# Redox and structural controls on tin isotopic fractionations among magmas

M. Roskosz, Quentin Amet, Caroline Fitoussi, N. Dauphas, Bernard Bourdon,

L. Tissandier, M.Y. Hu, A. Said, A. Alatas, E.E. Alp

► To cite this version:

M. Roskosz, Quentin Amet, Caroline Fitoussi, N. Dauphas, Bernard Bourdon, et al.. Redox and structural controls on tin isotopic fractionations among magmas. Geochimica et Cosmochimica Acta, 2020, 268, pp.42-55. 10.1016/j.gca.2019.09.036. insu-02410406

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Distributed under a Creative Commons Attribution 4.0 International License
Redox and structural controls on tin isotopic fractionations
among magmas

M. Roskosz$^{a,*}$, Q. Amet$^{b}$, C. Fitoussi$^{b}$, N. Dauphas$^{c}$, B. Bourdon$^{b}$, L. Tissandier$^{d}$, M.Y. Hu$^{e}$, A. Said$^{e}$, A. Alatas$^{e}$, E.E. Alp$^{e}$

a IMPMC, CNRS, UMR 7590, Sorbonne Universités, Université Pierre et Marie Curie, Muséum National d'Histoire Naturelle, CP 52, 57 rue Cuvier, Paris F-75231, France.

b Laboratoire de Géologie de Lyon, ENS de Lyon, UCBL and CNRS, 46 Allée d'Italie, Lyon, France.

c Origins Laboratory, Department of the Geophysical Sciences and Enrico Fermi Institute, The University of Chicago, 5734 South Ellis Avenue, Chicago, IL 60637, USA.

d Centre de Recherches Pétrographiques et Géochimiques (CRPG), Université de Lorraine, CNRS, UMR 7358, 15 Rue Notre Dame des Pauvres, F-54501 Vandoeuvre-lès-Nancy, France.

e Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA.

<footnote>* corresponding author: Mathieu.roskosz@mnhn.fr</footnote>
44 Abstract

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Recent analytical developments have made possible the determination of the isotopic composition of tin in igneous rocks. In order to establish a framework to interpret the mass-dependent tin isotopic signatures of planetary materials, seven geologically-relevant silicate glasses (basalt, rhyolite, enstatite and anorthite glasses) were synthesized with moderate amounts of ¹¹⁹Sn (on the order of a weight percent). Redox conditions were controlled during sample synthesis to set the redox ratio (Sn²⁺/Snₜot) from stannous (Sn²⁺) to stannic (Sn⁴⁺) glasses. The mean force constants of tin bonds in these glasses were determined by synchrotron nuclear resonant inelastic X-ray scattering (NRIXS) in order to determine the reduced isotopic partition function ratios (β-factors) of these glasses. Clues on the coordination chemistry and the valence state of tin in these glasses were also derived from synchrotron Mossbauer spectroscopy (SMS). The force constants of tin drastically increases from Sn²⁺-bearing to Sn⁴⁺-bearing glasses and varies significantly with the glass composition at a given redox state. The average coordination number of tin likely controls these variations with glass composition as suggested by SMS results. It is concluded that large isotope fractionation is expected between materials containing Sn²⁺ and Sn⁴⁺ respectively even at magmatic temperatures and that the coordination chemistry of tin in silicates strongly affect its isotope partitioning behavior. Our experimental data are finally used to interpret available Sn isotope data collected in terrestrial rocks. The incompatible behavior of Sn⁴⁺ in mantle minerals leads to an enrichment in heavy isotopes in mantle melts and to the depletion in heavy isotopes in solid residues of melting with a magnitude consistent with the isotope fractionation between Sn²⁺ and Sn⁴⁺ predicted by NRIXS data. Finally, we show that during fractional crystallization of basalt and considering the effect of tin coordination number in minerals and melts, the partitioning of Sn⁴⁺ into ilmenite leads to an enrichment in light isotopes in the residual melt.

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## 1. Introduction

Terrestrial planets are differentiated into metallic cores, and silicate mantles and crusts. The conditions that prevailed during planetary differentiation are difficult to establish because core formation is a protracted phenomenon that occurs concurrently with planetary growth such that core formation must take place over a range of conditions. Furthermore, only the very top layers of terrestrial planets are available for sampling, which have been tampered by several billions of years of geochemical evolution. Therefore, understanding the conditions (P, T, fO₂) that prevailed during planetary differentiation remains a major question in Earth sciences and planetology. An additional complication is that some siderophile elements are also volatile and have been partially lost during planetary accretion.

The behaviors of some moderately siderophile elements are influenced by the redox state of the source region during partial melting. For such elements, it is difficult to tell whether their low abundance in samples of planetary mantles reflects the fact that they have been scavenged into the core (e.g. Righter, 2011), redistributed by mantle processes (e.g. McDonough, 2003; Palme and O'Neill, 2003), or lost by vaporization or incomplete condensation, either during planetary collisions (Hin et al. 2017; Norris and Wood, 2017; Young et al., 2019) or solar nebula condensation (Cassen, 1996).

In the past decade, new insights into core formation conditions have been gained from the study of non-traditional isotope fractionation among samples from Earth's mantle (see Bourdon et al., 2018 for a review on this topic). Specifically, the isotopic compositions of Si, Cu and Fe of mantle-derived rocks have been interpreted to reflect the imprint of core formation (Fitoussi et al. 2009; Shahar et al., 2011; Hin et al., 2014; Savage et al., 2015; Polyakov, 2009). Alternative interpretations invoking volatility-controlled processes were also put forward, for instance in the case of Fe, Si and Mg (Poitrasson et al., 2004; Dauphas et al. 2015; Hin et al., 2017). The effect of partial melting, magma differentiation and redox state of the source region have also been pointed out in the case of Fe (Dauphas et al. 2009). To interpret natural isotopic variations in a quantitative framework, it is important to develop an accurate description of equilibrium isotopic fractionation between coexisting phases (namely, iron-based metal alloys, silicates, sulfides and gaseous species in particular). In this context, we have initiated a comprehensive experimental study of the isotope partitioning of tin, a versatile element that may either be siderophile, chalcophile, lithophile,
volatile, compatible and incompatible depending on the thermodynamical conditions prevailing in the reservoirs of interest (e.g., Heinrich, 1990; Witt-Eickschen et al., 2009).

Recent analytical developments have made possible the determination of the isotopic composition of tin in igneous rocks containing minute amounts of tin (Badullovich et al. 2017; Wang et al. 2018). As a moderately volatile element with a 50% condensation temperature of 704 K (Lodders, 2003), tin is depleted in the Earth's primitive mantle by a factor of 33±3 compared with Cl chondrites (Jochum et al., 1993). Therefore, Sn isotopes have the potential to carry important clues on the origin of volatile element depletion in Solar System materials (e.g., Davis and Richter, 2003; O'Neill and Palme, 2008; Norris and Wood, 2017). Tin has three oxidation states (Sn⁰, Sn²⁺, Sn⁴⁺) so its geochemical behavior is strongly influenced by oxygen fugacity. Since isotopic fractionation is controlled by differences in bond stiffness between two phases (e.g., Schauble, 2004; Dauphas et al. 2012), the diversity in Sn bonding environments could induce significant isotopic fractionation. In order to establish a framework to interpret the tin isotopic signatures of planetary materials, the effects of major igneous processes have first to be quantified.

The manner in which isotopes are fractionated between coexisting phases at equilibrium (β-factor or reduced partition function ratio) can in principle be determined experimentally (Bourdon et al., 2018) but it requires achieving equilibrium and measuring the isotopic compositions of coexisting phases with high precision, which is challenging at the high pressures and temperatures relevant to planetary differentiation (again see Bourdon et al., 2018 for a critical analysis of experimental results collected so far). Further complications arise, in the case of tin because of its high volatility that prevent experiments to be performed in a truly closed system, which can produce large isotopic fractionation that is difficult to control or account for (Bourdon et al., 2018). In the case of iron and tin, however, experimental difficulties can be overcome if the fractionation factors are measured using a synchrotron-based inelastic spectroscopy (Polyakov et al., 2005; Dauphas 2012, 2018). Over the past several years, this technique has seen considerable development (Dauphas et al., 2012, 2014; Blanchard et al., 2015; Dauphas et al., 2018) and it has already proven to be useful to address several first-order problems in iron isotope geochemistry (Dauphas et al., 2014; Roskosz et al., 2015; Shahar et al., 2016; Liu et al., 2017). The determination of these factors is achieved by measuring the phonon excitations using inelastic x-ray scattering on Mössbauer nuclei (Hu et al., 1999, and Hu et al. 2013). From this
phonon excitation, the mean force constants of the chemical bonds and projected partial phonon density of states (pDOS) can be calculated. These quantities can then be used to derive important properties such as specific heat, vibrational entropy, sound velocity, free energy and isotopic fractionation factors even under extreme P, T conditions (Hu et al., 2003; 2013). This technique is called nuclear resonant inelastic X-ray scattering (NRIXS). The use of the pDOS to derive isotope fractionation factors was initially proposed by Polyakov using available data on <sup>119</sup>Sn nuclear resonances (Polyakov et al., 2005; also see Dauphas et al. 2018 for a reevaluation of those results) before it became more extensively used with <sup>57</sup>Fe. A major result of this pioneering study was to show that the redox state of tin (Sn<sup>0</sup>, Sn<sup>2+</sup>, Sn<sup>4+</sup>) and the changes in coordination chemistry induced by a valence change strongly affect the partitioning of the Sn isotopes. This conclusion was reached based on the behavior of pure compounds yet (metal and oxides, namely pure Sn, SnO and SnO<sub>2</sub>).

