Iron isotope fractionation between pyrite (FeS2), hematite (Fe2O3) and siderite (FeCO3): A first-principles density functional theory study

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Geochimica et Cosmochimica Acta 73 (2009) 6565–6578 www.elsevier.com/locate/gca

Iron isotope fractionation between pyrite (FeS2), hematite (Fe2O3) and siderite (FeCO3): A first-principles density functional theory study Marc Blanchard a,*, Franck Poitrasson b, Merlin Me´heut a, Michele Lazzeri a, Francesco Mauri a, Etienne Balan a a

Institut de Mine´ralogie et de Physique des Milieux Condense´s (IMPMC), Universite´ Paris VI, CNRS UMR 7590, Universite´ Paris VII, IPGP, IRD UMR 206, Campus Boucicaut, 140 rue de Lourmel, 75015 Paris, France b Laboratoire d’e´tude des Me´canismes et Transfert en Ge´ologie, CNRS, Universite´ de Toulouse, IRD, 14 avenue Edouard Belin, 31400 Toulouse, France Received 12 September 2008; accepted in revised form 30 July 2009; available online 4 August 2009

Abstract In addition to equilibrium isotopic fractionation factors experimentally derived, theoretical predictions are needed for interpreting isotopic compositions measured on natural samples because they allow exploring more easily a broader range of temperature and composition. For iron isotopes, only aqueous species were studied by first-principles methods and the combination of these data with those obtained by different methods for minerals leads to discrepancies between theoretical and experimental isotopic fractionation factors. In this paper, equilibrium iron isotope fractionation factors for the common minerals pyrite, hematite, and siderite were determined as a function of temperature, using first-principles methods based on the density functional theory (DFT). In these minerals belonging to the sulfide, oxide and carbonate class, iron is present under two different oxidation states and is involved in contrasted types of interatomic bonds. Equilibrium fractionation factors calculated between hematite and siderite compare well with the one estimated from experimental data (ln a 57 Fe/54Fe = 4.59 ± 0.30& and 5.46 ± 0.63& at 20 °C for theoretical and experimental data, respectively) while those for Fe(III)aq-hematite and Fe(II)aq-siderite are significantly higher that experimental values. This suggests that the absolute values of the reduced partition functions (b-factors) of aqueous species are not accurate enough to be combined with those calculated for minerals. When compared to previous predictions derived from Mo¨ssbauer or INRXS data [Polyakov V. B., Clayton R. N., Horita J. and Mineev S. D. (2007) Equilibrium iron isotope fractionation factors of minerals: reevaluation from the data of nuclear inelastic resonant X-ray scattering and Mo¨ssbauer spectroscopy. Geochim. Cosmochim. Acta 71, 3833–3846], our iron b-factors are in good agreement for siderite and hematite while a discrepancy is observed for pyrite. However, the detailed investigation of the structural, electronic and vibrational properties of pyrite as well as the study of sulfur isotope fractionation between pyrite and two other sulfides (sphalerite and galena) indicate that DFT-derived b-factors of pyrite are as accurate as for hematite and siderite. We thus suggest that experimental vibrational density of states of pyrite should be re-examined. Ó 2009 Elsevier Ltd. All rights reserved.

1. INTRODUCTION *

Corresponding author. Tel.: +33 1 44 27 98 22; fax: +33 1 44 27 37 85. E-mail address: [email protected] (M. Blanchard). 0016-7037/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.gca.2009.07.034

The study of stable isotopes of iron and other transition elements benefited from the decisive impulse of important analytical developments involving plasma source mass spectrometry since the beginning of the decade (e.g. Mare´chal

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et al., 1999; Belshaw et al., 2000). This research field is receiving a growing attention because Fe isotopes may provide information complementary to the more conventional stable isotopes (e.g. O, C, S) to understand geological processes and study global biogeochemical cycles. The observed natural isotopic compositions indicate that the most significant iron isotope fractionations occur in low-temperature systems, including hydrothermal fluids and minerals precipitated through biologic or abiologic pathways (e.g. Dauphas and Rouxel, 2006; Johnson et al., 2008a). Among the numerous applications found in the literature, iron isotopes recorded in sedimentary rocks (banded iron formations, black shales) could help to trace the redox evolution of the ocean through geological times and thus to deduce the levels of atmospheric oxygen (Johnson et al., 2003, 2008b; Rouxel et al., 2005; Yamaguchi et al., 2005, 2007). Another application is the study of planetary formation processes such as accretion and mantle–core differentiation (e.g. Poitrasson et al., 2004, 2005, 2009; Schoenberg and von Blanckenburg, 2006; Williams et al., 2006). An essential basis for interpreting isotopic compositions in natural samples is to know the equilibrium isotopic fractionation factors (see, e.g. Johnson et al., 2003; Matthews et al., 2004). In this recent field of research, such data are rare and sometimes questionable. Equilibrium isotopic fractionation factors can be obtained through several approaches. Experimentally, the attainment of the equilibrium state or the extent of the isotopic exchange can be controlled using reactions of isotopic exchange or three-isotope exchange methods (e.g. O’Neil, 1986; Schuessler et al., 2007; Shahar et al., 2008). Equilibrium isotopic fractionation factors can also be determined theoretically from the computation of the vibrational properties. Concerning iron isotopes, only aqueous species were modeled by first-principles methods (Anbar et al., 2005; Hill and Schauble, 2008; Domagal-Goldman and Kubicki, 2008). Polyakov and co-workers (1997, 2000, 2007) developed an alternative approach, which permitted to predict the equilibrium iron isotope fractionation factors of a wide range of minerals either from Mo¨ssbauer or inelastic nuclear resonant X-ray scattering spectra (INRXS). The latter technique provides the phonon partial density of states for the resonant atom (i.e. Fe) and thus a more accurate estimation of the reduced partition functions. The work of Polyakov and co-workers (1997, 2000, 2007) provides a good understanding of the parameters controlling the relative order of the reduced partition functions from which fractionation factors are calculated. Overall, predicted equilibrium isotopic fractionation factors between minerals are in relatively good agreement with experimental values (e.g. Polyakov et al., 2007; Shahar et al., 2008). On the other hand, the agreement is not satisfactory for mineral-solution isotopic fractionation when Mo¨ssbauer- or INRXS-derived data are combined with the theoretical data for aqueous species (e.g. Skulan et al., 2002; Wiesli et al., 2004). To better understand this general discrepancy, minerals and aqueous species have to be treated within the same theoretical framework, what we do in the present study by combining new theoretical data for minerals with the consistent data of the literature for aqueous species.

