D E C Alp

C K Golden

F L H Sio

J Tissot

L Zhao

Gao

Marc Blanchard
email: marc.blanchard@impmc.upmc.fr

N Dauphas

M Y Hu

M Roskosz

E E Alp

D C Golden

C K Sio

F L H Tissot

J Zhao

L Gao

R V Morris

M Fornace

A Floris

M Lazzeri

E Balan

Reduced partition

INTRODUCTION

For decades now, the isotopic compositions of natural samples have been measured and found countless applications in all branches of geosciences (see for reviews, [START_REF] Valley | Stable isotope geochemistry[END_REF][START_REF] Johnson | Geochemistry of non-traditional isotopes[END_REF][START_REF] Eiler | Frontiers of stable isotope geoscience[END_REF]). In the meantime, isotope exchange experiments were performed to improve our understanding of the processes responsible for stable isotope fractionation. More recently, new approaches have emerged and are contributing to this field of research.

First-principles calculations give reduced partition function ratios (also called !factors) that can be combined for two phases in order to obtain the equilibrium isotope fractionation factor ("-factor), which is the quantity usually measured. These theoretical methods are also of great interest for investigating the mechanisms controlling the isotope fractionation at the molecular scale. For Mössbauer active elements (like iron), !-factors can also be obtained using Mössbauer spectroscopy through the measurement of the temperature dependence of the isomer shift [START_REF] Polyakov | The use of Mössbauer spectroscopy in stable isotope geochemistry[END_REF] or using Nuclear Resonant Inelastic X-ray Scattering (NRIXS, [START_REF] Polyakov | Determination of tin equilibrium isotope fractionation factors from synchroton radiation experiments[END_REF][START_REF] Polyakov | Equilibrium iron isotope fractionation factors of minerals: Reevaluation from the data of nuclear inelastic resonant X-ray scattering and Mössbauer spectroscopy[END_REF][START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF][START_REF] Dauphas | Magma redox and structural controls on iron isotope variations in Earth's mantle and crust[END_REF]. [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] and [START_REF] Hu | Moments in nuclear resonant inelastic x-ray scattering and their applications[END_REF] reported NRIXS data for 57 Fe-rich goethite FeO(OH), hydronium-jarosite (H 3 O)Fe 3 (SO 4 ) 2 (OH) 6 and potassium jarosite KFe 3 (SO 4 ) 2 (OH) 6 . From such measurements, one can deduce iron !-factors as a function of temperature. [START_REF] Polyakov | Equilibrium iron isotope fractionation factors of minerals: Reevaluation from the data of nuclear inelastic resonant X-ray scattering and Mössbauer spectroscopy[END_REF] had used projected partial phonon density of states (pDOS) obtained using this technique to calculate !-factors for various phases. [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] and [START_REF] Hu | Moments in nuclear resonant inelastic x-ray scattering and their applications[END_REF] used a different approach based on moment estimates of NRIXS scattering spectrum S(E), which simplifies evaluation of measurement uncertainties and potential systematic errors. The study of [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] was the first of its kind in measuring NRIXS spectra specifically for applications to isotope geochemistry. In doing so, they encountered a difficulty that had been unappreciated before concerning the baseline at low and high energies. Most previous studies in geosciences had focused on estimating the Debye sound velocity [START_REF] Hu | Measuring velocity of sound with nuclear resonant inelastic x-ray scattering[END_REF][START_REF] Sturhahn | Geophysical applications of nuclear resonant scattering[END_REF], from which compressional and shear wave velocities can be deduced if the bulk modulus and density of the phase are known. These estimates are derived from parts of the spectra that are close to the elastic peak for the nuclear transition of 57 Fe at 14.4125 keV. On the other hand, the force constant, which controls !-factors, is heavily influenced by details of the spectrum at the low-and high-energy ends of the spectrum. As a result, little attention had been paid to the accuracy of force constant measurements by NRIXS. [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] found that in some cases, significant counts were present even at high energies. The projected partial phonon density of states, g(E), and the scattering spectrum, S(E), never reached zero and as a result, the integrals that gave the force constants did not plateau for goethite and H-jarosite. These were interpreted to reflect the presence of multiple phonons at high energies. However, we were unable to replicate the measurements during another session of NRIXS measurements at the Advanced Photon Source synchrotron. This and other tests performed on other phases convinced us that the high counts in the tails are not from multiple phonons but rather reflect the presence of a non-constant baseline. To address this issue, Dauphas and collaborators have developed a software (SciPhon) that reliably corrects for non-constant baseline [START_REF] Dauphas | Magma redox and structural controls on iron isotope variations in Earth's mantle and crust[END_REF].

In the present study, the pDOS as well as the iron and oxygen !-factors of goethite are computed using first-principles calculations and compared to available experimental isotopic data. In parallel, the data published in [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] have been re-evaluated using SciPhon and we present revised estimates for the force constants of goethite and jarosite. To validate the approach and evaluate the long-term reproducibility of force constant measurements by NRIXS, we have analyzed the goethite sample two more times and the jarosite samples one more time. Those new results, together with a re-evaluation of previous data, are reported here.

MATERIALS AND METHODS

NRIXS spectroscopy

Nuclear resonant inelastic X-ray scattering is a nuclear spectroscopic technique that uses the nuclear transition of 57 Fe at 14.4125 keV to probe the vibration properties of iron [START_REF] Seto | Observation of nuclear resonant scattering accompanied by phonon excitation using synchrotron radiation[END_REF][START_REF] Sturhahn | Phonon density of states measured by inelastic nuclear resonant scattering[END_REF]. The method as implemented at sector 3-ID-B of the Advanced Photon Source at Argonne National Laboratory is briefly described hereafter. The incident beam is a pulsed X-ray beam of 70 ps duration and 153 ns interpulse duration. A monochromator restricts the energy spread of the incident beam to 1 meV. When the pulse hits the sample, X-rays are scattered by electrons and this electronic contribution is almost instantaneous. On the other hand, the excited 57 Fe nuclei have a finite lifetime of 141 ns and the electronic contribution can be eliminated from the signal by applying some time discrimination.

The signal from NRIXS is measured using Avalanche Photodiodes (APD). The energy is scanned around the nominal resonant energy over a typical interval of -150 to +150 meV. When the photon energy is higher than the resonance energy, the excess energy can be lost to excitation of phonon modes in the lattice and the nuclear excitation can still occur (phonon creation). When the energy is lower than the nominal resonance energy, the energy deficit can be provided by lattice vibrations and the nuclear excitation can still occur (phonon annihilation).