In this study we extend the range of Sn-bearing materials to more geologically-relevant silicate glasses synthesized with moderate amounts of tin (on the order of a weight percent). Redox conditions were controlled during sample synthesis to set the redox ratio (Sn<sup>2+</sup>/Sn<sub>tot</sub>) from stannous (Sn<sup>2+</sup>) to stannic (Sn<sup>4+</sup>) glasses. We also improved the data collection and processing using the recent developments on <sup>57</sup>Fe NRIXS. We show that large isotope fractionation has to be expected between materials containing Sn<sup>2+</sup> and Sn<sup>4+</sup>. We also demonstrate that at a given redox state, the coordination chemistry of tin in silicates must strongly affect its partitioning behavior. Our experimental data are finally used to interpret available Sn isotope data collected on mantle-derived rocks.

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## 2. Experimental methods

### 2.1 Sample synthesis and characterization

The starting glasses were the same as those studied by Dauphas et al. (2014). They were synthesized from mixtures of reagent grade $SiO_2$, $Al_2O_3$, $CaCO_3$, $MgO$, $Na_2CO_3$, $K_2CO_3$, $TiO_2$, and $Fe_2O_3$ that had been first fired in air from 300 to 1000 °C for several days. The starting material was melted in air at 1550 °C for 1 hour in a thin-walled Pt crucible, quenched on a copper plate, and then finely crushed. Approximately 2 wt% of pure $^{119}$SnO$_2$ (Cyclotron Instruments) was then added to the starting glass powders and crushed in ethanol for several minutes. The reason for that is $^{119}$Sn is the only Mossbauer isotope of Sn and working on enriched samples cuts on acquisition time. Two methodologies were used to prepare the samples depending on the oxygen fugacity that we wanted to achieve.

(1) Oxidized samples (labeled AIR) were prepared in a furnace in air. Glass droplets of approximately 25 mg were mounted (Ø ≈ 2 mm) on platinum wire loops. Glass powders were attached to wire loops using polyvinyl alcohol as a binder and were melted in air at 1550 °C for 10 min. A longer duration was found to induce a complete volatilization of tin. After 10 minutes, enough tin was preserved in glasses for NRIXS analysis (Table 1). Note that NRIXS is a method that probes the strength of the Sn bonds, so the fact that Sn isotopes could have been fractionated during sample preparation is inconsequential.

(2) Reduced glasses (labeled CCO) were prepared at the LMV (Clermont-Ferrand, France) in a non-endloaded piston-cylinder apparatus. Double capsules (platinum and graphite) were used to ensure buffered conditions slightly lower than the C/CO redox equilibrium (Medard et al., 2008) and to avoid any significant loss of tin during experiments. About 80 mg of the $SnO_2$-bearing glass powder was packed in a graphite capsule, which was then placed in a larger welded platinum capsule. A salt/pyrex/MgO assembly as described in Médard et al. (2006) was used to perform these syntheses. The reduced basalt and rhyolite were prepared at 1 GPa and 1550°C for 46 and 55 min respectively.

The chemical compositions of glasses were determined with an electron microprobe (Cameca SX 100 at the UMET laboratory; University of Lille) operating at 15 kV and 15 nA beam current. Albite, $MgO$, $Fe_2O_3$, $SnO_2$, orthose, wollastonite, $MnTiO_3$, Fe, Ni and Sn
190 standards were typically used to calibrate the microprobe. The SEM observations of the
191 silicate samples did not reveal any crystallization or heterogeneity.

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## 2.2 NRIXS data collection and processing

194 The NRIXS measurements were performed on Sector 30-ID-B at the Advanced Photon
195 Source (APS) at Argonne National Laboratory (USA). Details on the NRIXS technique are
196 available in Seto et al. (1995), Sturhahn et al. (1995), Chumakov and Sturhahn (1999), and
197 Dauphas et al. (2012, 2018). The storage ring was in the standard 24 bunch top-up mode of
198 APS. The phonon excitation spectrum is recorded by scanning the incident x-ray energy
199 (from -100 to +120 meV with 0.25 meV steps) with the high-resolution six-crystal,
200 cryogenically cooled monochromator (Toellner et al, 2011). The flux after the high-resolution
201 monochromator is measured to be 4 GHz/1 meV. Nuclear scattering was measured on small
glass chips using two Avalanche Photon Diode (APD) detectors positioned face to face and
202 perpendicular to the incident X-ray beam in order to optimize the signal collection. The
203 measurements were performed at room pressure and temperature (~298 K) with a
204 collection time of 2 seconds per step. The SciPhon software (version 1.0.4) was used for the
205 data reduction together with PHOENIX (Sturhahn, 2000). SciPhon is described in Dauphas et
206 al. (2018). It allows the user to visually remove the baseline and elastic peak.

208 NRIXS has been demonstrated to be a robust technique to characterize isotopic
209 fractionation among coexisting phases through the determination of the reduced partition
210 function ratio (so called the β factors). It was previously applied to Fe and Sn (Polyakov et al.,
211 2005; Dauphas et al., 2012; 2014; 2018; Roskosz et al., 2015, Shahar et al., 2016). Typically, β
212 can be approximated by:

$$1000 \times \ln \beta_{I/I*} = \frac{A_1}{T^2} + \frac{A_2}{T^4} + \frac{A_3}{T^6} \quad (1)$$

214 where *I* and *I*\* are two tin isotopes, *T* is temperature in K and *A*₁, *A*₂, *A*₃ are three parameters
215 that can either be calculated from the moment of the phonon excitation probability density
216 function *S*(*E*) (Dauphas et al. 2012; Hu et al. 2013) or from the moment of the projected
217 partial density of state, *g*(*E*) (Polyakov et al., 2005; Dauphas et al. 2012; Hu et al. 2013). At
218 high temperature, β factors can be calculated from the mean force constant *<F>* of the
219 bonds holding tin in position:

$$1000 \times \ln \beta_{I/I*} = 1000 \left( \frac{1}{M^*} - \frac{1}{M} \right) \frac{\hbar^2 \langle F \rangle}{8k^2 T^2} \quad (2)$$
where M and M* are the mass of two isotopes I and I*, $\hbar$ is the reduced Planck constant and k is the Boltzmann constant. For the ¹²²Sn/¹⁶⁶Sn ratio usually used to report Sn isotopic analyses, this formula takes the form,

$$1000 \times \ln \beta_{I/I*} = \frac{1865.6\langle F \rangle}{T^2}. \qquad (3)$$

The force constant can in turn either be deduced from g(E) or from S(E) because it is proportional to the third central moment of S(E):

$$\langle F \rangle = \frac{M}{\hbar} \int_{0}^{+\infty} E^2 g(E) dE = \frac{M}{\hbar} \int_{-\infty}^{+\infty} (E - E_R)^3 S(E) dE \qquad (4)$$

where E_R is the free recoil energy and M the mass of ¹¹⁹Sn. The equilibrium isotope fractionation coefficient $\alpha_{A-B}$ between two phases A and B can then be calculated as the ratio of their β factors:

$$1000 \times \ln \alpha_{A-B} = 1000 \times \ln \beta^A - 1000 \times \ln \beta^B \qquad (5)$$

## 2.3 SMS data collection and processing

Due to the relatively long lifetime of radioactive parent ¹¹⁹mSn (245 days) single line source embedded in CaSnO₃ or SnO₂, ¹¹⁹Sn Mossbauer spectroscopy is widely used and is the second most popular spectroscopy after ⁵⁷Fe. The lifetime of the first excited state is 18.3 ns and is long enough to be also observed upon excitation by a synchrotron pulse (Alp et al., 1993). The resonance energy is 23.880 keV, and the transition is between ground state of spin 1/2 to excited state of spin 3/2. Despite the high transition energy compared to iron, the recoil free fraction to observe the Mossbauer effect is relatively large.

