-transitional basaltic flood volcanism ended (25 My in the East) related to a decreasing magma flux, eruptions were limited to smaller volumes. These included highly alkaline lavas (lamprophyre, basanite to phonolite and trachyte series) and plutons of alkalic gabbro, syenite and granite [START_REF] Frey | Origin and evolution of a submarine large igneous province: The Kerguelen Plateau and Broken Ridge, southern Indian Ocean[END_REF][START_REF] Moine | Incompatible trace element and isotopic (D/H) characteristics of amphibole-and phlogopitebearing ultramafic to mafic xenoliths from Kerguelen Islands (TAAF, South Indian Ocean)[END_REF][START_REF] Weis | Kerguelen plutonic complexes: Sr, Nd, Pb isotopic study and inferences about their sources, age and geodynamic setting[END_REF][START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF]. The chemical interaction between the South-East-Indian-Ridge (SEIR) axis and the Kerguelen plume stopped prior to 25 My [START_REF] Doucet | Constraining the components of the Kerguelen mantle plume: A Hf-Pb-Sr-Nd isotopic study of picrites and high-MgO basalts from the Kerguelen Archipelago[END_REF]. The evolution in incompatible element concentration (La/Yb; fig 2b in [START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF]) in these alkaline magmatic series was interpreted as a varying extent of melting over the past 30 My [START_REF] Frey | Origin and evolution of a submarine large igneous province: The Kerguelen Plateau and Broken Ridge, southern Indian Ocean[END_REF][START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF][START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF] from relatively high in the Lower Miocene, low in the Upper Miocene to intermediate for the < 1 My Mount Ross lavas. [START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF] inferred that this temporal magmatic evolution is directly related to the evolutionary stages of plume-derived volcanism. The high magmatic supply generating the tholeiitic lavas forming the Kerguelen Plateau is derived from the plume-head (e.g., [START_REF] Campbell | Melting in an Archean mantle plume: heads it's basalts, tails it's komatiites[END_REF]) associated to the influence of the SEIR. Following dispersal of the plume head, magma supply decreased generating long-term magmatism derived from the plume-tail followed by a decrease in plume-derived volcanism due to a reduction of plume material creating the alkaline volcanism forming the Kerguelen Archipelago. Another possibility is that the Kerguelen Archipelago was not centered above the plume after 25 My or that the magma flux decreased due to the increasing lithospheric thickness beneath the archipelago [START_REF] Frey | Origin and evolution of a submarine large igneous province: The Kerguelen Plateau and Broken Ridge, southern Indian Ocean[END_REF][START_REF] Frey | Flood basalt from mont tourmente in the Central Kerguelen Archipelago: The change from transitional to alkalic basalt at ≈ 25 Ma[END_REF]. Nevertheless, the temporal isotopic and geochemical evolution of the Kerguelen plume-related lavas causes are still debated whether it results from a heterogeneous mantle source or not. Trace element and radiogenic isotope studies (Sr, Nd, Pb) performed on these three magmatic series [START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF] [START_REF] Barling | Extreme isotopic variations in Heard Island lavas and the nature of mantle reservoirs[END_REF], no assimilation of a geochemicallydistinct/continental lithospheric component was described [START_REF] Frey | Flood basalt from mont tourmente in the Central Kerguelen Archipelago: The change from transitional to alkalic basalt at ≈ 25 Ma[END_REF][START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF]. However, a study of picrites and high-MgO basalts from [START_REF] Doucet | Constraining the components of the Kerguelen mantle plume: A Hf-Pb-Sr-Nd isotopic study of picrites and high-MgO basalts from the Kerguelen Archipelago[END_REF] Furthermore, they define a unique chemical trend, in general, for Fe 2 O 3 , TiO 2 , P 2 O 5 and MgO content [START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF]. The MgO contents of the lavas for the three magmatic series are, except for 2 basaltic lavas, < 8 wt%. Seven of these samples were studied for Sr, Nd, Pb isotopes by [START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF] and the corresponding values are displayed in the Leyrit 1992, Weis et al. 1993 andthis study), as well as Sr, Nd, Pb isotopic ratios [START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF]) and GPS coordinates of the Kerguelen lavas.

Methods

Petrological studies

In order to characterize the different lavas of the Kerguelen Archipelago, petrographic observations were done by polarized microscopy, and different analyses were subsequently performed on available thin sections. Scanning-Electron Microscope (SEM) analyses using a JEOL JSM 6360LV housed at Géosciences Environnement Toulouse (GET) were first conducted. Back-scattered electron (BSE) images and reflected light microscopy photographs have been produced to facilitate the determination of the modal proportion of the iron-bearing mineral phases for each investigated lava.

Major element compositions of the bulk lavas are displayed in table (5.1). Most of the major elements compositions of the lavas were analyzed by [START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF] and Leyrit working distance: 11 mm. Analyses were calibrated using natural standards: albite for Na, Al, Si; almandine for Fe, Mg; diopside for Ca, orthoclase for K; spessartine for Mn and Ti metal for Ti and corrected by PhiRhoZ method.

Mass spectrometry

The whole rock samples were crushed and powdered in an agate mill at GET Toulouse.

Four to five mg of sample powder were digested at high temperature in closed Teflon beakers in a concentrated HCl-HNO3-HF solution. Separation of iron was performed using anion exchange chromatography with 0.5 ml of AG1-X4, 200-400 mesh in an HCl medium at GET Toulouse (Poitrasson and Freydier, 2005) or with 1 ml of AG1-X8, 200-400 mesh at the Laboratoire Magmas et Volcans (LMV) -Clermont-Ferrand [START_REF] Dauphas | Mass spectrometry and natural variations of iron isotopes[END_REF].

Iron isotope measurements were performed at CNRS-ENS Lyon on the national CNRS-INSU facility and at GET using in both cases a Thermo Scientific Neptune Plus MC-ICP-MS following the protocol described by Poitrasson and Freydier (2005). Purified samples were introduced into the system in 0.05M HCl via a Quartz Cyclonic Scott-Double Pass spray chamber with a 100 µL/min PFA nebuliser. The concentration of Fe and the internal standard doping element (Ni) were set respectively to a concentration of 2 ppm and 1.6 ppm when using the high mass resolution slits and 1 ppm and 0.8 ppm with Samples were dissolved in a HF-H 2 SO 4 solution with V 5+ in excess. Consequently, the ferrous iron released is fully oxidized shifting the equilibrium reaction to the righthand side under strongly acidic conditions. Then, the pH is increased to 5 in order to shift the equilibrium reaction to the left-hand side and regenerate Fe 2+ from the V 4+ .

The logic in performing such a back-titration is that the VO 2 species is less prone to accidental oxidation relative to Fe 2+ thereby minimizing experimental errors, particularly associated with H 2 SO 4 dissolution [START_REF] Bézos | The Fe3+/ΣFe ratios of MORB glasses and their implications for mantle melting[END_REF]. The colorimetric procedure is described in detail in Schuessler et al. (2009). Finally, Fe 3+ /Fe tot ratios were deduced from the measured Fe 2+ /Fe tot . [START_REF] Schuessler | Oxidation state of iron in hydrous phono-tephritic melts[END_REF] reported an external reproducibility of 0.20 wt% (2SD) for Fe 2+ (wt%) and Fe tot and of 0.03 % on the Fe 2+ /Fe tot ratio (for lavas containing between 0.8 and 12 wt% Fe tot ). In order to monitor data accuracy and precision, the USGS geostandard BHVO-2 was also analyzed. The BHVO-2 Fe tot measured by this colorimetric method yields 8.46 ± 0.12 wt%, which is indistinguishable from the USGS recommended values of 8.63 ± 0.14 wt%.

Results

Mineralogy

Detailed petrologic description of each lava of the LMS, UMS and MRS and the modal proportions of iron-bearing minerals therein are respectively displayed in Appendix C (Table 11 and19). The main iron-bearing minerals through all volcanic series are pyroxenes (between 30 and 5% by volume of the lavas) and oxides (between 14 and 1% by volume of the lavas). The microcrystalline mesostasis is composed of feldspar, pyroxene, olivine and oxide for the LMS and MRS and feldspars, nepheline, analcime, pyroxene and oxide in the UMS with evolving proportion and composition during the differentiation. Olivine phenocrysts are only present in the less differentiated basalts/basanites (RR08-88, ARC 590, ARC 321, ARC 172, RR08-64). X-Ray Diffraction analysis showed no evidence of clay minerals in the trachytes LVLK 127 and ARC 272 but the presence of amorphous mineraloids. EPMA analysis of this amorphous phase revealed that this iron-rich mineraloid corresponds to chlorophaeite, a common secondary amorphous phase in lavas [START_REF] Peacock | Chlorophaeite, sideromelane and palagonite from the Columbia River Plateau[END_REF]. The trachyte matrix consists of numerous feldspars, rare pyroxenes and Fe-Ti oxides microcrysts, as well as rare apatite inclusions. The feldspars around chlorophaeite are well-shaped with a clear contact between their rims and the mineraloids. Chlorophaeite represents 3% of LVLK 127 and 10% of Arc 272. This amorphous mineraloid mainly occurs in diktyatixic voids and as veinlets in the trachytes. Besides this amorphous chlorophaeite, there are no clay mineral identified by XRD. The presence of chlorophaeite in these trachytes is most likely due to deuteric alteration of interstitial volcanic glass and/or pyroxenes induced by endogenous fluids as proposed for lavas from Iceland and Ediacaran Volyn-Brest volcanics [START_REF] Peacock | Chlorophaeite, sideromelane and palagonite from the Columbia River Plateau[END_REF][START_REF] Środoń | Hydrothermal alteration of the Ediacaran Volyn-Brest volcanics on the western margin of the East European Craton[END_REF] due to the high volatile content of those trachytes.

Compositional evolution of the oxides

The FeO-Fe ). This magnetite appearance, followed by the crystallisation of titomagnetite (Mag 15-50 ) in the most evolved lavas, is associated with crystallization of ilmenite.

Compositional evolution of pyroxenes and amphibole

Electron (1997). The black dots represent the calculated δ 57 Fe from the model based on Rayleigh fractionation. Uncertainties of the models are calculated by propagation of the standard error of the initial isotopic composition of the liquid δ 0 and of the standard error of the fractionation factor ∆ 57 Fe mineral-melt used in the model (Rabin et al., 2021a)

Discussion

Petrologic evolution

Lower Miocene Series

The alkaline Lower Miocene basalts are characterized by the presence of olivine, pyroxene, plagioclase, sanidine and a small amount of oxides. Then, the magma evolves with more oxides crystallization together with alkaline feldspar and the appearance of chlorophaeite in the most evolved lavas. The presence of ulvöspinel-rich titanomagnetite together with ilmenite in the less evolved lavas followed by the sudden crystallization of small amounts (1%) of magnetite-rich titanomagnetite coincident with ilmenite disappearance is observed in this series. The compositional evolution of the oxides, from Fe 2+ -rich ulvöspinel and ilmenite to Fe 3+ -rich titanomagnetite in the last stage of magmatic differentiation, appears to be typical of a system open to oxygen exchange [START_REF] Toplis | Differentiation of ferro-basaltic magmas under conditions open and closed to oxygen: Implications for the skaergaard intrusion and other natural systems[END_REF]. The Fe 3+ /Fe tot in the whole rocks remains constant at roughly 0.5 over a large range of SiO 2 contents, from 48 -63 wt.%. This constancy is in spite of the variation in mineral composition and modal abundance, that otherwise result in a net increase in Fe 3+ /Fe tot over this range of SiO 2 as in the UMS and MRS suites.

Upper Miocene Series

Lavas of the UMS define a typical silica-undersaturated magmatic series, with a high proportion of sodium-rich feldspars, iron-rich pyroxene, felspathoids and oxides. In the early stage of differentiation, olivine, pyroxene, ulvöspinel and ilmenite are the crystallizing ironbearing phases, and incorporate mainly Fe 2+ because of the lack of appropriate crystal sites to accommodate ferric iron. Then, magnetite-rich titanomagnetite becomes the main iron-bearing phase (ARC 96), allowing considerable incorporation of Fe 3+ . The evolution of the oxide composition from ulvöspinel and ilmenite to magnetite-rich titanomagnetite and ilmenite disappearance shows that the system is open to oxygen exchange [START_REF] Toplis | Differentiation of ferro-basaltic magmas under conditions open and closed to oxygen: Implications for the skaergaard intrusion and other natural systems[END_REF]. These oxides systematics may be reconciled with the crystallization of predominantly ferrous iron-bearing minerals during the early stages of magmatic differentiation, which, by mass balance, leads to the increase of ferric iron in the remaining melt. Once the activity of ferric iron in the melt is sufficiently high, magnetite fractionates and incorporates largely ferric iron. This mineralogical evolution is supported by the observed increasing Fe 3+ /Fe tot ratio along magmatic differentiation of the UMS (fig 5.5).

Mont Ross Series

The mineralogical evolution of the silica-saturated Mont Ross series is similar to the LMS with only differences in proportion and composition. Chlorophaeite was not observed in the Mont Ross series. We observe the appearance of ulvöspinel-rich titanomagnetite with ilmenite in the less evolved lavas, followed by the sudden crystallization in small proportions of pure magnetite in MCP-99-01 (1.4 % ), concurrent with ilmenite disappearance.

As described for the LMS, compositional evolution of the oxides is typical of a system open to oxygen exchange [START_REF] Toplis | Differentiation of ferro-basaltic magmas under conditions open and closed to oxygen: Implications for the skaergaard intrusion and other natural systems[END_REF]. The Fe 3+ /Fe tot trend defined by the whole rocks is consistent with that of the oxides, because as the ferric iron content continues to increase as a function of increasing SiO 2 .

Mantle heterogeneity

Numerous studies previously underlined that iron isotopes could fractionate during magmatic differentiation (e.g., Poitrasson and Freydier, 2005;Schoenberg et al., 2009;Schuessler et al., 2009;Sossi et al., 2012;Williams et al., 2018;Zambardi et al., 2014). In order to correct for early olivine crystallization prior to eruption and recover the value of the primitive basalt (δ 57 Fe prim ) with an Mg# = 72, a Fe-Mg correction (see Sossi et al. (2016) for details) was performed on each basalt studied (Table 5.2). The ∆ 57 Fe olivine-basalt = -0.34 × 10 6 /T 2 and ∆ 57 Fe olivine-basanite = -0.20 × 10 6 /T 2 (at magmatic temperature) used in this correction were obtained by first-principle calculation (Rabin et al., 2021a) Fe 2+ (D=0.4) leads to an enrichment in Fe 3+ in the melt generated though partial melting [START_REF] Canil | Distribution of ferric iron in some upper-mantle assemblages[END_REF]Cottrell and Kelley, 2011;Mallmann and O'Neill, 2009) and therefore an enrichment in 57 Fe (e.g., Dauphas et al., 2014;Sossi and O'Neill, 2017). However, Gleeson et al. (2020) underlined that variation of f O 2 in the mantle can partly but not entirely explain the δ 57 Fe heterogeneity observed in the lavas. We performed here partial melting modelling (Fig. 5.7; Appendix C -Table 13 to 15) in order to assess whether a homogenous mantle source (i.e. mantle plume) could be responsible for the observed iron isotopes and chemical evolution though time in Kerguelen Archipelago or if multiple mantle sources are involved (e.g. metasomatized mantle, convecting asthenosphere). The equation of batch partial melting used, defined by Dauphas et al. (2014), takes into account isotopic fractionation between Fe 2+ -bearing melt and Fe 2+ -bearing minerals as well as isotopic fractionation between Fe 3+ -bearing melt and Fe 2+ -bearing melt (Appendix C -Table 13 to 15). The iron isotope fractionation factors between minerals and melt are taken from the first principles study of Rabin et al. (2021b) and the fractionation factors between Fe 3+ -melt and Fe 2+ -melt are taken from NRIX measurements of Dauphas et al. (2014). The temperature at depth during partial melting is taken to be ≈ 1300 • C. At such temperature associated to low pressure (< 2 GPa), clinopyroxene is the major contributor to partial melting. Iron isotopes fractionation factors between the minerals of interest (diopside, augite and forsterite) and melt (basalt, basanite) were taken from the first-principles study of Rabin et al. (2021b). Details of the partial melting models are displayed in Appendix C (Table 13 to 15) .

According to the model (Fig. 5.7), the low LMS δ 57 Fe prim (e.g. δ 57 Fe prim = -0.00 % for ARC 590) can only be explained by a highly oxidized mantle source (high f O 2 ) associated with low iron isotope signature. We assess, from our batch partial melting model that a mantellic source with a Fe 3+ /Fe tot = 0.1 and a δ 57 Fe of -0.20 ± 0.035 % is necessary to create such low isotopic signature in the LMS primitive basalts. Partial melting of metasomatized peridotites is the best candidate to explain such high f O 2 and low iron isotope signature. Indeed, Kerguelen Archipelago went thought multiple metasomatism events that impacted highly the upper-mantle (e.g., [START_REF] Delpech | In-situ geochemistry of sulfides in highly metasomatized mantle xenoliths from Kerguelen, southern Indian Ocean[END_REF][START_REF] Grégoire | Armalcolite-bearing, Ti-rich metasomatic assemblages in harzburgitic xenoliths from the Kerguelen Islands: Implications for the oceanic mantle budget of high-field strength elements[END_REF][START_REF] Grégoire | The lithospheric mantle beneath the Kerguelen Islands (Indian Ocean): Petrological and petrophysical characteristics of mantle mafic rock types and correlation with seismic profiles[END_REF][START_REF] Moine | Incompatible trace element and isotopic (D/H) characteristics of amphibole-and phlogopitebearing ultramafic to mafic xenoliths from Kerguelen Islands (TAAF, South Indian Ocean)[END_REF]. The metasomatism affecting Kerguelen Archipelago started before the eruption of the LMS as the Kerguelen harzburgites were generated most likely during the interaction of SEIR and the hot spot (e.g., [START_REF] Grégoire | Armalcolite-bearing, Ti-rich metasomatic assemblages in harzburgitic xenoliths from the Kerguelen Islands: Implications for the oceanic mantle budget of high-field strength elements[END_REF]. Metasomatism events also impacted the iron isotope composition of the peridotites, with δ 57 Fe as low as -0.806 ± 0.047 % in Kerguelen Archipelago [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF]. Furthermore, [START_REF] Mcguire | Metasomatic oxidation of upper mantle periodotite[END_REF] displays that metasomatized mantle mineral phases display on average 50-65% increase of Fe 3+ /Fe tot which by extrapolation could lead to a Fe 3+ /Fe tot whole-rock metasomatized peridotite up to 0.12. The other LMS basalts, that displays δ 57 Fe prim up to 0.159 % (Table 5.2), can be formed with δ 57 Fe mantle of about -0.04 ± 0.035 % and similar Fe 3+ /Fe tot = 0.1 (Fig. 5.7).

The Gleeson et al., 2020;[START_REF] Matthew | Crustal controls on apparent mantle py-roxenite signals in ocean-island basalts[END_REF][START_REF] Lambart | Markers of the pyroxenite contribution in the major-element compositions of oceanic basalts: Review of the experimental constraints[END_REF]Lambart et al., , 2016[START_REF] Lambart | Highly heterogeneous depleted mantle recorded in the lower oceanic crust[END_REF]Nebel et al., 2019;Williams and Bizimis, 2014). From an iron isotope perspective, pyroxenite have a δ 57 Fe between 0.06 ± 0.08 % and up to 0.27 ± 0.05 % [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF]Williams and Bizimis, 2014) and displays a higher Fe 3+ /Fe tot than fresh peridotites because of its higher clinopyroxene content [START_REF] Canil | Distribution of ferric iron in some upper-mantle assemblages[END_REF]. A pyroxenite component in the magmatic source associated with low degree of partial melting would explain why OIB displays significantly higher ferrous iron content compared to normal MORB ( Fe 3+ /Fe tot = 0.09 ± 0.02 ; Bézos et al. (2021)). A mantellic source of δ 57 Fe = 0.09 ± 0.059 % with a Fe 3+ /Fe tot = 0.09 is needed to reproduce the δ 57 Fe prim of 0.236 % (ARC 172). The other basalts from UMS can be formed with δ 57 Fe mantle of about -0.02 ± 0.059 % and similar Fe 3+ /Fe tot = 0.09 which implies a heterogenous mixing between pyroxenite and metasomatized peridotites in the mantle. Concerning the most recent magmatic series, MRS, the signature can also be explained by a mix of moderately metasomatized Archipelago is spread between -0.806 ± 0.047 % and 0.053 ± 0.064 % [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF] and the δ 57 Fe variations of pyroxenite is poorly constrained [START_REF] Poitrasson | On the iron isotope heterogeneity of lithospheric mantle xenoliths: Implications for mantle metasomatism, the origin of basalts and the iron isotope composition of the Earth[END_REF]Williams and Bizimis, 2014), therefore no precise proportion of each components in the source can be given. However, according to Monte Carlo mantle melting model for a pyroxenite source solution performed Gleeson et al. (2020), a proportion of 0.2 and 0.5 of enriched melt in the plume source would be necessary to explain the MRS and UMS iron isotopes signature, respectively.

Our partial melting models support the idea of the Kerguelen mantle source heterogeneity inferred in previous studies (e.g., [START_REF] Borisova | Lead isotope signatures of Kerguelen plume-derived olivine-hosted melt inclusions: Constraints on the ocean island basalt petrogenesis[END_REF][START_REF] Doucet | Constraining the components of the Kerguelen mantle plume: A Hf-Pb-Sr-Nd isotopic study of picrites and high-MgO basalts from the Kerguelen Archipelago[END_REF][START_REF] Doucet | Primitive neon and helium isotopic compositions of high-MgO basalts from the Kerguelen Archipelago, Indian Ocean[END_REF][START_REF] Frey | Flood basalt from mont tourmente in the Central Kerguelen Archipelago: The change from transitional to alkalic basalt at ≈ 25 Ma[END_REF]. The heterogeneity of the mantle could also be supported by the radiogenic isotopes (Sr-Nd-Pb) characteristics of the three magmatic series. Nebel et al. (2019) observed a co-variance between Sr, Nb, Pb isotopes and Fe isotopes variation for the Pitcairn OIB that are in favor of a mixing between the two physically distinct components (i.e. mantle plume and pyroxenite), supporting the hypothesis of an heterogenous mantellic source and the implication of a pyroxenite component. Looking at the little Sr-Nd-Pb isotope available data for our samples, it seems that these co-variances could also be observed in Kerguelen Archipelago. Further study would need to be pursued in order to confirm this hypothesis. Partial melting of different lithological source is needed to reproduce the LMS, UMS and MRS iron isotope basaltic compositions. This implies that the temporal isotopic variations of the Kerguelen lavas, is probably not due to a geodynamics evolution from the head to the plume tail [START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF], but from an heterogeneous sampling of distinct mantle domains (e.g. Mantle plume, metasomatized upper-mantle, pyroxenite) as inferred for the post-flood basalt volcanism on the Southeastern Ethiopian Plateau [START_REF] Nelson | Distinguishing plume and metasomatized lithospheric mantle contributions to post-flood basalt volcanism on the Southeastern Ethiopian Plateau[END_REF]. Our study shows that mantle heterogeneity at regional scale can also be underlined by iron isotope system. Williams et al. (2005) estimated indeed that many billions of years would be necessary the homogenize iron isotope mantle compositions with their surroundings (length-scales > 1km).

Magmatic differentiation: impact of fractional crystallization

The study of the three Kerguelen Archipelago magmatic series allows us to better constrain iron isotope fractionation along magmatic differentiation. The LMS trachyte exhibit similar δ 57 Fe and Fe 3+ /Fe tot as their associated basalt (Fig. 5.5a). Whereas both UMS and MRS displays heavier iron isotopic compositions and higher ferrous iron content in the most evolved lavas (Fig. 5.4b-c and 5.5a-b) as observed in other magmatic series such as in Cedar Butte Volcanoes and Red Hill (Sossi et al., 2012;Zambardi et al., 2014). The concomitant evolution of δ 57 Fe and Fe 3+ /Fe tot implies that the processes involved in these variations along magmatic differentiation might be similar (Fig. 5.5). Sossi and O'Neill (2017) evidenced that 57 Fe tends to form stiffer bonds to oxygen relative to 56 Fe or 54 Fe, and therefore to prefer the Fe 3+ state. In a magmatic system open to oxygen, where the Fe 3+ /Fe tot keep increasing during magmatic differentiation, δ 57 Fe is also expected to constantly increase accordingly.

In order to access whether fractional crystallization is the main processes involved in the heavier isotopic signature of the UMS and MRS lavas, Rayleigh distillation-based models (Sossi et al., 2012) were performed. The iron isotope composition of the liquid is defined as:

δ 57 F e = δ 0 + ∆ bulk ln(f F e ) (5.4)
Where δ 0 is the initial iron isotopic composition of the liquid, f F e the iron fraction remaining in the melt and ∆ bulk is the bulk isotope fractionation factor between mineral and melt, expressed as :

∆ bulk = ∆
57 F e mineral-melt × P n (5.5)

Where P n represents the modal proportion of iron bearing minerals of phase n and ∆ 57 Fe mineral-melt is the equilibrium fractionation factor between the mineral phase n and the melt that is dependent of temperature (10 6 /T 2 ). The absolute temperature is estimated using Sisson and Grove geothermometer [START_REF] Sisson | Temperatures and H2O contents of low-MgO high-alumina basalts[END_REF]. To constrain P n , Fe-bearing mineral proportions in the lavas were determined by petrographic observations (Appendix C -Table 16). The initial isotopic composition of the melt, δ 0 , corresponds to the δ 57 Fe prim of sample ARC 590 for the LMS, RR08-88 for the MRS and sample ARC 321 for the UMS.

The models use the set of equilibrium fractionation factor between minerals and melts calculated by Rabin et al. (2021b) except for the ∆ 57 Fe ilmenite-melt = 0.02 × 10 6 /T 2 obtained by combining the determination of Sossi and O'Neill (2017)) and Sossi et al. (2012).

The ∆ 57 Fe titomagnetite-melt used vary depending on the magnetite percentage (X mag ) in the titanomagnetite from Kerguelen lavas as fractionation factors are evolving depending on the concentration and oxidation state of iron (Schoenberg et al., 2009;Shahar et al., 2008;Sossi and O'Neill, 2017). We determined intermediate fractionation factors that are a function of X mag in titanomagnetite from the calculated ∆ 57 Fe ulvspinel-melt (Xmag =0)

and ∆ 57 Fe magnetite-melt (X mag = 1) by Rabin et al. (2021b) (Appendix C -Table 16). We also used the ∆ 57 Fe augite-melt t as an approximation for the rare amphibole presents in a Kerguelen phonolite as Schoenberg et al. (2009) showed that the ∆ 56 Fe Augite-Ferric-pargasite was about 0 % in an alkaline syenite. This is considered as a valid fractionation fac-tor as, in the Kerguelen phonolite (ARC 96), the main pyroxenes present are augite and the amphiboles are kaersutite which are similar to ferric-pargasite in term of similar low Fe 3+ /Fe tot content and similar (Na+K)/(Na+K+Ca) ratio of about 0.35. Also, iron isotope fractionation in plagioclase has been considered as negligible because the iron concentration in the plagioclase of the lavas studied is minor (Du et al., 2017). The different parameters as well as the ∆ 57 Fe mineral-melt used for the models are displayed in Appendix C (Table 16) and chapter 4. This self-consistent theoretical ∆ 57 Fe mineral-melt set is for the first time applied to natural samples. This new set allow us to account with precision for Fe 2+ -bearing minerals iron isotope fractionation that appeared to impact significantly the iron isotope composition of the lavas during magmatic differentiation (Rabin et al., 2021a). The UMS and MRS modelled iron isotope evolution displays an increase in δ 57 Fe in the most evolved lavas as observed in our studies (5.5). The model of crystal fractionation is fitting within uncertainties the δ 57 Fe evolution of both UMS and MRS meaning that fractional crystallization can be the process involved in these isotopic Significant δ 18 O variations in matrix and olivine within single eruption in Kilauea Volcano, another well-know OIB setting, were also interpreted as witnesses of hydrothermally altered basaltic crust assimilation [START_REF] Michael | Crustal contamination of Kilauea volcano magmas revealed by oxygen isotope analyses of glass and olivine from Puu Oo eruption lavas[END_REF]. However, we cannot preclude that the low δ 57 Fe of the LMS trachytes could also be due to the fact that the trachytes don't belong to the same magmatic series as the LMS basalt studied and their basaltic parental magma had lower δ 57 Fe and lower Fe 3+ /Fe tot .