In this paper, we determine the temperature dependence of the iron isotope fractionation factors for three minerals belonging to different mineral classes and that therefore involve different interatomic bond energies: a sulfide (pyrite, FeS2), an oxide (hematite, Fe2O3) and a carbonate mineral (siderite, FeCO3). The computational method, based on first-principles calculations, was previously applied to the stable isotopes of C and O in carbonates (Schauble et al., 2006) and to H, O and Si isotopes in silicates (Me´heut et al., 2007, 2009). This work, which uses for the first time ab initio methods for Fe isotopes in minerals, also provides an independent check on the Mo¨ssbauer- or INRXS-derived data. 2. METHODOLOGY 2.1. Calculation of equilibrium isotope fractionation factors In solids, equilibrium mass dependent isotope fractionation arises from the vibrational motions of atoms. The isotope fractionation factor of an element Y between two phases a and b, i.e. a(a, b, Y), is defined as the ratio of isotope ratios. It can also be written as the ratio of the reduced partition functions (also called b-factors): aða; b; Y Þ ¼

bða; Y Þ bðb; Y Þ

ð1Þ

where b(a, Y) is the Y isotope fractionation factor between the phase a and a perfect gas of Y atoms. Isotope fractionation factors are often reported in permil (&) and thus, we will adopt the usual notation: 103 ln a(a, b, Y) = 103 ln b(a, Y)  103 ln b(b, Y). Following the method described in more detail in Me´heut et al. (2007, 2009), the b-factors of each mineral were calculated from their harmonic vibrational frequencies using " #1=ðN q N Þ 3N Yat Y mq;i ehmq;i =ð2kT Þ 1  ehmq;i =ðkT Þ bða; Y Þ ¼ hmq;i =ðkT Þ m ehmq;i =ð2kT Þ i¼1 fqg q;i 1  e ð2Þ where mq,i are the frequencies of the phonon with wavevector q and branch index i = 1, 3Nat. Nat is the number of atoms in the unit-cell. mq,i and mq;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. The product is performed on a sufficiently large grid of Nq qvectors in the Brillouin zone. In Eq. (2), the three translational modes at the center of the Brillouin zone, with m0,i = 0, are not considered. Eq. (2) accounts for the fact that the isotope fractionation factor must be equal to 1 when temperature goes to infinity, i.e. “rule of the hightemperature product” or “Redlich–Teller rule” (Bigeleisen and Mayer, 1947). 2.2. Computational methods The b-factors were obtained from Eq. (2) using the phonon frequencies, mq,i, computed from first-principles meth-

Iron isotope fractionation between pyrite, hematite and siderite

ods based on the density functional theory (DFT). We used the generalized gradient approximation (GGA) to the exchange–correlation functional as proposed by Perdew et al. (1996). The ionic cores were described by ultrasoft pseudo-potentials that have been generated according to a modified Rappe–Rabe–Kaxiras–Joannopoulos scheme (Rappe et al., 1990) for iron, oxygen and carbon, and to a Vanderbilt scheme (Vanderbilt, 1990) for sulfur, zinc and lead. The wave-functions and the charge density were expanded in plane-waves with 40 and 480 Ry cutoffs, respectively, in all the cases. The Brillouin zone was sampled using a 4  4  4 k-point grid according to the Monkhorst–Pack scheme (Monkhorst and Pack, 1976). It was checked that increasing the wave-functions cutoff to 80 Ry and using a 8  8  8 k-point grid did not change the total energy by more than 15 meV/atom. Phonon frequencies have also reached a satisfactory convergence. For instance, increasing the wave-functions cutoff to 60 Ry and using a 6  6  6 k-point grid did not change the pyrite phonon frequencies by more than 0.2 cm1 at the center of the Brillouin zone. For hematite, all frequencies are converged within 6 cm1 with respect to a wave-functions cutoff of 80 Ry. Atomic relaxations were performed with the PWSCF code (Baroni et al., 2001; http://www.pwscf.org) until the resid˚ . The cell ual forces on atoms were less than 103 Ry/A parameters were also optimized. Calculations for hematite and siderite were spin-polarized and set up to the antiferromagnetic structure. Magnetic moments were free to relax. Phonon frequencies were calculated using the linear response method. They were derived from the second-order derivative of the total energy with respect to atomic displacements, using the PHONON code (Baroni et al., 2001; http://www.pwscf.org). The long-range character of the interatomic force constants was taken into account by computing the Born effective-charges and the electronic dielectric tensor (Baroni et al., 2001). As detailed in Section 3, phonon frequencies were computed at various q-vectors and on various q-point grids in order to make sure that the convergence of the b-factors is achieved. This fundamental criterion has been doubly checked by determining the vibrational properties on larger q-point grids from a Fourier interpolation of the force constants (see Me´heut et al., 2007, for details). 3. RESULTS 3.1. Structural and vibrational properties Hematite calculations have been carried out on the rhombohedral primitive cell (space group R3c), which contains 10 atoms. The antiferromagnetic configuration with the Fe3+ spins aligning along the [1 1 1] axis has been set to the most stable magnetic ordering (Rollmann et al., 2004). Theoretical results regarding the structural, electronic, magnetic and vibrational properties are in good agreement with the experimental data and are discussed in Blanchard et al. (2008). Like hematite, siderite presents a rhombohedral calcite structure (space group R3c) with a primitive cell containing 10 atoms. Siderite is known to be antiferromagnetic below the Ne´el temperature of