In order to determine the approximate valence state of Sn in different glasses, we used time-domain Synchrotron Mössbauer Spectroscopy (SMS) at beamline sector 30 (APS, USA). We measured two data sets: a first with glass samples of interest only, and a second set where different reference samples were placed together with the silicate glasses of interest. This procedure allows us to determine the relative isomer shift of the sample against the known value of the reference sample (Alp et al., 1995). Typical SMS measurements take from half an hour to a few hours of collection time. The main difficulty is to find the right thickness of the reference material so that the sample and the reference material have the same effective thickness defined as:

$$t_{eff} = f_{LM} \eta \sigma_0 d, \qquad (6)$$
where $f_{LM}$ is the recoil-free fraction of the Lamb-Mössbauer factor, $\eta$ is the number of
Mössbauer nuclei per unit volume, $\sigma_0$ is nuclear resonant cross-section (1.4 x 10$^{-18}$ cm$^2$ for
$^{119}$Sn) and $d$ is physical thickness of the absorber. The data analysis was performed using the
CONUSS software (Sturhan, 2000).

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259 **3. Results**

260

261 **3.1 Partial phonon density of states**

262

263 The partial phonon density of states $g(E)$ were derived from the phonon excitation
264 probability density function $S(E)$ measured by NRIXS, (Fig.1). The $S(E)$ previously collected on
265 $SnO_2$ by Hu et al., (1999) and on SnO by Giefers et al., (2006) were recently processed with
266 Sciphon, published elsewhere (see Dauphas et al., 2018 for an extensive discussion) and are
267 reproduced here. Systematic changes as a function of the valence state for a given glass
268 composition are expected. Reduced glasses exhibit a sharp and intense phonon peak at
269 about 10 meV followed by a featureless tail extending to 80 meV (Fig.1). Oxidized glasses
270 show a broader first phonon peak at slightly higher energy (around 15 meV) associated with
271 a second peak at 70 meV. In addition, the composition of the glass significantly affects $g(E)$.
272 This is particularly true in oxidized samples where the first phonon peak is found at lower
273 energy for rhyolite than for basalt (Fig. 1c). As discussed below, this feature has a significant
274 effect on the Sn isotope fractionation factors. Another important observation is that the
275 shape of $g(E)$ in glasses is relatively smooth compared to pure SnO and $SnO_2$ spectra (Fig.1a).
276 This is a clear indication that tin is readily accommodated in the glass structure and does not
277 form small crystallites of pure tin oxide or metal nuggets.

278

279 **3.2 Tin valence state in silicate glasses**

280 The isomer shift (IS) of tin dissolved in silicate glasses were extracted from the fit to
281 the Synchrotron Mössbauer spectra (SMS, Fig.2). From these IS (Table 2), the valence state
282 of the glass samples was estimated by comparison with existing data also compiled in Table
283 2. We note here that the SMS is less accurate and straightforward than conventional Sn
284 Mossbauer in determining the redox ratio of tin-bearing silicates. Nonetheless, it appears
285 that the glasses prepared in air have IS indistinguishable from the $SnO_2$ reference material. It
286 is thus reasonable to assume that these glasses only contain tetravalent tin (stannous tin,
287 $Sn^{4+}$). This result is consistent with the conventional Mössbauer study by Sitek et al. (1981),
288 which revealed the presence of only small amounts, restricted to less than 5%, of stannous
289 tin in granitic liquids prepared in air at 1600°C. This is also consistent with the fact that no
290 stannous tin was detected in granitic melts prepared at the magnetite/haematite fugacity
buffer (Durasova et al., 1997) and with the fact that such granitic glasses were systematically found to contain more stannous tin than basaltic glasses at a given oxygen fugacity (Durasova et al., 1984). The coordination chemistry of tin in these samples may be different though as suggested by the slightly different values of IS. Turning to reduced samples, the IS of the rhyolite is very similar to the IS of pure SnO, which strongly suggests that this sample contains only divalent tin (stannous tin). Finally, the IS of the reduced basalt is intermediate between the IS of SnO and SnO₂. This might be an indication that some Sn⁴⁺ is present in this glass. However, this glass, just as the reduced rhyolite, was prepared at high pressure and temperature in a graphite capsule. In these conditions, it is unlikely that significant amount of Sn⁴⁺ could be stabilized. Based on Ellingham diagrams (Lehmann, 2006), the Sn⁴⁺/Sn²⁺ equilibrium (defined from data on sodium silicate glasses studied by Johnston, 1965) and the SnO/SnO₂ equilibrium at 1550°C are found at oxygen fugacities several orders of magnitudes higher than the CCO buffer. Therefore, just as these glasses prepared in graphite capsules, do not contain measurable amounts of ferric iron (Roskosz et al., 2018), we can reasonably assume that the reduced basalt cannot contain measurable amount of stannic tin. This view is supported by previous conventional Mössbauer data suggesting that at oxygen fugacities lower than the Ni-NiO buffer no more than 5% of Sn⁴⁺ were found in granitic and basaltic glasses (Durasova et al., 1984, 1986, 1997). Consequently, the slightly different IS measured for basalt may suggest that Sn²⁺ dissolved in the basalt has a different coordination chemistry compared with tin present in the rhyolite. More experiments, involving tin conventional Mössbauer spectroscopy should be carried out in the future to confirm this interpretation.

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### 3.3 Tin force constants in silicate glasses

Table 2 compiles the valence state determined by SMS and the mean force constant $<F>$ measured by NRIXS. A more extensive compilation of physical properties calculated from the NRIXS spectra is also provided in Table 3. It includes estimates of the Lamb-Mössbauer factor, kinetic energy per atom, force constant, mean force constant at T=0 K, kinetic energy at T=0 K, vibrational specific heat, vibrational entropy, and critical temperature. Because the present paper is concerned with high temperature isotopic fractionation where 1000 lnβ scales as 1/T², only the force constant given by the 3rd moment of S(E) is discussed below but calculations at lower T can be done using parameters provided in Table 3.
The force constant of tin is a strong function of the valence state and of the composition of the silicate melts (Fig. 3). The derived values are also significantly different from those collected elsewhere for the pure compounds (Sn, SnO and SnO₂, see Dauphas et al., 2018). The force constant of oxidized glasses <F> is typically 200-250 N/m higher than those of the reduced glasses (for basalt and anorthite, it is twice higher). Whereas reduced basalts and rhyolites have similar force constants, which could be expected from the very similar pDOS of these glasses (Fig.3), the force constant of the reduced anorthite is significantly lower (by 50 N/m). The reduced anorthite glass has a force constant similar to pure SnO (Giefers et al., 2006), whereas the basalt and rhyolite have values higher than SnO. As tentatively discussed below, this is likely due to slightly different coordination of stannous tin in these glasses. Turning to oxidized samples, most glasses have force constants close to pure SnO₂, though generally slightly higher than this pure oxide. The force constant of rhyolitic glass is however lower by almost 100 N/m than that of pure SnO₂. This is related to the position of the first intense phonon peak of the oxidized rhyolite, which is measured at lower energy than for other samples (Fig. 3).

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# 4. Discussion

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## 4.1 Redox- and structural dependence of β-factors of Sn-bearing phases

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For all studied samples, $1000 \times \ln \beta_{122/116Sn}$ was calculated using the formalism detailed in Dauphas et al. (2012, 2018) with coefficients extracted from NRIXS data by the Sciphon software. Additional data collected on the same beamline for Sn, SnO and SnO$_2$ were included. These data represent an updated version of those used in the earlier work of Polyakov et al. (2005). For SnO and SnO$_2$, small differences (Table 2) are found as compared to the initial evaluation of Polyakov et al. (2009 and 413 N/m for SnO and SnO$_2$ respectively) and the reprocessing performed by Dauphas et al. (2018) and given in table 2. As emphasized by Dauphas et al. (2014) and Blanchard et al. (2015), great care must be exercised in handling the low and high-energy tails of NRIXS spectra for application to isotope geochemistry. Furthermore, Roskosz et al., (2015) demonstrated that such differences do not affect the equilibrium fractionation factors provided that self-consistent datasets are used to calculate them. In this sense, the present dataset should supersede original estimates of Polyakov et al. (2005) at least for applications to mantle and volcanic contexts because our samples are geochemically more relevant in these cases.

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The bond stiffness controls the isotope fractionation between coexisting silicate and oxide phases and it is well accepted that the higher the valence state, the stiffer the bonds. However, the trend of $<F>$ as a function of the valence state is not linear. While $<F>$ is very close for $Sn^0$ and $Sn^{2+}$, it is much higher for $Sn^{4+}$. The change in the valence state from metallic to ionic bonds does not necessarily involve significant changes in bond stiffness. This feature has already been observed for other isotopic systems. While the difference in $<F>$ between $Fe^0$ and $Fe^{2+}$ is small, the contrast in $<F>$ is higher between ferrous and ferric iron. This result implies that no significant Fe isotopes fractionation is expected at magmatic temperatures between metal and silicates (Poitrasson et al., 2009, Hin et al., 2012, Dauphas et al., 2014, Liu et al., 2017). A similar result was also found experimentally on Ni, for which no significant Ni isotope fractionation could be experimentally observed during metal-silicate and sulfide-silicate equilibrations at magmatic conditions (Lazar et al., 2012; Guignard et al., 2019). The increase of $<F>$ with the valence state at constant chemical composition also appears larger from $Sn^{2+}$ to $Sn^{4+}$-bearing glasses than those documented
between Fe²⁺- and Fe³⁺-bearing glasses (Dauphas et al., 2014). However, the force constants seem to exhibit the same trend as a function of the oxidation state of the cation of interest. From a purely Fe²⁺-bearing basalt to a purely Fe³⁺-bearing basalt <F> increases from 200 N/m to 350 N/m. From Sn²⁺-bearing basalt to Sn⁴⁺-bearing basalt, <F> increases from 250 N/m to 530 N/m. But this increase is comparable (~140-150 N/m) if it is expressed per valence unit (i.e. from (II) to (III) for Fe and from (II) to (IV) for tin). Therefore, the higher β-factors for tin isotopes as compared to Fe isotopes for a given glass composition reflects primarily the larger relative mass difference between the tin and iron isotopes respectively involved (Δm/m = 0.05 for ¹²²Sn/¹¹⁶Sn against 0.036 for ⁵⁶Fe/⁵⁴Fe).