From NRIXS spectra one can calculate !-factors by taking the moments of the scattering spectrum S(E) [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF][START_REF] Hu | Moments in nuclear resonant inelastic x-ray scattering and their applications[END_REF] or the projected partial phonon density of states g(E) [START_REF] Polyakov | Determination of tin equilibrium isotope fractionation factors from synchroton radiation experiments[END_REF][START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF]). In the present study, all samples were fine powders (isotropic) and the calculated pDOS represents an average from contributions of all crystallographic orientations. The pDOS is partial in the sense that NRIXS is only sensitive to 57 Fe. Using S(E), the formula that gives !-factors is [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF],

1000 ln " # 1000 M M $ %1 & ' ( ) * + 1 E R R 3 S 8k 2 T 2 % R 5 S %10R 2 S R 3 S 480k 4 T 4 + R 7 S + 210 R 2 S ( ) 2 R 3 S % 35R 3 S R 4 S % 21R 2 S R 5 S 20160k 6 T 6 , - . . / 0 1 1 , (Eq. 1)
where M and M* are the masses of the two isotopes considered (e.g., 56 and 54), E R is the free recoil energy (1.956 meV for 57 Fe), k is Boltzmann's constant, T is the temperature, and R i S is the i th centered moment of S given by

R i S = S(E)(E " E R ) i dE "# +#

$

. Equation (1) was derived by [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] and [START_REF] Hu | Moments in nuclear resonant inelastic x-ray scattering and their applications[END_REF] using two different mathematical approaches (expansions in powers of temperature vs. thermalized moments).

The !-factors can also be calculated from g(E) using the formula that is valid for E/kT < 2! [START_REF] Polyakov | Determination of tin equilibrium isotope fractionation factors from synchroton radiation experiments[END_REF][START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF],

1000 ln " # 1000 M M * $1 % & ' ( ) * m 2 g 8k 2 T 2 $ m 4 g 480k 4 T 4 + m 6 g 20160k 6 T 6 + , - . / 0 , (Eq. 2)
where m i g is the!i th !moment of g given by m i g = g 0 +" Polyakov et al. (2005) obtained this formula using perturbation theory and an expression of the kinetic energy while [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] obtained this formula using a Bernoulli expansion of the reduced partition function ratio. In general, !-factors can be expressed as, (Eq. 3) where the coefficients A 1 , A 2 , and A 3 can be calculated from either Eq. 1 (S) or Eq. 2 (g). The pDOS g is calculated from S using a Fourier-Log decomposition [START_REF] Johnson | Determination of the single-scattering probability distribution from plural-scattering data[END_REF][START_REF] Sturhahn | Phonon density of states measured by inelastic nuclear resonant scattering[END_REF][START_REF] Sturhahn | CONUSS and PHOENIX: Evaluation of nuclear resonant scattering data[END_REF][START_REF] Kohn | DOS: Evaluation of phonon density of states from nuclear resonant inelastic absorption[END_REF] and Eqs. 1 and 2 are mathematically equivalent. In practice, Eq. 1 is easier to use as errors are not correlated between different energy channels and it is more straightforward to assess the effects of the data reduction procedure (e.g., truncation in energy, baseline subtraction) on the estimates of the !-factor coefficients. The above-mentioned equations can also be written as [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF],

# (E)E i dE .
1000 ln " = A 1 T 2 + A 2 T 4 + A 3 T 6 ,
1000 ln 4 = B 1 F T 2 # B 2 F 2 T 4 , (Eq. 4)
where B 1 "#$%&, B 2 is a constant that depends on the shape of the pDOS and F is the mean force constant (in N/m) of the bonds holding iron in position,

F = M E R ! 2 R 3 S = M ! 2 m 2 g .
(Eq. 5)

The same samples as those initially measured by [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] were used in this study and details on the synthesis method can be found in [START_REF] Golden | Hydrothermal synthesis of hematite spherules and jarosite: Implications for diagenesis and hematite spherule formation in sulfate outcrops at Meridiani Planum, Mars[END_REF].

All samples were made starting with 57 Fe-rich metal (95 % vs 2.1 % natural abundance) as NRIXS is only sensitive to this Mössbauer isotope. The nature of the minerals analyzed was checked by X-ray powder diffraction (XRD). The Rietveld refinement (JADE software package, Materials Data Inc.) provided the following unit-cell parameters for the goethite sample: a = 4.58 Å, b = 9.94 Å, c = 3.02 Å (Pbnm space group). The particle sizes (equivalent sphere diameters of coherent domain sizes) were determined using the peak broadening of Rietveld-refined XRD, yielding 19.8 nm for goethite and 223.0 nm for K-Jarosite. Goethite particles are usually acicular. In Dauphas et al 2012, the powdered samples were mounted in compressed pellets, which could impart preferential orientation of the samples, as rightly noticed by [START_REF] Frierdich | Iron isotope fractionation between aqueous Fe(II) and goethite revisited: New insights based on a multi-direction approach to equilibrium and isotopic exchange rate modification[END_REF]. However, the latter measurements were done on the powdered samples mounted in vacuum grease at the tip of a kapton capillary, which should keep a random orientation of the particles. One or two APDs were mounted on the sides of the sample perpendicular to the incident beam, so as to capture the maximum solid angle of scattered X-rays. The forward signal was measured at the same time, providing an accurate estimate of the resolution function.

The data reduction was entirely done using a new software called SciPhon that is introduced briefly in [START_REF] Dauphas | Magma redox and structural controls on iron isotope variations in Earth's mantle and crust[END_REF] and will be the scope of a forthcoming publication. The main difference with the data reduction protocol used by [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] is the recognition that signal is present at the low and high energy ends of the spectrum at a level that is too high to be explained by the presence of multiple phonons. The approach used in SciPhon is to remove a linear baseline that is calculated by interpolating the data between the low and high-energy ends of the spectrum and truncating the data when the signal reaches a constant value. This is an effective method when a broad enough energy scan is acquired and one can clearly make the cut between what is signal and what is baseline. After truncation and baseline subtraction, the missing signal is reconstructed by calculating the contribution from the missing multiple phonons using a first estimate of g obtained from truncated S. When the baseline at the high-energy end is higher than at the lowenergy end (most common situation), the correction for non-constant baseline brings the force constant down. Conversely, when the baseline at the high-energy end is lower than at the high-energy end, the correction brings the force constant up. The other features of SciPhon are deconvolution of the resolution function using a steepest descent algorithm, removal of the elastic peak using a refined interpolation method, calculation of all parameters needed for application of NRIXS data to isotope geochemistry (Eqs. 1 and 2), and propagation of all uncertainties (not only counting statistics but also errors on baseline subtraction and energy scaling) on parameters derived from S. We applied the same algorithm to the new data reported here and to the raw data reported in [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF].

Computational methods

Goethite (!-FeOOH) has an orthorhombic unit cell (a = 4.598 Å, b = 9.951 Å, c = 3.018 Å, Pbnm space group, [START_REF] Yang | Goethite, !-FeO(OH), from singlecrystal data[END_REF], containing four formula units.

Calculations are done with the PWscf code [START_REF] Giannozzi | Quantum ESPRESSO: a modular and open-source software project for quantum simulations of materials[END_REF] http://www.quantum-espresso.org) using the density functional theory (DFT) and the generalized gradient approximation (GGA) to the exchange-correlation functional with the PBE parameterization [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]. The ionic cores are described by the ultrasoft pseudopotentials Fe.pbe-nd-rrkjus.UPF, O.pbe-rrkjus.UPF, H.pberrkjus.UPF, as in [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF][START_REF] Blanchard | First-principles study of the structural and isotopic properties of Al-and OH-bearing hematite[END_REF][START_REF] Blanchard | Infrared spectroscopic properties of goethite: anharmonic broadening, long-range electrostatic effects and Al substitution[END_REF]. The wave-functions and the charge density are expanded in plane-waves with 40 and 480 Ry cutoffs, respectively.