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38 K with the magnetic moments of Fe2+ atoms oriented parallel to the [1 1 1] axis (e.g. Jacobs, 1963). Our calculations have been set up with this antiferromagnetic configuration. The three sulfides, i.e. pyrite (FeS2), sphalerite (ZnS) and galena (PbS), crystallize in the cubic symmetry with four formula units per unit-cell. Their space groups are Pa3, F 43m and Fm3m, respectively. Experimentally, pyrite is diamagnetic indicating that Fe2+ atoms are in the low spin state (Miyahara and Teranishi, 1968). The structural, electronic and vibrational properties of this mineral have already been investigated by a similar first-principles method (Blanchard et al., 2005). Parameters of the fully optimized structures are displayed in Table 1. The unit-cell volumes compare well with experimental values, with only a small underestimation of 0.6% for pyrite and an overestimation of 0.6% and 1.2% for hematite and siderite, respectively. The internal degrees of freedom governing the atomic coordinates are in excellent agreement with experimental measurements. Results for sphalerite and galena are in agreement with those of von Oertzen et al. (2005) who investigated the electronic and optical properties of these sulfides using several quantum mechanical methods. Although the overestimation of unit-cell volumes is notable (2.4% and 3.9% for sphalerite and galena, respectively), the generalized gradient approximation (GGA) used here provides a reliable description of both minerals. The harmonic phonon frequencies of siderite and pyrite, computed at the center of the Brillouin zone (C-point), are Table 1 Lattice parameters and internal degrees of freedom (fractional unit). ˚) a (A xS Pyrite DFT Expa

Hematite DFT Expb

Siderite DFT Expc

5.407 5.418

0.383 0.384

˚) arhomb (A

arhomb (°)

xFe

xO

5.466 5.427

54.707 55.280

0.144 0.145

0.059 0.056

˚) arhomb (A

arhomb (°)

xO

5.790 5.795

48.149 47.722

0.525 0.524

˚) a (A Sphalerite DFT Expd

5.449 5.406 ˚) a (A

Galena DFT Expe a b c d e

6.001 5.924

Brostigen and Kjekshus (1969). Finger and Hazen (1980). Graf (1961). Swanson and Fuyat (1959). Wasserstein (1951).

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reported in Tables 2 and 3 (see Table 3 of Blanchard et al. (2008) for hematite). The pyrite results obtained here are comparable to the previous ones (Blanchard et al., 2005), even if the use of the present GGA functional instead of the LDA (local density approximation) functional increases the bond distances and therefore decreases the phonon frequencies by a few percents. This slightly improves the agreement with experimental data (see Table 2 of Blanchard et al., 2005). Sphalerite displays one triply degenerate Raman and IR-active vibrational mode, which splits into a doubly degenerate transverse component (TO) and a longitudinal component (LO). Our calculations indicate that these TO and LO modes are at 256 and 324 cm1 while Nilsen (1969) measured them at 271 and 352 cm1, respectively. Phonons investigation of galena is difficult. At the C-point, optical modes are Raman inactive and, although IR active, the LO mode mixes with the free carrier plasmons. However, it has been shown that TO and LO modes may be located around 68 and 215 cm1, respectively (Smith et al. 2002 and references therein). Our corresponding calculated frequencies are 50 and 216 cm1. In general, theoretical frequencies underestimate by a few percents the measured infrared and Raman frequencies. This systematic error is typical of the GGA approximation. We will discuss below how this bias can be taken into account in the computation of the reduced partition function. 3.2. Iron reduced partition functions (b-factors) and uncertainties The reduced partition functions, expressed in 103 ln b, were computed from Eq. (2). A special effort was put in checking the convergence as a function of the number of q-points used in Eq. (2). In particular, phonon frequencies Table 2 Theoretical and experimental frequencies of transverse optical modes of siderite (cm1). Mode *

4 5–6* 7–8  9–10* 11* 12–13* 14–15  16 17 18 19–20  21–22* 23* 24 25  26 27–28* 29–30  a

Symmetry

DFT

Exp.a

A2u Eu Eg Eu A2u Eu Eg A2g A1u A2g Eg Eu A2u A2g A1g A1u Eu Eg

183 183 196 208 224 278 282 284 320 331 683 693 783 787 1026 1028 1342 1355

201 185 186–187 224

287–300 269

730–736 730–741 861–885 1071–1100 1412–1422 1415–1434

Prinz et al. (1973), White (1974), Langille and O’Shea (1977), and Santilla´n and Williams (2004). * Infrared active modes.   Raman active modes.