In addition to the effect of the valence state, the coordination chemistry of cations in melts and minerals also plays an important control on the bond stiffness. At constant valence state, our data indicates that Sn⁴⁺-bearing basalt should be significantly enriched in heavy tin isotopes as compared to rhyolitic melts (by up to 0.2 ‰ at 1000K). By comparison, such a contrast was not observed for iron isotopes between ferric basalts and rhyolites, but a significant fractionation was resolved between ferrous rhyolites and basalts -rhyolites being enriched in heavy iron isotopes- (Dauphas et al., 2014). Similarly, the reduced anorthite glass shows a significantly lower <F> than other reduced glasses. This points to the entangled effects of the valence state and coordination chemistry. The coordination chemistry of stannous and stannic tin has been documented on pure minerals and glasses (e.g. Kossiakoff and Leavens, 1976; Seki et al., 1984; Giefers et al., 2007; Farges et al., 2006). It was found that Sn⁴⁺-bearing glasses have a typical EXFAS/XANES spectra very close to the aekerite (Ca₂SnAl₂Si₆O₁₈(OH)2.2H₂O), where tin is in octahedral position, while the environment of Sn²⁺ is closer to the romarchite (SnO), where tin is eight-fold coordinated. In details, the bond length Sn-O in oxidized glasses studied by Farges et al., (2006) was found to be slightly shorter than in crystalline cassiterite (though in both cases Sn is six-fold coordinated), which is consistent with the slightly higher <F> of most oxidized glasses studied here as compared to pure SnO₂ and with the fact that the higher the IS of the oxidized glass, the higher <F> (table2). Along the same line, 'Rhyolite Air' exhibits a mean force constant that is significantly lower than other samples and SnO₂. This must therefore point out either to a higher coordination number of Sn in this glass or to a longer Sn-O bond. Similarly, the isomer shift measured by Mössbauer spectroscopy is also lower for rhyolitic glass than it is for basaltic glass and cassiterite (Table 2). These observations could again
reflect a change in coordination of Sn⁴⁺, as argued by Johnson et al. (2005) whereby a higher coordination for Sn⁴⁺ in rhyolite would correspond to the lower isomer shift (-0.29 mm/s). The difference with the observations of Farges et al. (2006) who inferred a constant six-coordination of Sn(IV) may stem from the difference in melt composition. Turning to reduced glasses, no significant differences was found between rhyolitic and basaltic glasses. They both have force constants significantly higher than pure SnO, pointing out to a shorter Sn-O bond in glasses. Farges et al. (2006) demonstrated that Sn²⁺ coordination strongly depends on the aluminous content in haplogranitic melts. It was found that in peralkaline domain, the mean coordination number was higher than in metaluminous and peraluminous domains. In this context the lower force constant of anorthite could be explained by its metaluminous nature, whereas the peralkaline nature of other melts tends to lower the average Sn-O bond length in other glasses.

As a final remark, we note that no large (yet resolved) difference is found between the β-factors of pure α-Sn and SnO-bearing silicates (Fig.3). In other words, our results would suggest at first sight that no measurable tin isotopes fractionation during core-mantle differentiation should be analytically resolved (ΔSn<sub>M-S</sub> ~ -0.03‰). A detailed study of tin-bearing, geochemically relevant metal alloys at high pressure must however be carried out to confirm this preliminary conclusion.

422

423 4.3 Tin isotope fractionation during mantle melting

424

425 In the upper mantle, tin should exist both as Sn²⁺and Sn⁴⁺ (Durasova et al. 1986). As discussed before, owing to their different ionic radius and charge, Sn²⁺ and Sn⁴⁺ behave very distinctly during magmatic processes. During partial melting, as Sn⁴⁺ is more incompatible than Sn²⁺, Sn⁴⁺is enriched in the melt relative to the residual solid. The measurements of Sn force constant in the ‘basalt Air’ and ‘basalt CCO’ samples show that heavy isotopes will preferentially be associated with the high oxidation state (i.e. Sn⁴⁺), which is consistent with isotope fractionation theory (Schauble 2004). Hence, heavy Sn isotopes will preferentially be released in the melt during partial melting of a peridotite. Wang et al. (2018) measured the Sn isotopic composition of basalts and peridotites and found that peridotites show an enrichment in light Sn isotopes relative to basalts. This observation is best explained by the

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432

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434
higher incompatibility of Sn⁴⁺ relative to Sn²⁺ during partial melting which produces an enrichment of Sn⁴⁺ in the liquid. Because Sn⁴⁺ is associated with a high force constant, high δ¹²⁴Sn values (the ‰ deviation of the ¹²⁴Sn/¹¹⁶Sn relative to the NIST SRM 3161a standard) are observed in the liquid and the solid residue is enriched in light isotopes. Wang et al. (2018) also showed that the Sn isotope composition of peridotites becomes isotopically lighter as the degree of melting increases. Furthermore, they modeled the effect of partial melting using non-modal batch melting to interpret the Sn isotope measurements and found that the isotopic composition of peridotites could be explained by partial melting at ~3GPa and an isotope fractionation factor Δ¹²⁴Sn between Sn⁴⁺ and Sn²⁺ equal to 1.2‰ at 1200°C, where this latter value was fit to match the observations.

Here we have reexamined this data set in light of the NRIXS measurements of glasses. The Sn isotope fractionation factor between Sn⁴⁺ and Sn²⁺ determined with the NRIXS data is equal to 0.27‰ for a temperature of 1250 °C. Using the model derived in Wang et al. (2018) the calculated δ¹²⁴Sn of residual peridotites reaches a minimum value of -0.21‰, which is significantly higher than what is observed for an Sn depletion of 70% relative to the primitive mantle (Sn=180 ppb). This means that in light of the new NRIXS data, the non-modal batch-melting model presented in Wang et al. (2018) is not consistent with the Sn isotope data. For this purpose, based on the new value of Δ¹²⁴Sn between Sn⁴⁺ and Sn²⁺, we have developed a more sophisticated model based on fractional melting that is consistent with the new Sn isotope fractionation factors. We have modified the fractional melting model of Dauphas et al. (2009) for predicting the Fe isotope fractionation during partial melting (this model assumes that the Sn⁴⁺/Sn²⁺ ratio in each phase has a fixed value set by the redox state of Fe). This isotope system is similar to tin as there is an equilibrium isotopic fractionation between Fe³⁺ (incompatible) and Fe²⁺ (compatible). The new equations for Sn²⁺ and Sn⁴⁺ can be written:

$$460 \qquad \delta_s = \delta_0 + \ln(1-f) \frac{(Sn(IV)/Sn(II))_0}{[1+(Sn(IV)/Sn(II))_0]^2} (K_{Sn(IV)} - K_{Sn(II)}) \Delta_{Sn(II)}^{Sn(IV)} \qquad (7)$$

$$461 \qquad \delta_l = \delta_0 - \frac{(1-f)^{\frac{[K_{Sn(II)} + K_{Sn(IV)}(Sn(IV)/Sn(II))_0]}{[1+(Sn(IV)/Sn(II))_0]}}}{1-(1-f)^{\frac{[K_{Sn(II)} + K_{Sn(IV)}(Sn(IV)/Sn(II))_0]}{[1+(Sn(IV)/Sn(II))_0]}}} \ln(1-f) \frac{(Sn(IV)/Sn(II))_0}{[1+(Sn(IV)/Sn(II))_0]^2} (K_{Sn(IV)} - K_{Sn(II)}) \Delta_{Sn(II)}^{Sn(IV)} \qquad (8)$$

where,
$$
\begin{align}
463 \quad K_{Sn(IV)} &= [Sn(IV)]_l / [Sn(IV)]_s, \quad K_{Sn(II)} = [Sn(II)]_l / [Sn(II)]_s \\
464 \quad \Delta_{Sn(II)}^{Sn(IV)} &= \ln(\alpha_{Sn(IV)-Sn(II)}) \times 10^3 \tag{9}
\end{align}
$$