Increasing these energy cutoffs to 60 and 720 Ry does not modify significantly the vibrational frequencies (< 1%). For the electronic integration, the Brillouin zone is sampled according to the Monkhorst-Pack scheme [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF], using shifted 4 " 2 " 6 k-point grids. Increasing the number of k-points does not modify the structural and vibrational properties. Calculations are spin-polarized and set up to the antiferromagnetic structure. The spins are oriented along the c-axis of goethite with up and down spins in alternate chains of octahedra [START_REF] Cornell | The iron oxides: structure, properties, reactions, occurrences and uses[END_REF].

Magnetic moments are free to relax. Atomic positions are relaxed until the residual forces on atoms are less than 10 -4 Ry/a.u..

Additional calculations were performed using the GGA+U method since it is known that the addition of a Hubbard U correction on the Fe atom improves the description of the electronic and elastic properties of goethite by taking into account the strong on-site Coulomb repulsion of Fe 3d electrons (e.g. [START_REF] Otte | Pressure-induced structural and electronic transitions in FeOOH from first-principles[END_REF]). The value of the Hubbard U is determined using a linear response approach in an internally consistent way following the procedure proposed by [START_REF] Cococcioni | Linear response approach to the calculation of the effective interaction parameters in the LDA+U method[END_REF] and [START_REF] Kulik | Density functional theory in transition-metal chemistry: a self-consistent Hubbard U approach[END_REF]. Details about the practical procedure can be found in [START_REF] Blanchard | First-principles calculation of the infrared spectrum of hematite[END_REF]. The value of the Hubbard U is found equal to 3.34 eV, a value similar to the 3.30 eV found for hematite [START_REF] Blanchard | First-principles calculation of the infrared spectrum of hematite[END_REF]).

Following the method described in [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF], the !-factors were calculated from the harmonic vibrational frequencies using

"(a,Y ) = # q,i * # q,i e $h# q,i * 2kT ( ) 1$ e $h# q,i * kT ( ) 1$ e $h# q,i kT ( ) e $h# q,i 2kT ( ) q { } % i=1 3N at % & ' ( ( ) * + + 1 N q N ( ) (Eq. 6)
where # q,i are the frequencies of the phonon with wavevector q and branch index i = 1,3N at . N at is the number of atoms in the unit cell, while # q,i and # * q,i are the vibrational frequencies in two isotopologues. N is the number of sites for the Y atom in the unit cell, T is the temperature, h is the Planck constant and k is the Boltzmann constant.

Phonon frequencies were calculated within the harmonic approximation using the linear response theory [START_REF] Baroni | Phonons and related crystal properties from density-functional perturbation theory[END_REF][START_REF] De Gironcoli | Vibrational properties of MnO and NiO from DFT+U-based Density Functional Perturbation Theory[END_REF] as implemented in the PHonon code [START_REF] Giannozzi | Quantum ESPRESSO: a modular and open-source software project for quantum simulations of materials[END_REF]; http://www.quantum-espresso.org). Phonon frequencies were computed on shifted 2 " 2 " 2 q-point grids, for which the convergence of the !-factors is achieved.

The !-factors calculated from the pDOS (Eq. 2) are identical to the !-factors calculated directly from the harmonic vibrational frequencies (Eq. 6) providing that the highest energy of the pDOS is smaller than 2!kT. When it is not the case (i.e.

when OH stretching modes are considered in the pDOS) the formula derived from perturbation theory (Eq. 1-4 in [START_REF] Polyakov | Determination of tin equilibrium isotope fractionation factors from synchroton radiation experiments[END_REF]) must be used instead.

RESULTS

Reappraisal of NRIXS data

The two motivations for implementing a non-constant baseline subtraction procedure were that (1) replicate measurements of a given phase over several years yielded force constant values that were not reproducible and ( 2) the signal at the low and high energy ends of the spectrum often did not reach zero, so that the force constant integral did not converge. In Fig. 1, we show force constant determinations with or without baseline subtraction for goethite and jarosite. The values without baseline subtraction were processed in the same manner as in [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF], meaning that a constant background was subtracted, which is given by the average counts measured in a 10 meV window at the low-energy end of the spectrum (e.g., from -130 to -120 meV). The values without baseline subtraction differ slightly from those reported by [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] and [START_REF] Hu | Moments in nuclear resonant inelastic x-ray scattering and their applications[END_REF] because we now only use the phonon-creation side to calculate the force constant (the phonon-annihilation side is calculated from the detailed balance and the temperature). The phonon annihilation side is still measured to define the low-energy end of the baseline but this side of the spectrum often suffers from low counting statistics, which is the reason why it is not used to calculate the mean force constant. When no baseline is subtracted, the force constant values show variations from one sample to another that far exceed individual error bars (reduced $ 2 = 13.2). When the data are corrected for the presence of a nonconstant baseline, the average force constant does not change but the sample-to-sample scatter is very much reduced (reduced $ 2 = 3.1). The improvement in reproducibility (Fig. 1) and better consistency with force constant estimates from theory (see Sect. 3.2.1) justify our preference of the non-constant baseline data reduction algorithm [START_REF] Dauphas | Magma redox and structural controls on iron isotope variations in Earth's mantle and crust[END_REF]. Below, we discuss the results for each phase individually.

Goethite

Including the measurements published by [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF], goethite was analyzed by NRIXS three times. As explained in [START_REF] Dauphas | Magma redox and structural controls on iron isotope variations in Earth's mantle and crust[END_REF] and the method section, the main difference in the data reduction approach used here versus that used by [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] is the recognition that the non-zero signals at the low-and high-energy ends of the spectrum are not from multiple phonons but rather some non-constant baseline. [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] used the counts at the low energy end as baseline and interpreted the non-zero counts at the high-energy end to reflect the presence of multiple phonons. Because the signal never reached zero for goethite, a correction for missing multiple phonons was even applied, which exacerbated the problem. The new approach removes a linear baseline interpolated between the lowand high-energy ends of the spectrum using the SciPhon software [START_REF] Dauphas | Magma redox and structural controls on iron isotope variations in Earth's mantle and crust[END_REF]. Various quantities calculated from S(E) and g(E) are compiled in Table 1. The pDOS calculated for the three goethite samples are very reproducible and the average pDOS is plotted in Fig. 2.

As shown in Table 1 and later in Table 5 for jarosite samples, B 2 (Eq. 4) is almost the same for the phases investigated (~59,000), so the force constant is the main controlling factor in iron isotopic fractionation even at low temperature and we will focus on the values of the force constants in comparing the results obtained by [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] and [START_REF] Hu | Moments in nuclear resonant inelastic x-ray scattering and their applications[END_REF] with those presented here. The force constant values obtained from g are identical to those obtained from S, so we will only discuss the values obtained from S in the following.