Table 3 Theoretical and experimental frequencies of transverse optical modes of pyrite (cm1). Mode

Symmetry

DFT

Exp.a

4 5–7* 8–9 10–12* 13 14–15  16–18* 19–21  22–23 24  25–27* 28–30  31–33* 34–36 

Au Tu Eu Tu Au Eg Tu Tg Eu Ag Tu Tg Tu Tg

217 220 244 289 334 346 348 351 371 372 374 390 395 430

199 215 225 289 310 344 343 344 396 382 402 387 407 433

a *  

Bu¨hrer et al. (1993). Infrared active modes. Raman active modes.

were calculated at the C-point, at the “mean-value point” defined by Baldereschi (1973) and on 2  2  2 q-point grids centered or shifted with respect to the C-point (Fig. 1). When the Brillouin zone center (C-point) is considered, the Born effective-charge and electronic dielectric tensors must be known in order to take into account the longrange effects (which manifest themselves in the longitudinal optical–transverse optical (LO–TO) splitting; Baroni et al., 2001). In the current implementation of the PWSCF code and for spin-polarized systems (i.e. hematite and siderite), these dielectric quantities cannot be directly computed by linear response. In such a case, the Born effective-charges were obtained from finite differences of the bulk polarization induced by small atomic displacements away from the relaxed structure and the electronic dielectric constants were determined by fitting the LO–TO splittings to the experimental ones (Blanchard et al., 2008). For hematite, the use of the phonon frequencies at the C-point provides a b57/54 converged within 1.25& at 0 °C with respect to calculations with a larger number of qpoints (Fig. 1). A more accurate result is obtained by using the Baldereschi point (converged within 0.3& at 0 °C), while the use of a 2  2  2 q-point grid gives the same result as with the phonon frequencies obtained on a 8  8  8 q-point grid through Fourier interpolation. In agreement with the conclusion of Blanchard et al. (2008), the addition of a Hubbard U correction (i.e. on-site Coulomb repulsion treating electron correlation effects) does not change the results either. After optimization of the structure with either the GGA or GGA + U methods (U = 3.3 eV), b57/54 obtained from the phonon frequencies at the C-point, display a difference of only 0.13& at 0 °C, i.e. a relative uncertainty of 1.0%. For pyrite, the convergence of the b-factors is also achieved with a 2  2  2 q-point grid or the Baldereschi point. Also in this case, the Fourier interpolation on finer grids does not improve the convergence. In view of these results, for siderite, only frequencies at the Baldereschi point and on a 2  2  2 shifted grid were computed, giving b57/54 converged within 0.08& at 0 °C, between them.

Iron isotope fractionation between pyrite, hematite and siderite

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Fig. 1. Temperature dependence of the iron b-factors of pyrite, hematite and siderite obtained from phonon frequencies computed on various q-points of the Brillouin zone. Note that for the cubic structure of pyrite, the Baldereschi point exactly coincides with the q-point of a 2  2  2 shifted grid.

The two main sources of inaccuracy in the determination of the theoretical b-factors are related to the computation approach (DFT within the GGA approximation) and to the neglect of anharmonic effects. A detailed discussion of both effects can be found in Me´heut et al. (2007, 2009). As previously stated, the use of the GGA approximation is typically associated to a systematic underestimation of the phonon frequencies. This is expected to lead to a sys-

tematic error on the b-factor (see Section 3.2 and Appendix B of Me´heut et al., 2009). This kind of error can be accounted for and minimized by multiplying the theoretical frequencies by a scaling factor in order to match the experimentally measured frequencies. The scaling factors were quantified by taking the best linear-fit of the theoretical versus experimental frequencies. We found scaling factors of 1.015 ± 0.012, 1.083 ± 0.014 and 1.061 ± 0.012 for pyrite,

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Table 4 Fits of 103 ln b based on the function ax + bx2 + cx3, with x = 106/T2 (T in K) for 57Fe/54Fe isotope fractionation. Relative uncertainties on the b-factors were determined by propagating the standard deviation found on the scaling factors. Mineral Pyrite Hematite Siderite

a 1.2437 0.9940 0.5727

b

Relative uncertainty on 103 ln b (%)

c 3

4.8242  10 4.6883  103 2.3648  103

hematite and siderite, respectively (standard deviations correspond to 2r). The converged temperature dependence of the b57/54 of the three minerals is reported in Table 4 and Fig. 2. The difference between results obtained with or without scaling factors gives a hint on the accuracy of our calculated absolute b-factors (Fig. 2a). Given that the adjustments were made so as the computed infrared and Raman frequencies match experimental data, the uncertainty on the b-factors (i.e. error bars in Fig. 2) can be assessed by propagating the uncertainty found on the scaling factors.

5

2.6833  10 3.7137  105 3.6840  105

2.2 2.5 2.2

Iron b-factors of the three minerals are noticeably different; pyrite shows the highest value and siderite has the lowest one. Polyakov and Mineev (2000) and Polyakov et al. (2007) determined equilibrium reduced partition functions from Mo¨ssbauer spectroscopic data on the second-order Doppler shift. They also used inelastic nuclear resonant X-ray scattering (INRXS) to obtain the b-factor of hematite. All methods provide the same relative order with siderite, hematite and pyrite having increasing b-factors (Fig. 2b). As shown by Polyakov and Mineev (2000), two main parameters control this order: the oxidation state

Fig. 2. Temperature dependence of the iron b-factors of pyrite, hematite and siderite. (a) DFT results obtained from the raw phonon frequencies (dotted lines) and the scaled frequencies (continuous lines). Error bars were determined by propagating the standard deviation found on the scaling factors. (b) DFT results obtained from the scaled frequencies are compared with data from Polyakov and Mineev (2000) and Polyakov et al. (2007) reported as dashed lines. For these latter data, error bars of pyrite were taken from Polyakov et al. (2007) and we assign arbitrarily the same error bars to siderite given that no information is available for this mineral. As for hematite, the error bars correspond to the standard error determined by Polyakov et al. (2005) for the INRXS method.