Conclusion

Kerguelen Archipelago lavas display iron isotopic signatures ranging from δ 57 Fe = + 0.050 ± 0.037 % to 0.328 ± 0.121 % and Fe 3+ /Fe tot between 0.2 and 0.6. These variations enabled us to propose mechanisms involved in iron isotope fractionation during magmatic differentiation of the three magmatic series and to underline the hypothetical heterogeneity of the mantle beneath Kerguelen Archipelago.

The new set of precise Fe fractionation factor between minerals and melt from Rabin et al. (2021b) used, show that the δ 57 Fe evolution during magmatic differentiation of UMS and MRS lavas could be explained by fractional crystallization process. Iron isotopic evolution of the LMS suggested that the lavas might originate from different magmatic source or that iron isotopes fractionation could be sensitive to other processes such as assimilation of altered oceanic crust during magma residence or ascent. Further investigation on this magmatic series would be necessary to clarify to process generating the iron isotopic signature of the most evolved lavas.

Our study shows that the δ 57 Fe variation of the Kerguelen basalts cannot only be explained in term of partial melting degree and/or oxygen fugacity of a peridotite source.

The batch melting models suggest that an isotopically heavy mantle component in the mantle source (e.g. pyroxenite) is necessary in order to recreate the δ 57 Fe signature of the UMS basanites and MRS basalts. These results are consistent with recent studies that implied the necessity of a pyroxenite component in the mantellic source in order to reproduce δ 57 Fe of various OIB and MORB (e.g., Gleeson et al., 2020;Konter et al., 2016;Nebel et al., 2019). From an iron isotope perspective, the Kerguelen magmatism could originate from a heterogeneous mantellic source at small-scale over the last 25 Ma: from a metasomatized peridotites component in the mantle source leading the low δ 57 Fe LMS basalt compositions to the addition of a pyroxenite component in the mantle source generating the UMS and MRS basalt δ 57 Fe. We show that a simultaneous study of Fe 3+ /Fe tot and iron isotopes could be a powerful tracer for mantle lithological heterogeneity.

The last chapter is focused on the analytical development work performed during this thesis. We developed a methodology of in-situ iron isotopes analysis on silicate minerals using a fs-LA-MC-ICP-MS. This chapter details the methodology established in order to perform accurate, reproducible and precise Fe isotopes analysis on small silicates minerals, such as the olivine phenocrysts from Kerguelen Archipelago lavas. This chapter represents the beginning of a larger study that would ideally allow to determine in-situ Fe fractionation factors between silicates minerals and melts in natural lavas that could then be compared to the theoretical ones.

Introduction

Laser ablation system coupled to ICP-MS became a center of interest for the scientific community in the 80's. This new method of samples introduction in the plasma appears to have two major advantages over the conventional solution nebulization: i) It eliminates the preparation steps of the samples (dissolution, ion chromatography, dilution) that can be expensive, time consuming and also source of errors (e.g. fractionation during ion chromatography or contamination). ii) Laser ablation analyses are fast and only need really small amount (from femtograms to micrograms) of samples with high spatial resolution compared to solution nebulization that need much more material (milligrams).

In geosciences, laser ablation allows to analyze minerals directly within the rock without crushing it, handpicking minerals carefully and going through the all chemical steps as done in past (e.g., Sossi et al., 2012;Teng et al., 2011;Williams and Bizimis, 2014). The major limitation of laser ablation system coupled to (MC)-ICP-MS is the precision of the measurements that is poorer than the one obtained by solution nebulization due to the laser/matter interaction, the efficiency of aerosols transportation, the ionization of the produced aerosols in the plasma and the matrix effects. Much progress has been made in terms of uncertainties enhancement that enforce LA-(MC)-ICP-MS as one the most powerful and advanced analysis technique in laboratories, especially after the emergence of the femtosecond laser system. Since more than a decade, in situ iron isotopes measurements using both nanosecond and femtosecond LA-MC-ICP-MS were performed on magnetite to characterize banded iron formations (Czaja et al., 2013;Knipping et al., 2019;Li et al., 2013), on various oxides and hydroxides to characterize hydrothermal ore formation (Horn et al., 2006), on sulfides and iron meteorites (Košler et al., 2005) or on silicates minerals such as olivine in order to constrain kinetic isotope fractionation in magmatic settings (Collinet et al., 2017;Oeser et al., 2014Oeser et al., , 2015Oeser et al., , 2018)). In order to obtain precise isotopic ratios, many aspects of the in situ iron isotopes fractionation were investigated such as the matrix effect (Steinhoefel et al., 2009;Zheng et al., 2018) or the effect of the size and transportation of the aerosols generated in the ablation cells (D'Abzac et al., 2014;Košler et al., 2005). In 2014, Oeser et al. developed a method for in situ iron isotopes measurements using femtosecond laser coupled to MC-ICP-MS with a δ 56 Fe analytical reproducibility of 0.13 % based on replicate analyses of silicate reference glasses over one year, using a Ni-spiked solution such as for the solution nebulization method. This method is carried on since then at the Leibniz Universität in Hannover for different studies on olivine (Collinet et al., 2017;Oeser et al., 2015Oeser et al., , 2018)).

However, no other laboratory uses this methodology for in situ iron isotopes measurement. Therefore, there is no data from interlaboratory reproducibility for this method.

In 2016, the "Observatoire Midi-Pyrénées, Toulouse" purchased for the ICPMS laboratory a femtosecond laser ablation system that allowed us to start iron isotopes fs-La-MC-ICP-MS analysis, to set up in situ Fe isotope measurements for Earth and Environmental applications (Poitrasson and D'Abzac, 2017). This allowed us to test the methodology of Oeser et al.,2014 and adapt it to our laser ablation system and MC-ICP-MS conditions (nebulization and introduction of the Ni-spiked solution in the plasma, laser ablation parameters, spatial resolution). The main goal of this study is to produce accurate, precise and reproductible iron isotopes measurements with high spatial resolution on relatively small olivines crystals (and surrounding glasses) in various magmatic settings. The first part of this chapter is a generic part dedicated to the principles of laser ablation system, from nanosecond to femtosecond. This part underlines the major parameters that are essential to the use of the laser and the problematics that must be taken into account for accurate and precise La-(MC)-ICP-MS measurements whatever the elements/isotopes of interest. The second part summarizes the methodology used at the OMP for in situ iron isotopes measurements, description of the reference materials and samples as well as the details of the data post-treatment. The third and fourth parts focus on the results from the fs-La-MC-ICP-MS session of march 2019 and on the discussion of this results (reproducibility, spatial resolution ..) respectively.

Generalities

This part is dedicated to the generalities on laser ablation (principle of laser, major parameters during ablation, advantages and inconveniences ..), for complementary information see D'Abzac (2010), [START_REF] Claverie | Développement et applications d ' un système laser femtoseconde infra-rouge basse énergie et haute cadence de tir pour l ' analyse d ' éléments trace dans les solides par couplage ablation laser / ICPMS[END_REF] and [START_REF] Claverie | Laser ablation. Number Umr[END_REF]. For fundamental physics theory behind laser ablation see Demtröder (1981) and Svelto (2002). It can be a solid (rubis, Nd-YAG), a glass or a gas (He, ArF, XeCl..).

Principle of laser

The laser beam hence produced is characterized by its pulse duration (∆ t ), wave length (λ) and frequency (1/T), with T being the duration between two pulses. In the beginning of the laser technology, the laser beam produced slower impulsions than nanoseconds (10 -9 s) with beam light ranging from infrared to ultraviolet wavelength.

Pumping system

Active medium

Optical cavity

Laser beam Mirror Output mirror Figure 6.2: Schematic representation of a laser ablation system.

Laser ablation and ICP-MS: limitations

Despite its advantages, nanosecond laser ablation coupled to ICP-MS is confronted to a major limitation related to the chemical fractionation process. Many studies in the 90s demonstrated indeed that this method has a quantification bias dependent of the studied elements (e.g., [START_REF] Chen | Inter-element fractionation and correction in laser ablation inductively coupled plasma mass spectrometry[END_REF]Outridge et al., 1996Outridge et al., , 1997)). This accuracy issues can be explained by a laser ablation producing an aerosol with a heterogeneous particle size distribution which does not reflect the stoichiometry of the material of interest or by a modification of the elemental ratio during the creation of the crater. There is three potential sources of fractionation concerning nanoseconds laser ablation: 1) The preferential evaporation of the volatile elements in the carrier gas (e.g., Figg and Kahr, 1997); 2) The distribution size of the particles in the aerosol that are more or less efficiently ionized in the ICPMS plasma (e.g., Koch et al., 2005;Kuhn andGünther, 2003, 2004); 3) the isotopic fractionation induced by the heat generated during laser ablation of the sample (e.g., Jackson and Günther, 2003). Chemical fractionation can appear during the different step of LA-(MC)-ICPMS analysis. Therefore, particular attention to stoichiometry, aerosol transport efficacity and complete ionization in the plasma must be given towards obtaining accurate analysis. In order to compensate for chemical fractionation, also called matrix effect, the users are required to use standards with similar behavior (transparency, thermic conductivity, . . . ) and a chemical composition close to that of the samples of interest, in other term "matrix matching standards". This is often problematic for the analysis of natural samples that are rarely homogeneous.

Advantages of femtosecond vs nanosecond laser

For ablation purposes the femtosecond lasers constitute the last generation of ultrafast laser and deliver their energy in less than 10-15 seconds. Regarding laser-matter interaction, the laser ablation mechanisms differs when the threshold of picoseconds pulse duration is overcome. The main distinction is observable during the adsorption of the laser radiation by the solid of interest, when the delivered energy is converted into heat with a longer pulse. The heat production generates melting and evaporation of the material which induces ejection of electrons and ions in addition to the electrons formed by thermal ionization. This conducts to the formation of a micro-plasma at the surface of the sample, that expands and creates a shock wave due to the internal high pressure. This phenomenon induces microcracks and the damages propagate in a larger area than the laser impact zone (fig. 6.3a). The laser-matter interaction of about 10-9s during nanosecond laser ablation allows the creation of such plasma formation. The micro-plasma forms before the laser pulse ends. As a result, the plasma acts as a shield and the laser pulse loses energy. Moreover, thermal processes, such as partial melting, can occur and create massive fractionation. Concerning femtoseconds laser ablation, the energy delivery duration is short enough to prevent creation of plasma before the end of the pulse leading to very limited damage (fig. 6.3b). Therefore, there is no loss of energy and a very limited thermal effect. The femtosecond laser ablation is characterized as athermal. This is an essential parameter especially for measurements involving ICPMS owing to heat production in the material leads to fractionation. Another advantage of the ultrafast laser ablation is the quality of the aerosol generated. Femtosecond laser ablation produce finer particles, irrespective to their optical properties, than nanosecond ablation that produce micrometric sized particle (e.g., D'Abzac et al., 2012;Diwakar et al., 2014;Zheng et al., 2017Zheng et al., , 2018)).

In addition, femtosecond laser ablation produces an aerosol with homogeneous distribution of particle size that facilitate the aerosol transportation to the (MC)-ICP-MS and its ionization in the plasma. These properties strongly reduce the chemical fractionation possibility, and therefore matrix matching standards is not mandatory.

To resume, shorter laser pulse leads to greater benefits in terms of accuracy, precision and reproducibility of the analysis.

or long wavelength in term of analysis accuracy and precision. On the other side, Ikehata and Hirata (2013) reported slight improvement of δ 65 Cu measurement in sulfides using UV over IR fs-La-MC-ICP-MS. Only few studies are published on this subject and the importance of this parameter is often controverted as the comparisons between equipment is uneasy. However, it has been shown recently with a dual-wavelength femtosecond system that UV was often offering better statistics than IR in the same measurement conditions on zircon (D'Abzac et al., 2020) as well as for other matrices (d'Abzac and Chmeleff comm. Pers.)

The carrier gasses

The carrier gas is a major actor in term of data accuracy as efficient transportation of the aerosol to the MC-ICP-MS is one of the keys to prevent from chemical fractionation (part 6.2.2). Additionally, many studies demonstrated that addition of gas (He, O 2 , N 2 , H 2 , Xe) to the argon in the mass spectrometer plasma leads to higher analytical performances depending the elements studied (e.g., Durrant and Ward, 1994;Sesi et al., 1994). It generally allows to lower the limits of detections, matrix effect or even lower to potential interferences. Therefore, it is important to keep in mind that the choice of the carrier gas does not only affect the transportation processes but also the conditions of ionization in the plasma.

Different carrier gasses were tested in the literature (Ne, Kr, Xe, He, N) beginning by argon. However, Helium quickly appeared to be the best candidate because of its low atomic mass, low density, high potential of ionization (1.5 times larger than argon), and high thermal conductivity (allowing to cool the micro-plasma at the surface of the material; part 6.2.3). Helium reduce indeed drastically the number of particles on the sample surface (Günther and A. Heinrich, 1999a) and induce an increase in (MC)-ICP-MS signal by a factor 2 to 4 compared to argon (e.g., Eggins et al., 1998;Horn and Günther, 2003). Helium fulfills both requirements for the carrier gas: 1) limiting chemical fractionation by efficient aerosol transportation and restraining formation of plasma on the material surface; 2) Increasing the sensibility of the mass spectrometer.

Fluence

The fluence can be defined as the quantity of energy delivered by the laser on the sample surface impacted by the laser beam. This energy flux is expressed in J.cm -2 . The fluence controls the quantity of material ablated and therefore is directly related to analysis sensitivity and chemical fractionation. There is a positive linear relationship between signal intensity of the (MC)-ICP-MS and the fluence (Hergenröder et al., 2006). Depending on the fluence applied, two type of ablation are defined: at low fluence, vaporization dominates and at high fluence, explosion is the main ablation process. The fluence limit between the two type of ablation depends on the laser used and the sample analyzed. It is accepted that fluence impact the particles distribution size (Jeong et al., 1999;Koch et al., 2004;Russo et al., 2002) however no clear tendencies have been brought out as it highly depends on the laser and the ablated material. In order to obtain a high depth resolution with low ablation rate, it is recommended to work with lower fluences. It is important to note that the fluence decreases with the development of the crater, for both nanosecond and femtosecond pulse, because of the defocusing of the laser beam. This fluence evolution during crater formation is thus a cause of fractionation intensification.

Concerning isotopic ratio measurements, it has been inferred, for Cu and Si isotopic systems, that high fluence (> 2 J.m -2 for Cu, > 20 J.m -2 for Si) induce an increase of the heat-affected zone and generate isotopic fractionation during femtosecond laser ablation (Lazarov and Horn, 2015;Schuessler and Von Blanckenburg, 2014). Thus, when using high fluence during sample analysis matrix-matching standard are recommended. The influence of fluence on isotopic measurement during fs-LA-MC-ICP-MS still needs to be constrained.

Pulse repetition rate

The pulse repetition rate (expressed in Hz) has no impact on the depth of crater formation and consequently on the quantity and size of particles generated during laser ablation.

Pulse repetition rate controls however the quantity of material ablated per units of time.

This parameter is really important as it has a major impact on the signal intensity and the sensibility. An ablation with a high pulse repetition rate generates a high signal intensity on a short period of time leading to higher sensibility. On the other side, a low pulse repetition rate (< 5 Hz) induces an unstable signal leading to a lack of precision during measurements. The impact of pulse repetition rate on chemical fractionation is still debated. Pécheyran et al. (2005) did not distinguish any correlation between pulse repetition rate and chemical fractionation during fs laser ablation. Eggins et al.

(1998) and Horn et al. (2000) observed a correlation between degree of fractionation and pulse repetition rate during U/Pb analysis related to the transport of volatiles over refractory elements to the plasma. Others observed an increase in fractionation from the acquisition of Zn/Cu ratio (Koch et al., 2006) or melted crater borders that infer a heat accumulation during repeated and overlapping laser pulse due to lower ablation rate increase at high repetition rate (Gonzalez et al., 2008). The heat accumulation can be avoided by combining laser beam displacement speed and pulse repetition rate is order to diminish the pulse overlapping (forming a line instead of a point with increasing depth).

6.3 Iron isotopes measurements using fs-La-MC-ICP-MS 6.3.1 Femtosecond laser ablation -MC-ICP-MS

The analyses were performed using a commercial ESI "UC Femto dual-wavelength" femtosecond laser ablation system (Portland, USA) and a Thermofisher (Bremen, Germany)

"Neptune plus" MC-ICP-MS at the Observatoire Midi-Pyrénées facility (Table 6.1). The laser pulse amplification system delivers 280 fs pulses at the fundamental wavelength of 1030 nm and the unit can be operated as is. This wavelength can also be converted to a 257 nm beam, by fourth harmonic generation, with a gaussian shape (yield of 35% and a limited loss on the pulse width (≈ 1%)). This last setup was used for this study as it was easier to control than IR. The pulse repetition rate can vary between 1 to 250 Hz. The The measurements were performed in high resolution and static modes in order to measure 3 masses of iron (57 ,56 ,54), 2 masses of Ni (58, 60) and 53 Cr. The analyses were manually triggered and were composed of 60 cycles of 2 s including 8 cycles of background (gas blank) at the beginning of each analysis. The general tuning was performed using Fe-Ni-Cr solution as described in Chapter 1 and tuning of the intensity were performed using a standard called Puratronic, described below. Ablations of Puratronic were performed using a raster pattern with a spot size of 13 µm in diameter, a pulse repletion rate of 15

Hz, a fluence of ≈ 2.7 J/cm 2 and a speed of 5 µm/s. For both standards and samples analysis, the laser parameters were set up in order to obtain a fluence below 4 J.cm -2 .

Average signal intensities on olivine for 54 Fe, 56 Fe, 57 Fe were 0.83 V, 14 V and 0.32 V using 10 11 Ohm amplifiers.

Helium

Results

References materials

In order to evaluate the data quality with respect to the potential interferences, the reference standard analyses were plotted in a three isotopes plot, δ 56 Fe versus δ 57 Fe (fig.

6.5). The measurements in this plot draw a mass fractionation line with a slope of 1.438 ± 0.047 similar to the accepted values of the Fe mass-fractionation line from the literature (from 1.46 to 1.49; Beard and Johnson (2004); Craddock and Dauphas (2011);Malinovsky et al. (2003)). This indicates that the data acquired by fs-La-MC-ICP-MS are reliable and that the isobaric inference on 54 Fe were corrected efficiently. In-situ analyses of the four USGS glasses, BIR-1G, BHVO-2G, BCR-2 and NKT, exhibit δ 56 Fe values of 0.080 ± 0.19 % , 0.045 ± 0.09 % , 0.045 ± 0.09 % , 0.298 ± 0.13 % , respectively (Table 6.2; Annexe D, table 21 to 25). Our results for the references glasses are consistent with published δ 56 Fe values of the respective rocks powers and previous in situ analyses (fig. 3.6). Laser ablations analyses were spread all over the individual glass fragments, in order to assess the reference materials homogeneity. The iron isotopes variation within the USGS glasses are relatively small (considering the spatial resolution of ≈ 170 × 24 µm) and lie within the analytical uncertainty of this study ≤ 0.19 % for δ 56 Fe. Therefore, these USGS glasses can be considered as homogeneous with respect to the analytical uncertainties and suitable reference material for in situ iron isotopes measurements as inferred by Oeser et al. (2014) for BCR-2G, BHVO-2G and BIR-1G. Using Puratronic as a bracketing standard for BHVO-2 leads to accurate and precise iron isotopes measurements, confirming that femtosecond laser ablation does not require matrix matching standard (Steinhoefel et al., 2009;Zheng et al., 2018). However, BHVO-2 was used as the bracketing standard for the other glasses and olivine measurements in order to optimize the analyses session: BHVO-2 has similar/close ablation parameters to USGS glasses and olivine therefore the time between analyses is minimized.

Concerning San Carlos Olivine (SCO), a total of 104 analyses (over 2 days) were spread over 2 fragments of the olivine. Our in situ iron isotopes measurements revealed a significant heterogeneity within single olivine fragments, larger than the analytical uncertainty.

Iron isotopes variations in SCO are displayed in fig. 6.7. San Carlos Olivine is therefore not a suitable reference material for iron in situ analysis. Fe relative to the primary reference material used during the analytical session. δ 56 Fe (% IRMM-14) and δ 57 Fe(% IRMM-14) are the recalculated value relative to IRMM-014 with a value of δ 56 Fe BHVO-2/IRMM-014 = 0.114 ± 0.011 % , δ 57 Fe BHVO-2/IRMM-014 = 0.174 ± 0.016 % for BHVO-2 (Craddock and Dauphas, 2011) and δ 56 Fe Puratronic/IRMM-014 = 0.249 ± 0.032 % and δ 57 Fe Puratronic/IRMM-014 = 0.389 ± 0.057 % for Puratronic (Poitrasson pers. Comm.). Measurements uncertainties (2SD) of the δ-values represents the two standard deviation based on replicate analyses (n) within the analytical session (7 days). The uncertainties of the re-calculated δ 56 Fe (% IRMM-14) and δ 57 Fe (% IRMM-14) are propagated uncertainties of the uncertainties related to the laser ablation (SD LAS ) and of the uncertainties related to the solution nebulization (SD sol ) measurements of δ 57 Fe BHVO-2/IRMM-014 and δ 56 Fe BHVO-2/IRMM-014 where 2SE = 2 × SD 2 sol + SD 2 LAS .

Olivines profiles

Laser ablation profiles were carried out on two olivines from the RR08-88. The profiles are performed perpendicular to the rim, going from the olivine core to rim border (fig.

6.8a). Both lines (parallel to the rim) and spot ablations were performed in order to compare accuracy and reproducibility of both laser ablation methods. The iron isotopes profiles are displayed in figures 6.9 and 6.10. The δ 56 Fe uncertainties on single analysis are < 0.2 % for line measurements and < 0.4 % for spot measurements. In addition, multiple spot analyses were performed on the border of olivine 1, in order to assess the number of analyses statistically needed to obtain uncertainties similar to the line ablation methods (fig. 6.8b). δ 56 Fe (‰ vs IRMM-014)

Figure 6.10: Iron isotopes profiles across olivine 2 obtained with raster analysis.

Discussion

Precision and spatial resolution

The uncertainty of BIR-1G, assessed from 54 non-consecutive analyses over five days of analyses, can be considered as the long-term reproducibility achievable for iron isotopes using fs-LA-MC-ICP-MS. Our analytical uncertainties of 0.19 % for δ 56 Fe is slightly higher than the 0.13 % achieved by Oeser et al. (2014) obtained from raster ablation with different spatial resolution. Raster ablation on our BIG-1G standard is 170 × 24 µm against 200 × 50 µm for Oeser et al. (2014). Our poorer analytical uncertainties could be associated to the rather small spatial resolution used. However, the uncertainties given in this study is not strictly comparable to the ones provided by Oeser et al. (2014) as their uncertainties are based on repeated measurements of standard over one all year, consequently no strong conclusion can be made on this topic. Our study has the advantage to demonstrate that smaller spatial resolution still produces results with uncertainties < 0.2 % (on shorter time-scale) and can be useful for relatively small samples where large iron isotopic variations are expected (diffusion profiles in olivine or metal-silicates partition experimental samples for instance). The relationship between data precision and spatial resolution is clear looking at difference of uncertainties obtained with single spots (23 µm) and raster pattern analysis (150-170 × 20 µm). Looking at the olivine 1 profile (fig. 6.9), single spots analysis induced larger uncertainties (< 0.4 % ) than raster pattern (< 0.2% ). The large uncertainties during single spot analysis is induced by the fast decreasing signal intensity, due to the fluence variation during the crater formation as described in part 6.2.4.3. However, multiple single spots analysis (> 3) on parallel line can produce similar and even smaller uncertainties than raster analysis (fig. 6.8b).

Depending on the size or shape of the samples and the isotopic variations expected, both ablation methods are valid. However, when the samples size is not a limiting parameter raster pattern is preferable over single spot analysis.

Kerguelen Olivine isotopic profiles

The iron isotopes (raster) profile along the Kerguelen olivine 1 displays variation from core to rim (fig. 6.9). The olivine core displays a δ 56 Fe mean value of -0.05 ± 0.15 % down to -0.44 ± 0.14 % at the olivine rim. This δ 56 Fe variation along olivine crystal, already observed by Oeser et al. (2014), Oeser et al. (2015) and Collinet et al. (2017), are attributed to diffusion processes, in other term kinetic-driven fractionation. On the other side olivine 2, that is relatively smaller than olivine 1, displays a constantδ 56 Fe from core to rim of about -0.39 ± 0.15 % (fig. 6.10), a value similar to that of olivine 1 rim. This could mean that olivine 2 was completely affected by isotopic diffusion whilst olivine 1 core have been preserved. This indicates a possible re-homogenization of the olivine phenocrysts by the evolving melt Further isotopic studies of these olivine and the surrounding groundmass need to be performed in order to confirm or deny this hypothesis, however it seems that this sample is not suitable to obtain in situ iron equilibrium fractionation factor between olivine and the groundmass or will necessitate Fe isotopes diffusion modelling to disentangle the diffusion effect.

Conclusion

High precision iron isotopes ratios were determined successfully for USGS glasses and From the new set of fractionation factor, we were able to model precisely the effect of fractional crystallization during magmatic differentiation and to discuss the impact of partial melting on both iron and silicon isotope systems. It appeared that most of the heaviest Si and Fe isotopic signatures measured in evolved lavas could be explained by fractional crystallization, except for the highest δ 57 Fe signature of about 0.9 % . Moreover, our study supports the fact that partial melting is a viable process to explain the heavy Fe isotopes signatures of MORB compared to the mantle δ 57 Fe.

Thirdly, iron isotope measurements using ion-exchange chromatography and MC-ICP-MS were performed on natural lavas. In total twenty-trhee ocean island basalts from Kerguelen, belonging to three distinct magmatic series, were analyzed. The particularity of these OIB is their temporal chemical evolution, with high variation of alkaline content.

From the scope of iron isotopes, the variations of δ 57 Fe and Fe 3+ /Fe tot between the basalt samples of each magmatic series were inferred, thanks to various partial melting models, to originate from variations of mantle material sampling in the magmatic sources. Involvement of pyroxenite, with heavy Fe isotopes composition, in the mantle source was also proposed to explain the heavy Fe signatures of the most alkaline basalts. Our study supports the hypothesis proposed by previous studies that iron isotopes could be a good proxy to identify mantle heterogeneity. But we suggest that combining iron isotopes and Fe 3+ /Fe tot studies might be even more powerful. The evolution of iron isotopes signatures during magmatic differentiation is limited, with small increase of δ 57 Fe in the more evolved lavas of two of the series (UMS, MRS) while the δ 57 Fe evolution in the third series (LMS) is constant from basalt to trachyte compositions. From fractional crystallization modelling and using our new set of equilibrium fractionation factors between minerals and melts, we showed that the slight increase in δ 57 Fe in the Kerguelen magmatic series could be explained by fractional crystallization exclusively.