For goethite, [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] reported force constants of 307±9 N/m from S and 314±10 N/m from S after refinement. [START_REF] Hu | Moments in nuclear resonant inelastic x-ray scattering and their applications[END_REF] obtained similar values using the same data reduction approach. Significant signal remained at the high-energy end and as a result the force constant integral did not plateau (Fig. 6 of [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF], the new truncation-linear baseline subtraction protocol applied to the same data (using SciPhon) leads to a significant reduction of the force constant,

Jarosite

Including the measurements published by [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF], hydronium (H-) jarosite and potassium (K-) jarosite were analyzed twice each. Like for goethite, various quantities calculated from S(E) and g(E), using the new data reduction algorithm, are compiled in Table 2. The pDOS calculated for the two H-jarosite, and two K-jarosite samples are very reproducible and the average pDOS are plotted in In the case of H-jarosite, for which significant signal remained at the highenergy end and as a result the force constant integral did not converge (Fig. 6 of [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF], the new data-reduction leads to a significant reduction of the force constant. In the case of K-jarosite, the force constant integral plateaued and the estimated force constant does not change significantly with the new data reduction scheme used in SciPhon. Note that [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] had no satisfactory explanation as to why the force constant of H-jarosite was significantly higher than Kjarosite, other than invoking the presence of high-energy modes arising from hydrogen vibrations. The results presented here show that the force constants of Hjarosite and K-jarosite are indistinguishable (280±11 N/m vs. 290±12 N/m).

First-principles determination of isotopic fractionation properties

Iron !-factor of goethite

In first-principles calculations, the goethite cell parameters were fixed to their experimental values [START_REF] Yang | Goethite, !-FeO(OH), from singlecrystal data[END_REF] and atomic positions were relaxed using either GGA or GGA+U methods. GGA atomic positions are in excellent agreement with the experimental values (Table 3). Only the x atomic coordinate of H atoms shows a significant discrepancy revealing the difficulty of DFT to treat hydrogen bonds as well as the difficulty to determine experimentally the H positions. The GGA+U performs similarly as GGA for the description of interatomic distances (Table 4). This is also true for the vibrational properties. We observe a good correlation between experimental and theoretical frequencies for both GGA and GGA+U methods (Fig. 4). As already highlighted in [START_REF] Blanchard | Infrared spectroscopic properties of goethite: anharmonic broadening, long-range electrostatic effects and Al substitution[END_REF], the effect of the Hubbard U correction is mostly visible on the OH bending and stretching modes. The OH bending frequencies increase while the OH stretching frequencies decrease, consistently with lengthening of the O-H bond. The pDOS and in particular the iron contribution is calculated using a 8 " 4 " 12 q-points grid obtained through a Fourier (trigonometric) interpolation of the force constants (see [START_REF] Méheut | Equilibrium isotopic fractionation in the kaolinite, quartz, water system: Prediction from firstprinciples density-functional theory[END_REF] for details). We observe a general good agreement between the calculated and measured iron pDOS (Fig. 2a). In detail, small differences can be noted between the average measured pDOS and the calculated ones. These differences are of the same order of magnitude as the differences between the pDOS computed with GGA and GGA+U, and are distributed all along the energy range, which suggests that they are not related to any specific vibrational modes. However, these small divergences lead to a significant difference in the calculated iron force constants that are equal to 233 N/m and 247 N/m using GGA and GGA+U, respectively. The force constant is calculated from the pDOS g by using equation ( 5). The integral in the second moment of g explains why the OH vibration modes with their high energies contribute a little to the iron force constant while the corresponding pDOS signal is almost negligible (Fig. 2). The difference between the DFT (233 N/m for GGA, 247 N/m for GGA+U)

and NRIXS (268±8 N/m) force constants is already present before the OH vibration modes. Calculations show that the pDOS should be exactly zero over a certain energy range between the lattice modes and the OH bending modes (Fig. 2b), which is not the case for the NRIXS measurements and contributes to the overestimation of the iron force constant. This is likely due to the position of the baseline that should be slightly higher. The variability of the calculated 57 Fe/ 54 Fe !-factor is shown in Fig. 5, and the corresponding temperature dependences are reported in Tables 1 and5. Its variations follow variations in the iron force constant (Eq. 4). GGA and GGA+U results differ by ~8 % and it is not possible from the analysis of the vibrational properties to identify which computational method gives the best description of goethite. This difference should thus give an idea of the uncertainty intrinsic of DFT.

These theoretical values are lower than the NRIXS-derived !-factor but higher than the !-factor determined by [START_REF] Polyakov | The use of Mössbauer spectroscopy in stable isotope geochemistry[END_REF] from the Mössbauer measurements done by de Grave and Vanderberghe (1986). Therefore with the new NRIXS data reduction, the discrepancy between the various analytical approaches is reduced but remains significant.

A general scaling factor reflecting the systematic underestimation of vibrational frequencies by the GGA functional is sometimes applied (e.g. Schauble 2011) or a mineral-dependent scaling factor is taken, assuming an accurate preliminary assignment of the vibrational modes (e.g. [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF]). Here the theoretical !-factors were obtained by fixing the cell parameters to the experimental values and relaxing only the atomic positions. Following this procedure, we found that the interatomic bond lengths and therefore the vibrational frequencies are improved with respect to the fully optimized structure (i.e. relaxation of the cell parameters and atomic positions). No further frequency correction is needed. The iron !-factors of goethite calculated here are equal to 10.2 ‰ or 11.0 ‰ at 22°C, using either GGA or GGA+U methods. These values compare well with 10.5 ‰, which is the !-factor obtained from the fully relaxed structure and a frequency scale factor of 1.04 (quantified by taking the best linear-fit of the theoretical versus experimental frequencies). The same approach was applied to hematite, by keeping fixed the cell parameters of the rhombohedral primitive cell to the experimental values (a = 5.427 Å, ! = 55.28°, R3 c space group, [START_REF] Finger | Crystal structure and isothermal compression of Fe 2 O 3 , Cr 2 O 3 , and V 2 O 3 to 50 kbars[END_REF]. Unlike goethite, the addition of the Hubbard U clearly improves the atomic positions, interatomic distances and vibrational frequencies of hematite. Therefore, only GGA+U results are retained and it is found that the new GGA+U iron !-factor is undistinguishable from the !-factor previously published in [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF] where a frequency scale factor of 1.083 was used. In a similar way, we checked for pyrite that the iron !-factor obtained with the experimental cell parameters [START_REF] Brostigen | Redetermined crystal structure of FeS 2 (pyrite)[END_REF], i.e.

13.1 ‰ at 22°C, is close, within the expected uncertainty, to the value calculated in [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF], i.e. 13.6 ‰. However we would like to emphasize that fixing the cell parameters to the experimental values do not dispense with checking that the calculated frequencies are in good agreement with the measured values.