Iron isotope fractionation between pyrite, hematite and siderite

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and the degree of covalence of the chemical bonds. b-Factors for ferric ions are higher than those for ferrous ions in the case of minerals with ionic-type Fe bonds and the iron b-factors increases with increasing covalence of the chemical bonds. In detail, we note that differences exist between DFT and Mo¨ssbauer-derived results. For pyrite, a difference of 4.0& at 0 °C and 2.3& at 100 °C is observed while Mo¨ssbauer-derived data for hematite are slightly above the upper uncertainty limit of DFT calculations. Lastly, siderite results are in perfect agreement. 3.3. Sulfur reduced partition functions (b-factors) As we will see in the discussion part, the few iron isotopic data available do not permit to discuss the discrepancy observed for pyrite between DFT and Mo¨ssbauer-derived b-factors. However, one way to test the accuracy and validity of the calculated vibrational properties of pyrite is to look at the equilibrium sulfur isotope fractionation since iron and sulfur b-factors are dependent (Polyakov and Mineev, 2000; Polyakov et al., 2007). Fig. 3 and Table 5 report the temperature dependence of the b34/32 for the three sulfides studied here. These b-factors were determined from the phonon frequencies calculated on a 2  2  2 q-point shifted grid. Results do not change with the use of a denser q-point grid. The linear-fit of the phonon frequencies of sphalerite gave a scaling factor of 1.071 ± 0.020 (2r). On the other hand, experimental frequencies of galena are not accurate enough for determining the scaling factor. As a rough estimation, the scaling factor was assessed from the bond lengths accuracy. The relative overestimation of the bond length with respect to the experimental value is 1.64 times larger in galena (PbS bond length) than in sphalerite (ZnS bond length). Then a scaling factor of 1.116 ± 0.020 (2r) was used for the phonon frequencies of galena, which corresponds to a relative underestimation of the frequencies 1.64 times larger in galena than in sphalerite. Our theoretical results can be compared with data of Elcombe and Hulston (1975). These authors determined the sulfur b-factors of these two minerals from lattice dynamics calculations using a shell model obtained by fitting the calculated frequencies to the room temperature dispersion curves observed in three directions. Considering the uncertainties of both methods, sphalerite results are in agreement while DFT gives a larger b-factor than the lattice dynamics study for galena (Fig. 3). For pyrite, Polyakov et al. (2007) estimated the sulfur b-factor from the experimental heat capacity of Ogawa (1976) and the iron b-factor that they derived using Mo¨ssbauer measurements. At 0 °C, this procedure gives a result that is 4.5& lower than the DFT result, i.e. 24.72 ± 0.55&.

Fig. 3. Temperature dependence of the sulfur b-factors of pyrite, sphalerite and galena. b-Factors obtained from the scaled DFT frequencies are compared with data from Polyakov et al. (2007) and data from lattice dynamics obtained by Elcombe and Hulston (1975).

compared with the available experimental data in the following paragraphs. 4.1.1. Iron isotope fractionation between hematite and siderite In Fig. 4, we compare the theoretical equilibrium iron isotope fractionations between hematite and siderite with the available experimental data, which actually correspond to a combination of experimental data for Fe(III)aq-hematite (Skulan et al., 2002), for Fe(II)aq-siderite (Wiesli et al., 2004) and for Fe(II)aq–Fe(III)aq (Welch et al., 2003). Skulan et al. (2002) and Welch et al. (2003) used 57Fe-enriched tracer experiments to deduce the equilibrium fractionation factors whereas Wiesli et al. (2004) performed abiotic precipitation experiments at 20 °C. Considering the uncertainties inherent to each method, the overall agreement between theoretical, Mo¨ssbauer- and INRXS-derived data, and experimental data is good. The fact that DFT predictions fall in the lower end of this data range might be due to a slight underestimation of the hematite b-factor by DFT (Fig. 2), even though the reason of such an underestimation remains unclear in light of the reliable description of the hematite properties. It has also been checked that the frequency shifts of Raman modes with 18O substitution are in good agreement with experimental measurements, i.e. within the experimental uncertainty (Massey et al., 1990).

4. DISCUSSION 4.1. Iron isotope fractionation between minerals The temperature dependence of the iron isotope fractionation between the three minerals, calculated from the rescaled phonon frequencies, are reported in Table 6 and

4.1.2. Iron isotope fractionation for the mineral pairs pyrite– hematite and pyrite–siderite For the two mineral pairs pyrite–hematite and pyrite– siderite, the equilibrium fractionation factors predicted by DFT are significantly different from those derived from Mo¨ssbauer and/or INRXS data (Fig. 5). At 0 °C, DFT

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Table 5 Fits of 103 ln b based on the function ax + bx2 + cx3, with x = 106/T2 (T in K) for 34S/32S isotope fractionation. Relative uncertainties on the b-factors were determined by propagating the standard deviation found on the scaling factors. Mineral Pyrite Sphalerite Galena

a

b

1.9617 1.3907 0.5953

Relative uncertainty on 103 ln b (%)

c 3

9.5397  10 4.8719  103 0.8524  103

5

6.0390  10 2.2996  105 0.2135  105

Table 6 Fits of 103 ln a based on the function ax + bx2 + cx3, with x = 106/T2 (T in K) for

2.3 3.7 3.6

57

Fe/54Fe isotope fractionation.

Minerals

a

b

c

Relative uncertainty on 103 ln a (%)

Pyrite–hematite Pyrite–siderite Hematite–siderite

2.4975  101 6.7096  101 4.2204  101

1.3591  104 2.4594  103 2.5198  103

1.0304  105 1.0007  105 1.1122  105

15.0 4.6 6.5

Fig. 4. Temperature dependence of the iron isotope fractionation between hematite and siderite. Equilibrium fractionation factors from Polyakov et al. (2007) (dashed line) and from the scaled DFT frequencies (continuous line) are compared with experimental data (see text).

results are about 3& and 4& lower for pyrite–hematite and pyrite–siderite, respectively. This discrepancy comes from the pyrite b-factor (Fig. 2) and its validity cannot be directly evaluated since no experimental equilibrium Fe isotope fractionation factor is known for these minerals. In the following, we discuss in detail the quality of the DFT model with respect to available measurements of vibrational and vibration-dependent properties of pyrite including phonon dispersion, heat capacity and the relation between iron and sulfur b-factors. Lastly we look at the sulfur isotope fractionation among sulfides.