Finally, a methodology for in situ iron isotope measurements on silicate minerals (e.g. olivine) was developed using a femtosecond laser ablation system coupled to a MC-ICP-MS. Multiple USGS silicate glasses were analyzed as well as San Carlos olivine during the analytical development. We demonstrated that USGS glasses are homogeneous but that San Carlos olivines are heterogenous regarding their iron isotope composition.

Consequently, we inferred that San Carlos olivines are not suitable standards for in situ Fe isotopes fs-LA-MC-ICP-MS measurements. We also concluded from this work that it is possible to measure δ 56 Fe variation, with a precision < 0.2% and a resolution of × 24 µm 2 , within olivine crystals, with our setup at the OMP.

Conclusion générale (French Version)

L'objectif de cette étude était de comprendre de manière approfondie les processus im-

Perspectives

First-Principles calculations

The pursuit of the investigation of equilibrium Fe and Si isotope fractionation at hightemperature via first-principles calculations can be divided in two distinct work areas:

1) The study of more complex minerals (i.e. minerals from the amphibole and mica groups) that are common in the last stage of magmatic differentiation. Indeed, the presence of OH -ions in the amphibole structures could induce significant isotope fractionation compared to the pyroxene, as we showed in this thesis that the local environment of the atoms of interest is one of the main parameters influencing the Si and Fe β-factors. Also, the micas group displays drastically distinct crystal structure (i.e. phyllosilicates) compared to olivine (i.e. nesosilicate), pyroxene and amphibole (i.e. inosilicates). Therefore, it is likely that the micas display substantial Fe and Si isotopic fractionation.

2) A thorough study of the isotopic properties of silicate melts by investigating a wider range of compositions, especially melts with high SiO 2 content (> 70%) as there were assessed to strongly fractionate compared to basaltic melts (Dauphas et al. 2014). This will be useful to determine if other processes than fractional crystallization are necessary to explain the heaviest δ 57 Fe observed in granites of about 0.9% . Furthermore, investigation of varying Fe 3+ /Fe tot content in the melts could be studied in order to confirm the strong relationship between iron oxidation state and Fe isotope fractionation in glasses inferred by NRIXS measurements (Dauphas et al. 2014).

Characterization of mantle heterogeneities

In order to confirm that Fe could be a good proxy for the determination of mantle heterogeneities, a further investigation of basaltic rocks should be carried out. The thorough investigation of OIB is essential as they display the widest Fe isotopes variation, especially the more alkaline ones. Moreover, additional investigation of available pyroxenites would be useful to better constrain their isotopic signatures and to assess if partial melting of these rocks can really be responsible for the heavy Fe isotope signatures observed in basalts. Table 7: Calculated and experimental lattice parameters (in Å) for aegirine [START_REF] Redhammer | Singlecrystal X-ray diffraction and temperature dependent 57Fe Mössbauer spectroscopy on the hedenbergite-aegirine (Ca,Na)(Fe2+,Fe3+)Si2O6 solid solution[END_REF] and forsterite [START_REF] Böstrom | Single-crystal X-ray diffraction studies of synthetic Ni-Mg olivine solid solutions[END_REF]

A- B- C- D- E- F- G- H- I- J- K- L- M- N- FINAL - A- B- 1 1 1 2 1 1 1 1 A - B - D- E- C - F- G- A - B - D - C -

  isotopes compositions of various geological reservoirs against the IRMM-014 reference standard. The gray line represents the mean ffi 56 Fe of basalts with an average value of 0.105 ± 0.006% (n=43). Modified from Li et al. (2017) and references therein. . . . . . . . . . . . . . . . . . . . . . . . . .

xi 1 . 4

 14 Structural arrangement of the SiO 4 tetrahedra in nesosilicate (a), sorosilicate (b), cyclosilicate (c), tectosilicate (d), single chain (e) and double chain (f) inosilicate and phyllosilicate (g) minerals. Oxygen and silicon atoms are represented by large spheres and smaller red dots, respectively.

2. 2

 2 Evolution of the ground-state energy (Ry) calculated of a diopside in function of the energy cut-off G c chosen during scf calculations using Quantum Espresso code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Schematic 4 x 4 k-points grid (white circles) for a 2d square lattice (black circles). Figures a and b are equivalent with a centered (one point at the origin) grid and 6 inequivalent k-points in the Brillouin Zone. Figures C

  are identical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Illustration of spin states in a 2d periodic material. Circles correspond to individual atoms and the dotted lines display a single supercell.

Figure

  (a) corresponds to a diamagnetic material, as all electrons are paired on each atoms. Figure (b) and (c) display configurations with single unpaired electron on each atom.

  Figure (b) corresponds to a ferromagnetic configuration with all electron spins alined in one direction. Figure (c) represents an antiferromagnetic state with electron spins alternating in direction on adjacent atoms. Modified from, Sholl and Steckel (2009). . . . . . . . . . . xii 2.6 Figure (a) represents a calculated DOS for bulk Si

  is a schematic illustration of the organisation of the electronic bands, with E f corresponding to the Fermi energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 High temperature dependence of the reduced partition function ratio for a) iron isotopes and b) silicon isotopes. For clarity, only forsterite with iron in M 1 site is displayed in these figures. . . . . . . . . . . . . . . . . . . . . 3.2 10 3 lnβ as a function of the average force constant for iron (a) and silicon

3. 3

 3 Relationship between the average force constants and the average bond lengths for Fe (a) and Si (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Iron force constant as a function of three parameters: a) Volume of the Fe octahedron, b) iron valence as given by the Bader charges (Henkelman et al., 2006) and c) iron concentration in the minerals. Dashed lines are only guides for the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Evolution of the Fe (a) and Si (b) force constant in the olivine solid-solution as a function of the number of iron atoms in the neighboring crystal sites

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Population analysis of the cation-oxygen coordination numbers for the different simulated melts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 High-temperature dependence of the reduced partition function ratio for a) iron isotopes and b) silicon isotopes. . . . . . . . . . . . . . . . . . . . . 4.3 10 3 lnβ as a function of the average force constant for iron (a) and silicon (b) at T = 1500K. Minerals in plain symbols 10 3 lnβ are taken from the first-principles study of Rabin et al. (2021). The linear fit for iron is f (x) = 0.0019x for melts and minerals. The linear fit for silicon is f (x) = 0.0046x for melts and minerals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Relationship between the average force constant and the average coordination number for Fe (a) and Si (b). . . . . . . . . . . . . . . . . . . . . . . . 4.5 Relationship between the average force constant and the average bond length for Fe (a) and Si (b). . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 4.6 Comparison of iron force constant of glasses (circles) under different oxygen fugacity measured experimentally by Dauphas et al. (2014) with our calculated iron force constant for the different melts (triangles) versus SiO 2 . 141 4.7 Relationship between the degree of polymerization (NBO/T) and the Fe force constant (a) and Si force constant (b) of the melts. . . . . . . . . . . 142 4.8 Relationship between the average Fe-cation distance and the Fe force constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.9 Relationship between the average iron force constant and the SiO 2 (a), Al 2 O 3 (b), FeO (c), MgO (d), CaO (e) and Na 2 0 (f) content of the melts. . 144 4.10 Relationship between the average silicon force constant and the SiO 2 (a),

  with starting andesite composition (SiO 2 = 58.23, TiO 2 = 0.94, Al 2 O 3 = 16.94, Cr 2 O 3 = 0.03, FeO = 7.31, MgO = 3.99, CaO = 7.08, Na 2 O = 3.47, K 2 O=1.57, P 2 O 5 = 0.24, H 2 O

  180 5.2 Total Alkali versus Silica (TAS) diagram showing the evolution of the Lower Miocene Series (LMS), the UpperMiocene Series (UMS) and the Mont Ross Series (MRS). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 xv 5.3 FeO-TiO 2 -Fe 2 O 3 ternary diagram showing the evolution of the Ulvospinel-Magnetite solid solution through magmatic differentiation in a) LMS, b) UMS and c) MRS. Ca-Mg-Fe Ternary diagram showing the nature of the clinopyroxnene present in the lavas for d) LMS, e) UMS and f) MRS. . . . 5.4 Measured variations of whole-rocks δ 57 Fe (% ) versus SiO 2 (wt %) for a) LMS, c) UMS and versus Zr b) MRS. The dashed lines represent the mean δ 57 Fe of the lavas with SiO 2 content below 58 wt% (a and c) or below 1000 ppm Zr (c) and the light grey areas represent the 2 standards error of the mean using the Student t-correcting factor Platzner (1997). The black dots represent the calculated δ 57 Fe from the model based on Rayleigh fractionation. Uncertainties of the models are calculated by propagation of the standard error of the initial isotopic composition of the liquid δ 0 and of the standard error of the fractionation factor ∆ 57 Fe mineral-melt used in the model (Rabin et al., 2021a) leading to 2 SE of 0.07 % for LMS model, 0.11 % for UMS and 0.06 % for MRS. . . . . . . . . . . . . . . . . . . . . 5.5 Fe 3+ /Fe tot versus an index of differentiation (SiO 2 or Zr) showing the evolution of oxygen fugacity along differentiation path for LMS and MRS (a) and UMS (b) series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Recalculated iron isotopes signature of the primitive basalts (δ 57 Fe prim ) for each series versus their calculated Fe 3+ /Fe tot . The dashed line represents the mean δ 57 Fe prim of all the lavas (except for sample ARC 172, and ARC 590 that display δ 57 Fe extreme values) and the light grey area represents the 2 standards error of the mean using the Student t-correcting factor (Platzner, 1997). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Modelled Iron isotopes fractionation and Fe 3+ /Fe tot during partial melting with varying degree of partial melting (f) assuming different initial δ 57 Fe mantle and oxygen fugacity of the Kerguelen plume source for the LMS (a,b), UMS (c,d) and MRS (e). Red dashed lines represent the Fe 3+ /Fe tot evolution in the melt and the residue. Blue lines represent the δ 57 Fe evolution and the orange lines represent the residue δ 57 Fe evolution and associated orange and blue areas represent the 2 standard errors of the model related to the ∆ 57 Fe mineral-melt values (Rabin et al., 2021b): 0.035 % for the ∆ 57 Fe mineral-basalt (LMS and MRS) and 0.059 % for the ∆ 57 Fe mineral-basanite (UMS). The grey windows represent the range of partial melting degree generally expected for alkaline OIB generation. . . . . . . . . . . . . . . . . xvi 6.1 Schema of the three different ways for photons to interacts with electrons (grey sphere): (a) Absorption of radiation, (b) Spontaneous emission and finally (c) Stimulated emission that is the principle behind laser beam generation. E 0 represent the ground state of energy, E1 an excited state. . . . 6.2 Schematic representation of a laser ablation system. . . . . . . . . . . . . . 6.3 Schematic effects of the laser-matter interactions during (a) nanosecond and (b) femtosecond pulses from (from Micromachining handbook, Clark-MXR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Schematic drawing of the set up permitting the addition of a Ni standard solution to the plasma during laser ablation. . . . . . . . . . . . . . . . . . 6.5 Three-isotopes plot for iron of the United States Geological Survey (USGS)

  . . . 6.7 Iron isotopes compositions (δ 56 Fe) of the San Carlos Olivine. . . . . . . . . 6.8 Transmitted light microscopy image of Olivine 1 from Kerguelen Archipelago after femtosecond laser ablation MC-ICP-MS analyses: (a) represents the raster (blue arrow) and single spot (green arrow) analyses profiles. Numbers corresponds to the successful analyses: results are displayed in fig. 6.9. (b) represents single spot analysis performed to assess the number of analysis statistically needed to obtain uncertainties as small as with raster line measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Iron isotopes profiles across olivine 1: (a) obtained with raster analysis and (b) single spot analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Iron isotopes profiles across olivine 2 obtained with raster analysis. . . . . xvii 1 Schematic projected representation (001) of the cationic environments of fayalite, forsterite, augite, enstatite and ulvopsinel. Each non-equivalent position for iron of the studied minerals are displayed here. 1 corresponds to M 1 sites and 2 to M 2 sites except for Ulvospinel were 1 corresponds to the T sites and 2 to the M sites. . . . . . . . . . . . . . . . . . . . . . . . . 2 Optimized structures of the Fe-bearing minerals studied. . . . . . . . . . . 3 Mean Square Displacement versus time (in picoseconds) of the modelled basalt melt (100 atoms). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Mean Square Displacement versus time (in picoseconds) of the modelled basalt melt (183 atoms). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Mean Square Displacement versus time (in picoseconds) of the modelled basalt_HP melt at 120 kbar. . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Mean Square Displacement versus time (in picoseconds) of the modelled basalt_ID melt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Mean Square Displacement versus time (in picoseconds) of the modelled basanite melt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Mean Square Displacement versus time (in picoseconds) of the modelled trachyte melt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Mean Square Displacement versus time (in picoseconds) of the modelled phonolite melt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Time-convergence analysis of calculated cumulative pressure for each simulated melt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Iron force constant calculated on 10 snapshots from the basalt trajectory (blue triangles) associated to the cumulated mean of the iron force constant (yellow triangles). The dashed line represents the mean Fe force constant of the basalt modelled. The evolution of the cumulative mean of the Fe force constant allows us to say that a least 6 snapshots are needed to obtain accurate iron force constant for our modelled melts. . . . . . . . . . . . . . 12 Radial distribution functions of the basalt (100 atoms, "Basalt" model) computed from the VASP simulations . . . . . . . . . . . . . . . . . . . . . 13 Radial distribution functions of the basalt (183 atoms, "Basalt_183" model) computed from the VASP simulations . . . . . . . . . . . . . . . . . . . . . 14 Radial distribution functions of the basalt at high pressure (120 kbar, "Basalt_HP" model) computed from the VASP simulations. . . . . . . . . 15 Radial distribution functions of the iron-depleted basalt ("Basalt_ID" model) computed from the VASP simulations. . . . . . . . . . . . . . . . . . . . . xviii 16 Radial distribution functions of the basanite computed from the VASP simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Radial distribution functions of the trachyte computed from the VASP simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Radial distribution functions of the phonolite computed from the VASP simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Theoretical iron (a) and silicon (b) isotope fractionation factors between minerals and basalt (solid lines). The corresponding polynomial fits are inTable 4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Theoretical iron (a) and silicon (b) isotope fractionation factors between minerals and basalt_ID (solid lines). The corresponding polynomial fits are in Table 4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Theoretical iron (a) and silicon (b) isotope fractionation factors between minerals and basanite (solid lines). The corresponding polynomial fits are in Table 4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Theoretical iron (a) and silicon (b) isotope fractionation factors between minerals and trachyte (solid lines). The corresponding polynomial fits are in Table 4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1. 2

 2 Iron β-factor (lnβ 57 / 54 Fe) of minerals relevant to the Earth upper-mantle and the crust from ab-initio, NRIXS, and experimental studies.(High pressure minerals are excluded). . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Iron β-factor (lnβ 57 / 54 Fe) of minerals relevant to the Earth upper-mantle and the crust from ab-initio, NRIXS, and experimental studies.(High pressure minerals are excluded). . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Silicon β-factor (1000lnβ 30 Si/ 28 Si) of minerals relevant to the Earth uppermantle and crust from ab-initio studies under the form Ax 3 + Ax 2 + Cx

2. 1

 1 Comparaison of calculation duration (in seconds) in order to perform 1 step of molecular dynamics (100 atoms, 7 elements) using Quantum Espresso code and various architecture on CALMIP supercalculator. The most favorable configuration for this calculation is 1 CPU core associated to 1 GPU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 3.1 Calculated and experimental lattice parameters (in Å) and space group of the studied minerals as well as the modeling parameters used in Quantum Espresso. c, o and m stand for cubic, orthorhombic and monoclinic crystal systems. References for the experimental lattice parameters are: 1 -Hexiong and Subratra (1995); 2 -Bosi et al. (2009); 3 -Redhammer et al. (2006); 4 -Clark et al. (1969); 5 -Zhang et al. (1997); 6 -Böstrom, D.(1987); 7 -Brown and Prewitt (1973); 8 -Smyth (1975); 9-Haavik et al. (2000). Uncertainties are in brackets at the end of each experimental lattice parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Polynomial fits of the reduced partition function ratios of 57 Fe/ 54 Fe and 30 Si/ 28 Si and average force constants, <F> in N/m. Polynomial fits use equation of type B 1 x 2 + B 2 x for 10 β 57 Fe/ 54 Fe and Ax 3 + Bx 2 + Cx for 10 3 lnβ 30 Si/ 28 Si with x = 10 6 /T 2 for T between 0 and 1500 K. . . . . . . . 4.1 Characteristics of the modeled melts. The pressure given for each melt represents the mean pressure of each FPMD trajectories. The temperature is the target temperature of the FPMD simulations. Basalt_IR_183 and Basalt_IR_HP have the same composition as the iron-rich basalt Basalt_IR with either a larger simulation cell (183 atoms) or a higher pressure (12 GPa). Basalt_ID is the iron depleted basalt. Basalt_HP is the basalt modeled at 12 GPa. . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Composition in (wt %) and degree of polymerization (NBO/T) of the modeled melts as well as the mean valence of iron atoms in the melts. . . . . . 4.3 Cation-oxygen coordination number as well as the average cation-oxygen bond length of the modeled melts. . . . . . . . . . . . . . . . . . . . . . . . 4.4 Iron-cation and silicon-cation coordination numbers and their corresponding average distance in the modeled melts. . . . . . . . . . . . . . . . . . . 4.5 Polynomial fits of the reduced partition function ratios for 57 Fe/ 54 Fe and 30 Si/ 28 Si. Polynomial fits use equation of type Ax 2 + Bx for 10 3 lnβ 57 Fe-54 Fe and Ax 3 + Bx 2 + Cx for 10 3 lnβ 30 Si-28 Si with x = 10 6 /T 2 for T between 0 and 1500 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Fe and the calculated δ 57 Fe prim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Analytical conditions for femtosecond Laser ablation -MC-ICP-MS iron isotopes measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fe isotope compositions of USGS references glasses.δ 56 Fe (% LAS) and δ 57 Fe (% LAS) represent the δ 56 Fe and δ 57 Fe relative to the primary reference material used during the analytical session. δ 56 Fe (% IRMM-14) and δ 57 Fe(% IRMM-14) are the recalculated value relative to IRMM-014 with a value of δ 56 Fe BHVO-2/IRMM-014 = 0.114 ± 0.011 % , δ 57 Fe BHVO-2/IRMM-014 = 0.174 ± 0.016 % for BHVO-2 (Craddock and Dauphas, 2011) and δ 56 Fe Puratronic/IRMM-014 = 0.249 ± 0.032 % and δ 57 Fe Puratronic/IRMM-014 = 0.389 ± 0.057 % for Puratronic (Poitrasson pers. Comm.). Measurements uncertainties (2SD) of the δ-values represents the two standard deviation based on replicate analyses (n) within the analytical session (7 days). The uncertainties of the re-calculated δ 56 Fe (% IRMM-14) and δ 57 Fe (% IRMM-14) are propagated uncertainties of the uncertainties related to the laser ablation (SD LAS ) and of the uncertainties related to the solution nebulization (SD sol ) measurements of δ 57 Fe BHVO-2/IRMM-014 and δ 56 Fe BHVO-2/IRMM-014 where 2SE = 2 × SD 2 sol + SD 2 LAS . . . . . . . . . . xxiii Differences of total energies of the different mineral composition compared with corresponding spin configuration. The name of the configuration corresponds to the name displays in Figure 1. In the case of Hortonolite, the 14 non equivalent positions of iron were tested in a ferromagnetic (F) configuration, then the configuration with the lowest total energies (configuration B) was tested afterwards with an antiferomagnetic (AFM) configuration. AFM configuration were taken to perform phonon calculation on hortonolite.269 Silicon β-factor (10 3 lnβ 30 Si/ 28 Si). . . . . . . . . . . . . . . . . . . . . . . . 271 Iron β-factor (10 3 lnβ 57 Fe/ 54 Fe). . . . . . . . . . . . . . . . . . . . . . . . . 272 Iron force constants (<F>), iron concentration (%at) and valence of iron in the different minerals studied. . . . . . . . . . . . . . . . . . . . . . . . . 273 Iron force constants (<F>), mean Fe-O bond lengths, and volumes of the Fe polyhedron for each M1, M2 and T sites where iron is present in the minerals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Details of the fractional crystallization models, including the Fe-bearing minerals proportion, the SiO 2 content, the fraction of iron in the melt f (Fe), the temperature and the δ 57 Fe/ 54 Fe calculated in model 1 (considere cpx and opx to have the same iron isotope fractionation factors as forsterite) and model 2 (takes in account cpx and opx iron isotope fractionation)).

1 Introduction 1 .

 11 . . . . . . . . . . . . . 280 Iron force constants (<F>) as well as the Fe β-factors, obtained using the « VAF » method, of the different melts modelled . . . . . . . . . . . . . . 288 Modeling of δ 57 Fe values during fractional crystallization using our new set of fractionation factor between minerals and melt. More details available : here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Modelling of δ 30 Si values during fractional crystallization using our new set of fractionation factor between minerals and melt. More details available : here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 Petrographic description from microscope observations and location of the Kerguelen Archipelago lavas. . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Modal proportions of the iron bearing phases in the Kerguelen lavas studied.295 Details of the partial melting model used to recreate the signature of the δ 57 Fe LMS basalt values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 xxiv CHAPTER

Figure 1 . 1 :

 11 Figure 1.1: Iron isotopes compositions of various geological reservoirs against the IRMM-014 reference standard. The gray line represents the mean δ 56 Fe of basalts with an average value of 0.105 ± 0.006% (n=43). Modified from Li et al. (2017) and references therein.

Figure 1 . 2 :

 12 Figure 1.2: Iron isotopes compositions of various terrestrial and extraterrestrial hightemperature samples against the IRMM-014 reference standard. The grey line represents the mean δ 57 Fe value for chondrites = -0.015 ± 0.009 2SE % (Johnson et al., 2020). Data are from : Craddock and Dauphas (2011a); Needham et al. (2009); Poitrasson et al. (2005); Schoenberg and von Blanckenburg (2006); Wang et al. (2013, 2014) for chondrites; Craddock et al. (2013); Huang et al. (2011); Poitrasson et al. (2013); Weyer and Ionov (2007); Williams et al. (2005); Zhao et al.(2010, 2012, 2015) for peridotites;Gleeson et al. (2020);Konter et al. (2016);[START_REF] Nebel | Iron isotopic evidence for convective resurfacing of recycled arc-front mantle beneath back-arc basins[END_REF]Nebel et al. ( , 2018Nebel et al. ( , 2019));Sossi et al. (2016);Teng et al. (2013) for the basalts ;Poitrasson et al. (2004);[START_REF] Schoenberg | Modes of planetary-scale Fe isotope fractionation[END_REF];Wang et al. (2012);Weyer et al. (2005) for the Moon;Poitrasson et al. (2004);Sossi et al. (2016);Wang et al. (2012) for Mars ;Poitrasson et al. (2004);[START_REF] Schoenberg | Modes of planetary-scale Fe isotope fractionation[END_REF];Wang et al. (2012);Weyer et al. (2005) for angrites.

Figure 1 . 4 :

 14 Figure 1.4: Structural arrangement of the SiO 4 tetrahedra in nesosilicate (a), sorosilicate (b), cyclosilicate (c), tectosilicate (d), single chain (e) and double chain (f) inosilicate and phyllosilicate (g) minerals. Oxygen and silicon atoms are represented by large spheres and smaller red dots, respectively. After Poitrasson (2017) and Lameyre (1986).

Figure 2 . 1 :

 21 Figure 2.1: Schematic representation of the self-consistent iterative resolution of Kohn-Sham equations (modified from Martin (2004)).

Figure 2 .

 2 2 givens anexample of a test done on diopside, where different energy cut-off were applied. It is clear that an energy cut-off of 40 Ry is enough for this system. In order to compare quantities from multiple crystals, all calculations need to have the same energy cut-off.

Figure 2 . 2 :

 22 Figure 2.2: Evolution of the ground-state energy (Ry) calculated of a diopside in function of the energy cut-off G c chosen during scf calculations using Quantum Espresso code.

Figure 2 . 3 :

 23 Figure 2.3: Schematic 4 x 4 k-points grid (white circles) for a 2d square lattice (black circles). Figures a and b are equivalent with a centered (one point at the origin) grid and 6 inequivalent k-points in the Brillouin Zone. Figures C represents ascenario with a shifted grid from the origin leading to only 3 inequivalent k-points[START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF].

Figure 2 . 5 :

 25 Figure 2.5: Illustration of spin states in a 2d periodic material. Circles correspond to individual atoms and the dotted lines display a single supercell.Figure (a) corresponds to a diamagnetic material, as all electrons are paired on each atoms. Figure (b) and (c) display configurations with single unpaired electron on each atom.Figure (b) corresponds to a ferromagnetic configuration with all electron spins alined in one direction.Figure (c) represents an antiferromagnetic state with electron spins alternating in direction on adjacent atoms. Modified from, Sholl and Steckel (2009).

Figure 2

 2 Figure 2.6: Figure (a) represents a calculated DOS for bulk Si[START_REF] Sholl | Density Functional Theory: A Practical Introduction[END_REF], where valence and conduction bands are clearly distinguishable. The band gap is located around 5 eV. Figures (b) is a schematic illustration of the organisation of the electronic bands, with E f corresponding to the Fermi energy.