Oxygen !-factor of goethite and hematite

Beside iron !-factors, first-principles calculations provide the 18 O/ 16 O !-factor of goethite as a function of temperature (Fig. 6 and Table 6). For goethite, GGA and GGA+U results differ by ~4%. In the ideal goethite structure, half of the oxygen atoms are hydroxylated and the others are not. Calculations show that these two oxygen populations can be distinguished isotopically. At thermodynamic equilibrium, heavier isotopes will concentrate preferentially in hydroxylated sites. The oxygen isotope fractionation of goethite corresponds then to the average of all oxygen atoms of the system. Similar calculations were also performed on hematite, using the same model as that previously used by [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF]. As explained in section 3.2.1, only GGA+U results are considered. With ~62 ‰ at 25 °C, the oxygen !-factor of hematite is found to be very close to the one of goethite, but with a slightly more linear temperature dependence (Fig. 6).

DISCUSSION

Iron !-factor of goethite: intercomparison of the different methods

In principle, DFT, NRIXS and Mössbauer spectroscopy should lead to statistically undistinguishable iron !-factors. In the case of hematite, an excellent agreement is found between DFT-derived 57 Fe/ 54 Fe !-factor (i.e. 10.9 ‰ at 22 °C, [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF]) and NRIXS-derived 57 Fe/ 54 Fe !-factor (i.e. 11.3 ± 0.4 ‰ at 22 °C, [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF]). In the case of pyrite, the apparent discrepancy between DFT and Mössbauer results that was reported in [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF] and in [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], could be resolved by using a better constrained temperature dependence of the Mössbauer spectra [START_REF] Blanchard | Comment on "New data on equilibrium iron isotope fractionation among sulfides: Constraints on mechanisms of sulfide formation in hydrothermal and igneous systems[END_REF]). The first NRIXS-based estimation has recently confirmed our pyrite iron !-factor [START_REF] Polyakov | Fe !-factors for sulfides from NRIXS synchrotron experiments[END_REF]. In light of these previous studies, it is essential to investigate the origin of the scattering of the goethite iron !-factors obtained from the three analytical techniques.

The first obvious difference between goethite and hematite is the presence of hydrogen atoms, and it is well known that the accurate description of hydrogen bonding using DFT remains a concern. However, it is important to note that the lengths of the Fe-O and Fe-OH bonds are well described (Table 4) and that all vibrational frequencies below 700 cm -1 (i.e. excluding OH bending and stretching modes) compare well with the experimental values (Fig. 4). These vibrational modes are the main ones that contribute to the iron !-factor (Fig. 2c). The main difference between GGA, GGA+U and NRIXS comes from the high-energy end of the lattice modes (65 -90 meV or 525 -725 cm -1 ) and the fact that the NRIXS pDOS does not go down to zero between the lattice modes and OH bending modes (Fig. 2).

The nature and quality of the samples may also have an impact on iron isotope composition. On one hand, a 57 Fe-rich sample with particle sizes of ~20 nm was synthesized for the NRIXS measurements, and on the other hand, Mössbauer spectroscopy was done on natural sample from the Harz mountains, well-crystallized with particle size of ~1 "m (de [START_REF] De Grave | 57 Fe Mössbauer effect study of wellcrystalized goethite (!-FeOOH)[END_REF]. Some kind of internal disorder is however commonly observed in goethite, using e.g. differential thermal analysis, infrared or magnetic measurements [START_REF] Cornell | The iron oxides: structure, properties, reactions, occurrences and uses[END_REF]. This internal disorder may correspond to crystal defects and/or iron vacant sites that are compensated for by extra, non-stoichiometric protons. All this amounts to introducing distortions with respect to the ideal crystal structure. In "real" goethite crystals, there will be a larger variation in the length of Fe-O bonds, which will lead to a larger variation of the local iron !-factors (i.e. !-factors associated with each iron atom). In a similar but simpler way, the ideal goethite structure displays already two oxygen populations with specific isotopic signatures and the bulk oxygen !-factor corresponds to the average of these two local !-factors (Fig. 6). The prediction of the Iron force constant and !-factor can also be affected by surface sites, the contribution of which should depend on the size of the crystals. Unlike for defects in bulk, the molecular relaxation in vicinity of the surfaces will more likely give rise to a specific isotopic signature that will affect the overall isotopic composition. [START_REF] Beard | Iron isotope fractionation between aqueous ferrous iron and goethite[END_REF] and [START_REF] Frierdich | Iron isotope fractionation between aqueous Fe(II) and goethite revisited: New insights based on a multi-direction approach to equilibrium and isotopic exchange rate modification[END_REF] investigated the isotopic exchange between aqueous Fe(II) and goethite, using two sizes of goethite. Their results demonstrate that the equilibrium isotopic properties of nano-scale minerals may be distinct from micron-scale or larger minerals. They found that iron surface sites are enriched in heavy isotopes compared to bulk goethite. This fractionation is consistent with the fact that the NRIXS-derived !-factor (for a nano-scale sample) is higher than the Mössbauer-derived !-factor (for a micron-scale sample). However the difference observed in !-factors (Fig. 5) is large compared to the difference in equilibrium fractionation measured by [START_REF] Frierdich | Iron isotope fractionation between aqueous Fe(II) and goethite revisited: New insights based on a multi-direction approach to equilibrium and isotopic exchange rate modification[END_REF] for various particle sizes. [START_REF] Rustad | Prediction of iron-isotope fractionation between hematite (!-Fe 2 O 3 ) and ferric and ferrous iron in aqueous solution from Density Functional Theory[END_REF] examined iron isotope fractionation between hematite and aqueous iron, and found almost no difference between bulk and surface !-factors. This conclusion applies to the (012) hematite surface with molecularly and dissociatively adsorbed water, using DFT calculations and the embedded cluster approach. Only few experimental and theoretical data exist on the topic. More studies are needed with other mineral surfaces and more structurally-complex surfaces.

Isotopic fractionations between minerals ("-factors)

The experimental iron isotope fractionation factors between goethite, hematite and pyrite in condition of thermodynamic equilibrium can be determined by combining the measurements from [START_REF] Skulan | Kinetic and equilibrium Fe isotope fractionation between aqueous Fe(III) and hematite[END_REF], [START_REF] Welch | Kinetic and equilibrium Fe isotope fractionation between aqueous Fe(II) and Fe(III)[END_REF], [START_REF] Syverson | Experimental determination of equilibrium Fe isotopic fractionation between pyrite and dissolved Fe under hydrothermal conditions[END_REF] and [START_REF] Frierdich | Iron isotope fractionation between aqueous Fe(II) and goethite revisited: New insights based on a multi-direction approach to equilibrium and isotopic exchange rate modification[END_REF] (Table 7). Keeping in mind that these mineralmineral isotopic fractionations do not represent direct measurements and involve approximations like temperature extrapolations, we can compare them with the equilibrium fractionation factors estimated from NRIXS, Mössbauer or DFT (Fig. 7).