465 $\delta_s$ and $\delta_l$ are respectively the isotopic composition of the solid and the melt, $f$ is the degree
466 of melting, $(Sn(IV)/Sn(II))_0$ is the initial relative abundance of $Sn^{4+}$ and $Sn^{2+}$. With the
467 force constant calculated for the 'Basalt Air' and 'Basalt CCO' samples in hands, it is possible
468 to estimate the $\Delta_{Sn(II)}^{Sn(IV)}$ as a function of temperature. In our model, we assume a melting of a
469 peridotite consisting of 15% clinopyroxene, 25% orthopyroxene, 60% olivine. Clinopyroxene
470 melts first and after 15% of melting the solid residue has 30% orthopyroxene and 70%
471 olivine. Finally, at 30% of melting, the solid residue is only composed of olivine. The
472 parameters used in the model (degree of melting of peridotites and partition coefficients of
473 $Sn^{2+}$ and $Sn^{4+}$) are taken from Wang et al. (2018). In that study, the pressure was shown to
474 have a major influence on the Sn isotope fractionation during partial melting. Indeed, at
475 higher pressures, $Sn^{4+}$ becomes more incompatible ($D_{cpx}^{Sn(IV)} \approx 10^{-2}$) while $Sn^{2+}$ becomes more
476 compatible ($D_{cpx}^{Sn(II)} \approx 1$). Since $Sn^{4+}$ is enriched in heavy isotopes compared with $Sn^{2+}$, the net
477 consequence of an increasing pressure is to produce a more pronounced depletion in light
478 Sn isotopes in residual peridotites. With an increasing degree of melting, $Sn^{4+}$ is released in
479 the liquid, which leads to a depletion in heavy isotopes in the solid residue (Figure 5). At
480 equilibrium, $\Delta_{Sn(II)}^{Sn(IV)}$ decreases as temperature increases. This explains why partial melting at
481 1250°C produces lower $\delta^{124}Sn$ values in the solid. Using the newly determined force
482 constants for $Sn^{2+}$ and $Sn^{4+}$, the Sn isotopic signature for peridotites is best explained by a
483 fractional melting at 3GPa around 1350°C-1450°C which is consistent with melting
484 conditions under such pressure (Dasgupta et al. 2007, Mallman and O'Neill 2009). We
485 considered the partial melting of a spinel peridotite. Note here that the melting of a
486 peridotite in the garnet stability field would produce little to no Sn isotope fractionation
487 between the liquid and the solid since both Sn(II) and Sn(IV) are compatible in the garnet
488 (Wang et al. 2018). Therefore, the presence of garnet in the source would be inconsistent
489 with data from Wang et al. (2018). Figure 6 shows that the liquid is enriched in heavy
490 isotopes due to a higher $Sn^{4+}$ abundance in the melt. The $Sn^{4+}$ dilution during partial melting
491 induces a decrease of $\delta^{124}Sn$. Similarly to Figure 5, a decrease in temperature leads to a
492 higher isotopic fractionation relative to the Bulk Silicate Earth (BSE). Conversely, an increase
of pressure does not significantly influence the magnitude of the isotopic fractionation but mostly affects the value of partition coefficients. As shown in Wang et al. (2018), at higher pressures, Sn⁴⁺ which is located in the M1 site becomes more incompatible in clinopyroxene, thereby modifying the overall compatibility of Sn during melting. As the majority of Sn is in the form of Sn⁴⁺ in the upper mantle, most of the tin budget will be in the melt (Sn⁴⁺/Sn²⁺ = 7.3), leading to a less marked tin enrichment in heavy isotopes (<0.2‰).

## 4.4 Tin isotope fractionation during fractional crystallization

Badullovich et al. (2017) have shown that fractional crystallization of basalts leads to a large enrichment in tin in the residual melt as tin is incompatible in the crystallizing mineral assemblage. Interestingly, there is enrichment in light Sn isotopes in melts with an MgO below 8 wt%. The calculated bulk tin mineral/melt partition coefficient indicates that the oxidation state of tin must be predominantly Sn⁴⁺ during magma differentiation, as Sn²⁺ would be far more compatible and Sn concentrations would not fit the trend in Badullovich et al. (2017). Altogether, these observations were suggested to result from the incorporation of isotopically heavy Sn in ilmenite. The Sn isotope fractionation was interpreted in terms of a difference in coordination of Sn⁴⁺ between the melt (where tin would be eightfold-coordinated) and the ilmenite where it is sixfold-coordinated (Badullovich et al. 2017). However, as discussed above, Sn⁴⁺ was reported to be six-fold coordinated both in silicate melts (Farges et al. 2006) and in ilmenite (Klemme et al. 2006). Most of our spectroscopic results are consistent with the conclusions of Farges et al. (2006). Nonetheless, the dependence of <F> against the glass composition determined in our study may help to better evaluate the hypothesis of Badullovich et al. (2017) regarding the effect of structural properties of tin in magmas on Sn isotope fractionation.

In order to reexamine this data set, we have attempted to model the Sn isotope composition of the Kilauea Iki lava lake using the force constants determined in the NRIXS data. In this context, we modeled the effect of successive steps of ilmenite fractional crystallization in a magma chamber. The δ¹²⁴Sn value in the melt at step n, δ_l^n was calculated from the δ¹²⁴Sn value at step n-1:

$$ \delta_l^n = \delta_l^{n-1} - \phi_s^n \Delta_{\text{Rhyolite Air}}^{\text{SnO}_2} \qquad (10) $$
$$
\begin{array}{ll}
\text{523} & \text{where } \phi_s^n = \frac{f_s[Sn]_s}{f_s[Sn]_s + f_l[Sn]_l^n}, \quad [Sn]_l^n = \frac{[Sn]_l^{n-1} - f_s[Sn]_s}{f_l} \\
\text{524} & \delta_s^n = \Delta_{\text{Rhyolite Air}}^{\text{SnO}_2} + \delta_l^n \qquad (11) \\
\text{525} & \text{with the following initial conditions:} \\
\text{526} & \delta_l^0 = 0.6\text{‰}, [Sn]_s = 7.6 \text{ ppm}, [Sn]_l^0 = 1 \text{ ppm} \qquad (12) \\
\text{527} & \text{and considering } \Delta_{\text{Rhyolite Air}}^{\text{SnO}_2} = 0.09\text{‰} \text{ at } 1100^{\circ}\text{C. The rationale here is that the force} \\
\text{528} & \text{constant of } \text{SnO}_2 \text{ is a proxy for that of ilmenite since } \text{Sn}^{4+} \text{ is likely to substitute to } \text{Ti}^{4+}, \text{ where} \\
\text{529} & \text{Fe}^{2+}-\text{O, } \text{Ti}^{4+}-\text{O}_2 \text{ becomes Fe}^{2+}-\text{O, } \text{Sn}^{4+}-\text{O}_2. \text{ As shown before, } \text{SnO}_2 \text{ has a higher force constant} \\
\text{530} & \text{than rhyolite glass (unlike basalt and other synthetic glasses studied here), thereby} \\
\text{531} & \text{suggesting that the Sn isotope composition of ilmenite could be heavier than the residual} \\
\text{532} & \text{liquid, which is qualitatively consistent with the observations of Badullovich et al. (2017).} \\
\text{533} & \text{Because samples from Badullovich et al. (2017) are characterized by an intermediate} \\
\text{534} & \text{composition between a basalt and rhyolite, both composition could a priori be considered} \\
\text{535} & \text{for the residual liquid composition. In this model, at each step, ilmenite enriched in heavy tin} \\
\text{536} & \text{isotopes crystallizes and is removed from the system thereby inducing an enrichment of light} \\
\text{537} & \text{isotopes in the residual liquid (Fig 7). Figure 7 shows that the lowest } \delta^{124}\text{Sn observed in} \\
\text{538} & \text{Badullovich et al. (2017) (i.e. the most fractionated sample) could be explained by a} \\
\text{539} & \text{crystallization of } 10.4 \pm 0.07 \text{ wt\% of ilmenite in the magma chamber. This number is roughly} \\
\text{540} & \text{consistent with observed modal abundances of ilmenite in Hawaiian basalts (Peck et al.} \\
\text{541} & \text{1966). In order to further verify this hypothesis, we modeled the expected Ti and Sn} \\
\text{542} & \text{abundances during fractional crystallization. We observed that crystallization of 10\%} \\
\text{543} & \text{ilmenite should decrease the } \text{TiO}_2 \text{ abundance in the melt to 1.3\%. However, the basaltic} \\
\text{544} & \text{andesite analyzed by Badullovich et al. (2017) show a } \text{TiO}_2 \text{ of 2.59 \%wt which reflect that the} \\
\text{545} & \text{proportion of crystallized ilmenite is } \sim 1.5\% \text{ which is far less than the 10\% required to explain} \\
\text{546} & \text{the Sn isotope composition. Thus, although the idea of ilmenite crystallization is in} \\
\text{547} & \text{qualitative agreement with the observations, the observed fractionation would require a} \\
\text{548} & \text{much higher Sn isotope fractionation factor between ilmenite and melt than what we} \\
\text{549} & \text{considered in this model. We note here that the Sn isotope observations for Kilauea Iki} \\
\text{550} & \text{cannot be rationalized in terms of kinetic isotope fractionation with incorporation of light Sn} \\
\text{551} & \text{isotopes in the minerals because this would leave a melt with enrichment in heavy Sn} \\
\text{552} & \text{isotopes, which is the opposite of what is observed (Badullovich et al., 2017).} \\
\text{553}
\end{array}
$$
554

## 5. Conclusions

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563

Seven geologically-relevant silicate glasses were synthesized with moderate amounts of ¹¹⁹Sn. Redox conditions were controlled during sample synthesis to set the redox ratio (Sn²⁺/Snₜot) from stannous (Sn²⁺) to stannic (Sn⁴⁺) glasses. The mean force constants of tin bonds in these glasses were determined by synchrotron nuclear resonant inelastic X-ray scattering (NRIXS) in order to determine the reduced isotopic partition function ratios (β-factors) of these glasses. The coordination chemistry and the valence state of tin in these glasses were also derived from synchrotron Mossbauer spectroscopy (SMS). For all studied samples, 1000 × ln β₁₂₂/₁₁₆Sn was calculated.