A

  . H. Wilson and R. H. Fowler. The theory of electronic semi-conductors. -II. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical andPhysical Character, 134(823):277-287, 1931. doi: 10.1098/rspa.1931.0196.The lack of knowledge concerning Si and Fe isotopes partitioning between silicate minerals at magmatic temperatures was underlined in the introduction chapter. This prevents the scientific community to disentangle the processes involved in Fe and Si fractionation during magmatic differentiation. Therefore, the third chapter of this thesis is focused on the determination of the Fe and Si fractionation factors of oxide and silicate minerals composing the mantle and crust. The determination of such fractionation factors was performed theoretically using first-principles calculation.CHAPTER 3First-principles calculation of iron and silicon isotope fractionation between Fe-bearing minerals at magmatic temperatures: The importance of second atomic neighbors Ségolène RABIN, Marc BLANCHARD, Carlos PINILLA, Franck POITRASSON, Michel GRÉGOIRE S. Rabin, M. Blanchard, C. Pinilla, F. Poitrasson, M. Gregoire, First-principles calculation of iron and silicon isotope fractionation between Fe-bearing minerals at magmatic temperatures: The importance of second atomic neighbors, Geochimica et Cosmochimica Acta, Volume 304, 2021, Pages 101-118. :https: // doi. org/ 10. 1016/ j. gca. 2021. 03. 028

  (i.e. the local chemical composition around Fe atoms). This conclusion is also valid for silicon isotopes. In the investigated nesosilicates and inosilicates, silicon isotope reduced partition function ratios (also called β-factors) show no correlation with the average Si-O bond length, which remains almost constant, but Si β-factors are correlated with the local chemical composition of the minerals. Fractional crystallization is one of the mechanisms, which could explain the evolution of iron isotopic compositions during magmatic differentiation. Using the present theoretical set of equilibrium fractionation factors allows us to assess the impact of inter-mineral isotopic fractionations, and shows that pyroxene appears to be the main mineral phase driving the isotopic evolution to a heavier signature in the most evolved lavas.

  isotopes is tested through different silicate compositions. The goal of the present study is to produce a consistent data set of iron and silicon β-factors representing the main Fe-bearing minerals of lavas. Four clinopyroxenes are investigated: aegirine, hedenbergite, augite and diopside as well as the most common orthopyroxene, i.e. enstatite. The olivine solid-solution is studied through four distinct compositions: fayalite, hortonolite and two forsterites (i.e. Fe-bearing and Fe-free forsterites). In addition, pure magnetite and ulvospinel were added to the study in order to model the fractionation factor between oxide and silicate minerals. This set of iron and silicon fractionation factors is obtained from first-principles quantum mechanical calculations. These data enable us to discuss the effect of different parameters (e.g. coordination number (CN), iron oxidation state, Fe-O and Si-O bond lengths, average or local chemical compositions) on the inter-mineral fractionation factors and to discuss the role of inter-silicate isotopic fractionation on the process of fractional crystallization that could explain the heavy Fe signatures of evolved lavas during magmatic differentiation.

  in Appendix A). Concerning Fe-bearing forsterite, the unique iron atom (in M 1 or M 2 site) is high spin. In hedenbergite and aegirine, iron is only located in M 1 site, half of the Fe M1 are set up with spin up polarization and the other half with a spin down polarization. In augite, where iron atoms are distributed between M 1 and M 2 sites, the spin polarization has been set up in order to get an antiferromagnetic ordering occurs between corner-sharing octahedra (Fig.1in Appendix A). As structure with 2 Fe M1 and 1 Fe M2 with spin up polarization and the others as spin down polarization (Fig.1in Appendix A , conf. C). In enstatite (config.F in Fig 1in Appendix A) and diopside, iron is arranged in high-spin state.

Figure 3 . 1 :

 31 Figure 3.1: High temperature dependence of the reduced partition function ratio for a) iron isotopes and b) silicon isotopes. For clarity, only forsterite with iron in M 1 site is displayed in these figures.

Figure 3 . 2 :

 32 Figure 3.2: 10 3 lnβ as a function of the average force constant for iron (a) and silicon (b)at T =1280 K. The linear fit between 10 3 lnβ 57 Fe/ 54 Fe and the iron force constant is : f (x) = 0.0026x and is in total agreement with the equation ofDauphas et al. (2014). The linear fit between 10 3 lnβ 30 Si/ 29 Si and the silicon force constant is: f (x) = 0.0062x.

  Figure 3.3: Relationship between the average force constants and the average bond lengths for Fe (a) and Si (b).

Figure 3 . 4 :

 34 Figure 3.4: Iron force constant as a function of three parameters: a) Volume of the Fe octahedron, b) iron valence as given by the Bader charges (Henkelman et al., 2006) and c) iron concentration in the minerals. Dashed lines are only guides for the eye.

Figure 3

 3 Figure3.5: Evolution of the Fe (a) and Si (b) force constant in the olivine solid-solution as a function of the number of iron atoms in the neighboring crystal sites sharing an edge or a corner with the central Fe or Si polyhedron (small symbols). For comparison, large circles correspond to the mean force constant evolution with the Fe molar concentration in the mineral M sites (upper axis). For example, in panel a, fayalite (red large circle) contains only iron and no magnesium in its M1 and M2 sites. Therefore, the corresponding iron concentration in M sites is 100 mol% Fe, and the mean Fe force constant (F = 99.6 N/m) is an average of the Fe force constants of the four Fe atoms in M1 sites (88.1, 88.3, 87.3 and 87.3 N/m) and of the four Fe atoms in M2 sites(109.8, 109.8, 112.9 and 112.9 N/m).

  are displayed in Fig. 3.5b. It is clear that the Si tetrahedron sharing edges and corners with only iron atoms in fayalite displays a lower Si force constant compared to the one sharing edges and corners with only magnesium atoms in forsterite, with intermediate compositions in between. Aegirine and hedenbergite are also good examples as they have the same structure (Appendix A -Fig 2) and differ only by the substitution of Na (aegirine) to Ca (hedenbergite) in M 1 sites. In these clinopyroxenes, a Si atom is surrounded by 9 atoms: 2 Na/Ca atoms sharing an edge with the Si tetrahedron and 7 atoms (2 Si, 2 Ca/Na and 3 Fe) sharing the corners. As a result, the substitution of the Ca by Na in aegirine in the local environment of Si modifies the average force constant from 654.9 N/m to 647.8 N/m. The nature and number of second neighbors has thus definitely an impact on Si force constant and therefore on Si β-factor (Fig. 3.5b).

Figure 3 . 6 :

 36 Figure 3.6: Comparison of the 10 3 lnβ 57 Fe/ 54 Fe (a) and 10 3 lnβ 30 Si/ 29 Si (b) calculated in the present study at T = 1280 K, with previous data computed from NRIXS measurements(Dauphas et al., 2012; Dauphas et al., 2014), Mössbauer spectroscopy(Polyakov and Mineev, 2000), experimental petrology(Sossi and Foden, 2017), and DFT calculations(Huang et al., 2014, Qin et al. 2016). The uncertainties are ≤ 0.05 % for NRIXS-derived data(Dauphas et al., 2014(Dauphas et al., , 2012;; Polyakov et al. 2007, Mineev et al. 2007) and ≤ 0.10 % for Mössbauer-derived data(Polyakov and Mineev, 2000).

  these studies,Huang et al. (2014) andQin et al. (2016) calculated from first-principles methods a large variety of silicate mineral β-factors such as for olivine, clinopyroxene and orthopyroxene with a limited cationic environment (Ca, Mg). Our results agree with their previous calculations showing the evolution olivine > orthopyroxene > clinopyroxene in terms of 30 Si enrichment (Fig.3.6b). However, the absolute values of the β-factors exhibit systematic differences (Fig.3.6b) that can be explained by different reasons. Firstly, the calculations ofHuang et al. (2014) were performed at zero-pressure and T = 300K, whereasQin et al. (2016) performed calculations at zero-pressure and T = 0K, using both the same LDA functional. Since the volume expands with increasing temperature, the β-factors calculated byHuang et al. (2014) shifts to lower values with respect to

  3.6) Mineral β-factors are taken from the present study. In order to reproduce the evolution of the melt β-factor during magmatic differentiation, the iron β-factor of the melt was calculated from the sigmoidal equation of the iron force constant proposed by Dauphas et al. (2014) from NRIXS measurements of synthetic glasses:

2 .Figure 3 . 7 :

 237 Figure3.7: Models of δ 57 Fe evolution during magmatic differentiation: the black line represents model 1 where a bulk fractionation factor α f orsterite-melt is used with forsterite representing all Fe 2+ -bearing minerals, and the red line represents model 2 where the differences between the relevant Fe 2+ -bearing minerals are taken into account (i.e., α enstatite-melt , α augite-melt ). Models were obtained using the MCS software(Bohrson and Spera, 2007) with starting andesite composition (SiO 2 = 58.23, TiO 2 = 0.94, Al 2 O 3 = 16.94, Cr 2 O 3 = 0.03, FeO = 7.31, MgO = 3.99, CaO = 7.08, Na 2 O = 3.47, K 2 O=1.57, P 2 0 5 = 0.24, H 2 O = 0.20) with an FMQ buffer(Dauphas et al., 2014). The circles in the background correspond to δ 57 Fe from the literature(Du et al., 2017; Foden et al., 2015; He et al., 2017; Poitrasson and Freydier, 2005; Schuessler et al., 2009; Sossi et al., 2012; Telus et al., 2012; Teng et al., 2013; Williams et al., 2018; Xia et al., 2017; Zambardi et al., 2014).

  This work was supported by the ECOS-NORD/COLCIENCIAS French-Colombian cooperation program (Project number: C17U01). CP also acknowledges funding from COLCIENCIAS through research grants No. 2015-710-51568. This work was also supported through a grant to FP from the Programme National de Planétologie (PNP) of CNRS/INSU, co-funded by CNES. Calculations were performed using the HPC resources from CALMIP(Grant 2019. SR deeply thanks CALMIP staff for their support.In chapter 3, the Fe and Si fractionation factors of oxides and silicates minerals have been theoretically determined at magmatic temperatures. We displayed that there is significant isotope fractionation between silicates minerals. The process involved in such fractionation after temperature and redox effects appears to be governed by the local atomic environment (i.e. nature and number of atoms around the isotopes of interest). From this discovery, many questions emerged including two major problematics: 1) Are the Fe and Si isotopic signatures of silicate melts similar to their mineral counterparts? 2) In which extent the Fe and Si isotopic composition of silicate melts can vary when their composition evolves? In this chapter, first-principles molecular dynamics (FPMD) calculations are performed on silicate melts to access their Fe and Si isotopic compositions. For the first time, theoretical mineral-melt Si and Fe fractionation factors were determined. From these theoretical mineral-melt fractionation factors, models were performed in order to evaluate the role of fractional crystallization during magmatic differentiation.

GRÉGOIRE 4 . 1 Abstract

 41 Determination of the silicate melts iron and silicon isotopes signature is one of the major challenges of high-temperature isotope geochemistry, consequently melts are often approximated by silicate glasses. Calculation of precise equilibrium Si and Fe isotopes fractionation factors between minerals and melt would indeed allow us to distinguish equilibrium fractionation from diffusion-driven kinetic fractionation involved in the iron and silicon isotopes signatures on Earth and other planets. In this study, we use for the first time, first-principles molecular dynamics based on density functional theory to determine iron and silicon isotopes compositions of different silicate melts, namely: iron-rich basalt, iron-depleted basalt, basanite, trachyte and phonolite. The 57 Fe/ 54 Fe β-factors of the different melts span over a 1 % range at 1000 K while 30 Si/ 28 Si β-factors are much less influenced by the melt composition with a 0.5 % fractionation range at the same temperature. The main parameters controlling iron isotope fractionation in silicate melts having similar iron oxidation state, after temperature, are the Fe-O bond lengths as well as the nature and number of iron second neighbors. Silicon isotopes fractionation seems also affected, but to a lesser extent, by its local chemical composition with decreasing Si-Fe distances leading to slightly heavier Si β-factor in the melt. From these Fe and Si β-factors of melts, a new set of equilibrium fractionation factors between a variety of minerals and melts has been calculated. These new ∆ 57 Fe min-melt and ∆ 30 Si min-melt sets allow us to discuss whether processes such as fractional crystallization, partial melting and diffusion could be responsible for the documented Fe and Si isotopes variations in igneous rocks. Our results suggest that: 1) fractional crystallization can explain Fe and Si isotopic evolution during magmatic differentiation, at least up to δ 57 Fe = 0.65 % and δ 30 Si = -0.1 % respectively, 2) partial melting of the upper mantle can produce the MORB iron

  tions are often limited by the lack of available equilibrium fractionation factors between coexisting phases, especially between minerals and silicate melts. Hence, the models used in the literature are under-constrained and precise quantification of the processes involved in Fe and Si isotopic fractionation in igneous rocks is hampered. It became then essential to obtain Si and Fe isotope compositions of various melts in order to quantify the processes associated to the isotopic variations in magmatic settings. By now, only few experimental and in-situ studies were carried out to obtain iron isotope composition of melts, using glasses as a proxy.Dauphas et al. (2014) displayed significant variations in Fe isotope fractionation factors with varying composition of melts. Nevertheless, the mechanisms involved in isotopic fractionation in melts are still under-constrained. The degree of mass-dependent equilibrium fractionation between minerals has been associated to the oxidation state (for Fe; e.g.Dauphas et al. (2014);Polyakov and Mineev (2000)), the bonding environment (i.e. bond length and coordination number; e.g.Sossi and O'Neill (2017)), and the local chemical environments of the atoms of interests (nature and number of the second neighbors ;Rabin et al. (2021),Macris et al. (

  5, e max is the maximum energy of the vibrational spectrum, and E(e/k B T) is the Einstein function of the vibrational energy of a single harmonic oscillator at frequency ν = e/h, expressed as:

Figure 4 . 1 :Figure 4 . 2 :

 4142 Figure 4.1: Population analysis of the cation-oxygen coordination numbers for the different simulated melts.

Figure 4 .

 4 Figure 4.3: 10 3 lnβ as a function of the average force constant for iron (a) and silicon (b) at T = 1500K. Minerals in plain symbols 10 3 lnβ are taken from the first-principles study ofRabin et al. (2021). The linear fit for iron is f (x) = 0.0019x for melts and minerals. The linear fit for silicon is f (x) = 0.0046x for melts and minerals.

  4.4a). Trachyte that has a Fe-O coordination number of 3.74,

Figure 4 . 4 :

 44 Figure 4.4: Relationship between the average force constant and the average coordination number for Fe (a) and Si (b).

Figure 4 . 5 :

 45 Figure 4.5: Relationship between the average force constant and the average bond length for Fe (a) and Si (b).

Figure 4 . 6 :

 46 Figure 4.6: Comparison of iron force constant of glasses (circles) under different oxygen fugacity measured experimentally by Dauphas et al. (2014) with our calculated iron force constant for the different melts (triangles) versus SiO 2 .

4. 9 Figure 4 . 7 :Figure 4 . 8 :

 94748 Figure 4.7: Relationship between the degree of polymerization (NBO/T) and the Fe force constant (a) and Si force constant (b) of the melts.

Figure 4 . 9 :

 49 Figure 4.9: Relationship between the average iron force constant and the SiO 2 (a), Al 2 O 3 (b), FeO (c), MgO (d), CaO (e) and Na 2 0 (f) content of the melts.

Figure 4 .

 4 Figure 4.10: Relationship between the average silicon force constant and the SiO 2 (a), Al 2 O 3 (b), FeO (c), MgO (d), CaO (e) and Na 2 0 (f) content of the melts.

Figure

  Figure 4.11: Relationship between the average Si-cation distance and the Si force constant.

  ) estimates for terrestrial basalts under IW+0.3 f O 2 conditions (See Fig. 4 of Prissel et al. (2018)) and lies within our range of values for terrestrial basalts. Dauphas et al. (2014) and Prissel et al. (2018) inferred that there is no Fe isotope fractionation between forsterite and melts of various planetary composition. Our calculated fractionation factor between forsterite and the basalt with low iron content (Basalt_ID) ∆ 57 Fe forsterite-basalt I D = -0.01 % at 1500 K would agree with such a conclusion. However, we find a fractionation factor between forsterite and the basalt with high iron content (FeO = 13.1 wt%) ∆ 57 Fe forsterite-basalt = -0.15 % at 1500K.

4. 5 . 4 . 1

 541 Parameters controlling Si isotope fractionation.

  3+ (K d = C melt /C solid ≈ 4.5; Mallmann and O'Neill (2009), Cottrell and Kelley (2011)) relative to Fe 2+ (K d = 1; Mallmann and O'Neill (2009); Davis et al. (2013)) suggests that iron isotope fractionation could occur during partial melting of upper-mantle to create oceanic crust. Many studies using different models (i.e. batch-melting, non-modal melting; e.g. Dauphas et al. (2014); Sossi et al. (2016); Weyer and Ionov (2007); Williams and Bizimis (2014); Williams et al. (2009)),

(

  2017);Oeser et al. (2015Oeser et al. ( , 2018));Prissel et al. (2018);Sio et al. (2018);[START_REF] Sio | Thermal and crystallization histories of magmatic bodies by Monte Carlo inversion of Mg-Fe isotopic profiles in olivine[END_REF];Sio et al. (2013);Teng et al. (2011). As discussed part 4.5.5.1 and underlined byNebel et al. (2018), although diffusion in olivine can create large isotopic variations at the crystal scale, it is not expected to affect significantly (> 0.03 % ) the melt composition during magmatic differentiation as proposed byLundstrom (2009) andZambardi et al. (2014).

  high-magma by fractional crystallization (Fig. 5.2). The Mont Ross series follow an alkalienrichment trend intermediate to that of the Upper and Lower Miocene Series (Fig 5.2).

Figure 5 . 2 :

 52 Figure 5.2: Total Alkali versus Silica (TAS) diagram showing the evolution of the Lower Miocene Series (LMS), the UpperMiocene Series (UMS) and the Mont Ross Series (MRS).

(

  1992). Two lavas (ARC 272; ARC 590) were analyzed in this study at CNRS-CRPG (Nancy, France) with a JY70 ICP-AES after alkaline fusion and acid dissolution. The major element composition of all mineral phases was determined with the electron probe micro-analyzer (EPMA) SXFive -CAMECA at the Centre de microcaractérisation Raimond Castaing, Toulouse. The EPMA was operated at a beam current of 20 nA and an accelerating voltage of 15 kV. At least 10 analyses per mineral phase were carried out for each sample. Nine elements were systematically measured and are reported herein as oxides: Na 2 O, SiO 2 , K 2 O, CaO, MnO, FeO, MgO, Ti 2 O and Cr 2 O 3 . The standards used were: albite (Na, TAP), wollastonite (Ca, Si, PET), sanidine (K, PET), manganese titanate (Mn, LLIF; Ti, LPET), aluminium oxide (Al, TAP), hematite (Fe, LLIF), magnesium oxide (Mg, TAP) and chromium oxide (Cr, LPET). The counting time was 10 s on the peak for each element and 5 s on each side of the peak for the background.The presence of clay minerals was suspected in two samples after the petrographic and EPMA studies: LVLK 127 and ARC 172. X-Ray Diffraction (XRD) measurements were performed at the Institut de Chimie des Milieux et Matériaux de Poitiers. Acquisitions were made on a Bruker D8 Advance diffractometer (40 kV and 40 mA) coupled with a copper source of wavelength (Kα1) = 1.540598 Å. The analytical conditions for XRD acquisition on oriented preparations were: 2 -30 • 2θ, step size of 0.01 • 2θ, acquisition time per step of 0.5 s. X-ray acquisition has been done in air dried (AD) state and after saturation with ethylene glycol (EG) as vapor to point out the occurrence of expandable phases. SEM observations of amorphous phase found by XRD analysis were conducted on a JEOL JSM-IT500 SEM equipped with a BRUKER linxeye Energy dispersive spectrometer EDX associated with SPIRIT software at the Institut de Chimie des Milieux et Matériaux de Poitiers. The analytical conditions were as follows: 15kV; 1nA; counting time: 50 s; and

2 O 3 -

 23 TiO 2 ternary diagram permits classification of the oxides based on EPMA analyses (Fig 5.3 a-c). In the LMS and MRS from the basalt to the trachyandesite, the oxides are mostly intercumulus ulvöspinel-rich titomagnetite (Mag 25 ) with minor ilmenite. The crystallization of almost pure magnetite (Mag 90 ) in the first trachyte (MPC 99-01) in the Mont Ross series and magnetite-rich ulvospinel (Mag 40-60 ) in the LMS trachytes (LVLK 127 and ARC 272) coincide with the end of ilmenite crystallization. Concerning the UMS, there is a continuous evolution of the oxide compositions from titomagnetite (Mag 50 ) in basanite to the almost pure magnetite in the tephri-phonolite ARC 356 (Mag 80

  Probe microanalyser analysis permits the classification of the pyroxene groups in the pseudo-quadrilateral of a CaO-FeO-MgO diagram (Fig 3d-f). Clinopyroxenes are abundant in the LMS, UMS and MRS suites, and they dictate their fractional crystallization histories. Indeed, pyroxenes represent between 44% (ARC 590) and 83% (ARC 391) of the iron-bearing mineral phases in the lavas. In the LMS and MRS from basalts to trachyandesites the clinopyroxenes are augite and diopside (Wo 45-50 En 40-25 Fs 15-25 ). Pyroxene composition shifts to hedenbergite (Wo 40-50 En 0-10 Fs 50-60 ) in the most evolved rocks. There is neither amphibole in the LMS nor in MRS. Concerning the UMS, the main clinopyroxene phases are hedenbergite (Wo 50 En 0-20 Fs 30-50 ) and diopside (Wo 45-50 En 25-40 Fs 10-30 ) with the crystallisation of augite (Wo 25-45 En 5-20 Fs 45-70 ) in the most evolved lavas. Augite-Aegirine also occurs in the most evolved phonolites in sample ARC 411. Some aenigmatite (ARC 411, ARC 414), are also present in the phonolites and some amphibole, ferrokärsutite (ARC 96). Their evolution shows that the most evolved phonolites (ARC 414, ARC 411) contain Fe 2+ -rich minerals (aenigmatite, augite).

Fe2O3

Figure 5 2 Figure 5 . 4 :

 5254 Figure 5.3: FeO-TiO 2 -Fe 2 O 3 ternary diagram showing the evolution of the Ulvospinel-Magnetite solid solution through magmatic differentiation in a) LMS, b) UMS and c) MRS. Ca-Mg-Fe Ternary diagram showing the nature of the clinopyroxnene present in the lavas for d) LMS, e) UMS and f) MRS.

  Figure 5.5: Fe 3+ /Fe tot versus an index of differentiation (SiO 2 or Zr) showing the evolution of oxygen fugacity along differentiation path for LMS and MRS (a) and UMS (b) series.

  peridotites and small amount pyroxenite component with Fe 3+ /Fe tot = 0.06 and δ 57 Fe = -0.06 ± 0.035 % (Fig 7c). Partial melting of silica-depleted pyroxenites component can also explain the low SiO 2 , high FeO and high CaO/Al 2 O 3 content of the Upper-Miocene silica-undersaturated series and the intermediate alkaline Mont Ross series[START_REF] Lambart | Markers of the pyroxenite contribution in the major-element compositions of oceanic basalts: Review of the experimental constraints[END_REF]. The δ 57 Fe and the Fe 3+ /Fe tot of the mantellic source proposed in our models allow us to give an order of magnitude of the expected mantellic variations needed to create the isotopic and geochemical variations observed in the different alkaline series of Kerguelen Archipelago. The δ 57 Fe variation of the metasomatized peridotites beneath Kerguelen

  Figure5.7: Modelled Iron isotopes fractionation and Fe 3+ /Fe tot during partial melting with varying degree of partial melting (f) assuming different initial δ 57 Fe mantle and oxygen fugacity of the Kerguelen plume source for the LMS (a,b), UMS (c,d) and MRS (e). Red dashed lines represent the Fe 3+ /Fe tot evolution in the melt and the residue. Blue lines represent the δ 57 Fe evolution and the orange lines represent the residue δ 57 Fe evolution and associated orange and blue areas represent the 2 standard errors of the model related to the ∆ 57 Fe mineral-melt values(Rabin et al., 2021b): 0.035 % for the ∆ 57 Fe mineral-basalt (LMS and MRS) and 0.059 % for the ∆ 57 Fe mineral-basanite (UMS). The grey windows represent the range of partial melting degree generally expected for alkaline OIB generation.

  evolutions. Concerning the LMS, the model predicts heavier iron isotopes composition for the trachytes than what is actually observed. This low δ 57 Fe signature of the trachytes (ARC 272, LVLK 127) cannot be related to hydrothermal alteration, asRouxel et al. (2003) demonstrated that altered lavas tend to have higher δ 57 Fe than fresh basalts due to the preferential leaching of 54 Fe. The discrepancy between the modeled and measured LMS δ 57 Fe values could eventually be explained by assimilation of Kerguelen basaltic crust and/or peridotite xenolith with low δ 57 Fe during the magma residence or ascent.[START_REF] Borisova | Lead isotope signatures of Kerguelen plume-derived olivine-hosted melt inclusions: Constraints on the ocean island basalt petrogenesis[END_REF] inferred indeed from a Pb isotopic perspective, that a large part of the recent Kerguelen OIB (≤ 25 Ma) has been affected by basaltic crust assimilation generated by the high thermal flux related to the Kerguelen Plume.[START_REF] Gagnevin | Open-system processes in the genesis of silica-oversaturated alkaline rocks of the Rallier-du-Baty Peninsula, Kerguelen Archipelago (Indian Ocean)[END_REF] also demonstrated that open system processes associated with assimilation of hydrothermallyaltered oceanic crust are required to generate the silica-oversaturated alkaline trachytes from the Rallier-du-Baty Peninsula located in the South-West of Kerguelen archipelago.

  The acronym L.A.S.E.R stands for "Light Amplification by Stimulated Emission of Radiation". The operation of the laser is based on the quantum physics theories and can be resume in terms of the interaction of photon on a surface composed of atoms that are electronically neutral. There is three ways for photons to interacts with atoms, two naturals called absorption of radiation and spontaneous emission and an artificial one called stimulated emission (fig.6.1).

Figure 6 . 1 :

 61 Photon

  polished samples and standards are placed in a two volumes cell from ESI, permitting a high sensitivity on rather large samples (thin sections or plots). A beam imaging system controls the spot size delivered on the samples: from 1 to 60 µm. The ablated material is transported with 180 to 250 ml.min -1 of Helium into a 2.5 m PTFE-coated tubing and is mixed with argon using a Y-shape connector. Finally, a spiked Ni-solution (10 ppm) is added to the mix via a 50 µl.m -1 PTFE nebulizer and a cyclonic spray chamber and using an in-house glass T-connector (Verres Vagner, Toulouse, Fr) that is directly connected to the MC-ICP-MS torch injector (fig.6.4). Injection of a liquid in the plasma (wet plasma condition) allows us to improve stability of the ICP signal and diminishes the mass-bias fractionation drift. The concomitant injection of Ni-spiked solution to the aerosol, enables us to correct for mass-bias fractionation during the session of analysis as demonstrated byOeser et al. (2014) for laser ablation and byPoitrasson and Freydier (2005) for solution nebulization. Data treatment steps are detailed in the section 6.3.4.Regarding the MC-ICP-MS, the total argon gas flow added to the torch gas flow (14 L.min -1 ) is composed of the auxiliary Ar flow of 0.59 L.min -1 , the sample Ar flow of ≈ 0.62 L.min -1 and the Ar flow of 0.46 L.min -1 added in order to nebulize the Ni-spiked solution.

Figure 6 . 4 :

 64 Figure 6.4: Schematic drawing of the set up permitting the addition of a Ni standard solution to the plasma during laser ablation.