The range of iron !-factors obtained for goethite (Fig. 5) leads to a significant spread of iron "-factors for pyrite-goethite and hematite-goethite (Fig. 7). DFT "-factor is in good agreement with the experimental pyrite-hematite value (0.62 ‰ compared to # 57 Fe = 0.44 ± 1.0 ‰ at 350 °C). For pyrite-goethite, DFT "-factors (GGA and GGA+U) are within the error bar of the experimental data. For hematite-goethite, DFT curves are 1-2 ‰ lower than the experimental points at 98 °C. The Mössbauerderived !-factor of goethite [START_REF] Polyakov | The use of Mössbauer spectroscopy in stable isotope geochemistry[END_REF] seems to be in better agreement with experimental points, while NRIXS data fall in the lowest part of the range (i.e., little iron isotopic fractionation between goethite and hematite). The same conclusions could also be reached looking at the isotopic fractionation between aqueous Fe(II) and goethite [START_REF] Frierdich | Iron isotope fractionation between aqueous Fe(II) and goethite revisited: New insights based on a multi-direction approach to equilibrium and isotopic exchange rate modification[END_REF]. However this would require to combine the present mineral !-factors with a Fe(II) aq !-factor obtained from a different technique (for instance, the DFT values from Rustad et al., 2010, currently considered as the most reliable, and based on an aperiodic model, B3LYP exchange-correlation functional and localized basis sets). This practice should be considered with caution and the combination of !-factors determined by a single methodology is always preferable.

It often happens that the iron !-factors derived from Mössbauer are slightly different and lower than the ones derived from NRIXS, but usually when the "-factors are determined the agreement between these two methods and with the experimental data is satisfactory. This is shown, for instance, for the isotopic fractionation between molten silicate, FeS and metal [START_REF] Dauphas | Magma redox and structural controls on iron isotope variations in Earth's mantle and crust[END_REF].

For the oxygen isotopes, DFT results (GGA and GGA+U) indicate a small equilibrium fractionation between hematite and goethite (between -3.1 and +0.8 ‰ over the whole temperature range, Fig. 8). These theoretical estimations are consistent with the study of [START_REF] Yapp | Oxygen isotopes in iron (III) oxides, 1. Mineral-water fractionation factors[END_REF], which suggests, from synthesis experiments conducted in the temperature range 25-120 °C, that goethite and hematite are isotopically indistinguishable at equilibrium. Several experimental studies also investigated oxygen isotope fractionation in hematite-water and goethite-water systems (e.g. [START_REF] Bao | Oxygen isotope fractionation in ferric oxide-water systems: Low temperature synthesis[END_REF][START_REF] Yapp | Oxygen isotopes in synthetic goethite and a model for the apparent pH dependence of goethite-water 18 O/ 16 O fractionation[END_REF]. The hematite-goethite fractionation curve that can be derived from [START_REF] Bao | Oxygen isotope fractionation in ferric oxide-water systems: Low temperature synthesis[END_REF] shows a similar temperature-dependence as our GGA results but is slightly more positive (Fig. 8). In this study, experiments of goethite synthesis were performed at a pH higher than 14. However Yapp's results (2007) suggest that, in addition to temperature, pH can affect the measured oxygen isotope fractionation between goethite and water. He found for goethite synthesized at low pH (~1 to 2) a curve that differs significantly from the ones obtained at high pH (>14). Using this low-pH curve for goethite-water from [START_REF] Yapp | Oxygen isotopes in synthetic goethite and a model for the apparent pH dependence of goethite-water 18 O/ 16 O fractionation[END_REF] and the hematite-water oxygen fractionation from [START_REF] Bao | Oxygen isotope fractionation in ferric oxide-water systems: Low temperature synthesis[END_REF], the hematite-goethite fractionation curve falls in the range predicted by DFT methods but with an stronger temperature-dependence (Fig. 8). According to [START_REF] Yapp | Oxygen isotopes in synthetic goethite and a model for the apparent pH dependence of goethite-water 18 O/ 16 O fractionation[END_REF], data measured for goethite crystallized at low pH may approach isotopic equilibrium values. These data are therefore the ones that must be compared preferentially with the DFT results. In conclusion, theoretical estimations for oxygen isotopes are consistent with experimental measurements, even if the exact temperature-dependence remains uncertain.

Our data display a significant discrepancy with the results of the semi-empirical approach of [START_REF] Zheng | Calculation of oxygen isotope fractionation in metal oxides[END_REF][START_REF] Zheng | Oxygen isotope fractionation between hydroxide minerals and water[END_REF], i.e. the modified increment method where the equilibrium oxygen isotope fractionation factors of oxides are assessed with respect to a reference mineral (quartz) by considering the bond-type in the crystal structure (e.g., bond strength, effect of mass on isotopic substitution). [START_REF] Blanchard | First-principles study of the structural and isotopic properties of Al-and OH-bearing hematite[END_REF] already reported a discrepancy for the oxygen isotope fractionation between hematite and corundum, suggesting that the modified increment method cannot be used to reliably predict isotopic fractionation factors. (triangles) infrared and Raman frequencies [START_REF] Cambier | Infrared study of goethites of varying crystallinity and particle size: I. Interpretation of OH and lattice frequencies[END_REF][START_REF] De Faria | Heated goethite and natural hematite: Can Raman spectroscopy be used to differentiate them?[END_REF]. This highlights the energy gap existing between lattice modes and OH bending modes. (c, d) The force constant is calculated from the pDOS by using the formula

F = M ! 2 g(E)E 2 0
# dE , with the upper limit ! = ! (Eq. 5). The term g(E)E 2 is plotted in (c), while the force constant integral for different values of the integration upper-limit is plotted in (d). frequencies of goethite [START_REF] Cambier | Infrared study of goethites of varying crystallinity and particle size: I. Interpretation of OH and lattice frequencies[END_REF][START_REF] De Faria | Heated goethite and natural hematite: Can Raman spectroscopy be used to differentiate them?[END_REF]. [START_REF] Skulan | Kinetic and equilibrium Fe isotope fractionation between aqueous Fe(III) and hematite[END_REF][START_REF] Welch | Kinetic and equilibrium Fe isotope fractionation between aqueous Fe(II) and Fe(III)[END_REF][START_REF] Syverson | Experimental determination of equilibrium Fe isotopic fractionation between pyrite and dissolved Fe under hydrothermal conditions[END_REF][START_REF] Frierdich | Iron isotope fractionation between aqueous Fe(II) and goethite revisited: New insights based on a multi-direction approach to equilibrium and isotopic exchange rate modification[END_REF]. For Mössbauer, pyrite, hematite and goethite data are taken from [START_REF] Blanchard | Comment on "New data on equilibrium iron isotope fractionation among sulfides: Constraints on mechanisms of sulfide formation in hydrothermal and igneous systems[END_REF], [START_REF] Polyakov | Equilibrium iron isotope fractionation factors of minerals: Reevaluation from the data of nuclear inelastic resonant X-ray scattering and Mössbauer spectroscopy[END_REF] and [START_REF] Polyakov | The use of Mössbauer spectroscopy in stable isotope geochemistry[END_REF], respectively. For GGA and GGA+U, pyrite and hematite data are from [START_REF] Blanchard | Iron isotope fractionation between pyrite (FeS 2 ), hematite (Fe 2 O 3 ) and siderite (FeCO 3 ): a first-principles density-functional theory study[END_REF] while goethite data correspond to this study. For NRIXS, hematite data are from [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF] and goethite data are from this study. No NRIXS iron !-factor is published yet for pyrite. (1991,1998) are shown for comparison, as well as the following experimental data:

Exp. 1 corresponds to data from [START_REF] Bao | Oxygen isotope fractionation in ferric oxide-water systems: Low temperature synthesis[END_REF] where goethite was synthesized at high pH (>14), and Exp. 2 combines the hematite-water isotopic fractionation factors from [START_REF] Bao | Oxygen isotope fractionation in ferric oxide-water systems: Low temperature synthesis[END_REF] with goethite-water data at low pH (~1 to 2) from [START_REF] Yapp | Oxygen isotopes in synthetic goethite and a model for the apparent pH dependence of goethite-water 18 O/ 16 O fractionation[END_REF].

  i.e. 277±15 N/m. The larger error bar reflects the fact that additional sources of uncertainties associated with baseline subtraction and energy scaling are propagated in the calculation of the force constant[START_REF] Hu | Moments in nuclear resonant inelastic x-ray scattering and their applications[END_REF][START_REF] Dauphas | Magma redox and structural controls on iron isotope variations in Earth's mantle and crust[END_REF]. The two new goethite measurements yield force constants of 264±13 N/m and 267±13 N/m, averaging to 268±8 N/m for the three goethite measurements.

Fig. 3 .

 3 Fig. 3.

  iron !-factor of a goethite crystal containing a certain amount of defects would require the accurate characterization of the sample at the molecular scale, which is beyond the scope of the present paper. It is noteworthy that the presence of defective sites is not necessarily expressed in the overall isotopic composition because compensation of local !-factors may occur. For instance,[START_REF] Blanchard | First-principles study of the structural and isotopic properties of Al-and OH-bearing hematite[END_REF] investigated the isotopic properties of hematite with iron vacant sites compensated by protons. The local iron !-factors display variations over a significant range (1.1 ‰ at 0 °C), but extreme values compensate each other, resulting for the model investigated, in a negligible effect of these cationic vacancies on the iron isotope composition of hematite. Similarly, we built a 2 " 1 " 2 supercell of goethite containing one iron vacancy compensated by three protons. The overall 57 Fe/ 54 Fe !-factor is almost unchanged (11.99 ‰ instead of 11.94 ‰ at 0 °C using GGA), even if the local !factors display values ranging from 11.22 ‰ to 13.61 ‰ at 0 °C (the highest value, i.e. 13.61 ‰, corresponds to an iron atom surrounded by two vacancies because of the periodic repetition of the simulation cell).

Figure Captions Figure 1 .

 Captions1 Figure Captions

Figure 2 .

 2 Figure 2. (a) Average measured (NRIXS) and calculated (GGA, GGA+U) phonon

Figure 3 .

 3 Figure 3. Average phonon density of states of iron atoms in hydronium-jarosite and

Figure 4 .

 4 Figure 4. Calculated (GGA and GGA+U) versus experimental infrared and Raman

Figure 5 .

 5 Figure 5. Temperature dependence of the iron !-factor of goethite.

Figure 6 .

 6 Figure 6. Temperature dependence of the oxygen !-factor of goethite and hematite.

Figure 7 .

 7 Figure 7. Temperature dependence of the iron "-factor between pyrite and goethite,

Figure 8 .

 8 Figure 8. Temperature dependence of the oxygen "-factor between hematite and

  Figure 1.

  Figure 7.

Table 1 .

 1 Goethite properties derived from NRIXS, based on the scattering spectrum, S(E), or the projected partial phonon density of states, g(E). 726

		Goethite 1	Goethite 2	Goethite 3	Mean for Goethite
	Parameters from S
	Temperature from detailed balance (K)	287	302	296	295
	Lamb-Mössbauer factor from S	0.7741 ± 0.0026	0.7548 ± 0.0020	0.7604 ± 0.0020	0.7614 ± 0.0012
	Mean square displacement <z 2 > from S (Å 2 )	0.00481 ± 0.00006	0.00528 ± 0.00005	0.00514 ± 0.00004	0.00511 ± 0.00003
	Internal energy/atom from S (meV)	29.00 ± 0.73	29.17 ± 0.67	28.95 ± 0.67	29.04 ± 0.40
	Kinetic energy/atom from S (meV)	14.50 ± 0.37	14.58 ± 0.34	14.48 ± 0.34	14.52 ± 0.20
	Force constant from S (N/m)	277 ± 15	264 ± 13	267 ± 13	268 ± 8
	(without truncation/baseline subtraction)	(319 ± 17)	(250 ± 16)	(260 ± 14)
	56 Fe/ 54 Fe ! coefficients from S
	1000 ln ! =A 1 /T 2 +A 2 /T 4 +A 3 /T 6 (T in K)
	A 1	7.898E+05 ± 4.384E+04 7.537E+05 ± 3.755E+04 7.605E+05 ± 3.724E+04 7.659E+05 ± 2.264E+04
	A 2	-5.074E+09 ± 6.634E+08 -5.382E+09 ± 7.678E+08 -5.474E+09 ± 6.729E+08 -5.301E+09 ± 4.024E+08
	A 3 1000 ln ! =B 1 <F>/T 2 -B 2 <F> 2 /T 4 (T in K)	7.516E+13 ± 1.866E+13 1.154E+14 ± 3.225E+13 1.111E+14 ± 2.430E+13 9.318E+13 ± 1.345E+13
	B 1	2853	2853	2853	2853
	B 2	55630	59269	60154	58351
	Parameters from g
	Lamb-Mössbauer factor from g	0.7740	0.7546	0.7602	0.7629
	Mean square displacement <z 2 > from g (Å 2 )	0.00481	0.00528	0.00515	0.00508
	d<z 2 >/dT (Å 2 /K)	1.45E-05	1.60E-05	1.56E-05	1.54E-05
	Critical temperature (K)	1295	1170	1203	1222
	Resilience (N/m)	95	86	88	90
	Internal energy/atom from g (meV)	29.76	29.92	29.70	29.80
	Kinetic energy/atom from g (meV)	14.88	14.96	14.85	14.90
	Vibrational entropy (kb/atom)	1.00	1.05	1.03	1.03
	Helmholtz free energy (meV)	3.89	2.60	2.86	3.12
	Vibrational specific heat (kb/atom)	0.87	0.89	0.88	0.88
	Lamb-Mössbauer factor at T=0	0.92	0.92	0.92	0.92
	Kinetic energy/atom at T=0 (meV)	7.91	7.68	7.69	7.76
	Force constant from g (N/m)	276	262	266	268

Table 2 .