Fig. 5. Temperature dependence of the iron isotope fractionation between pyrite and siderite and between pyrite and hematite. Equilibrium fractionation factors predicted by Polyakov et al. (2007) are in dashed lines and DFT calculations with scaled frequencies are in continuous lines.

4.1.2.1. Vibrational properties of pyrite. We have investigated the phonon dispersion of pyrite along three symmetry directions (Fig. 6) as well as its vibrational density of states (VDOS; Fig. 7). The phonon dispersion compares well with the dispersion that Bu¨hrer et al. (1993) measured by coherent inelastic neutron scattering (see their Fig. 3). These authors analyzed their experimental data with a rigid ion model involving short-range valence forces and derived from it several properties of which the total VDOS and the eigenvectors weighted VDOS to separate the contributions of sulfur and iron (Fig. 5 in Bu¨hrer et al., 1993). The same features are present in our calculated VDOS. We clearly see the separation of sulfur and iron contributions into the high and low frequency regions, respectively (Fig. 7). The heat capacity at constant volume, which de-

Iron isotope fractionation between pyrite, hematite and siderite

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Fig. 6. Calculated phonon dispersion curves of pyrite along the [1 0 0], [1 1 0] and [1 1 1] directions.

pends mainly on the phonon frequencies, has also been calculated in the harmonic approximation using the following equation (Kittel, 1976)  2 hm eðhm=kT Þ C V ¼ 3N at k ð3Þ ðhm=kT Þ  1 Þ2 kT ðe Symbols are the same as in Eq. (2). DFT results using either the raw phonon frequencies or the scaled frequencies are in excellent agreement with the experimental measurements of Ogawa (1976) (Fig. 8).

4.1.2.2. Iron and sulfur b-factors of pyrite. It is important to note that, at any temperature, the b-factor values determined by Polyakov et al. (2000, 2007) for 57Fe/54Fe and 34 32 S/ S are about the same while DFT tells that the b34/32 is 1.6 times higher than the b57/54 (Figs. 2 and 3). Polyakov et al. (2000, 2007) estimated the sulfur b-factor from the experimental heat capacity (Ogawa, 1976) and the iron b-factor derived from Mo¨ssbauer measurements. This approach is based on the first-order thermodynamic perturbation theory (Landau and Lifshits, 1980), where b-factors are related to the kinetic energy of atoms using the following equation:   mj  mj K j 3 ln b ¼  ð4Þ mj zRT 2 where mj is the atomic mass, Kj is the kinetic energy (per mole) of the sublattice of atoms of interest (j), z is the multiplicity of isotope substitution. Second-order Doppler shift

Fig. 7. Calculated vibrational density of states of pyrite: iron and sulfur contributions (bottom) to the total one (top), which is vertically shifted.

Fig. 8. Heat capacity of pyrite at constant volume. DFT results obtained from the raw phonon frequencies (6  6  6 q-point grid) and the scaled frequencies are compared with the experimental data (polynomial fit of Ogawa’s measurements (1976) in the range 200– 350 K).

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in Mo¨ssbauer spectra provides the iron kinetic energy (KFe) whereas experimental heat capacity is related to the total kinetic energy of the crystal (KTotal). In the harmonic approximation, the heat capacity at constant volume is indeed half the first derivative of the total kinetic energy with respect to the temperature. The excellent agreement observed between the theoretical and experimental heat capacity (Fig. 8) indicates that the kinetic energy of the total crystal lattice (KTotal) predicted by DFT is similar to the one that Polyakov et al. (2007) obtained from Ogawa’s experimental data (1976). The sulfur kinetic energy is then deduced from the total and iron kinetic energies, KS = KTotal  KFe, and the sulfur b-factor is obtained using Eq. (4). Using this method, Polyakov et al. (2000, 2007) found that the kinetic energy per iron atom is larger than the kinetic energy per sulfur atom (KFe = 1.02  KS at 0 °C) while in our case it is in the opposite direction (KFe = 0.95  KS at 0 °C). Note that the DFT b-factors are the same when calculated using raw vibrational frequencies interpolated on q-point grids (Eq. (2)) or first-order thermodynamic perturbation theory (Eq. (4)). In this latter case, kinetic energies are calculated from the theoretical partial vibrational density of states. The use of the local density approximation (LDA) instead of GGA or the use of the experimental structure without relaxation does not change the relative order of the theoretical sulfur and iron kinetic energies. Given that DFT properly describes the structural, electronic, vibrational and thermodynamic properties of pyrite, we are confident that this theoretical approach also provides a reliable partition of the iron and sulfur kinetic energies. This would mean that the iron b-factor of Polyakov et al. (2007) is overestimated while the sulfur b-factor is underestimated. The measurement of the iron phonon density of states of pyrite by INRXS synchrotron radiation experiments could be an independent verification of this prediction. 4.1.2.3. Sulfur isotope fractionation among sulfides. The temperature dependence of the sulfur b-factor for pyrite– sphalerite–galena calculated using the DFT approach can be compared with previous theoretical and experimental data (Fig. 9 and Table 7). For the 34S/32S fractionation factor between sphalerite and galena, Elcombe and Hulston (1975) assessed that the uncertainty of the results obtained by lattice dynamic is ±5% at room temperature and becomes worse as temperature increases. Hence, both theoretical results are in good agreement. They also match the experimental data of Kajiwara et al. (1969) and Smith et al. (1977), with the exception of the values measured by Kajiwara et al. (1969) at the lowest temperatures. This would indicate that equilibrium was not fully reached in these low-temperature experiments. DFT results for the 34 32 S/ S fractionation factors between pyrite and galena and between pyrite and sphalerite also show a satisfactory agreement with experimental data. However we note that the theoretical curves are in the higher range of the experimental measurements. The isotopic fractionation sphalerite–galena being well reproduced, this small discrepancy would come from an overestimation by a few percent of the b34/32 of pyrite. As it has been demonstrated in the previous paragraph, iron and sulfur b-factors are dependent.