  Figure 6.8: Transmitted light microscopy image of Olivine 1 from Kerguelen Archipelago after femtosecond laser ablation MC-ICP-MS analyses: (a) represents the raster (blue arrow) and single spot (green arrow) analyses profiles. Numbers corresponds to the successful analyses: results are displayed in fig. 6.9. (b) represents single spot analysis performed to assess the number of analysis statistically needed to obtain uncertainties as small as with raster line measurements

  natural olivine by in situ femtosecond laser ablation MC-ICP-MS for the first time at the Laboratoire Géosciences Environnement Toulouse using the Observatoire Midi-Pyrénées ICPMS laboratory. The interlaboratory comparison allows to confirm the robustness of the in situ iron isotopes fs-La-MC-ICP-MS methodology developed at the Leibniz Universität in Hannover byOeser et al. (2014), with a different femtosecond laser ablation7.1 General conclusions (English Version)The purpose of this work was to better understand the processes involved in Fe and Si isotopes fractionation during magmatic differentiation. A multidimensional approach, based on ab-initio calculations as well as bulk and in-situ analysis of natural samples, was set up in order to highlight the mechanisms involved in equilibrium Fe and Si fractionation at high-temperature.Firstly, the equilibrium Fe and Si isotopic properties of various silicate minerals and oxides were investigated through first-principles calculations. This allowed us to obtain a large set of silicon and iron equilibrium fractionation factors between the Fe-bearing minerals relevant to the crust formation. Ours results essentially show that Fe and Si isotope fractionations among Fe 2+ -bearing minerals are not negligible, even at high temperature.The joint study of Fe and Si isotopic properties and their local environments allowed us to bring out the crystal chemical properties controlling equilibrium isotope fractionation. In this study, a slight correlation between Fe-O and Si-O bond length and their associated Fe and Si β-factor was observed. The coordination number appeared to play a major role on Fe isotope fractionation with four-fold coordinated iron displaying heavier ß-factor than six-fold coordinated iron in minerals. The charge of iron in the minerals emerged to have also a great impact on iron isotope fractionation with the Fe 3+ -bearing minerals presenting heavy isotopes signatures compared to the Fe 2+ -rich minerals. Iron oxidation state and coordination number can indeed explain to a certain extent Fe isotopic fractionation.However, we demonstrated that, after temperature, the main parameter influencing the Si and Fe β-factors of silicate minerals containing Fe and Si atoms with similar coordination number and charge is the local crystal chemical environment (i.e. number and nature of the second atomic neighbor). Moreover, this work allowed us to underline the significant impact of Fe 2+ -bearing minerals (i.e. pyroxene in particular) on iron isotope fractionations during magmatic differentiation via simple fractional crystallization models.Secondly, first-principles molecular dynamics calculations were performed in order to model silicate melts and investigate their Fe and Si isotopic properties. This was the most challenging part of this thesis work. Indeed, the isotopic properties of silicate melts were never explored by first-principles calculations before. Liquids are disordered and evolve with time, consequently they necessitate a statistical treatment and the potential anharmonicity of the systems makes the β-factor calculation complicated. The complex chemical compositions of the modelled melts, close to the natural ones, with 7 to 8 elements was also a challenge as large number of atoms in the simulation cell leads to time-consuming calculation. Two different statistical methods were used to calculate the melts β-factors: i) The first method is based on the periodic sampling of the molecular dynamics trajectory. The β-factors were then calculated, from the snapshots sampled, at zero temperature in the harmonic approximation. ii) The second method is a recent method based on the determination of the β-factors via the kinetic energy of the molecular dynamics trajectories. We demonstrated that in the case of silicate melts, both methods give similar Fe β-factor results. As for the minerals, the melt structural properties controlling Fe and Si isotopes fractionation were investigated. It appears that for the range of SiO 2 content investigated here, there is no significant change of silicon isotopic properties. Concerning iron, we exhibit that there is a relationship between Fe-O bond lengths and the Fe β-factors of the melt. While silicon isotopic properties are not dependent of the composition of the melts, iron isotopic properties appeared to be correlated to iron and magnesium concentrations. Moreover, a strong correlation with Fe-Fe and Fe-Mg distances were underlined in this study: with short Fe-Fe distances (and inversely long Fe-Mg distances) leading to an enrichment in heavy Fe isotopes in the melts. Consequently, as inferred for the silicate minerals, the iron isotope fractionation in melts is also controlled by its local environment. This work allowed us to determine for the first time Fe and Si isotopes fractionation factors between minerals and various silicate melts.

Figure 1 :

 1 Figure 1: Schematic projected representation (001) of the cationic environments of fayalite, forsterite, augite, enstatite and ulvopsinel. Each non-equivalent position for iron of the studied minerals are displayed here. 1 corresponds to M 1 sites and 2 to M 2 sites except for Ulvospinel were 1 corresponds to the T sites and 2 to the M sites.

Figure 9 :Figure 10 :

 910 Figure 9: Mean Square Displacement versus time (in picoseconds) of the modelled phonolite melt.

Figure 11 :Figure 18 :

 1118 Figure 11: Iron force constant calculated on 10 snapshots from the basalt trajectory (blue triangles) associated to the cumulated mean of the iron force constant (yellow triangles). The dashed line represents the mean Fe force constant of the basalt modelled. The evolution of the cumulative mean of the Fe force constant allows us to say that a least 6 snapshots are needed to obtain accurate iron force constant for our modelled melts.

Figure 19 :

 19 Figure 19: Theoretical iron (a) and silicon (b) isotope fractionation factors between minerals and basalt (solid lines). The corresponding polynomial fits are in Table4.7.

Figure 20 :Figure 21 :Figure 22 :

 202122 Figure 20: Theoretical iron (a) and silicon (b) isotope fractionation factors between minerals and basalt_ID (solid lines). The corresponding polynomial fits are in Table 4.7.

  Model calculation detailed in Supp-data of Dauphas et al. 2014 Proportion of mineral melting = 80% of diopside, 20 % of enstatite δ 57 Fe mantle = -0.2 % δ 57 Fe mantle = -0melt Fe 3+ /Fe tot δ 57 Fe melt Fe 3+ /Fe tot δ 57 Fe melt Fe 3+ /Fe tot δ 57 Fe melt Fe 3+ /Fe tot 0Details of the partial melting model used to recreate the signature of the δ 57 Fe LMS basalt values.

  The authors are: Ségolène Rabin, Franck Poitrasson, Paolo Sossi, Maud Boyer, Hervé Leyrit and Michel Grégoire.The chapter 6, refers to the development of iron isotopes in situ measurements using fs-

LA-MC-ICP-MS at the Observatoire Midi-Pyrénées (OMP). The first part of the chapter is a state of the art on the principle of laser ablation and the essential parameters to care about during analytical development. The second part is a detailed explanation of the methods developed in order to acquire accurate and precise iron isotopes compositions of silicate glasses geostandards and Fe isotopes profiles in olivines of Kerguelen Archipelago.

Finally, the results obtained so far are presented and discussed. The in situ Fe isotope measurement development by fs-LA-MC-ICP-MS at the OMP were performed by myself and François-Xavier D'abzac with the help of Jérôme Chmeleff and Franck Poitrasson.

  Le chapitre 5 est une étude de la composition isotopique du fer de trois séries magmatiques ayant des teneurs en alcalins différentes, provenant des Iles Kerguelen. La méthodologie concernant la chromatographie ionique et les mesures faites au MC-ICP-MS est détaillée dans ce chapitre. L'étude de ces séries magmatiques, ainsi que les coefficients de fractionnement entre les minéraux et les liquides silicatés calculés au chapitre 4, permettent de discuter l'évolution de la composition isotopique du fer lors de la différenciation magmatique. également, la possibilité d'utiliser la composition en isotopes du fer des basaltes, ainsi que leurs rapports Fe 3+ /Fe tot , pour retracer l'hétérogénéité du manteau supérieur sous l'archipel de Kerguelen est abordée.

	L'objectif de cette thèse est d'approfondir nos connaissances concernant les mécan-
	ismes impliqués dans le fractionnement isotopique à l'équilibre du fer et du silicium dans
	les minéraux et les liquides silicatés. Cette étude est basée sur une approche multidi-
	mensionnelle combinant des calculs ab initio de fractionnement isotopique du fer et du
	silicium entre les minéraux et les liquides silicatés, l'analyse des isotopes du fer dans des
	laves naturelles, ainsi que des mesures in situ de la composition isotopique en fer d'olivines
	grâce à un laser femtoseconde couplé à un MC-ICP-MS. Le chapitre 6 porte sur le développement d'une méthode d'analyse in situ des isotopes
	Ce premier chapitre a pour but d'introduire les concepts fondamentaux du fraction-du fer à l'Observatoire Midi-Pyrénées (OMP) grâce à un laser femtoseconde couplé à

. La compréhension incomplète des mécanismes à l'origine du fractionnement isotopique du fer et du silicium à l'équilibre et hors équilibre (i.e. fractionnement cinétique) empêche à l'heure actuelle la communauté scientifique de conclure quant aux processus contrôlant les variations isotopiques en fer et silicium observées durant la différenciation magmatique. nement isotopique ainsi que l'état de l'art concernant le fractionnement isotopique du fer et du silicium lors de la différenciation magmatique. Dans le chapitre 2, les méthodes de calculs théoriques et numériques, basées sur la théorie de la fonctionnelle de la densité (DFT) et utilisées afin d'obtenir les propriétés isotopiques du fer et du silicium dans différents minéraux, sont présentées. Dans une seconde partie, la méthode de calcul par dynamique moléculaire ab-initio utilisée pour caractériser les liquides silicatés, est détaillée. Nous abordons dans le troisième chapitre, le calcul des propriétés isotopiques du fer et du silicium au sein d'une variété de minéraux présents dans le manteau supérieur et la croûte. Les différents paramètres contrôlant le fractionnement isotopique du fer et du silicium au sein des minéraux riche en Fe 2+ sont détaillés dans ce chapitre. Cet article a été accepté sous condition de révisions mineures à Geochimica and Cosmochimica Acta. Les auteurs sont : Ségolène Rabin, Marc Blanchard, Carlos Pinilla, Franck Poitrasson and Michel Grégoire. Dans la continuité des calculs sur les phrases solides, nous présentons dans le chapitre 4 le calcul des propriétés isotopiques du fer et du silicium au sein de liquides silicatés. Différents liquides ont été modélisés avec des compositions en éléments majeurs proches de celles des liquides silicatés naturels afin de quantifier l'influence des alcalins, de la teneur en SiO 2 et en fer. Les paramètres contrôlant le fractionnement isotopique du fer et du silicium sont discutés dans ce chapitre. Pour la première fois, des coefficients de fractionnement entre différents minéraux et différents liquides silicatés sont présentés. A partir de ces nouveaux coefficients de fractionnement, l'impact de la fusion partielle et de la cristallisation fractionnée sur les signatures isotopiques des roches magmatique naturelles est discuté grâce à des modèles. Cette étude n'a pas encore été soumise pour publication à l'heure actuelle. Les auteurs sont : Ségolène Rabin, Marc Blanchard, Carlos Pinilla, Franck Poitrasson and Michel Grégoire. un MC-ICP-MS. La première partie du chapitre est un état de l'art des principes fondamentaux de l'ablation laser ainsi que des paramètres essentiels à prendre en compte durant le développement analytique. Dans un second temps, la méthode utilisée permettant d'obtenir des résultats justes, précis et reproductibles sur des verres silicatés et des olivines est présentée. Enfin, les premiers résultats sur des olivines de Kerguelen sont exposés. Les analyses in situ des isotopes du fer ont été effectuées par François-Xavier d'Abzac et moi-même, avec le soutien de Jérôme Chmeleff et de Franck Poitrasson.

  Wt% SiO 2 . This was also supported by the model produced by Foden

	between olivine and melt and magnetite and melt obtained by NRIXS, shows that
	FC could explain the heavy Fe isotopes compositions in the silicic magmatic rocks
	above 68
	inferred that crystallisation of mag-
	netite, in a system open to oxygen, should drive the Fe isotopes values of the more
	evolved rocks to low δ 57 Fe if fractional crystallization was the only process involved
	in iron isotope fractionation during magmatic differentiation. On the other side,
	Dauphas et al. (2014) model, using their set of equilibrium fractionation factors

Table 1 .

 1 2: Iron β-factor (lnβ 57 / 54 Fe) of minerals relevant to the Earth upper-mantle and the crust from ab-initio, NRIXS, and experimental studies.(High pressure minerals are excluded).

	The determination of Fe β-factor for silicate melt is one of the biggest challenge in
	high-temperature Fe isotope geochemistry field. As mentioned part 1.4.3.2, Dauphas
	et al. (2014) approximated Fe isotopes compositions of terrestrial silicate melts by the
	mean of synthetic glasses as Prissel et al. (2018) did for lunar melts. Dauphas et al.
	(2014) demonstrated that redox conditions have a major impact on the melt β-factor
	but they also implied that coordination environnement of Fe may have an impact on the
	melt iron isotopes signatures. Later, Sossi and O'Neill (2017) also observed Fe isotope
	fractionation between minerals displaying similar coordination number.
	Johnson et al. (2020) stated that "Consistency between different methods is evidence
	that the inferred fractionation is likely to reflect equilibrium isotope partitioning and does
	not suffer from laboratory-induced fractionation or imperfections in assumed bonding or
	vibrational energies." Therefore, it is essential to obtain more Fe isotopes fractionation
	factors between minerals and melts using various methods in order to obtain a unified set
	of fractionation factors that could be used in order to understand geological processes.

Table 1 .

 1 4 displays the Si β-factors available in the literature, for the Earth's mantle and crust minerals that we are interested in this phD thesis.

		0
			Poitrasson and Zambardi (2015) Zambardi et al. (2014)	y = 0.0069x -0.638 y = 0.0069x -0.638
			Savage et al. (2012)
		-0.1	Savage et al. (2011)
	δ 30 Si (‰ vs NBS-28)	-0.3 -0.2	y = 0.0059x -0.591 y = 0.0059x -0.591
		-0.4
		-0.5 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82
			SiO2 (Wt %)
	Figure 1.6: Compilation of the silicon isotopes compositions of crustal igneous rocks versus
	SiO 2 (in wt.%) as an index of differentiation (Poitrasson and Zambardi, 2015; Savage et al.,
	2011, 2012; Zambardi et al., 2014)

Table 2 .

 2 1 displays some of the tests that have been performed to determine the impact of the supercalculator architecture on calculation duration. The goal here, is not to go into much details but to underline the importance of knowing the configuration of the supercalculator in order to run efficient calculations.

	Number of CPU cores	Number of GPU	0	1
	1		-	118.1 s
	18		-	179.9 s
	36		198.2 s	-

Table 2 .

 2

1: Comparaison of calculation duration (in seconds) in order to perform 1 step of molecular dynamics (100 atoms, 7 elements) using Quantum Espresso code and various architecture on CALMIP supercalculator. The most favorable configuration for this calculation is 1 CPU core associated to 1 GPU.

  Si 2 O 6 ) as the sample analyzed by NRIXS spectroscopy inDauphas et al. (2012). Four compositions of the olivine solid-solution were studied: the two end-members, fayalite (Fe2SiO4) and forsterite (Mg 2 SiO 4 ), as well as two intermediate compositions, hortonolite (FeMgSiO 4 ) and Fe-bearing forsterite (Fe 0.125 Mg 0.875 SiO 4 ) in order to better assess the impact of iron concentration on the olivine solid solution. Concerning the oxides, pure ulvospinel (Fe 2 TiO 4 ) as well as magnetite (Fe 3 O 4 ) were studied.

	Minerals	Formula	sym	This study	Exp.	Space group	k-space
	Enstatite	Fe 2 Mg 14 Si 16 O 48	o	a= 18.456	a 1 = 18.2747(21)	Pbca	4 × 4 ×
				b= 8.994	b 1 = 8.8729(9)
				c= 5.273	c 1 = 5.1988(5)
	Ulvospinel	Fe 16 Ti 8 O 32	c	a= 8.483	a 2 = 8.4875(4)	Fd3m	2 × 2 ×
	Aegirine	Fe 4 Na 4 Si 8 O24	m	a= 9.857	a 3 = 9.7412(8)	C 2/c	4 × 4 ×
				b= 8.927	b 3 = 8.9086(7)
				c= 5.368	c 3 = 5.2776(5)
				γ= 107.764	γ 3 = 106.221(5)
	Augite	(Ca 2 Fe 2 )(MgFe 3 )Si 8 O 24	m	a= 10.007	a 4 = 9.699(1)	C 2/c	4 × 4 ×
				b= 9.108	b 4 = 8.884(1)
				c= 5.284	c 4 = 5.272(1)
				γ= 107.525	γ 4 = 106.97(2)
	Diospide	Ca 4 (FeMg 3 )Si 8 O 24	m	a= 9.909	a 5 = 9.749(2)	C 2/c	4 × 4 ×
				b= 9.051	b 5 = 8.922(1)
				c= 5.330	c 5 = 5.248(8)
				γ= 106.211	γ 5 = 106.04(1)
	Hedenbergite	Fe 4 Ca 4 Si 8 O 24	m	a= 10.039	a 5 = 9.838(2)	C 2/c	4 × 4 ×
				b= 8.986	b 5 = 9.021(7)
				c= 5.251	c 5 = 5.242(7)
				γ= 104.14	γ 5 = 104.797(8)
	Forsterite	Fe 1 Mg 7 Si 4 O 1 6	o	a= 4.818	a 6 = 4.755(3)	Pnma	2 × 2 ×
				b= 10.366	b 6 = 10.198(6)
				c= 6.042	c 6 = 5.979(4)
	Forsterite	Mg 8 Si 4 O 16	o	a= 4.803	a 6 = 4.755(3)	Pnma	2 × 2 ×
				b= 10.328	b 6 = 10.1985(6)
				c= 6.046	c 6 = 5.9792(4)
	Hortonolite	Fe 4 Mg 4 Si 4 O 16	o	a= 4.874	a 7 = 4.775(1)	Pnma	2 × 2 ×
				b= 10.443	b 7 = 10.280(1)
				c= 5.986	c 7 = 6.016(1)
	Fayalite	Fe 8 Si 4 O 16	o	a= 4.957	a 8 = 4.818(2)	Pnma	2 × 2 ×
				b= 10.411	b 8 = 10.471(3)
				c= 5.943	c 8 = 6.086(2)
	Magnetite	Fe 6 O 8	c	a= 8.400	a 9 = 8.397	Fd3m	6 × 6 ×
	Table 3.1: Calculated and experimental lattice parameters (in Å) and space group of the
	studied minerals as well as the modeling parameters used in Quantum Espresso. c, o
	As described above, pyroxenes are not restricted to a single chemical composition in and m stand for cubic, orthorhombic and monoclinic crystal systems. References for the
	nature: proportion of the different cations such as iron, magnesium, sodium and calcium experimental lattice parameters are: 1 -Hexiong and Subratra (1995); 2 -Bosi et al.
	(2009); 3 -Redhammer et al. (2006); 4 -Clark et al. (1969); 5 -Zhang et al. (1997); 6 bound to the Si-O structure can evolve from 0 to 100%. In order to reproduce common -Böstrom, D. (1987); 7 -Brown and Prewitt (1973); 8 -Smyth (1975); 9-Haavik et al.

O 4 where T is the tetrahedral site preferentially occupied by Si and occasionally Al or Ti and M 1 and M 2 are octahedral sites occupied mainly by Fe 2+ , Mg, Mn, Ni. Pyroxenes are part of the inosilicate group with the general formula [M 2 ][M 1 ](T) 2 O 6 where the six-coordinated M 1 site is occupied mainly by Mg, Fe 2+ , Fe 3+ , Al, Ti, and the five to eight coordinated M 2 sites by Mg, Fe 2+ , Na, Ca. The tetrahedral sites are occupied by Si and minor Al, Ti or even Fe 3+ . Substitution of Mg 2+ by Fe 2+ in pyroxene can create distortion of M 1 octahedra (Cameron and Papike, 1981), as iron ions are larger. The general formula of the spinel group is AB 2 O 4 . In magnetite, A cations represents Fe 2+ in the octahedral M sites and B represents Fe 3+ in tetrahedral T site and octahedral M sites. A substantial amount of titanium can enter in the magnetite structure, forming a continuous solid solution between magnetite and ulvospinel. In ulvospinel, A represents tetravalent (Ti 4+ ) and B the divalent (Fe 2+ ) cations. In this study, A cations occupy the octahedral M sites and B cations occupy both octahedral M and tetrahedral T sites (Barth and Posnjak, 1932). Fe-bearing clinopyroxene compositions of various magmatic rocks, we modeled the endmembers aegirine (FeNaSi 2 O 6 ) and hedenbergite (FeCaSi 2 O 6 ) as well as intermediate compositions such as augite (Mg 0.25 Fe 1.25 Ca 0.5 Si 2 O 6 ) and diopside (Mg 0.75 Fe 0.25 )CaSi 2 O 6 . For the most common orthopyroxene, enstatite, we chose the same composition ((Fe 0.125 Mg 0.875 ) 2 (2000). Uncertainties are in brackets at the end of each experimental lattice parameters.

  .1). Primitive cells were sufficient to proceed to multiple Fe-Mg substitutions among the different silicate compositions and to have a realistic representation of mineral structures. Space groups for each mineral are displayed in Table3.1. All non-equivalent iron positions as well as the spin state and magnetic ordering, were tested by doing self-consistent field calcula-

tions in order to find the model having the lowest energy (Fig

1

and Table in Appendix

A), on which phonon calculations were performed. Concerning Fe-bearing forsterite, two iron non-equivalent positions were tested and the model where iron is in M 1 site displays the lowest energy by about 7.02 kJ/mol. This energy difference can be interpreted as iron being preferentially in M 1 sites compared to M 2 sites. Following the same approach as

[START_REF] Umemoto | A first-principles investigation of hydrous defects and IR frequencies in forsterite: The case for Si vacancies[END_REF]

, we assessed the percentage of iron in each site. Our results

Table 3 .

 3 2: Polynomial fits of the reduced partition function ratios of 57 Fe/ 54 Fe and 30 Si/ 28 Si and average force constants, <F> in N/m. Polynomial fits use equation of type B 1 x 2 + B 2 x for 10 β 57 Fe/ 54 Fe and Ax 3 + Bx 2 + Cx for 10 3 lnβ 30 Si/ 28 Si with x = 10 6 /T 2 for T between 0 and 1500 K.

	30 Si/ 28 Si

Méheut et al 2009, a

systematic relative uncertainty of about 5% can be applied to each β-factor, and force constant calculated in this study. At 1000 K, the whole set of 10 3 lnβ 57 Fe/ 54 Fe spans in a range of 0.5 % and the 10 3 lnβ 30 Si/ 28 Si are within a range of 1 % . All studied minerals are enriched in 57 Fe and in 30 Si with respect to fayalite. The highest iron β-factors are found in the Fe 3+ -bearing minerals, aegirine and magnetite. The highest silicon β-factors 57 Fe/ 54 Fe are found in forsterite and enstatite. It is noteworthy that forsterite displays different 10 3 lnβ 57 Fe/ 54 Fe when the sole iron of the unit cell is in M 1 or M 2 site. Indeed, the forsterite-M1 configuration, that is the most energetically favorable (part 3.4.1), displays a higher 10 3 lnβ 57 Fe/ 54 Fe of about 0.094 % at 980 K compared to the Forsterite-M2 configuration.

Table 4 .

 4 1: Characteristics of the modeled melts. The pressure given for each melt represents the mean pressure of each FPMD trajectories. The temperature is the target temperature of the FPMD simulations. Basalt_IR_183 and Basalt_IR_HP have the same composition as the iron-rich basalt Basalt_IR with either a larger simulation cell (183 atoms) or a higher pressure (12 GPa). Basalt_ID is the iron depleted basalt. Basalt_HP is the basalt modeled at 12 GPa.

	Basalt	100	1341.5	1.64	1.1	4000
	Basalt_183	183	2489.52	1.39	0.9	4000
	Basalt_HP	100	1079.03	1.98	11.7	4000
	Basalt_ID	100	1289.51	1.66	0.7	4000
	Basanite	110	1366.48	1.78	1.2	4000
	Trachyte	104	1355.82	1.68	1.8	4000
	Phonolite	102	1248.44	1.82	0.9	4000

Table 4 .

 4 These silicate minerals, already studied in Rabin et al. (2021), are aegirine (FeNaSi 2 O 6 ) and forsterite (Fe 0.25 Mg 1.75 SiO 4 ). The only purpose of these calculations is to assess the potential bias related to the use of different pseudopotentials on the Fe and Si β-factors of silicate minerals. This will allow us to provide a more accurate set of fractionation factors between silicate minerals and melts using mineral data from Rabin et al. (2021).

		Basalt_IR Basalt_ID Basanite Trachyte Phonolite
	SiO 2	49.09	50.54	48.75	62.20	56.48
	Al 2 O 3	13.89	14.30	16.55	16.83	23.15
	FeO T	13.05	6.72	14.57	6.79	3.60
	MgO	10.98	15.07	7.19		0.00
	CaO	10.18	10.48	9.10	2.66	0.00
	Na 2 O	2.81	2.90	3.87	5.81	10.29
	K 2 O				5.71	6.48
	Fe mean valence	2.3	2.37	2.36	2.33	2.28
	NBO/T	1	1	0.71	0.29	0.28

2: Composition in (wt %) and degree of polymerization (NBO/T) of the modeled melts as well as the mean valence of iron atoms in the melts.

Table 4

 4

	4.1). The ambiguous role

coordinated iron atoms increase by about 15%, meaning that iron at high pressure behaves mainly as a network modifier in the melt. Effective charge of iron is roughly similar in the different melts and ranges between 2.27 and 2.36 (Table

4

.2). The magnesium structural parameters appear composition independent, with similar Mg-O coordination .4). The iron cationic neighbors are at a distance > 2.9 Åfor all elements and all melts, except in iron-rich melts where the average distance between two iron atoms can be as low as 2.25 Å. Silicon is the most represented neighbor with an average Fe-Si coordination number between 3.02 and 4.30. The other elements display a Fe-cation coordination number < 2, except for Fe-Al coordination number in the phonolite (3.09). The silicon cationic neighbors are at a distance > 3 Åfor all elements and all melts. The silicon close environment is more varied, with no clear preference in term of nature of second neighbors (Si, Al, K). The average number of network formers (Si+Al) around Si are always > 3.8 except for basalt_ID that has an average number of 1.87. This would suggest that iron-depleted basalts (basalt_ID), unlike the other melts, cannot form a proper tetrahedral network with agrees with

Table 4 .

 4

		92 2.28 6.55 2.30	-	-	3.91 1.76 5.31 1.99 4.57 1.77 4.11 1.64
	Basalt_IR_183 6.88 2.26 6.43 2.28	-	-	3.95 1.78 5.25 1.97 4.49 1.77 4.07 1.64
	Basalt_IR_HP 9.53 2.24 9.27 2.24	-	-	5.08 1.80 7.12 1.95 5.45 1.76 4.64 1.64
	Basalt_ID	6.95 2.28 6.24 2.35	-	-	4.37 1.94 5.27 1.99 4.59 1.77 4.09 1.63
	Basanite	7.48 2.31 7.76 2.30	-	-	4.73 1.90 5.32 1.98 4.75 1.77 4.17 1.64
	Trachyte	6.49 2.25 6.80 2.32 10.28 2.68 3.74 1.85	-	-	4.66 1.81 4.08 1.63
	Phonolite	-	-	7.71 2.25 11.42 2.65 5.11 1.63	-	-	4.73 1.78 4.23 1.64
	Table 4.3: Cation-oxygen coordination number as well as the average cation-oxygen bond
	length of the modeled melts.

The degree of polymerization is an important parameter that is directly associated to the composition of the silicate melts. According to

[START_REF] Mysen | Structure and properties and structure of aluminosilicate melts[END_REF]

, a melt can be described in term of degree of polymerization, designated as NBO/T where NBO represents 4: Iron-cation and silicon-cation coordination numbers and their corresponding average distance in the modeled melts.

Table 4 .

 4 10 3 lnβ 57 Fe/ 54 Fe 10 3 lnβ 30 Si/ 28 Si 5: Polynomial fits of the reduced partition function ratios for 57 Fe/ 54 Fe and 30 Si/ 28 Si. Polynomial fits use equation of type Ax 2 + Bx for 10 3 lnβ 57 Fe -54 Fe and Ax 3 + Bx 2 + Cx for 10 3 lnβ 30 Si -28 Si with x = 10 6 /T 2 for T between 0 and 1500 K.