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569

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573

574

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576

577

Reduced glasses exhibit a sharp and intense phonon peak at about 10 meV followed by a featureless tail extending to 80 meV. Oxidized glasses show a broader first phonon peak at slightly higher energy associated with a second peak at 70 meV. The force constant of tin is a strong function of the valence state and of composition of the silicate melts. The values are also significantly different from those collected for pure crystalline compounds. Whereas reduced basalts and rhyolites have similar force constants, the force constant of the reduced anorthite is significantly lower. Concerning oxidized samples, most glasses have force constants close to pure SnO₂ though slightly higher. The force constant of rhyolitic glass is however significantly lower by almost 100 N/m than that of pure SnO₂. At constant valence state, our data indicates that Sn⁴⁺-bearing basalt should be significantly enriched in heavy tin isotopes as compared to rhyolitic melts (by up to 0.2 ‰ at 1000K). The effect of Al and possibly K likely explains the β-factors of outliers (reduced anorthite and oxidized rhyolite) as compared to other samples because these cations affect either Sn-O bond length and/or the coordination number of tin in the melts.

578

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581

582

583

584

585

From our dataset it is concluded that the incompatible behavior of Sn⁴⁺ in mantle minerals leads to an enrichment in heavy isotopes in mantle melts and to the depletion in heavy isotopes in solid residues of melting with a magnitude consistent with the isotope fractionation between Sn²⁺ and Sn⁴⁺ predicted here. During fractional crystallization of basalt and considering the effect of tin coordination number in minerals and melts, the partitioning of Sn⁴⁺ into ilmenite would lead to an enrichment in light isotopes in the residual melt. As a final remark, we note that no large difference is found between the β-factors of pure α-Sn (measured elsewhere) and SnO-bearing silicates. Our results would suggest, at first sight that
586 no measurable tin isotopes fractionation during core-mantle differentiation should be
587 analytically resolved in natural samples. A detailed study of tin-bearing, geochemically
588 relevant metal alloys at high pressure must however be carried out to confirm this
589 preliminary conclusion.

590

591

592
593 Acknowledgements:

594 This manuscript benefited from thoughtfull remarks and comments from the associate
595 editor: J. Prytulak and from A. Burnham and two anonymous reviewers. We deeply thank
596 Séverine Bellayer (UMET, Lille, France), Béatrice Doisneau and Sylvain Pont (IMPMC, Paris,
597 France) for EMPA and SEM observations. This research used resources of the Advanced
598 Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated
599 for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-
600 06CH11357. This work benefited from the FACCTS 2015-173-N0 grant to E.E.A. and M.R. This
601 work was also supported by NASA grants NNX17AE86G (LARS), NNX17AE87G (Emerging
602 Worlds), and NSSC17K0744 (Habitable Worlds) to N.D.

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735 Lehmann, B. (2006). Metallogeny of tin. Springer.

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737 Liu, J., Dauphas, N., Roskosz, M., Hu, M.Y., Yang, H., Bi, W., Zhao, J., Alp, E.E., Hu, J.Y., Lin, J.-F., 2017. Iron isotopic fractionation between silicate mantle and metallic core at high pressure. Nature Comm. http://dx.doi.org/10.1038/ncomms14377.

740
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744 Mallmann, G., O’neill, H. S. C., 2009. The crystal/melt partitioning of V during mantle melting
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748 McDonough, W.F., 2003. Compositional model for the Earth's core. In: Carlson, R.W. (Ed.),
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752 Médard, E., Schmidt, M.W., Schiano, P., Ottolini, L. 2006 Melting of Amphibole bearing wehrlites: an experimental study on the origin of ultra-calcic nepheline normative melts. J. Petrology, 47, 481-504.

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756 Médard, E., McCammon, C.A., Barr, J.A., Grove, T.L. Oxygen fugacity, temperature reproducibility, and H2O contents of nominally anhydrous piston-cylinder experiments using graphite capsules. Amer. Miner. 93, 1838-1844.

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760 Norris, C.A., Wood, B.J., 2017. Earth's volatile contents established by melting and vaporization. Nature, 549, 507-510.

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763 O'Neill, H.St.C., Palme, 2008 Collisional erosion and non-chondritic composition of the terrestrial planets. Philos. Trans. A Math. Phys. Eng. Sci. 366, 4205-4238.

765
766 Palme, H., O'Neill, H.St.C., 2003. Cosmochemical estimates of mantle composition. In: Carlson, R.W. (Ed.), Treatise on Geochemistry, vol. 2. Elsevier-Pergamon, Oxford, pp. 1-38.

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769 Peck, D. L., Wright, T. L., Moore, J. G., 1966. Crystallization of tholeiitic basalt in Alae lava lake, Hawaii. Bulletin Volcanologique, 29(1), 629-655.

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772 Poitrasson, F., Halliday, A.N., Lee, D.C., Levasseur, S., Teutsch, N., 2004. Iron isotope differences between Earth, Moon, Mars and Vesta as possible records of contrasted accretion mechanisms. Earth Planet. Sci. Lett. 223, 253-266.

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776 Poitrasson, F., Roskosz, M., Corgne, A., 2009. No iron isotope fractionation between molten alloys and silicate melt to 2000 °C and 7.7 GPa: experimental evidence and implications for planetary differentiation and accretion. Earth Planet. Sci. Lett. 278, 376-385.

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780 Polyakov, V.B., 2009. Equilibrium iron isotope fractionation at core-mantle boundary conditions. Science 323, 912-914.

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783 Polyakov, V.B., Mineev, S.D., Clayton, R.N., Hu, G., Mineev, K.S., 2005. Determination of tin equilibrium isotope fractionation factors from synchrotron radiation experiments. Geochim. Cosmochim. Acta 69, 5531-5536.

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787 Righter, K. 2011. Prediction of metal-silicate partition coefficients for siderophile elements: an update and assessment of PT conditions for metal-silicate equilibrium during accretion of the Earth. Earth Planet. Sci. Lett. 304, 158-167.

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791 Roskosz, M., Deloule, E., Ingrin, J., Depecker, C., Laporte, D., Merkel, S., Remusat, L., Leroux, H., 2018. Kinetic D/H fractionation during hydration and dehydration of silicate glasses, melts and nominally anhydrous minerals. Geochim. et Cosmochim. Acta 233, 14-32.

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795 Roskosz, M., Sio, C.K., Dauphas, N., Bi, W., Tissot, F.L.H., Hu, M.Y., Zhao, J., Alp, E.E., 2015. Spinel-olivine-pyroxene equilibrium iron isotopic fractionation and applications to natural peridotites. Geochim. Cosmochim. Acta 169, 184-199.
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800 Copper isotope evidence for large-scale sulphide fractionation during Earth's differentiation.
801 Geochemical Perspectives Letters 1, 53-64.

802
803 Schauble, E.A., 2004. Applying stable isotope fractionation theory to new systems. *Rev
804 Mineral Geochem* **55**, 65-111.

805
806 Seki, H., Ishizawa, N., Mizutani, N., Kato, M., 1984. High temperature structures of the rutile-
807 type oxides, TiO₂ and SnO₂. *Yogyo Kyokai Shi* (J. Ceramic Assoc.) **92**, 219-223.

808 Seto, M., Yoda, Y., Kikuta, S., Zhang, X.W., Ando, M. 1995. Observation of nuclear resonant
809 scattering accompanied by phonon excitation using synchrotron radiation. Phys. Rev. Lett.
810 74, 3828.

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812 temperature Si isotope fractionation between iron metal and silicate. Geochim. Cosmochim.
813 Acta 75, 7688-7697.