Fig. 9. Temperature dependence of the sulfur isotope fractionation in the system pyrite–sphalerite–galena. Theoretical fractionation factors are compared with experimental data (Kajiwara et al., 1969; Smith et al., 1977). Elcombe and Hulston (1975) estimated an uncertainty of ±5% at room temperature for their lattice dynamics results.

Therefore Fig. 9 suggests that the calculated iron b-factor of pyrite is reliable within this error bar. Polyakov et al. (2007) used the same approach to test the validity of their iron b-factor of pyrite. They combined their sulfur b-factor for pyrite with the theoretical data of Elcombe and Hulston (1975) for sphalerite and galena. The agreement with the experimental data is better than in our case but this might be fortuitous because the combined b-factors were obtained from different methods. The approach of Polyakov et al. (2007) thus cumulates the uncertainties due to the determi-

Iron isotope fractionation between pyrite, hematite and siderite Table 7 Fits of 103 ln a based on the function ax + bx2 + cx3, with x = 106/T2 (T in K) for

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34

S/32S isotope fractionation.

Minerals

a

b

c

Relative uncertainty on 103 ln a (%)

Pyrite–galena Pyrite–sphalerite Sphalerite–galena

1.3664 0.5710 0.7954

8.6873  103 4.6677  103 4.0195  103

5.8254  105 3.7394  105 2.0860  105

3.7 12.2 7.0

nation of the iron b-factor from Mo¨ssbauer measurements, uncertainties for the measurements of pyrite heat capacity and uncertainties coming from the use of a lattice dynamical model for sphalerite and galena In conclusion, nothing indicates that DFT does not describe properly the pyrite properties. The iron b-factor of pyrite is as accurate as for the other minerals studied here. Therefore, iron isotope fractionation factors obtained by DFT for the mineral pairs pyrite–hematite and pyrite–siderite are reliable. At the opposite to INRXS-derived b-factors, b-factors determined from Mo¨ssbauer measurements must be considered with caution because of the several sources of error inherent to the method (Polyakov et al., 2007). 4.2. Iron isotope fractionation between minerals and aqueous species of iron

give a 57Fe/54Fe fractionation factor of +1.46 ± 0.30& at 100 °C. The consistency between the iron b-factor values for hematite derived by two different methods (Fig. 2b) suggests that the theoretical Fe(III)aq and/or the equilibrium experimental data have to be reassessed. 4.2.2. Iron isotope fractionation between Fe(II)aq and siderite Wiesli et al. (2004) have assessed the equilibrium iron isotope fractionation factor between Fe(II)aq and siderite by performing abiotic synthesis experiments. These authors obtained the equivalent of a 57Fe/54Fe fractionation of +0.71 ± 0.32& at 20 °C. The theoretical prediction is higher (Fig. 11). Combining our assessment of the iron bfactor for siderite with that for Fe(II)aq from DomagalGoldman and Kubicki (2008) we found a 57Fe/54Fe fractionation factor of +2.39 ± 0.42& at the same temperature.

Several quantum mechanical works predicted the equilibrium isotopic fractionation between FeðH2 OÞ6 3þ and FeðH2 OÞ6 2þ (Anbar et al., 2005; Domagal-Goldman and Kubicki, 2008; Hill and Schauble, 2008). Here we combined our data, or those of Polyakov et al. (2007), with the b-factors of FeðH2 OÞ6 3þ and FeðH2 OÞ6 2þ calculated by Domagal-Goldman and Kubicki (2008), in order to obtain the theoretical equilibrium fractionation factors between mineral and dissolved iron. These authors performed molecular orbital/density functional theory calculations and employed the integrated equation formalism of polarized continuum model (IEFPCM) to treat the solvent effects. The disagreements observed between their results and previous works are discussed in Domagal-Goldman and Kubicki (2008). They mainly arise from the use or not of scaled frequencies to get the b-factors. 4.2.1. Iron isotope fractionation between Fe(III)aq and hematite To our knowledge, only one experimental study has investigated the iron isotope fractionation between FeðH2 OÞ6 3þ and hematite (Skulan et al., 2002). This work, performed under conditions allowing both dissolution and precipitation of hematite at 98 °C, has provided both a kinetic (for a 12 h precipitation experiment) and equilibrium fractionation factors (+1.95 ± 0.18& and 0.15 ± 0.30&, respectively, when converted to 57Fe/54Fe). The idea that no iron isotope fractionation occurs between Fe(III)aq and hematite at equilibrium, is also supported by the preferred fractionation factor that Johnson et al. (2003) obtained from natural data. However, results derived from both DFT calculations and Mo¨ssbauer measurements indicate a significant positive fractionation (Fig. 10). With our assessment of the iron b-factor for hematite, calculations

Fig. 10. Temperature dependence of the iron isotope fractionation between Fe(III)aq and hematite. The solid and empty circles are experimental data that Skulan et al. (2002) obtained for equilibrium and kinetic fractionation, respectively. Lines are the theoretical equilibrium fractionation factors obtained by combining the bfactors of Fe(III)aq of Domagal-Goldman and Kubicki (2008) (IEFPCM model), with the b-factors of hematite. Hematite bfactors are from Polyakov et al. (2007) (dashed line) and from DFT calculations with scaled frequencies (continuous line).