		A	B	Value at 1500 K	A	B	C	Value at 1500 K
	Basalt_IR -0.0059 1.0332	0.461 ± 0.031	0.0028 -0.1474 6.9342	3.073 ± 0.058
	Basalt_ID -0.0035 0.7269	0.324 ± 0.015	0.0031 -0.1569 7.2036	3.202 ± 0.056
	Basanite	-0.0061 0.8986	0.406 ± 0.057	0.0031 -0.1578 7.1926	3.191 ± 0.062
	Trachyte -0.0044 0.8178	0.365 ± 0.041	0.0033 -0.165 7.4202	3.290 ± 0.047
	Phonolite -0.0155 1.8063	0.813 ± 0.084	0.0033 -0.1643 7.334	3.254 ± 0.081

Table

	Basalt_IR	243 ± 17	235 ± 24	669 ± 11
	Basalt_ID	171 ± 8	193 ± 19	695 ± 13
	Basanite	196 ± 13	196 ± 23	687 ± 8
	Phonolite	403 ± 44	290 ± 99	700 ± 15
	Trachyte	191 ± 20	159 ± 38	716 ± 10
	Basalt_IR_HP	-	269 ± 33	-
	Basalt_IR_183	-	222 ± 12	-

  ∆A-B 57 Fe/ 54 Fe

							∆A-B 29 Si/ 30 Si
		C1	C2	at 1500 K		C1	C2	C3	at 1500 K
	∆Aeg irine-Basalt_IR	0.0013	-0.0081 -0.003 ± 0.04	∆Aegirine-Basalt_IR	-0.0003 0.0128 -0.1088 -0.046 ± 0.15
	∆Enstatite-Basalt_IR	0.0028	-0.32	-0.142 ±0.03	∆Enstatite-Basalt_IR	0.0001 -0.0029	0.229	0.101 ± 0.16
	∆Forsterite-Basalt_IR	0.0036	-0.3385 -0.150 ± 0.03	∆Forsterite-Basalt_IR	0.0004 -0.0035 0.2951	0.130 ± 0.16
	∆Augite-Basalt_IR	0.0038	-0.4266 -0.189 ± 0.03	∆Augite-Basalt_IR	-0.0002 0.0077	0.0022	0.002 ± 0.15
	∆Hortonolite-Basalt_IR	0.0039	-0.4042 -0.179 ± 0.03	∆Hortonolite-Basalt_IR	-0.0003	0.01	0.074	0.035 ± 0.15
	∆Hedenbergite-Basalt_IR	0.0043	-0.4329 -0.192 ± 0.03	∆Hedenbergite-Basalt_IR	-0.0002 0.0067 -0.0376 -0.015 ± 0.15
	∆Fayalite-Basalt_IR	0.0047	-0.5601 -0.248 ± 0.03	∆Fayalite-Basalt_IR	-0.0007 0.0268	-0.515 -0.224 ± 0.14
	∆Diopside-Basalt_IR	0.0039	-0.3884 -0.172 ± 0.03	∆Diopside-Basalt_IR	0.0004 0.000009 0.1026	0.046 ± 0.16
	∆Ulvospinel-Basalt_IR	0.0033	-0.3104 -0.137 ± 0.03	∆Forsterite-Mg-Basalt_IR 0.0001 -0.0092 0.4385	0.193 ± 0.16
	∆Magnetite-Basalt_IR	0.0023	-0.0476 -0.021 ± 0.04	∆Quartz-Basalt_IR	0.0062 -0.0694 0.7804	0.334 ± 0.17
					∆Anorthite-Basalt_IR	0.0059	-0.066	0.7351	0.314 ± 0.17
					∆Albite-Basalt_IR	0.0055 -0.0609 0.6467 0.276 ± 0.171
	∆Aegirine-Basalt_ID	-0.0011	0.2982	0.132 ± 0.03	∆Aegirine-Basalt_ID	-0.0006 0.0223 -0.3782 -0.164 ± 0.15
	∆Enstatite-Basalt_ID	0.0004	-0.0137 -0.006 ± 0.02	∆Enstatite-Basalt_ID	-0.0002 0.0066 -0.0404 -0.017 ± 0.16
	∆Forsterite-Basalt_ID	0.0012	-0.0322 -0.014 ± 0.02	∆Forsterite-Basalt_ID	-0.0003	0.006	0.0257	0.013 ± 0.16
	∆Augite-Basalt_ID	0.0014	-0.1203 -0.053 ± 0.02	∆Augite-Basalt_ID	-0.0005 0.0172 -0.2672 -0.115 ± 0.15
	∆Hortonolite-Basalt_ID	0.0015	-0.0979 -0.043 ± 0.02	∆Hortonolite-Basalt_ID	-0.0006 0.0195 -0.3434 -0.149 ± 0.15
	∆Hedenbergite-Basalt_ID	0.0019	-0.1266 -0.056 ± 0.02	∆Hedenbergite-Basalt_ID	-0.0005 0.0162	-0.307 -0.133 ± 0.15
	∆Fayalite-Basalt_ID	0.0023	-0.2538 -0.112 ± 0.02	∆Fayalite-Basalt_ID	-0.001	0.0363 -0.7844 -0.342 ± 0.14
	∆Diopside-Basalt_ID	0.0015	-0.0821 -0.036 ± 0.02	∆Diopside-Basalt_ID	-0.0003 0.0095 -0.1668 -0.072 ± 0.15
	∆Ulvospinel-Basalt_ID	0.0009	-0.0041 -0.002 ± 0.02	∆Forsterite-Mg-Basalt_ID -0.0002 0.0003	0.1691	0.075 ± 0.16
	∆Magnetite-Basalt_ID	-0.000056 0.258656 0.115 ± 0.03	∆Quartz-Basalt_ID	0.0059 -0.0599	0.511	0.216 ± 0.17
					∆Anorthite-Basalt_ID	0.0052 -0.0514 0.3773	0.158 ± 0.17
					∆Albite-Basalt_ID	0.0056 -0.0565 0.4657	0.196 ± 0.17
	∆Aegirine-Basanite	0.0015	0.1265	0.057 ± 0.06	∆Aegirine-Basanite	-0.0006 0.0232 -0.3672 -0.159 ± 0.16
	∆Enstatite-Basanite	0.003	-0.1854 -0.082 ± 0.06	∆Enstatite-Basanite	-0.0002 0.0075 -0.0294 -0.012 ± 0.17
	∆Forsterite-Basanite	0.0038	-0.2039 -0.090 ± 0.06	∆Forsterite-Basanite	-0.0003 0.0069	0.0367	0.018 ± 0.17
	∆Augite-Basanite	0.004	-0.292	-0.129 ± 0.06	∆Augite-Basanite	-0.0005 0.0181 -0.2562 -0.110 ± 0.16
	∆Hortonolite-Basanite	0.0041	-0.2538 -0.112 ± 0.06	∆Hortonolite-Basanite	-0.0006 0.0204 -0.3324 -0.144 ± 0.16
	∆Hedenbergite-Basanite	0.0045	-0.2983 -0.132 ± 0.06	∆Hedenbergite-Basanite	-0.0005 0.0171	-0.296 -0.128 ± 0.16
	∆Fayalite-Basanite	0.0049	-0.4255 -0.188 ± 0.06	∆Fayalite-Basanite	-0.001	0.0372 -0.7734 -0.336 ± 0.15
	∆Diopside-Basanite	0.0041	-0.2538 -0.112 ± 0.06	∆Diopside-Basanite	-0.0003 0.0104 -0.1558 -0.067 ± 0.16
	∆Ulvospinel-Basanite	0.0035	-0.1758 -0.077 ± 0.06	∆Forsterite-Mg-Basanite	-0.0002 0.0012	0.1801	0.080 ± 0.17
	∆Magnetite-Basanite	0.0025	0.087	0.039 ± 0.06	∆Quartz-Basanite	0.0059	-0.059	0.522	0.221 ± 0.18
					∆Anorthite-Basanite	0.0052 -0.0505 0.3883	0.163 ± 0.18
					∆Albite-Basanite	0.0056	-0.056	0.4767	0.201 ± 0.18
	∆Aegirine-Trachyte	-0.0002	0.2073	0.092 ± 0.05	∆Aegirine-Trachyte	-0.0008 0.0304 -0.5948 -0.258 ± 0.15
	∆Enstatite-Trachyte	0.0013	-0.1046 -0.046 ± 0.04	∆Enstatite-Trachyte	-0.0004 0.0147	-0.257 -0.111 ± 0.16
	∆Forsterite-Trachyte	0.0021	-0.1231 -0.054 ± 0.04	∆Forsterite-Trachyte	-0.0005 0.0141 -0.1909 -0.082 ± 0.16
	∆Augite-Trachyte	0.0023	-0.2112 -0.093 ± 0.04	∆Augite-Trachyte	-0.0007 0.0253 -0.4838 -0.210 ± 0.16
	∆Hortonolite-Trachyte	0.0024	-0.1888 -0.083 ± 0.04	∆Hortonolite-Trachyte	-0.0008 0.0276	-0.56	-0.244 ± 0.15
	∆Hedenbergite-Trachyte	0.0028	-0.2175 -0.096 ± 0.04	∆Hedenbergite-Trachyte	-0.0007 0.0243 -0.5236 -0.228 ± 0.16
	∆Fayalite-Trachyte	0.0032	-0.3447 -0.153 ± 0.04	∆Fayalite-Trachyte	-0.0012 0.0444	-1.001 -0.436 ± 0.15
	∆Diopside-Trachyte	0.0024	-0.173	-0.076 ± 0.04	∆Diopside-Trachyte	-0.0005 0.0176 -0.3834 -0.167 ± 0.16
	∆Ulvospinel-Trachyte	0.0018	-0.095	-0.042 ± 0.04	∆Forsterite-Mg-Trachyte	-0.0004 0.0084 -0.0475 -0.019 ± 0.17
	∆Magnetite-Trachyte	0.0008	0.1678	0.075 ± 0.05	∆Quartz-Trachyte	0.0057 -0.0518 0.2944	0.121 ± 0.17
					∆Anorthite-Trachyte	0.005	-0.0433 0.1607	0.063 ± 0.17
					∆Albite-Trachyte	0.0054 -0.0484 0.2491	0.102 ± 0.17

Table 4 .

 4 7: Polynomial fits of iron and silicon fractionation factors between various minerals and melts. Polynomial fits use equation of type C 1 x 2 + C 2 x for ∆ 57 Fe/ 54 Fe and C 1 x 3 + C 2 x 2 + C 3 x for ∆ 30 Si/ 28

Table 4 .

 4 al (2019) ∆ melt-mineral 30 Si/ 28 Si Basalt_IR Trachyte MORB-like melt Granite-like melt 8: Comparison between ∆ melt-mineral 30 Si/ 28 Si calculated in this study and the ones estimated in Deng et al. (2019) at 1123 K.

	Diopside	-0.08	0.29	0.25	0.38
	Orthoenstatite	-0.18	0.19	0.18	0.31
	Olivine	-0.34	0.03	0.12	0.25
	Anorthite	-0.48	-0.11	-0.07	0.06
	Albite	-0.54	-0.17	-0.14	-0.01

  suggested a homogeneous source for Kerguelen Archipelago lavas during the last ≈ 25 My with relatively high 87 Sr/ 86 Sr

(> 0.7050), low 143 Nd/ 144 Nd (< 0.5126) and low 206 Pb/ 204 Pb (< 18.27) signature and, unlike Heard Island

Table 5 .

 5 Table 5.1.

			05'58"E	46'08"E	02'29"E	03'50"E	46'04"E	46'21"E	46'48"E		10'31"E	06'23"E	06'57"E	07'20"E	08'44"E	10'18"E	15'57"E	16'25"E		38'19"E	34'41"E	32'40"E	34'32"E	34'46"E	32'29"E	36'11"E	20'48"E
	Pb Localisation		49 • 34'56"S 70 •	49 • 34'56"S 69 •	49 • 41'17"S 70 •	49 • 40'26"S 70 •	49 • 38'44"S 69 •	49 • 33'09"S 69 •	49 • 33'42"S 69 •		49 • 34'40"S 70 •	49 • 34'40"S 70 •	49 • 34'24"S 70 •	49 • 34'43"S 70 •	49 • 32'10"S 70 •	49 • 35'37"S 70 •	49 • 32'37"S 70 •	49 • 33'20"S 70 •		49 • 34'55"S 69 •	49 • 33'02"S 69 •	49 • 33'45"S 69 •	49 • 32'17"S 69 •	49 • 33'01"S 69 •	49 • 33'27"S 69 •	49 • 36'06"S 69 •	49 • 30'51"S 69 •
	Samples SiO TiO Al O FeO T MnO MgO CaO Na2O K O P O LOI Total 87 Sr/ 86 Nd/ 144 Pb/ 204 Nd 206 Sr 143	LMS	18.440 0.51265(2) LVLK 87 Basalt 46.96 3.93 15.89 12.86 0.18 5.37 8.41 3.16 1.33 0.39 98.48 0.70513	ARC 590 Basalt 47.92 3.21 16.8 10.42 0.19 3.55 8.79 3.55 1.67 0.49 1.82 99.55	18.379 0.51261(6) LVLK 123 Basalte 48.96 3.05 14.95 11.72 0.18 4.89 10.01 3.13 1.19 0.37 98.45 0.70521	18.543 0.51270(7) LVLK 108 Basalte 49.37 3.18 15.26 11.57 0.17 4.8 9.46 3.25 1.4 0.42 98.88 0.70511	18.435 0.51265(4) LVLK 131 Trachyte 62.59 0.52 15.67 6.88 0.24 0.24 2.84 5.25 4.82 0.11 99.16 0.70510	ARC 272 Trachyte 63.12 0.38 15.69 5.82 0.18 0.18 1.9 5.55 5.7 <L.D. 1.36 100.53	18.427 0.51261(5) LVLK 127 Trachyte 63.04 0.32 16.95 5.04 0.16 0.13 1.93 5.9 5.89 0.06 -99.71 0.70566	UMS	18.074 0.51249(1) ARC 321 Basanite 43.52 4.11 14.72 12.55 0.2 5.8 9.4 3.17 3.57 1.88 -100.32 0.70573	ARC 172 Basanite 44 2.44 12.24 11.75 0.17 9.53 9.46 3.17 1.39 0.68 3.09 99.24	ARC 361 Phono-tephrite 48.3 2.37 16.59 8.26 0.18 2.57 6.02 4.64 4.67 0.88 3.27 98.67	ARC 356 Tephri-phonolite 51.48 1.71 18.07 6.06 0.14 1.85 4.56 4.62 5.53 0.48 4.08 99.26	ARC 391 Phonolite 54.03 0.86 19.1 5.88 0.19 0.92 2.32 9.84 4.97 0.32 0.93 100.02	18.084 0.51256(2) ARC 96 Phonolite 56.32 0.31 22.1 2.74 0.15 0.17 1.04 10.63 6.04 0.04 0.70538 -99.8	ARC 411 Phonolite 58.23 0.38 19.89 3.58 0.15 0.17 1.7 6.98 6.73 0.05 1.26 99.52	ARC 414 Phonolite 57.9 0.44 20.18 3.66 0.15 0.09 1.87 7.28 6.99 0.06 0.63 99.65	MRS	RR08-88 Basalt 45.03 3.29 13.01 12.30 0.16 10.17 9.07 2.27 1.43 0.49 0.98 97.22	RR08-64 Trachy-basalt 48.61 2.96 17.52 10.08 0.17 4.15 7.45 4.07 2.89 0.9 0.53 98.79	RR08-78 Trachy-basalte 47.81 3.38 17.31 10.56 0.16 4.68 8.04 3.72 2.71 0.88 -0.46 99.25	RR08-31 Trachy-andesite 52.16 2.14 18.07 8.48 0.16 2.83 5.79 5.05 3.75 0.75 0.4 99.17	RR08-63 Trachy-andesite 54.63 1.24 17.65 7.07 0.16 1.5 3.55 5.11 5.02 0.48 1.02 96.41	RR08-85 Trachyte 57.78 0.47 18.21 6.46 0.16 0.47 2.21 5.49 6.07 0.33 1.02 97.65	RR08-74 Trachyte 60.35 0.11 18.08 4.26 0.14 0 1.13 6.91 5.6 0.05 2.32 96.63	MCP 99-01 Trachyte 61.1 0.06 18.94 4.14 0.18 0.13 1.08 7.87 5.61 0.04 0.79 99.22

1: Major element compositions (from

Table 5 .

 5 leading to 2 SE of 0.07 % for LMS model, 0.11 % for UMS and 0.06 % for MRS. Fe (% ) 2 SE Repetions δ 57 Fe prim (% ) ∆ 57 Fe raw-corr 2: Whole-rock Iron isotope measurements from LMS, UMS and MRS relative to the IRMM-014 standard data as well as the corrected basalt iron isotope signature of the initial parental meltsδ 57 Fe prim (seeSossi et al. 2016). ∆ 57 Fe raw-corr represents the difference between the measured δ 57 Fe and the calculated δ 57 Fe prim .

	Sample δ 57 Silica under-saturated series Type
	UMS
	ARC 321	Basanite	0.167	0.095	6	0.133	0.034
	ARC 172	Basanite	0.244	0.094	6	0.236	0.008
	ARC 356	Tephri-phonolite	0.231	0.055	6	0.116	0.114
	ARC 361	Phono-tephrite	0.163	0.064	6
	ARC 391	Phonolite	0.308	0.053	6
	Arc 96	Phonolite	0.328	0.121	6
	Arc 411	Phonolite	0.206	0.031	6
	ARC 414	Phonolite	0.197	0.098	5
	Silica undersaturated series
	LMS
	LVLK 87	Basalt	0.215	0.063	6	0.159	0.056
	ARC 590	Basalt	0.066	0.060	6	-0.002	0.068
	LVLK 123 Basalt	0.162	0.082	6	0.095	0.066
	LVLK 108 Basalt	0.156	0.058	6	0.105	0.051
	LVLK 131 Trachyte	0.160	0.073	6
	ARC 272	Trachyte	0.050	0.037	6
	LVLK 127 Trachyte	0.107	0.017	9
	MRS
	RR08-88	Basalt	0.138	0.047	6	0.112	0.025
	RR08-64	Trachy-basalt	0.166	0.062	6	0.086	0.080
	RR08-78	Trachy-basalte	0.210	0.025	9	0.135	0.075
	RR08-31	Trachy-andesite	0.141	0.102	6
	RR08-63	Trachy-andesite	0.201	0.085	3
	RR08-85	Trachyte	0.209	0.067	9
	RR08-74	Trachyte	0.265	0.053	8
	MCP 99-01 Trachyte	0.266	0.062	9
	Standards
	BHVO-2	Basalt	0.176	0.026	6
	BIR-1A	Basalt	0.173	0.021	9
	CG-2	Nephelinic syenite 0.25	0.083	6

  . The Figure5.6: Recalculated iron isotopes signature of the primitive basalts (δ 57 Fe prim ) for each series versus their calculated Fe 3+ /Fe tot . The dashed line represents the mean δ 57 Fe prim of all the lavas (except for sample ARC 172, and ARC 590 that display δ 57 Fe extreme values) and the light grey area represents the 2 standards error of the mean using the Student t-correcting factor (Platzner, 1997).δ 57 Fe prim corrected from olivine and clinopyroxene fractionation, range between -0.002 and +0.236% (Table5.2). Most of the lavas display a δ 57 Fe of about 0.103 ± 0.02 % close to the normal MORB average values (δ 57 Fe = 0.100 ± 0.01% ;Sossi et al. (2016)); except for ARC 172 (δ 57 Fe = 0.236 % ) and ARC 590 (δ 57 Fe = -0.002 % ). However, the lavas from the 3 magmatic series display different Fe 3+ /Fe tot which signify that the mantle source went through different level of oxidation. The less compatible nature of Fe 3+ (D= 1) over

		0.5
		LMS LMS			UMS UMS		MRS MRS
		0.45	ARC 590		ARC 321	RR08-88
		0.35 0.4	LVLK 108 LVLK 87 LVLK 123		ARC 172 ARC 361	RR08-64 RR08-78
		0.3
	δ 57 Fe prim (‰)	0.15 0.2 0.25					δ 57 Fe = 0.114 ± 0.02 ‰
		0.1
		0.05
		0
		-0.05
		-0.1
		0	0.1	0.2	0.3	0.4	0.5	0.6	0.7
					Fe 3+ / Fe tot

  UMS displays slightly heavierδ 57 Fe prim than LMS with extreme value of + 0.236 % for ARC 172 associated with Fe 3+ /Fe tot of about 0.35. This heavier isotope signature implies the participation of a heavy iron isotope enriched component in the mantle associated with the metasomatized peridotites. Pyroxenite, that is present in Kerguelen Archipelago at greater depth than harzburgite (> 30 km: Grégoire et al. (2001)), appears as a good candidate. Many recent studies suggested that pyroxenite component is mandatory to explain MORB and OIB geochemical and isotopic signature (e.g.,Aulinas et al., 2010;

  56 Fe (% LAS) 2 SD δ 57 Fe (% LAS) 2 SD δ 56 Fe (% IRMM-14) 2 SE δ 57 Fe (% IRMM-14) 2 SE n Primary reference material

				NKT-1G	BCR-2G	BIR-1G	BHVO-2G
				0.184	-0.069	-0.034	-0.200
	1.6	Slope Slope : 1.438 ± 0.047		0.13	0.09	0.19	0.12
				0.196	-0.119	-0.033	-0.375
	δ 57 Fe (‰ vs IRMM-014)			0.17 0.298	0.21 0.045	0.32 0.080	0.31 0.045
							BHVO-2
							BIR-1
							OSC
							BHVO-2
				0.13	0.09	0.19	0.12	BCR-2
	-0.4 -0.3 -0.2 -0.1	0	0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9	1
				δ 56 Fe (‰ vs IRMM-014)
	Figure 6.5: Three-isotopes plot for iron of the United States Geological Survey (USGS) 0.014 0.141 0.055 0.370
	reference and the San Carlos Olivine. The line represents the mass-dependent fractiona-
	tion line with a slope of 1.438 ± 0.047, that is indistinguishable from previously measured
	isotopic mass-dependent fractionation trends in natural and references samples of 1.466 ±
	0.013 (Malinovsky et al., 2003), 1.475 ± 0.022 (Craddock and Dauphas, 2011), and from theoretical prediction at equilibrium (1.475). 0.31 12 0.32 54 0.21 21 0.18 18
				BHVO-2G	BHVO-2G	BHVO-2G	Putratronic

δ

Table 6 .

 6 2: Fe isotope compositions of USGS references glasses.δ 56 Fe (% LAS) and δ 57 Fe (% LAS) represent the δ 56 Fe and δ 57

  Nous avons démontré principalement que le fractionnement isotopique entre les minéraux contenant du Fe 2+ est significatif et ne peut donc être négligé. L'étude conjointe des propriétés isotopiques du Fe et Si ainsi que de leurs environnements locaux a permis de déterminer les propriétés cristallochimiques contrôlant le fractionnement du fer et du silicium à l'équilibre dans les minéraux. Dans cette étude, une faible corrélation entre les distances Fe-O et Si-O et leurs β-facteurs associés a été observée. La coordinance joue un rôle majeur sur le fractionnement isotopique du fer dans les minéraux. En effet, le fer en coordinance quatre montre un enrichissement en iso-observées au sein des minéraux. Cependant, nous avons démontré, qu'après la température, le paramètre ayant une influence majeure sur le fractionnement isotopique des minéraux contenant exclusivement du Fe 2+ , et ayant des coordinances similaires, est l'environnement cristallochimique local (i.e. le nombre et la nature des seconds voisins). Dans un deuxième temps, des calculs de dynamique moléculaire ab initio ont été faits afin de modéliser des liquides silicatés et de déterminer leurs propriétés isotopiques (Fe, Si). Cette étude est la partie la plus complexe et ambitieuse de ce travail de thèse. En effet, les propriétés isotopiques des liquides silicatés n'avaient jamais été déterminées par calcul théorique auparavant. Les liquides sont désordonnés et évoluent avec le temps, c'est pourquoi il est nécessaire de les étudier de manière statistique. L'anharmonicité potentielle des systèmes liquides rend également le calcul des facteurs ß plus compliqué. De plus, la composition des liquides silicatés modélisés, proche de celle de liquides silicatés SiO 2 étudiée, il n'y a pas de variations isotopiques significatives en silicium. Le système isotopique du fer quant à lui, montre une corrélation entre les β-facteurs du fer et les distances Fe-O. Alors que le système isotopique du silicium apparait comme indépendant de la composition chimique globale des liquides silicatés, les variations isotopiques en fer apparaissent corrélées à la concentration en fer et en magnésium. De plus, un lien fort est observé entre les distances Fe-Fe et Fe-Mg et le β-facteur du fer : les distances Fe-Fe les plus courtes (ou inversement les distances Fe-Mgles plus longues) correspondant à un enrichissement en isotopes lourds dans les liquides Grâce à cette nouvelle gamme de coefficients de fractionnement, nous avons modélisé l'effet de la cristallisation fractionnée sur les signatures isotopiques du fer et du silicium lors de la différenciation magmatique et également discuté l'impact de la fusion partielle sur les isotopes du fer. Il est apparu que la majorité des signatures isotopiques en fer et silicium pendant la différenciation magmatique peuvent être expliquées par la cristallisation fractionnée, à l'exception de certaines valeurs extrêmes (δ 57 Fe ≤ 0.9 % ). Notre étude soutient également l'hypothèse que la fusion partielle est un processus viable pour expliquer la signature enrichie en isotopes lourds des MORB par rapport au manteau.Dans un troisième temps, les signatures isotopiques δ 57 Fe de 23 laves de type OIB provenant des Iles Kerguelen ont été mesurées. Ces laves font partie de 3 séries magmatiques distinctes et sont caractérisées par leurs évolutions compositionnelles temporelles et plus particulièrement leurs teneurs variables en alcalins. Nous avons suggéré que les variations en δ 57 Fe et Fe 3+ /Fe tot observées dans les basaltes ont pour origine une hétérogénéité de la source magmatique et que ces laves ne reflètent donc pas uniquement la signature du plume mantellique. Nous suggérons également l'implication d'un type de roche man-57 Fe des laves des Iles Kerguelen peut en grande partie être expliquée par la cristallisation fractionnée. de 170 × 24 µm 2 au sein de cristaux d'olivine, en suivant le protocole mis en place à l'OMP.