814
815 Shahar, A., Schauble, E.A., Caracas, R., Gleason, A.E., Reagan, M.M., Xiao, Y., 2016. Pressure-
816 dependent isotopic composition of iron alloys. Science **352**, 580-582.

817
818 Sitek, J., Stemprok, M., Voldan, J., CicàSkovà, O., 1981, Mossbauer spectra of tin containing
819 granite glasses. Silikàty 24, 243-249.

820 Sturhahn, W. 2000 CONUSS and PHOENIX: Evaluation of nuclear resonant scattering data.
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822 Sturhahn, W., Toellner, T.S., Alp, E.E., Zhang, X., Ando, M., Yoda, Y., Kikuta, S., Seto, M.,
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824 Resonant Scattering. Phys. Rev. Letters 74, 3832.

825 Toellner, T.S., Alatas, A., Said, A.H., 2011 Six-reflection meV-monochromator for
826 synchrotron radiation. J. Synchrotron Rad. 18, 605.

827
828 Wang, X., Amet, Q., Fitoussi, C., Bourdon, B., 2018. Tin isotope fractionation during
829 magmatic processes and the isotope composition of the bulk silicate Earth. Geochimica
830 Cosmochimica Acta, 228, 320-335.

831
832 Witt-Eickschen, G., Palme, H., O'Neill, H.S.C., Allen, C.M., 2009. The geochemistry of the
833 volatile trace elements As, Cd, Ga, In and Sn in the Earth's mantle: new evidence from in situ
834 analyses of mantle xenoliths. Geochimica et Cosmochimica Acta 73, 1755-1778.

835
836 Young, E.D., Shahar, A., Nimmo, F., Schlichting, H.E., Schauble, E.A., Tang, H., Labidi, J. 2019.
837 Near-equilibrium isotope fractionation during planetesimal evaporation. Icarus, 323, 1-15.

838
839
840 **Figure captions:**

841
842
843 **Figure 1:** Partial projected density of states of Sn in the studied samples. Spectra of SnO and SnO₂ are from Dauphas et al., 2018.
844
845

846 **Figure 2:** Typical SMS spectra collected with and without references for rhyolitic and basaltic samples. Red lines are the best fits to the data.
847
848

849 **Figure 3:** Mean force constants of the studied samples as a function of their average valence states. Typical error of 5% are applied to the SMS-based determination of the valence state of the samples.
850
851
852

853 **Figure 4:** 1000 × ln β₁₂₂/₁₁₆Sₙ calculated for all the analyzed samples as a function of temperature.
854
855

856 **Figure 5:** δ¹²⁴Sn in the solid residue as a function of the degree of partial melting of a peridotite (data for peridotites from Wang et al. 2018 and were measured as ¹²⁴Sn/¹¹⁶Sn relative to the NIST SRM 3161a standard). As Sn⁴⁺ is preferentially associated with heavy Sn isotopes, δ¹²⁴Sn decreases as partial melting proceeds due to the release of more incompatible Sn⁴⁺ to the liquid. At high pressure (3 GPa), Sn⁴⁺ is more incompatible than at low pressure (1.4 GPa) which enhances the isotope fractionation in the solid residue. cpx: clinopyroxene. The curves at 3 GPa were drawn for various temperatures as shown by the color code.
857
858
859
860
861
862
863

864 **Figure 6:** δ¹²⁴Sn of the melt fraction during partial melting (data for basalts are from Wang et al. 2018 and were measured as ¹²⁴Sn/¹¹⁶Sn relative to the NIST SRM 3161a standard). During partial melting, δ¹²⁴Sn decreases in the melt due to Sn⁴⁺ dilution. This explains why OIBs, which form by lower degrees of partial melting than MORBs, show a heavier isotopic composition than the latter. Unlike the solid residue, a low pressure leads to a larger isotopic fractionation in the liquid relative to the BSE. Note that samples with δ¹²⁴Sn higher than 0.2‰ (BHVO-1/2, Koo2) are not explained by this model. This could reflect the contribution of a Pyroxenite component in the melting source as discussed by Wang et al. (2018).
865
866
867
868
869
870
871

872 **Figure 7:** Effect of ilmenite crystallization on the Sn isotopic composition of the residual liquid and ilmenite. Both ilmenite and the residual liquid become isotopically lighter as magmatic differentiation progresses. The lowest δ¹²⁴Sn observed in Badullovich et al. (2017) (0.48 ± 0.07‰) is explained by 10.4 ± 0.07 wt% of ilmenite crystallization. Measurements of Badullovich et al. (2017) were collected as ¹²²Sn/¹¹⁸Sn relative to the Sn_IPGP in house standard, they are here recalculated in the interest of homogeneity of notation throughout this study, using a δ¹²⁴Sn notation (¹²⁴Sn/¹¹⁶Sn), still relative to the IPGP standard.
873
874
875
876
877
878

879
<table><caption>Table 1: Sample composition determined by EMPA (Wt%)</caption><thead><tr><th>Sample</th><th>SiO2 (Wt%)</th><th>Al2O3 (Wt%)</th><th>MgO (Wt%)</th><th>CaO (Wt%)</th><th>FeO (Wt%)</th><th>Na2O (Wt%)</th><th>K2O (Wt%)</th><th>TiO2 (Wt%)</th><th>SnO (Wt%)</th></tr></thead><tbody><tr><td>Rhyolite-CCO</td><td>72.50 ± 0.41</td><td>10.02 ± 0.08</td><td>-</td><td>-</td><td>-</td><td>4.11 ± 0.17</td><td>4.97 ± 0.17</td><td>4.67 ± 0.09</td><td>-</td><td>1.98 ± 0.10</td></tr><tr><td>Rhyolite-Air</td><td>75.73 ± 0.30</td><td>10.38 ± 0.40</td><td>-</td><td>-</td><td>-</td><td>3.79 ± 0.17</td><td>4.31 ± 0.23</td><td>4.55 ± 0.09</td><td>-</td><td>0.65 ± 0.37</td></tr><tr><td>Basalt-CCO</td><td>44.82 ± 0.16</td><td>13.98 ± 0.07</td><td>17.54 ± 0.15</td><td>10.89 ± 0.07</td><td>7.45 ± 0.09</td><td>1.97 ± 0.12</td><td>-</td><td>-</td><td>0.81 ± 0.01</td><td>1.69 ± 0.04</td></tr><tr><td>Basalt-Air</td><td>44.69 ± 0.18</td><td>13.94 ± 0.07</td><td>17.26 ± 0.19</td><td>10.89 ± 0.03</td><td>7.49 ± 0.08</td><td>1.90 ± 0.08</td><td>-</td><td>-</td><td>0.82 ± 0.02</td><td>2.12 ± 0.13</td></tr><tr><td>Anorthite-CCO</td><td>42.68 ± 0.17</td><td>35.89 ± 0.09</td><td></td><td>19.41 ± 0.06</td><td></td><td></td><td></td><td></td><td></td><td>1.83 ± 0.08</td></tr><tr><td>Anorthite-Air</td><td>42.72 ± 0.24</td><td>35.97 ± 0.11</td><td></td><td>19.63 ± 0.13</td><td></td><td></td><td></td><td></td><td></td><td>1.62 ± 0.11</td></tr><tr><td>Enstatite-Air</td><td>59.19 ± 0.16</td><td></td><td>39.14 ± 0.18</td><td></td><td></td><td></td><td></td><td></td><td></td><td>1.73 ± 0.14</td></tr></tbody></table>

880

881

882

883

884

885

886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

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902

903

904
905
906

<table><caption>Table 2: SMS and force constant data of the analyzed samples.</caption><thead><tr><th colspan="2">Sample</th><th>Isomer shift (mm/s)</th><th>Tin valence state</th><th colspan="3">Force constant (N/m)</th></tr></thead><tbody><tr><td rowspan="2">Basalt</td><td>Basalte Air</td><td>0.043</td><td>4</td><td>509</td><td>±</td><td>22</td></tr><tr><td>Basalte CCO</td><td>1.23</td><td>2</td><td>254</td><td>±</td><td>16</td></tr><tr><td rowspan="2">Rhyolite</td><td>Rhyolite Air</td><td>-0.29</td><td>4</td><td>412</td><td>±</td><td>21</td></tr><tr><td>Rhyolite CCO</td><td>2.4</td><td>2</td><td>250</td><td>±</td><td>14</td></tr><tr><td rowspan="3">Synthetic glasses</td><td>Anorthite Air</td><td>0.37</td><td>4</td><td>536</td><td>±</td><td>42</td></tr><tr><td>Anorthite CCO</td><td>-</td><td>2</td><td>204</td><td>±</td><td>13</td></tr><tr><td>Enstatite Air</td><td>0.53</td><td>4</td><td>558</td><td>±</td><td>68</td></tr><tr><td rowspan="3">Reference material</td><td>Sn</td><td>2.56</td><td>0</td><td>155</td><td></td><td></td></tr><tr><td>SnO</td><td>2.54</td><td>2</td><td>181</td><td>±</td><td>16</td></tr><tr><td>SnO2</td><td>0 (Reference)</td><td>4</td><td>492</td><td>±</td><td>28</td></tr></tbody></table>