 2 Hydronium-jarosite and potassium-jarosite properties derived from NRIXS, based on the scattering spectrum, S(E), or the projected 737

	738	partial phonon density of states, g(E).
			H-Jarosite 1	H-Jarosite 2	Mean for H-Jarosite	K-Jarosite 1	K-Jarosite 2	Mean for K-Jarosite
		Parameters from
		S
		Temperature from
		detailed balance
		(K)	300		297		298		302		304		303
		Lamb-Mössbauer
		factor from S	0.6576 ± 0.0023	0.6415 ± 0.0022	0.6492 ± 0.0016	0.6920 ± 0.0016	0.6724 ± 0.0018	0.6832 ± 0.0012
		Mean square
		displacement
		<z 2 > from S (Å 2 )	0.00787 ± 0.00006	0.00833 ± 0.00006	0.00810 ± 0.00004	0.00691 ± 0.00004	0.00745 ± 0.00005	0.00715 ± 0.00003
		Internal
		energy/atom from
		S (meV)	29.58 ± 0.74	29.59 ± 0.74	29.59 ± 0.52	29.70 ± 0.73	29.92 ± 0.73	29.81 ± 0.52
		Kinetic
		energy/atom from
		S (meV)	14.79 ± 0.37	14.80 ± 0.37	14.79 ± 0.26	14.85 ± 0.37	14.96 ± 0.37	14.90 ± 0.26
		Force constant
		from S (N/m)	273 ± 15	289 ± 17	280 ± 11	279 ± 16	305 ± 18	290 ± 12
		(without
		truncation/baselin
		e subtraction)	(316 ± 17)	(241 ± 20)			(281 ± 13)	(294 ± 17)
		56 Fe/ 54 Fe !
		coefficients from
		S
		1000 ln !
		=A 1 /T 2 +A 2 /T 4 +A 3 /T 6 (T in K)
			7.778E+	4.156E+	8.254E+	4.881E+	7.979E+	3.164E+	7.951E+	4.551E+	8.705E+	5.071E+	8.287E+	3.387E+
		A 1	05 ±	04	05 ±	04	05 ±	04	05 ±	04	05 ±	04	05 ±	04
			-		-		-		-		-		-
			4.864E+	5.725E+	7.939E+	1.150E+	5.475E+	5.126E+	5.208E+	9.136E+	8.989E+	1.444E+	6.289E+	7.720E+
		A 2	09 ±	08	09 ±	09	09 ±	08	09 ±	08	09 ±	09	09 ±	08
		A 3	6.032E+ ± 1.438E+	2.269E+ ± 5.105E+	7.257E+ ± 1.384E+	8.178E+ ± 3.445E+	2.941E+ ± 7.686E+	1.173E+ ± 3.143E+

Table 3 .

 3 Theoretical and experimental atomic coordinates [±(x,y,3/4),

	±(1/2+x,1/2+y,1/4)] of goethite. Values in brackets give the differences with respect
	to the experimental values.
		xFe	yFe	xH	yH	xO	yO	xOh	yOh
	GGA	0.0503	0.8523	0.4041	0.0855	0.6978	0.2001	0.1989	0.0528
		(0.001)	(-0.001)	(0.026)	(0.004)	(-0.008)	(0.001)	(0.000)	(0.000)
	GGA+U 0.0603	0.8559	0.4011	0.0940	0.6798	0.1946	0.1959	0.0570
		(0.011)	(0.002)	(0.023)	(0.012)	(-0.026)	(-0.005)	(-0.003)	(0.004)
	Exp.*	0.0489	0.8537	0.3781 0.0817	0.7057	0.1991	0.1987 0.0530
	*Yang et al. (2006)

Table 4 .

 4 Theoretical and experimental interatomic distances (Å). Values in bracketsgive the differences with respect to the experimental values.

		Fe-O	Fe-Oh	O-H
	GGA	1.8963 (-0.037)	2.1091 (0.010)	0.9975 (0.125)
		1.9725 (0.017)	2.1167 (0.011)
	GGA+U	1.9484 (0.015)	2.0958 (-0.004)	1.0130 (0.140)
		1.9889 (0.033)	2.1018 (-0.004)
	Exp.*	1.9332	2.0994	0.8729
		1.9560	2.1059
	*Yang et al. (2006)

Table 5 .

 5 Fits of the calculated 10 3 ln ! based on the function ax + bx 2 + cx 3 , with x =

	10 6 /T 2 (T on K) for 57 Fe/ 54 Fe isotope fractionation in goethite.
		a	b	c
	GGA	9.384 ! 10 -1	-4.7789 ! 10 -3	4.9888 ! 10 -5
	GGA+U	10.248 ! 10 -1	-5.8539 ! 10 -3	6.8122 ! 10 -5

Table 6 .

 6 Fits of the calculated 10 3 ln ! based on the function ax + bx 2 + cx 3 , with x = 10 6 /T 2 (T on K) for 18 O/ 16 O isotope fractionation in goethite and hematite.

	a	b	c

Table 7 .

 7 Mineral-mineral iron isotope fractionations determined from experimental measurements. The extrapolation to higher temperature is made assuming a linear dependence as a function of 10 6 T 2 and a fractionation equal to zero at infinite temperature. ! 56 Fe shown here are subsequently converted in ! 57 Fe for comparison in Fig.6.

	Pyrite-Fe(II) aq	Fe(II) aq -Goethite	Pyrite-Goethite
	Syverson et al. (2013) Frierdich et al. (2014)
	0.99 ± 0.3 ‰	-0.23 ± 0.1 ‰	0.76 ± 0.3 ‰
	at 350 °C	-0.27 ± 0.1 ‰*	0.72 ± 0.3 ‰
		extrapol. to 350 °C	at 350 °C

Hematite-Fe(III) aq Fe(III) aq -Fe(II) aq Fe(II) aq -Goethite Hematite-Goethite

	Skulan et al. (2002)	Welch et al. (2003)	Frierdich et al. (2014)
	0.1 ± 0.2 ‰	1.87 ± 0.2 ‰	-0.66 ± 0.1 ‰	1.31 ± 0.3 ‰
	at 98 °C	extrapol. to 98 °C	-0.77 ± 0.1 ‰*	1.2 ± 0.3 ‰
			extrapol. to 98 °C	at 98 °C
	Pyrite-Fe(II) aq	Fe(II) aq -Fe(III) aq	Fe(III) aq -Hematite	Pyrite-Hematite
	Syverson et al. (2013)	Welch et al. (2003)	Skulan et al. (2002)
	0.99 ± 0.3 ‰	-0.66 ± 0.2 ‰	-0.04 ± 0.2 ‰	0.29 ± 0.4 ‰
	at 350 °C	extrapol. to 350 °C	extrapol. to 350 °C	at 350 °C
	* smaller particles

ACKNOWLEDGMENTS L. Paulatto is acknowledged for his technical support to the computational work. This work was performed using HPC resources from GENCI-IDRIS (Grant 2014-i2014041519). This work has been supported by the French National Research Agency (ANR, projects 11-JS56-001 "CrIMin" and 2011JS56 004 01 "FrIHIDDA"), grants from NSF (EAR 1144429) and NASA (NNX12AH60G). Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357.