	Cette étude a également permis de souligner l'impact des minéraux riches en Fe 2+ (i.e.
	surtout les pyroxènes) sur le fractionnement isotopique du fer durant la différenciation
	magmatique grâce à des modèles de cristallisation fractionnée.
	Enfin, le développement d'une méthode de mesure des isotopes du fer utilisant un laser
	femtoseconde couplé à un MC-ICP-MS a été réalisé à l'OMP afin d'effectuer des analyses
	in situ de minéraux silicatés (i.e. olivines). Dans ce but, plusieurs verres certifiés USGS
	ainsi que l'olivine de San Carlos ont été utilisés durant le développement analytique.
	Nous avons démontré que les verres USGS utilisés ont des compositions en isotopes du

pliqués dans le fractionnement des isotopes du fer et du silicium lors de la différentiation magmatique. Dans ce but, une approche multidimensionnelle combinant des calculs ab initio, l'analyse de roches naturelles ainsi que l'analyse in situ d'olivines, a été mise en place afin de comprendre les mécanismes contrôlant le fractionnement du fer et du silicium à l'équilibre. Dans un premier temps, les propriétés isotopiques théoriques du fer et du silicium dans les minéraux silicatés et les oxydes ont été déterminées. Cette étude a permis d'obtenir un large ensemble de coefficients de fractionnement entre de nombreux minéraux présents dans la croûte terrestre. topes lourds par rapport au fer en coordinance six. L'état d'oxydation du fer a également un rôle important sur le fractionnement isotopique du fer, les minéraux contenant du Fe 3+ ont un β-facteur plus élevé que les minéraux contenant uniquement du Fe 2+ . L'état d'oxydation du fer ainsi que la coordinance peut expliquer une partie des variations iso-topiques naturels, avec 7 à 8 éléments différents rend les calculs très couteux en temps. Deux méthodes statistiques ont été utilisés afin d'obtenir les β-facteurs des liquides silicatés : i) La première méthode est basée sur l'échantillonnage périodique de la trajectoire de dynamique moléculaire. Le β-facteur est alors calculé à partir des compositions instantanées échantillonnées, à T = 0 K et en se plaçant dans l'approximation harmonique. ii) La seconde méthode est une méthode récente basée sur la détermination des β-facteurs à partir de l'énergie cinétique de la trajectoire de dynamique moléculaire. Nous avons démontré dans cette étude que pour les liquides silicatés, les deux méthodes donnent des β-facteurs similaires. Comme pour les minéraux silicatés, les propriétés isotopiques du fer et du silicium ainsi que les paramètres structuraux des liquides ont été explorés. Il est apparu que pour la gamme de silicatés. Ce qui signifie que, comme pour les minéraux, les compositions isotopiques en fer dans les liquides silicatés sont contrôlées par l'environnement local. Ce travail nous a également permis d'obtenir pour la première fois des coefficients de fractionnement entre des minéraux et des liquides silicatés. tellique ayant une signature δ 57 Fe élevée, les pyroxenites, dans la source magmatique des roches les plus alcalines. Notre étude soutient l'hypothèse déjà émise dans la littérature que le système isotopique du fer peut être un outil puissant pour déterminer les éventuelles hétérogénéités du manteau. Nous allons plus loin en montrant que la combinaison d'analyses des signatures isotopiques δ 57 Fe et d'analyses de l'état d'oxydation du fer dans les roches peut être encore plus efficace. L'évolution isotopique lors de la différenciation magmatique est limitée dans les laves de Kerguelen, mais un léger enrichissement en isotopes lourds dans les roches les plus différenciées est observé dans deux des séries (UMS, MRS) tandis que la troisième série possède un δ 57 Fe constant. La modélisation de la variation isotopiques δ 57 Fe durant la cristallisation fractionnée, utilisant les coefficients de fractionnement entre les minéraux et les liquides silicatés calculés, a montré que l'évolution en δ fer homogènes contrairement à l'olivine San Carlos. Nous déconseillons donc l'utilisation de l'olivine San Carlos comme standard lors de l'analyse des isotopes du fer par ablation laser couplé à un MC-ICP-MS. Ce travail de développement nous permet de conclure qu'il est possible de mesurer les signatures isotopiques δ 56 Fe avec une précision < 0.2 % , et une résolution

Table 1 :

 1 Differences of total energies of the different mineral composition compared with corresponding spin configuration. The name of the configuration corresponds to the name displays in Figure1. In the case of Hortonolite, the 14 non equivalent positions of iron Fe 1 Mg 7 Si 4 O 12

		Configuration Spin configuration ∆E (Ry)
	Augite
		A	AFM	0.00104
			F	0.00161
		B	AFM	0.00090
			F	0.00103
		C	AFM
			F	0.00007
		D	AFM	0.00413
			F	0.00078
	Hortonolite
		A	F	0.02245
		B	F	-
		C	F	0.00479
		D	F	0.00238
		E	F	0.00627
		F	F	0.00089
		G	F	0.01321
		H	F	0.05401
		I	F	0.01074
		J	F	0.01375
		K	F	0.04600
		L	F	0.01110
		M	F	0.01705
		N	F	0.01320
		B	AFM	-0.00082
	Forsterite	M 1	F	-
		M 2	F	0.00537
		A	F	0.00072
	Enstatite	B	F	0.01191
		C	F	0.01185
		D	F	0.01216
		E	F	0.01242
		F	F	-
		G	F	0.01075

were tested in a ferromagnetic (F) configuration, then the configuration with the lowest total energies (configuration B) was tested afterwards with an antiferomagnetic (AFM) configuration. AFM configuration were taken to perform phonon calculation on hortonolite. Legend : FAYALITE Fe 16 Ti 8 O 32 ULVOSPINEL Fe 2 Mg 14 Si 16 O 48 ENSTATITE Figure 2: Optimized structures of the Fe-bearing minerals studied.

Table 2 :

 2 Silicon β-factor (10 3 lnβ 57 F e/ 54 F e 10 6 /T 2 T (K) Enstatite Forsterite M 1 Forsterite M 2 Ulvospinel Aegirine Augite Hedenbergite Diopside Fayalite Hortonolite Magnetite

	Si).
	Si/
	lnβ

10

Table 3 :

 3 Iron β-factor (10 lnβ Fe/ Fe).

	Minerals	< F > Fe % at Valence of iron
	Aegerine	216.4	0.242	1.786
	Enstatite	150.5	0.066	1.363
	Forsterite-M 1 146.4	0.094	1.331
	Augite	127.8	0.281	1.370
	Hortonolite	134.3	0.324	1.377
	Hedenbergite 126.4	0.225	1.401
	Forsterite M 2 125.1	0.094	1.331
	Fayalite	99.5	0.548	1.389
	Diopside	135.9	0.062	1.395
	Ulvospinel	151.8	0.500	1.379
	Magnetite	207.5	0.724	1.511

Table 4 :

 4 Iron force constants (<F>), iron concentration (%at) and valence of iron in the different minerals studied.

		<F>	Fe-O bond length Polyhedron volume
	M 1 sites
	M 1 -Augite	127.381	2.163	13.245
	M 1 -Hedenbergite 126.356	2.152	13.207
	M 1 -Diopside	135.864	2.146	13.078
	M 1 -Enstatite	150.547	2.149	13.686
	M 1 -Aegerine	216.375	2.063	11.428
	M 1 -Hortonolite	135.982	2.184	13.067
	M 1 -Fayalite	95.810	2.184	13.277
	M 1 -Ulvospinel	114.115	2.122	12.527
	M 1 Forsterite	146.418	2.163	12.878
	M-Magnetite	178.6	2.0626	11.6754
	M 2 sites
	M 2 -Augite	128.437	2.050	3.388
	M 2 -Hortonolite	122.899	2.165	12.749
	M 2 -Fayalite	115.853	2.160	12.930
	M 2 -Forsterite	125.118	2.191	13.397
	T sites
	T-Ulvospinel	189.555	1.933	3.695
	T-Magnetite	265.4	1.8849	3.4367

Table 5 :

 5 Iron force constants (<F>), mean Fe-O bond lengths, and volumes of the Fe polyhedron for each M1, M2 and T sites where iron is present in the minerals. Mean Square Displacement versus time (in picoseconds) of the modelled basalt melt (100 atoms). Mean Square Displacement versus time (in picoseconds) of the modelled basalt_HP melt at 120 kbar. Mean Square Displacement versus time (in picoseconds) of the modelled basalt_ID melt. Mean Square Displacement versus time (in picoseconds) of the modelled basanite melt.

		Ca
		Na
		Fe
		Mg
	19.5	Al Si
	3.9
	MSD
	0.8
	0.2
		0.2	0.3	0.6	1.3	2.6	5.1	10.2
					Time (ps)
	Figure 3:

Figure 4: Mean Square Displacement versus time (in picoseconds) of the modelled basalt melt (183 atoms).

Table 8 :

 8 Iron force constants (<F>) as well as the Fe β-factors, obtained using the « VAF » method, of the different melts modelled

				Force constant (N/m) 10000 K 6000 K 4000 K 3000 K 1000K 500 K
	Basalt	4000K_120kbar	E_1	261	0.011	0.031	0.070	0.124	1.103	4.261
			E_2	300	0.013	0.036	0.080	0.143	1.267	4.868
			E_3	244	0.010	0.029	0.065	0.116	1.033	3.978
			Mean	269	0.012	0.032 0.072 0.128 1.134 4.369
			SD	29	0.001	0.003	0.008	0.014	0.120 0.455
			2 SE	33
				Force constant (N/m) 10000 K 6000 K 4000 K 3000 K 1000K 500 K
	Basalt	4000K_10kbar	A_1	209	0.009	0.025	0.056	0.099	0.889	3.468
			A_2	229	0.010	0.027	0.061	0.109	0.968	3.739
			A_3	267	0.011	0.032	0.071	0.127	1.127	4.337
			A_4	235	0.010	0.028	0.063	0.112	0.992	3.803
			Mean	235	0.010	0.028 0.063 0.112 0.994 3.837
			SD	24	0.001	0.003	0.006	0.011	0.099 0.364
			2 SE	24
				Force constant (N/m) 10000 K 6000 K 4000 K 3000 K 1000K 500 K
	Basalt	4000K_10kbar_183atomes D_1	215	0.009	0.026	0.058	0.102	0.911	3.520
			D_2	228	0.010	0.027	0.061	0.108	0.961	3.693
			Mean	222	0.009	0.026 0.059 0.105 0.936 3.606
			SD	9	0.000	0.001	0.002	0.004	0.036 0.122
			2 SE	12
				Force constant (N/m) 10000 K 6000 K 4000 K 3000 K 1000K 500 K
	Trachytes		J_1	189	0.008	0.022	0.051	0.090	0.801	3.110
			J_2	165	0.007	0.020	0.044	0.078	0.698	2.705
			J_3	125	0.005	0.015	0.033	0.059	0.535	2.136
						0.028	0.063	0.111	1.000	4.000
			Mean	159	0.007	0.019 0.043 0.076 0.678 2.650
			SD	33	0.001	0.004	0.009	0.015	0.134 0.489
			2 SE	38
				Force constant (N/m) 10000 K 6000 K 4000 K 3000 K 1000K 500 K
	Basanite		K_1	168	0.007	0.020	0.045	0.080	0.712	2.775
			K_2	223	0.000	0.027	0.060	0.106	0.943	3.656
			K_3	200	0.009	0.024	0.053	0.095	0.845	3.270
			K_4	195	0.008	0.023	0.052	0.092	0.823	3.183
			Mean	196	0.006	0.023 0.052 0.093 0.831 3.221
			SD	23	0.004	0.003	0.006	0.011	0.095 0.361
			2 SE	23
				Force constant (N/m) 10000 K 6000 K 4000 K 3000 K 1000K 500 K
	Basalt_H		H_1	212	0.009	0.025	0.057	0.101	0.897	3.469
			H_2	182	0.008	0.022	0.049	0.087	0.770	2.983
			H_3	186	0.008	0.022	0.050	0.088	0.786	3.048
			Mean	193	0.008	0.023 0.052 0.092 0.818 3.167
			Ecart_type	16	0.001	0.002	0.004	0.008	0.069 0.264
			2 SE	19
				Force constant (N/m) 10000 K 6000 K 4000 K 3000 K 1000K 500 K
	Phonolite		I_1	189	0.008	0.023	0.051	0.090	0.802	3.125
			I_2	424	0.018	0.050	0.113	0.201	1.786	6.832
			I_3	255	0.011	0.030	0.068	0.121	1.082	4.233
			I_4	292	0.013	0.035	0.078	0.139	1.236	4.776
			Mean	290	0.012	0.034 0.078 0.138 1.227 4.741
			Ecart_type	99	0.004	0.012	0.026	0.047	0.414 1.554
			2 SE	99

Table 4

 4 .7.

	Temp (C • ) Temp (K) 10 6 /T 2 SiO2	fef f SiO2 opx (%) cpx (%) quartz (%) Albite (%) Anorthite (%) δ 30 Si
	1150.42969	1423.580	0.493	58.668 92.25	0.31		0.272	0.425	-0.212
	1145.42969	1418.580	0.497	58.913 88.75	0.31		0.280	0.419	-0.198
	1140.42969	1413.580	0.500	59.151 85.55	0.31		0.287	0.414	-0.197
	1135.42969	1408.580	0.504	59.413 82.48	0.34		0.276	0.391	-0.225
	1130.42969	1403.580	0.508	59.676 79.60	0.34		0.282	0.383	-0.225
	1125.42969	1398.580	0.511	59.934 76.95	0.34		0.290	0.378	-0.224
	1120.42969	1393.580	0.515	60.187 74.50	0.34		0.298	0.373	-0.223
	1115.42969	1388.580	0.519	60.435 72.22	0.33		0.310	0.370	-0.222
	1110.42969	1383.580	0.522	60.681 70.10	0.33		0.317	0.365	-0.221
	1105.42969	1378.580	0.526	60.924 68.12	0.33		0.322	0.363	-0.220
	1100.42969	1373.580	0.530	61.165 66.27	0.33		0.330	0.357	-0.219
	1095.42969	1368.580	0.534	61.405 64.53	0.32		0.338	0.352	-0.218
	1090.42969	1363.580	0.538	61.646 62.90	0.32		0.346	0.346	-0.217
	1085.42969	1358.580	0.542	61.890 61.35	0.32		0.347	0.347	-0.216
	1080.42969	1353.580	0.546	62.139 59.89	0.32		0.356	0.342	-0.215
	1075.42969	1348.580	0.550	62.406 58.50	0.31		0.366	0.338	-0.213
	1070.42969	1343.580	0.554	62.825 57.10	0.28		0.384	0.347	-0.210
	1065.42969	1338.580	0.558	63.426 55.68	0.25		0.409	0.355	-0.194
	1060.42969	1333.580	0.562	64.016 54.37	0.24		0.424	0.354	-0.191
	1055.42969	1328.580	0.567	64.582 53.17	0.24		0.431	0.353	-0.189
	1050.42969	1323.580	0.571	65.125 52.07	0.23		0.442	0.347	-0.186
	1045.42969	1318.580	0.575	65.646 51.04	0.23		0.448	0.346	-0.185
	1040.42969	1313.580	0.580	66.148 50.09	0.23		0.455	0.343	-0.183
	1035.42969	1308.580	0.584	66.630 49.21	0.22		0.466	0.337	-0.181
	1030.42969	1303.580	0.588	67.094 48.38	0.22		0.468	0.339	-0.179
	1025.42969	1298.580	0.593	67.541 47.60	0.21		0.478	0.332	-0.177
	1020.42969	1293.580	0.598	67.971 46.88	0.21		0.485	0.329	-0.176
	1015.42969	1288.580	0.602	68.397 46.20	0.23		0.482	0.316	-0.181
	1010.42969	1283.580	0.607	68.823 45.58	0.27		0.466	0.293	-0.194
	1005.42969	1278.580	0.612	69.234 44.98	0.26		0.476	0.292	-0.191
	1000.42969	1273.580	0.617	69.630 44.42	0.26		0.488	0.286	-0.189
	995.429688	1268.580	0.621	70.012 43.88	0.25		0.499	0.280	-0.186
	990.429688	1263.580	0.626	70.381 43.37	0.25		0.502	0.282	-0.184
	985.429688	1258.580	0.631	70.737 42.88	0.24		0.513	0.276	-0.182
	980.429688	1253.580	0.636	71.081 42.41	0.24		0.529	0.272	-0.179
	975.429688	1248.580	0.641	71.414 41.96	0.24		0.540	0.266	-0.177
	970.429688	1243.580	0.647	71.736 41.53	0.23		0.544	0.264	-0.175
	965.429688	1238.580	0.652	72.047 41.12	0.23		0.553	0.260	-0.173
	960.429688	1233.580	0.657	72.348 40.72	0.22		0.563	0.253	-0.171
	955.429688	1228.580	0.663	72.639 40.34	0.22		0.571	0.248	-0.169
	950.429688	1223.580	0.668	72.922 39.97	0.22		0.582	0.249	-0.167
	945.429688	1218.580	0.673	73.195 39.62	0.22		0.592	0.242	-0.165
	940.429688	1213.580	0.679	73.460 39.28	0.21		0.600	0.237	-0.163
	935.429688	1208.580	0.685	73.716 38.95	0.21		0.604	0.235	-0.162
	930.429688	1203.580	0.690	73.965 38.64	0.21		0.621	0.230	-0.159
	925.429688	1198.580	0.696	74.197 38.29	0.19	0.05	0.592	0.216	-0.151

Table 10 :

 10 Modelling of δ 30 Si values during fractional crystallization using our new set of fractionation factor between minerals and melt. More details available : here.

	Mantle Fe 3+ /Fe tot 0.10
	1000lnβFe 2+ melt 0.344 F= 199 (Dauphas et al. 2014)	Partial melting model -Lower Miocene Series
	∆ 57 Fe diopside-basalt

Table 14 :

 14 Details of the partial melting model used to recreate the signature of the δ 57 Fe UMS basalt values. Proportion of mineral melting = 80% of diospide, 20 % of enstatite Model calculation detailed in Supp-data of Dauphas et al. 2014 δ 57 Fe mantle = -0.06 % Fe 3+ /Fe tot δ 57 Fe mantle Fe 3+ /Fe tot

	1.00 -0.060 0.060	0.90 -0.044 0.061	0.80 -0.029 0.062	0.70 -0.012 0.065	0.60 0.004 0.071	0.50 0.020 0.079	0.48 0.024 0.081	0.46 0.028 0.083	0.44 0.031 0.085	0.42 0.035 0.088	0.4 0.039 0.092	0.38 0.043 0.095	0.36 0.047 0.098	0.34 0.050 0.102	0.32 0.054 0.106	0.3 0.058 0.110	0.28 0.062 0.115	0.26 0.066 0.120	0.24 0.070 0.126	0.22 0.075 0.132	0.2 0.079 0.139	0.18 0.083 0.147	0.16 0.088 0.155	0.14 0.093 0.164	0.12 0.098 0.175	0.1 0.103 0.186	0.08 0.108 0.199	0.05 0.116 0.221	0.02 0.125 0.248	0.01 0.129 0.259	f δ 57 Fe mantle	Melt	DFe 3+ 4.5	DFe 2+ 1	Temperature (K) 1573	Mantle Fe 3+ /Fe tot 0.07
	-0.169	-0.159	-0.148	-0.137	-0.127	-0.117	-0.114	-0.112	-0.110	-0.108	-0.105	-0.103	-0.101	-0.099	-0.097	-0.095	-0.092	-0.090	-0.088	-0.086	-0.083	-0.081	-0.079	-0.077	-0.074	-0.072	-0.070	-0.066	-0.062	-0.061
	0.013	0.014	0.016	0.017	0.019	0.022	0.022	0.023	0.024	0.024	0.025	0.026	0.027	0.027	0.028	0.029	0.030	0.031	0.033	0.034	0.035	0.037	0.038	0.040	0.042	0.044	0.047	0.051	0.056	0.058		Residue	∆2s-3s 0.1	∆2l-3l 0.263	1000lnβFe 2+ melt 0.344 F= 199 (Dauphas et al. 2014) 1000lnβFe 3+ melt 0.606 F = 351 (Dauphas et al. 2014)	Partial melting model -Mont Ross series
																																		∆2l-2S		∆ 57 Fe enstatite-basalt	∆ 57 Fe diopside-basalt
																																		0.151		0.129	0.156

Table 15 :

 15 Details of the partial melting model used to recreate the signature of the δ 57 Fe MRS basalt values.

			δ 57 Fe	-0.002	0.137	0.181			Titano-magnetite δ 57 Fe	0.25 0.133	0.30 0.151	0.36 0.258	0.34 0.265	0.25 0.314
	Melt : basalt	Minerals nature Proportion of Fe-bearing phases	Mg # Pyroxene Olivines Xmag Pyroxene Olivine Illmenite Titano-magnetite	0.60 Augite Hortonolite 25 0.44 0.24 0.12 0.21	0.38 Hedenbergite -50 0.83 0.00 0.00 0.17	0.37 Hedenbergite -50 0.83 0.00 0.00 0.17	Melt : basanite	Minerals nature Proportion of Fe-bearing phases	Mg # Pyroxene 1 Olivines Xmag Pyroxene 1 Olivine Aegirine Illmenite	0.59 --50 0.75 0.00 0.00 0.00	0.45 Hedenbergite Hortonolite 50 0.67 0.03 0.00 0.00	0.36 --50 0.64 0.00 0.00 0.00	0.28 Diopside -50 0.63 0.00 0.00 0.03	0.31 Hedenbergite -50 0.63 0.00 0.00 0.00
	δ 57 Fe prim = -0.0021 %		F f(fe)	1.00 1.00	0.57 0.49	0.51 0.35	δ 57 Fe prim = 0.133 %		F f(fe)	1.00 1.00	0.77 0.82	0.50 0.35	0.42 0.30	0.38 0.19
	LMS		Sample name Temp. (K) MgO (wt%) FeO (wt%)	ARC 590 1417 10.17 12.29	ARC 272 1369 3.54 10.40	LVLK 127 1390 2.83 8.47	UMS		Sample name Temp. (K) MgO (wt%) FeO (wt%)	ARC 172 1337 11.74 9.53	ARC 321 1435 12.54 5.80	ARC 361 1291 8.25 2.57	ARC 356 1246 8.47 1.85	ARC 391 1331 6.05 1.50	arc 96 1340 0.17

Table 16 :

 16 Details of the fractional crystallization models, including the mineral proportion of the iron bearing phases, FeO and MgO contents, fraction of iron in the melt f (Fe), the temperature and the δ 57 Fe/ 54 Fe calculated. More details available : here Samples δ 57 Fe (% ) 2 SE MC-ICP-MS facilities Protocol of Fe purification

	ARC 321	0.203	a	Poitrasson and Freydier,
		0.125	a	Poitrasson and Freydier,
		0.146	a	Poitrasson and Freydier,
		0.332	b	Poitrasson and Freydier,
		0.079	b	Poitrasson and Freydier,
		0.118	b	Poitrasson and Freydier,
	mean	0.167	0.095
	ARC 172	0.213	a	Poitrasson and Freydier,
		0.164	a	Poitrasson and Freydier,
		0.133	a	Poitrasson and Freydier,
		0.311	b	Poitrasson and Freydier,
		0.367	b	Poitrasson and Freydier,
		0.275	b	Poitrasson and Freydier,
	mean	0.244	0.094
	ARC 356	0.224	a	Poitrasson and Freydier,
		0.204	a	Poitrasson and Freydier,
		0.233	a	Poitrasson and Freydier,
		0.310	b	Poitrasson and Freydier,
		0.154	b	Poitrasson and Freydier,
		0.260	b	Poitrasson and Freydier,
	mean	0.231	0.055
	ARC 361	0.170	a	Poitrasson and Freydier,
		0.095	a	Poitrasson and Freydier,
		0.123	a	Poitrasson and Freydier,
		0.121	b	Poitrasson and Freydier,
		0.218	b	Poitrasson and Freydier,
		0.250	b	Poitrasson and Freydier,
	mean	0.163	0.064
	ARC 391	0.317	a	Poitrasson and Freydier,
		0.313	a	Poitrasson and Freydier,
		0.329	a	Poitrasson and Freydier,
		0.252	b	Poitrasson and Freydier,
		0.385	b	Poitrasson and Freydier,
		0.253	b	Poitrasson and Freydier,
	mean	0.308	0.053
	Arc 96	0.358	a	Poitrasson and Freydier,
		0.308	a	Poitrasson and Freydier,
		0.298	a	Poitrasson and Freydier,
		0.151	b	Poitrasson and Freydier,
		0.345	b	Poitrasson and Freydier,
		0.508	b	Poitrasson and Freydier,
	mean	0.328	0.121
	Arc 411	0.225	a	Poitrasson and Freydier,
		0.210	a	Poitrasson and Freydier,
		0.225	a	Poitrasson and Freydier,
		0.214	b	Poitrasson and Freydier,
		0.216	b	Poitrasson and Freydier,
		0.147	b	Poitrasson and Freydier,
	mean	0.206	0.216
	ARC 414	0.152	a	Poitrasson and Freydier,
		0.169	a	Poitrasson and Freydier,
		0.138	a	Poitrasson and Freydier,
		0.370	b	Poitrasson and Freydier,
		0.120	b	Poitrasson and Freydier,
		0.235	b	Poitrasson and Freydier,
	mean	0.197	0.098

Table 17 :

 17 Details of the δ 57 Fe measurements (in % ) of the Kerguelen UMS lavas. The Neptune Plus MC-ICP-MS facilities used for these measurements are : a) Lyon or b) Toulouse. δ 57 Fe (% ) 2 SE MC-ICP-MS facilities Protocol of Fe purification

	Samples
	RR08-88	0.087	a	Poitrasson and Freydier,
		0.164	a	Poitrasson and Freydier,
		0.144	a	Poitrasson and Freydier,
		0.083	b	Poitrasson and Freydier,
		0.200	b	Poitrasson and Freydier,
		0.148	b	Poitrasson and Freydier,
	mean	0.138	0.060
	RR08-64	0.086	a	Poitrasson and Freydier,
		0.139	a	Poitrasson and Freydier,
		0.128	a	Poitrasson and Freydier,
		0.197	b	Poitrasson and Freydier,
		0.246	b	Poitrasson and Freydier,
		0.203	b	Poitrasson and Freydier,
	mean	0.166	0.062
	RR08-78	0.22356919	b	Dauphas and Rouxel, 2006
		0.22293443	b	Dauphas and Rouxel, 2006
		0.16664762	b	Dauphas and Rouxel, 2006
		0.2052678	b	Dauphas and Rouxel, 2006
		0.20507467	b	Dauphas and Rouxel, 2006
		0.15443923	b	Dauphas and Rouxel, 2006
		0.25010079	b	Dauphas and Rouxel, 2006
		0.21541022	b	Dauphas and Rouxel, 2006
		0.24662533	b	Dauphas and Rouxel, 2006
	mean	0.210	0.025
	RR08-31	0.178	a	Poitrasson and Freydier,
		0.137	a	Poitrasson and Freydier,
		0.218	a	Poitrasson and Freydier,
		0.258	b	Poitrasson and Freydier,
		0.048	b	Poitrasson and Freydier,
		0.009	b	Poitrasson and Freydier,
	mean	0.141	0.102
	RR08-63	0.239	b	Poitrasson and Freydier,
		0.172	b	Poitrasson and Freydier,
		0.191	b	Poitrasson and Freydier,
	mean	0.201	0.069
	RR08-85	0.140	b	Dauphas and Rouxel, 2006
		0.224	b	Dauphas and Rouxel, 2006
		0.263	b	Dauphas and Rouxel, 2006
		0.220	b	Dauphas and Rouxel, 2006
		0.076	b	Dauphas and Rouxel, 2006
		0.173	b	Dauphas and Rouxel, 2006
		0.267	b	Dauphas and Rouxel, 2006
		0.146	b	Dauphas and Rouxel, 2006
		0.371	b	Dauphas and Rouxel, 2006
	mean	0.209	0.067
	RR08-74	0.34718022	b	Dauphas and Rouxel, 2006
		0.28090811	b	Dauphas and Rouxel, 2006
		0.25675878	b	Dauphas and Rouxel, 2006
		0.18538468	b	Dauphas and Rouxel, 2006
		0.33156342	b	Dauphas and Rouxel, 2006
		0.19796815	b	Dauphas and Rouxel, 2006
		0.31061223	b	Dauphas and Rouxel, 2006
		0.20901345	b	Dauphas and Rouxel, 2006
	mean	0.265	0.053
	MCP 99-01	0.284	a	Poitrasson and Freydier,
		0.254	a	Poitrasson and Freydier,
		0.211	a	Poitrasson and Freydier,
		0.221	b	Poitrasson and Freydier,
		0.307	b	Poitrasson and Freydier,
		0.457	b	Poitrasson and Freydier,
		0.224	b	Poitrasson and Freydier,
		0.183	b	Poitrasson and Freydier,
		0.253	b	Poitrasson and Freydier,
	mean	0.266	0.062

Table 19 :

 19 Details of the δ 57 Fe measurements (in % ) of the Kerguelen MRS lavas. The Neptune Plus MC-ICP-MS facilities used for these measurements are : a) Lyon or b) Toulouse.