907
908
909
910
911
<table><caption>Table 3: Compilation of parameters derived from NRIXS spectra.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Basalt air</th><th rowspan="2">Basalt CCO</th><th rowspan="2">Rhyolite air</th><th rowspan="2">Rhyolite CCO</th><th rowspan="2">Anorthite Air</th><th rowspan="2">Anorthite CCO</th><th rowspan="2">Enstatite Air</th></tr><tr><th>-119.502 to 149.449</th><th>-119.14 to 149.073</th></tr></thead><tbody><tr><td>Total energy range:</td><td>-119.52 to 148.833</td><td>-119.192 to 147.916</td><td>-118.847 to 148.957</td><td>-119.419 to 148.933</td><td>-119.14 to 149.449</td><td>-119.14 to 149.073</td><td>-16.142 to 149.579</td></tr><tr><td>Energy cutoff (left and right in meV):</td><td>19.2 and 11.8</td><td>38.8 and 18.8</td><td>29.8 and 27.6</td><td>34.4 and 30.6</td><td>21.2 and 26</td><td>45.6 and 48.8</td><td>23 and 19</td></tr><tr><td>Baseline subtracted:</td><td>linear</td><td>linear</td><td>linear</td><td>linear</td><td>linear</td><td>linear</td><td>linear</td></tr><tr><td>a=</td><td>5.30E-03</td><td>± 1.62E-03</td><td>5.86E-03</td><td>± 1.14E-03</td><td>3.90E-03</td><td>± 1.8E-03</td><td>± 8.82E-04</td></tr><tr><td>b=</td><td>1.70E-00</td><td>± 2.01E-01</td><td>2.06E+00</td><td>± 1.30E-01</td><td>2.19E+00</td><td>± 1.54E-01</td><td>± 1.15E+00</td><td>± 1.04E-01</td></tr><tr><td>Input temperature (K):</td><td>300</td><td>300</td><td>300</td><td>300</td><td>300</td><td>300</td><td>300</td></tr><tr><td>Temperature from detailed balance (K):</td><td>311</td><td>310</td><td>303</td><td>312</td><td>308</td><td>310</td><td>299</td></tr><tr><td>From S</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>lamb-mossbauer factor from S:</td><td>0.5309</td><td>± 0.0013</td><td>0.3580</td><td>± 0.0018</td><td>0.3435</td><td>± 0.0021</td><td>± 0.0021</td></tr><tr><td>Mean square displacement &lt;r²&gt; from S (A²):</td><td>4.36E-03</td><td>± 1.25E-05</td><td>7.08E-03</td><td>± 2.49E-05</td><td>7.37E-03</td><td>± 3.03E-05</td><td>9.87E-03</td><td>± 4.39E-05</td></tr><tr><td>Internal energy/atom from S (meV):</td><td>29.21</td><td>± 0.63</td><td>27.56</td><td>± 0.59</td><td>28.60</td><td>± 0.63</td><td>27.54</td><td>± 0.60</td></tr><tr><td>Kinetic energy/atom from S (meV):</td><td>14.60</td><td>± 0.31</td><td>13.78</td><td>± 0.30</td><td>14.30</td><td>± 0.31</td><td>13.77</td><td>± 0.30</td></tr><tr><td>Force constant from S (N/m):</td><td>509</td><td>± 22</td><td>254</td><td>± 16</td><td>412</td><td>± 21</td><td>250</td><td>± 14</td></tr><tr><td>From g</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>lamb-mossbauer factor from g:</td><td>0.52E-03</td><td></td><td>0.3479</td><td></td><td>0.3348</td><td></td><td>0.2276</td><td></td></tr><tr><td>Mean square displacement &lt;r²&gt; (A²):</td><td>4.42E-03</td><td></td><td>7.28E-03</td><td></td><td>7.54E-03</td><td></td><td>1.02E-02</td><td></td></tr><tr><td>α<sup>2</sup>-2/α<sup>2</sup>+2 (A²/K):</td><td>1.40E-05</td><td></td><td>2.35E-05</td><td></td><td>2.44E-05</td><td></td><td>3.33E-05</td><td></td></tr><tr><td>Critical temperature (K):</td><td>491</td><td></td><td>293</td><td></td><td>292</td><td></td><td>207</td><td></td></tr><tr><td>Resilience (N/m):</td><td>98</td><td></td><td>59</td><td></td><td>57</td><td></td><td>41</td><td></td></tr><tr><td>Internal energy/atom from g (meV):</td><td>29.23</td><td></td><td>27.59</td><td></td><td>28.68</td><td></td><td>27.62</td><td></td></tr><tr><td>Kinetic energy/atom from g (meV):</td><td>14.62</td><td></td><td>13.80</td><td></td><td>14.34</td><td></td><td>13.81</td><td></td></tr><tr><td>Virbational entropy (kb/atom):</td><td>1.19</td><td></td><td>1.55</td><td></td><td>1.34</td><td></td><td>1.59</td><td></td></tr><tr><td>Heimholtz free energy (meV):</td><td>-1.50</td><td></td><td>-12.37</td><td></td><td>1.34</td><td></td><td>-13.56</td><td></td></tr><tr><td>Virbational specific heat (kb/atom):</td><td>0.89</td><td></td><td>0.94</td><td></td><td>0.91</td><td></td><td>0.94</td><td></td></tr><tr><td>lamb-mossbauer factor at T=0 from g:</td><td>0.8695</td><td></td><td>0.8220</td><td></td><td>0.8402</td><td></td><td>0.8040</td><td></td></tr><tr><td>Kinetic energy/atom at T=0 from g (meV):</td><td>6.95</td><td></td><td>4.70</td><td></td><td>6.14</td><td></td><td>4.63</td><td></td></tr><tr><td>Force constant from g (N/m):</td><td>510</td><td></td><td>255</td><td></td><td>413</td><td></td><td>251</td><td></td></tr><tr><td>beta-value coefficients from S</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>1000 In beta 1225n/1165n=A1/T²+A2/T⁴+A3/T⁶ (T in K)</td><td>982115</td><td>± 42294</td><td>490338</td><td>± 30368</td><td>794520</td><td>± 39635</td><td>481906</td><td>± 26319</td></tr><tr><td>A1:</td><td>-8.25E-09</td><td>± 9.74E-08</td><td>-3.31E+09</td><td>± 6.99E-08</td><td>-5.45E+09</td><td>± 7.63E+08</td><td>-2.93E+09</td><td>± 4.7E-08</td></tr><tr><td>A2:</td><td>1.769E-14</td><td>± 4.15E-13</td><td>6.853E+13</td><td>± 2.86E+13</td><td>8.328E+13</td><td>± 2.63E+13</td><td>5.186E+13</td><td>± 1.6E-13</td></tr><tr><td>A3:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>1000 In beta 1225n/1165n=B1+F/T2-B2+F/T2/T⁴ (T in K)</td><td>1925</td><td></td><td>1929</td><td></td><td>1929</td><td></td><td>1929</td><td></td></tr><tr><td>B1:</td><td>24460</td><td></td><td>39763</td><td></td><td>26822</td><td></td><td>37981</td><td></td></tr><tr><td>B2:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>beta-value coefficients from g</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>1000 In beta 1225n/1165n=A1/T²+A2/T⁴+A3/T⁶ (T in K)</td><td>983125</td><td></td><td>491262</td><td></td><td>796442</td><td></td><td>483783</td><td></td></tr><tr><td>A1:</td><td>-8.28E-09</td><td></td><td>-3.32E+09</td><td></td><td>-5.48E+09</td><td></td><td>-2.95E+09</td><td></td></tr><tr><td>A2:</td><td>1.797E-14</td><td></td><td>6.933E+13</td><td></td><td>8.492E+13</td><td></td><td>5.302E+13</td><td></td></tr></tbody></table>
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>
Basalt_Air

<figure><img src="image_5.png" /></figure>

Basalt_Air + SnO

<figure><img src="image_6.png" /></figure>

Rhyolite_Air

<figure><img src="image_7.png" /></figure>

Rhyolite_Air + SnO

<figure><img src="image_8.png" /></figure>

Basalt_CCO

<figure><img src="image_9.png" /></figure>

Basalt_CCO + SnO

<figure><img src="image_10.png" /></figure>

Rhyolite_CCO

<figure><img src="image_11.png" /></figure>

Rhyolite_CCO + SnO

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>
<figure><img src="image_14.png" /></figure>

● Pure Sn compounds
● Anorthite
● Enstatite
● Basalt
● Rhyolite
<figure><img src="image_15.png" /></figure>

T (K)
<figure><img src="image_16.png" /></figure>
<figure><img src="image_17.png" /></figure>
<figure><img src="image_18.png" /></figure>

Minimum $\delta^{124}\text{Sn}$ observed in KI samples