		Milhas hemtite δ 57 Fe (% )
	07 and 08 april 2017 06 to 08 december 2018 20 to 22 july 2020
	0.716	0.737	0.740
	0.741	0.704	0.777
	0.620	0.704	0.837
	0.744	0.791	0.752
	0.720	0.813	0.718
	0.713	0.673	0.704
	0.922	0.707	0.741
	0.726	0.637	0.706
	0.606	0.830	0.911
	0.741	0.726	0.822
	0.737	0.723	0.775
	0.811	0.791	0.795
	0.733	0.740	0.909
	0.806	0.739	0.776
	0.737	0.734	0.711
	0.735	0.902
	0.756	0.700
	0.705	0.748
	0.783	0.778
	0.752	0.820
	0.761	0.676
	Average δ 57 Fe	0.753 ± 0.034 %

Table 20 :

 20 δ 57 Fe measurements (in % ) of the in-house reference standard (Milhas hematite) for each analytical sessions.The Neptune Plus MC-ICP-MS facilities used for these measurements are: a) Lyon in 2017 or b) Toulouse in 2018 and 2020.

		BIR-1
	n δ 56 Fe [% ] 2 σ δ 57 Fe [% ] 2 σ
	1 -0.02	0.13 0.01	0.28
	2 -0.04	0.16 -0.06	0.30
	3 -0.01	0.12 0.20	0.27
	4 0.19	0.14 0.17	0.26
	5 0.15	0.14 -0.04	0.36
	6 0.05	0.15 -0.03	0.33
	7 0.06	0.21 0.02	0.28
	8 0.14	0.14 0.04	0.29
	9 0.00	0.18 -0.10	0.29
	10 0.08	0.17 0.25	0.25
	11 0.05	0.20 0.06	0.27
	12 0.10	0.18 0.22	0.28
	13 0.02	0.18 0.03	0.26
	14 -0.05	0.18 -0.03	0.29
	15 0.07	0.22 0.09	0.29
	16 0.02	0.16 0.23	0.27
	17 -0.08	0.15 0.09	0.28
	18 -0.02	0.20 0.35	0.33
	19 0.02	0.13 0.08	0.19
	20 -0.10	0.16 0.19	0.30
	21 -0.08	0.14 -0.04	0.25
	22 -0.13	0.14 -0.16	0.23
	23 -0.19	0.20 -0.30	0.33
	24 -0.27	0.21 -0.32	0.28
	25 -0.11	0.15 -0.07	0.29
	26 -0.14	0.14 -0.25	0.32
	27 -0.25	0.17 -0.39	0.32
	28 -0.05	0.17 0.07	0.29
	29 -0.11	0.18 -0.12	0.31
	30 -0.07	0.18 -0.09	0.31
	31 -0.03	0.17 -0.23	0.32
	32 -0.05	0.16 -0.12	0.22
	33 -0.05	0.18 -0.02	0.29
	34 -0.04	0.13 0.00	0.29
	35 -0.02	0.13 0.14	0.25
	36 -0.09	0.12 -0.19	0.24
	37 -0.12	0.15 -0.16	0.27
	38 -0.23	0.11 -0.19	0.29
	39 -0.19	0.09 -0.23	0.28
	40 -0.08	0.11 -0.12	0.30
	41 0.00	0.10 0.01	0.26
	42 -0.08	0.11 0.11	0.28
	43 -0.10	0.14 0.03	0.34
	44 -0.01	0.13 -0.18	0.31
	45 0.07	0.10 0.15	0.31
	46 -0.04	0.15 -0.16	0.34
	47 -0.05	0.13 -0.27	0.31
	48 0.05	0.13 -0.11	0.32
	49 0.08	0.17 0.02	0.31
	50 -0.04	0.17 -0.02	0.30
	51 0.03	0.20 -0.05	0.35
	52 -0.01	0.16 -0.17	0.35
	53 -0.05	0.16 -0.24	0.33
	54 0.02	0.17 0.12	0.34

Table 23 :

 23 δ 56 Fe and δ 57 Fe values of the USGS BIR-1 glass measured by fs-LA-MC-ICP-MS (raster analysis). OSC n δ 56 Fe [% ] 2 σ δ 57 Fe [% ] 2 σ n δ 56 Fe [% ] 2 σ δ 57 Fe [% ] 2 σ

	-0.02	0.12	-0.14	0.27	53	-0.04	0.17	-0.22	0.29
	-0.18	0.14	-0.42	0.31	54	-0.10	0.17	0.06	0.32
	-0.11	0.12	-0.25	0.30	55	-0.21	0.14	0.02	0.28
	-0.06	0.17	-0.34	0.33	56	-0.13	0.17	-0.01	0.27
	-0.12	0.15	-0.31	0.38	57	0.14	0.16	0.36	0.33
	-0.17	0.18	-0.15	0.34	58	-0.04	0.16	-0.06	0.32
	-0.07	0.19	-0.16	0.34	59	0.00	0.17	0.21	0.34
	-0.12	0.18	-0.25	0.32	60	0.42	0.16	0.72	0.39
	-0.17	0.18	-0.45	0.37	61	0.14	0.12	0.34	0.37
	-0.14	0.20	-0.10	0.36	62	0.27	0.16	0.56	0.38
	-0.09	0.20	-0.14	0.37	63	0.38	0.15	0.85	0.40
	-0.08	0.19	-0.41	0.32	64	0.14	0.18	0.37	0.43
	-0.15	0.15	-0.11	0.36	65	0.10	0.14	0.13	0.35
	-0.20	0.15	-0.13	0.29	66	0.61	0.16	0.76	0.34
	-0.08	0.11	-0.10	0.30	67	0.31	0.16	0.54	0.34
	-0.07	0.13	-0.12	0.28	68	0.56	0.21	0.64	0.33
	0.00	0.15	-0.07	0.31	69	-0.16	0.16	-0.16	0.35
	-0.12	0.18	-0.28	0.35	70	-0.07	0.12	0.03	0.32
	-0.08	0.14	-0.08	0.32	71	-0.01	0.15	-0.19	0.32
	-0.22	0.13	-0.36	0.35	72	0.00	0.11	-0.15	0.30
	-0.09	0.18	-0.07	0.30	73	-0.10	0.13	-0.18	0.29
	-0.10	0.16	-0.23	0.32	74	-0.14	0.14	-0.30	0.33
	-0.13	0.15	-0.12	0.31	75	0.14	0.13	-0.05	0.30
	-0.16	0.14	-0.04	0.26	76	0.03	0.14	-0.01	0.31
	-0.05	0.19	0.01	0.29	77	0.07	0.14	0.14	0.28
	-0.07	0.13	-0.05	0.29	78	0.12	0.13	0.26	0.33
	0.05	0.21	0.01	0.40	79	0.03	0.18	-0.10	0.32
	0.13	0.17	0.40	0.35	80	0.01	0.15	-0.08	0.30
	-0.17	0.15	-0.24	0.34	81	0.51	0.19	0.62	0.34
	0.04	0.17	0.27	0.38	82	0.56	0.20	0.58	0.40
	-0.05	0.16	0.23	0.38	83	0.48	0.24	0.81	0.47
	0.07	0.22	0.16	0.36	84	0.01	0.19	-0.12	0.34
	-0.11	0.17	0.01	0.27	85	-0.06	0.18	-0.13	0.32
	-0.10	0.16	-0.15	0.30	86	-0.10	0.19	-0.15	0.34
	0.05	0.17	0.25	0.22	87	-0.08	0.18	-0.18	0.35
	-0.02	0.14	0.29	0.37	88	0.03	0.16	-0.06	0.34
	0.13	0.15	0.52	0.39	89	-0.08	0.18	-0.19	0.33
	0.11	0.17	0.43	0.37	90	-0.04	0.20	-0.09	0.33
	0.25	0.15	0.50	0.43	91	-0.01	0.19	-0.03	0.30
	0.17	0.17	0.34	0.41	92	-0.01	0.19	-0.17	0.30
	0.20	0.16	0.36	0.44	93	-0.08	0.15	-0.24	0.23
	0.18	0.20	0.10	0.38	94	0.11	0.16	0.12	0.22
	0.19	0.16	0.22	0.33	95	0.01	0.16	0.06	0.27
	0.13	0.16	0.25	0.32	96	-0.17	0.16	-0.13	0.27
	0.18	0.20	0.36	0.34	97	-0.02	0.14	0.03	0.29
	0.03	0.19	0.19	0.33	98	-0.04	0.11	-0.06	0.29
	0.08	0.21	0.18	0.38	99	-0.11	0.15	-0.16	0.32
	0.33	0.17	0.50	0.30	100	0.02	0.17	-0.05	0.33
	0.10	0.19	0.23	0.25	101	-0.04	0.17	0.10	0.27
	0.16	0.20	0.36	0.27	102	0.02	0.16	0.19	0.26
	-0.23	0.16	-0.24	0.26	103	0.16	0.18	0.35	0.25
	0.03	0.19	0.15	0.29	104	0.26	0.13	0.29	0.26

Table 25 :

 25 δ 56 Fe and δ 57 Fe values of the San Carlos Olivine measured by fs-LA-MC-ICP-MS (raster analysis).

Acknowledgements

We are grateful to B.N. Moine for providing samples of the Kerguelen Archipelago. We deeply thank P. Patrier and A. Meunier for their help on XRD data acquisition and expertise on alteration minerals. SR thanks Jan Schuessler for the advices concerning the colorimetric procedure. We also thank M. Henry for maintaining the clean lab., P. Telouk and J. Chmeleff for maintaining the MC-ICP-MS of respectively Lyon and Toulouse. This work was supported by the Programme National de Planétologie (PNP) of CNRS/INSU, co-funded by CNES, to FP.

 Dauphas et al. (2014). The grey circles in the background correspond to δ 57 Fe (a) and δ 30 Si (b) from the literature (Du et al., 2017;Foden et al., 2015;He et al., 2017;Poitrasson and Freydier, 2005;Poitrasson and Zambardi, 2015;Savage et al., 2011Savage et al., , 2012;;Schuessler et al., 2009;Sossi et al., 2012;Telus et al., 2012;Teng et al., 2013;Williams et al., 2018;Xia et al., 2017;Zambardi et al., 2014). The fractionation factor ∆ 57 Fe min-melt (∆ 57 Fe enstatite-trachyte ; ∆ 57 Fe diopside-trachyte ; ∆ 57 Fe ulvospinel-trachyte ; ∆ 57 Fe magnetite-trachyte and ∆ 30 Si min-melt (∆ 30 Si enstatite-trachyte ; ∆ 30 Si diopside-trachyte ; ∆ 30 Si anorthite-trachyte ; ∆ 30 Si albite-trachyte ; ∆ 30 Si quartz-trachyte ) used in these models are displayed in Table 4.7. The linear trends in b) represent the linear fit of bulk rock analyses obtained by Savage et al. (2011) in black and Zambardi et al. (2014) in green from natural magmatic rocks analysis as well as the linear trend of our fractional crystallization modelled value (in red).

CHAPTER 5

Insights from iron isotopes into magmatic differentiation and mantle heterogeneity of the

Kerguelen Archipelago

Ségolène RABIN, Franck POITRASSON, Paolo SOSSI, Maud BOYER, Michel GRÉ-GOIRE, Hervé LEYRIT lighted Pb and Hf isotopic variations that was interpreted as the presence of small-scale heterogeneities within the Kerguelen plume source. Additionally, [START_REF] Doucet | Primitive neon and helium isotopic compositions of high-MgO basalts from the Kerguelen Archipelago, Indian Ocean[END_REF]; [START_REF] Mattielli | Hf Isotope Evidence for a Miocene Change in the Kerguelen Mantle Plume Composition[END_REF] emphasized that the Kerguelen hotspot source has a heterogeneous noble gas isotope composition as reflected by variable 4 He/ 3 He and higher 21 Ne/ 22 Ne of the Kerguelen basalts compared to other OIB (Iceland). Later, [START_REF] Borisova | Lead isotope signatures of Kerguelen plume-derived olivine-hosted melt inclusions: Constraints on the ocean island basalt petrogenesis[END_REF] confirmed that the youngest lavas of Kerguelen Archipelago (≤ 24 My) originate from heterogeneous mantle sources on the basis of variable Pb isotope ( 207 Pb/ 206 Pb) signature of olivine-hosted melt inclusions.

The studied areas correspond to the Gallieni Peninsula located in the South of the Kerguelen Archipelago and the South-East Province composed of the Jeanne d'Arc and Ronarc'h Peninsula (Fig. 5.1). Although the lavas from the South-East Province all have alkaline affinity, they have been divided into 2 chronological groups: Lower Miocene Series (LMS) and Upper Miocene Series (UMS). The LMS is characterized by alkali basalts to trachytes ranging in age from ≈ 22 to 24 My. In order to be consistent with previous publications on this magmatic series (e.g., [START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF][START_REF] Weis | Geochemical characteristics of the youngest volcano (Mount Ross) in the Kerguelen Archipelago: inferences for magma flux, lithosphere assimilation and composition of the Kerguelen plume[END_REF], we continue to use the term "LMS" even if it appears that some lavas are from the late oligocene [START_REF] Nicolaysen | 40Ar/39Ar geochronology of flood basalts from the Kerguelen Archipelago, southern Indian Ocean: Implications for Cenozoic eruption rates of the Kerguelen plume[END_REF]. On the other hand the UMS is characterized by highly alkalic basanites to phonolites dated from ≈ 6.6 to 10.2 My [START_REF] Weis | Kerguelen Archipelago revisited: geochemical and isotopic study of the Southeast Province lavas[END_REF] and is mainly located on Ronarc'h Peninsula. The main feature of the Gallieni Peninsula is Mount Ross, which is the youngest (<1 My) and highest (1850m) volcanic edifice of the Archipelago. The Mount Ross series (MRS) is a silica-saturated alkaline series characterized by abundant evolved lavas, namely trachytes.

Sample description

Twenty-three samples from the LMS, UMS and MRS were analyzed for their whole rock iron isotopic composition. Localities of the samples are provided on the geological map (Fig. 5.1) and in [START_REF] Leyrit | Kerguelen : Cartographie et magmatologie des presqu'iles Jeanne d'Arc et Ronarc[END_REF].

at medium mass resolution slits. The international reference material for Fe isotopes, IRMM-14, which was made up to the same Fe and Ni concentration as the samples was analyzed between each sample to allow sample-standard bracketing. Between each sample and standard, the system was rinsed with 0.05M HCl for 60 seconds. Blanks were measured in 10 cycles of 8.369 s each, whereas the sample analysis consisted of 25 cycles of 8.369 s each in static mode (no amplifier rotation among the Faraday cups), constituting one analysis. All samples were analyzed at least five times (i.e., five replicate analyses;

Appendix C -Table 17 and19). The iron isotope results are expressed as the delta value (in % ) relative the isotopic reference standard IRMM-014 (Equation 5.1).

δ 57 F e( ) = 57 F e/ 54 F e sample 57 F e/ 54 F e IRM M -014 -1 × 1000 (5.1)

Mass bias correction used the Ni-spiking method described by Poitrasson and Freydier (2005) for the samples analyzed in high mass resolution. The uncertainties given in Table 5.2 represents the reproducibility of the replicates with a 95% confidence intervals (Platzner, 1997) :

Where t is the student-t correction factor for 95% confidence, n the number of analysis and SD is the standard deviation. Data quality control was assured by the periodic analysis of the Milhas hematite in house reference material (δ 57 Fe = 0.753 ± 0.034 % , 2SD, n = 57 from April 2017 to July 2020) every six samples and also by the analysis of three purified geostandards: BIR-1 (δ 57 Fe = 0.107 ± 0.065 % ), CG-2 (δ 57 Fe = 0.253 ± 0.073% ) and BHVO-2 (δ 57 Fe = 0.176 ± 0.026% ), all with uncertainties reported as 2SE. There is no literature data on CG-2 for iron isotopes but BIR-1 as well as BHVO-2 are regularly used as a standard for iron isotopes studies and displays results congruent with the literature compilation of Craddock and Dauphas (2011). Milhas hematite was both analyzed in Lyon and Toulouse laboratories and display really similar δ 57 Fe values (Appendix C -Table 20). Therefore, no inter-laboratory bias were observed.

Fe 3+ /Fe tot determination

Ferrous iron contents were determined by the colorimetric method of Wilson (1960) adapted by [START_REF] Schuessler | Oxidation state of iron in hydrous phono-tephritic melts[END_REF]. The principle of this colorimetric method follows the reaction:

(5.3)

Iron isotope compositions

The δ 57 Fe of 23 whole-rock samples and three geostandards (CG-2, BIR-1, BHVO-2) are displayed in Table 5.2 and illustrated in Figure 5.4. The whole rock isotopic compositions measured vary from 0.050 ± 0.037 % to 0.155 ± 0.057 % for the silica saturated lower Miocene series, from 0.163 ± 0.064 % to 0.328 ± 0.121 % for the silica undersaturated upper Miocene series and from 0.132 ± 0.068 % to 0.272 ± 0.062 % for the silica-saturated series from the Mount Ross (Table 5.2). Each series stretch within the range reported for igneous rocks on Earth (e.g., [START_REF] Dauphas | Iron isotopes may reveal the redox conditions of mantle melting from Archean to Present[END_REF]Foden et al., 2015;Poitrasson and Freydier, 2005;Sossi et al., 2012). In figure 5.4, the 57 Fe of the LMS and MSR undersaturated lavas are plotted against SiO 2 , an index of differentiation whereas the UMS lavas δ 57 Fe are plotted against Zr concentration as an index of differentiation for highly alkaline series [START_REF] Jeffery | Peralkaline Felsic Magmatism of the Atlantic Islands[END_REF].

The LMS does not exhibit an increase of δ 57 Fe with increasing SiO 2 , considering the 2SE. This is similar to what is observed for Hekla volcanic rocks (Schuessler et al., 2009) characterized by a constant δ 57 Fe when SiO 2 increases from 40 wt% to 67 wt% SiO 2 .

The UMS shows constant δ 57 Fe for lavas with Zr concentration under 1000 ppm followed with a hint of an increase of δ 57 Fe with increasing Zr concentration, although remaining undistinguishable within uncertainties (Fig. 5.4). The MRS suite displays constant δ 57 Fe values with increasing SiO 2 within uncertainties from the least evolved basalt, RR08-88, to the trachyte RR08-67. Above 58 wt% SiO 2 , the more evolved trachytes of MRS, MCP 99-01 and RR08-74, displays heavier iron isotopes composition than the basalt.

Ferric-ferrous iron ratios

Results of the colorimetric method are shown in Fig. 5.5 and Fig. the Fe 3+ /Fe tot ratio of sample RR08-88 from MRS is typical of OIB with values near 0.2 (Cottrell and Kelley, 2014). Thereafter, the MSR displays a constant increase of ferric iron concentration throughout the differentiation series, a change that is mirrored in

Essentials parameters during lasers ablation

The wavelength

The wavelength of the laser beam has a major role on laser-matter iteration as it drives the way the beam will penetrate into the sample. The wavelength choice highly depends on the opacity/transparency of the sample (e.g., Alexander et al., 1998;[START_REF] Chenery | Determination of rare earth elements in single mineral grains by laser ablation microprobe-inductively coupled plasma mass spectrometry-preliminary study[END_REF]Guillong et al., 2003;Günther and A. Heinrich, 1999b;Russo et al., 2000). Indeed, when the light is not absorbed but transmitted, the beam penetrates in the heart of the material releasing heat leading to micro-cracks and fusion of matter during ablation as described part 6.2.3. These thermally induced damage can occur using both nanosecond and femtosecond laser (e.g., Seydoux-Guillaume et al., 2010). The wavelength is therefore responsible for the quantity and size of the particles in the aerosol produced. A high wavelength in the IR spectrum associated to a transparent sample will lead to high quantity and thicker particles than in the far UV. Studies demonstrated indeed that the use of laser beam with short wavelength (far-UV) produces thin particles, independently of the material transparency, facilitating transport and ionization of the sample. It consequently reduces the chemical fractionation effect observed leading to more accurate and precise analyses.

Concerning the femtosecond laser ablation, the impact of wavelength choice is still debated whether working with UV over IR is still advantageous as for the nanosecond laser. Freydier et al. (2008) determined that there were no benefits of working with short

Reference materials

In order to assess the accuracy, precision and reproductivity of our method of iron isotopes analysis with the fs-La-MC-ICP-MS, multiple standards were analyzed:

Puratronic

Puratronic is a ferric iron oxides metal from Alfa Aesar. It is composed of 99.998 % of iron. Nebulization solution measurements of puratronic purified at the GET and analyzed at CNRS-ENS Lyon, following the procedure displayed in chapter 5, display a δ 56 Fe value of 0.249 ± 0.032 % and a δ 57 Fe of 0.389 ± 0.057 % (F. Poitrasson, pers. comm.).

San Carlos olivine

San Carlos olivine is a natural forsterite-rich olivine of mantle xenolith origin. It is used in multiple geoscience electron probe studies as a standard for Mg, Si and Fe.

Original olivine sample analyzed by Jarosewish in 1980 contains about 10.61 wt% of (Fe 2 O 3 ) tot . However, a later study evidenced a variability in iron concentration depending on the grain from Fo 89.2 to Fo 91.1 (Fournelle, 2011).

NKT-1G

NKT-1G is a nephelinite from Knippa, Texas, USA. Iron concentration ((Fe

of this sample is 13.55 ± 0.2 wt%.

BHVO-2G

BHVO-2 stands for Basalt Hawaiian Volcanic Observatory. This standard originates from a pahoehoe lava erupted in fall 1919 from the Halemaumau crater. Iron concentration ((Fe 2 O 3 ) tot ) of this sample is 12.3 ± 0.2 wt%.

BIR-1G

BIR-1 is an Icelandic basalt collected from the interglacial lavas flows of the Reykjavik dolerites. Iron concentration ((Fe 2 O 3 ) tot ) of this sample is 11.3 ± 0.12 wt%.

BCR -2G

BCR-2 stands for Basalt Columbia River. This basalt originates from the Bridal Veil Flow Quarry, near Portland, Oregon, USA. This basalt is particularly rich in iron with an iron concentration ((Fe 2 O 3 ) tot ) of about 13.8 ± 0.2 wt%.

The four USGS synthetic glasses were generated by melting kilogram aliquots of powered USGS standard at 1540 • C under a nitrogen atmosphere (Wilson and Taggard, unpublished) to create homogeneous standards suitable to laser ablation. Ablation of the USGS glasses were performed using lines with a spot size of 24 µm in diameter, a repetition rate of 20 Hz, a fluence of ≈ 2 J/cm -2 and a laser beam speed of 8 µm/s.

Samples

The two olivines analyzed in this chapter originate from the South of the Kerguelen Archipelago. Olivine phenocrysts are part of the basalt RR08-88, studied in chapter 5.

The RR08-88 thin-section is polished and 33 µm thick. Inclusions and fractures within studied crystals are rare. The olivine 1 measures ≈ 1mm × 600 µm and olivine 2 ≈ 400 × 150 µm. The large olivine phenocrysts, like olivine 1, in the basalt present an iron chemical evolution from the core to the rim. The rim (50 µm thick) displays generally an enrichment in iron, from 18 wt% Fe in the core up to 24 wt% Fe at the rim. Ablation of olivine 1 and 2 were performed using lines with spot size diameter of 20 µm, a repetition rate of 20 Hz, a fluence of ≈ 2.5 J/cm -2 and a laser beam speed of 8 µm/s.

Mass-bias correction and uncertainties calculation

Instrumental mass bias during iron isotopes measurement is corrected using 58 Ni/ 60 Ni following Oeser et al. (2014) procedure which can be summarized in 3 steps.

First, the isobaric interference of 58 Ni on 58 Fe correction is performed. Using the measured 57 Fe -56 Fe ratio, assuming a natural ratio of 0.023095 (Taylor et al., 1992) and applying the exponential law, the fractionation coefficient β 57/56 is calculated for each cycle acquired and is expressed as: The calculated 58 Fe intensity was subtracted to the measured signal intensity of mass 58 to obtain the 58 Ni corr , and finally the interference-corrected 58 Ni/ 60 Ni were calculated:

Second, the isobaric interference of 54 Cr on 54 Fe is corrected by calculating a fractionation coefficient β 58/60 using the corrected 58 Ni/ 60 Ni ratio as well as the 60 Ni/ 58 Ni certified ratio of 0.385199 (Gramlich et al., 1989): were taken from (Craddock and Dauphas, 2011) and the value of δ 56 Fe Puratronic/IRMM-014 from solution nebulization MC-ICP-MS (See above).

Uncertainties of individual analysis of San Carlos olivine and Kerguelen olivine are calculated by propagating the analytical errors of the bracketing standard with those of the samples (Horn et al., 2006).

δ 56 Fe (‰ vs IRMM-014)

(

(3)

(5)

(3)

(10)

(

(3) (This study) (This study) -0.

Acknowledgement

We are grateful to Ingo Horn and Martin Oeser for their precious help during the development of the in situ iron isotopes MC-ICP-MS methods at the GET. We deeply thank Jérôme Chmeleff for its support through the analytical development and for maintaining the femtosecond laser and Neptune Plus at the OMP. This work was supported by the Programme National de Planétologie (PNP) of CNRS/INSU, co-funded by CNES to FP.

CHAPTER 7

Conclusions and perspectives

fs-LA-MC-ICP-MS measurements

The analytical developments performed before and during this thesis represent the start of many possibilities of Fe isotope measurements in the geosciences fields. Concerning our problematic, the first investigation should be carried out on olivine in order to assess in-situ iron isotope fractionation factors between olivine and glasses in natural samples that could be compared to the ab-initio calculated ones. A post-processing, including diffusion models, would be most likely necessary for most samples in order to disentangle equilibrium Fe isotope fractionation from kinetic-driven fractionation. From there, other minerals could be analyzed such as pyroxenes and oxides.

Abstract

Iron and silicon isotope measurements in high-temperature geological reservoirs display significant isotopic fractionations, especially within the Earth's crust. However, the mechanisms involved in these fractionations are not fully understood. Their knowledge and that of the fractionation factors in equilibrium conditions is central to interpret the measurements carried on natural rocks. To address this issue, this work is a multidimensional approach based on first principles calculation of Fe and Si isotope fractionations in minerals and silicate melts, on Fe isotope measurements of lavas from Kerguelen Archipelago and on analytical developments of in situ Fe isotope measurements of glasses and olivines using a femtosecond laser coupled to MC-ICP-MS. We provide the theoretical Fe and Si isotope properties of various minerals relevant to Earth crust formation and, for the first time, of silicate melts. Results suggest that significant Fe and Si isotope fractionations are expected between Fe 2+ -bearing minerals at high temperature. We also show that, besides temperature and redox changes, iron and silicon second neighbors have a major impact on isotopic fractionations. From our set of equilibrium fractionation factors between minerals and melts, we suggest that fractional crystallization could be the main process involved in Fe and Si isotope evolution during magmatic differentiation. In addition, iron isotope composition of Kerguelen oceanic basalts underlines the complexity of the hotspot evolution, with potential source heterogeneity. Finally, the analytical developments performed on MC-ICP-MS with femtosecond laser, allow in situ Fe isotope measurements of silicate glasses and minerals with an uncertainty better than 0.2% .

Résumé