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Obviously, the excellent agreement observed between the DFT- and Mo¨ssbauer-derived iron b-factor of siderite is also visible here. Considering the uncertainty possibly introduced by the difficulty to model iron aqueous complexes (e.g. geometry not well constrained, simulations done on a static geometry, solvent effects) and the experimental difficulty to control and to identify the iron dissolved species, we can at least conclude that the equilibrium iron isotope fractionation between Fe(II)aq and siderite is positive. For the two aqueous–mineral fractionation experiments investigated here, the predicted fractionations are higher (Figs. 10 and 11) whereas the experimentally measured aqueous–aqueous (Anbar et al., 2005; Domagal-Goldman and Kubicki, 2008) and mineral–mineral (Figs. 4 and 9; Shahar et al., 2008) fractionations are in better agreement with the predicted fractionations when the same model is used for each aqueous species or minerals. But we must keep in mind that this agreement does not mean that the absolute values of the calculated b-factors are correct. A wide range of b-factors is reported for Fe(II)aq and Fe(III)aq species (Anbar et al., 2005; Hill and Schauble, 2008; Domagal-Goldman and Kubicki, 2008), depending on the model employed or the method applied to get the b-factors from the calculated frequencies. Although the modeling of crystals is relatively straightforward, modeling

of the dynamical properties of solutions is more difficult and requires additional approximations. So b-factors of aqueous species may not be precise enough to be combined with those calculated for minerals. If we assume that the bfactors calculated for minerals are reliable then we can assess the b-factor values of the aqueous species needed for matching the experimental isotopic fractionation data. At 22 °C, the b57/54 of Fe(III)aq would be 10.86 ± 0.27& or 11.72 ± 0.29& if we take the hematite b-factor derived by DFT or from INRXS measurements (Polyakov et al., 2007), respectively. At the same temperature, the b57/54 of Fe(II)aq would be 7.03 ± 0.18&. In this case, only one value is obtained since the siderite b-factor is exactly the same for both methods. It is noteworthy that these b-factor values are lower than the calculated values reporter in the literature (Anbar et al., 2005; Hill and Schauble, 2008; Domagal-Goldman and Kubicki, 2008). These hypothetical values lead to a 57Fe/54Fe fractionation factor between Fe(III)aq and Fe(II)aq that is equal to 3.83 ± 0.32& considering the DFT b-factors for minerals and 4.69 ± 0.34& considering the data of Polyakov et al. (2007). Experimentally, Welch et al. (2003) measured the equivalent of a 57 Fe/54Fe fractionation between Fe(III)aq and Fe(II)aq of 4.50 ± 0.35&. This would suggest that the hematite b-factor derived from accurate INRXS measurements (Polyakov et al., 2007) is slightly more precise than the DFT value, which is 0.85& lower at 22 °C (Fig. 2b). The same conclusion was made by looking at the fractionation between hematite and siderite (Fig. 4). 5. CONCLUSIONS

Fig. 11. Temperature dependence of the iron isotope fractionation between Fe(II)aq and siderite. The experimental data (solid circle) corresponds to the fractionation factor determined by Wiesli et al. (2004). Lines are the theoretical equilibrium fractionation factors obtained by combining the b-factors of Fe(II)aq of DomagalGoldman and Kubicki (2008) (IEFPCM model), with the b-factors of siderite. Siderite b-factors are from Polyakov et al. (2007) (dashed line) and from DFT calculations with scaled frequencies (continuous line).

Only few experimental data are available for assessing the accuracy of theoretical predictions of equilibrium iron isotope fractionation among pyrite, siderite and hematite, and significant effort should therefore be put in this direction in the future. Previous theoretical works on other stable isotopes (Me´heut et al., 2007, 2009) and the detailed verification of the structural, electronic and vibrational properties show that the isotopic fractionation factors between minerals computed with the theoretical framework of DFT are reliable and should provide sufficient accuracy to interpret isotopic measurements on natural samples. Our calculated iron b-factors for siderite and hematite are in reasonable agreement with those derived from Mo¨ssbauer or INRXS measurements (Polyakov and Mineev, 2000; Polyakov et al., 2007). For pyrite, results suggest that the iron b-factor of Polyakov et al. (2007) is overestimated while the sulfur b-factor is underestimated. INRXS experiments would be necessary to get the iron phonon density of states and thus an independent and more accurate estimation of the iron b-factor of pyrite. At the same time, more experimental studies will be needed to expand the currently limited database, with a particular effort on demonstrating isotopic equilibrium, e.g. by using reversals of isotopic exchange, or three-isotope methods (e.g. O’Neil, 1986; Schuessler et al., 2007; Shahar et al., 2008). For isotopic fractionation factors between solid and aqueous phases, theoretical predictions are larger than the experimental values. Computational efforts will have to be pursued to make

Iron isotope fractionation between pyrite, hematite and siderite

the ab initio values for aqueous species directly comparable to the values for solid phases. ACKNOWLEDGMENTS Calculations were performed at the IDRIS Institute of CNRS (Project No. i20080411519). C.A. Heinrich is thanked for his valuable comments. We are grateful to S. Domagal-Goldman, V.B. Polyakov and E. Schauble for helpful reviews. We also thank C.M. Johnson (AE) whose detailed comments helped improving the manuscript. This work has been supported by the French National Research Agency (ANR, project “SPIRSE”) to E.B., a 3F program (CNRS-INSU) to F.P. and an “Ope´ration Scientifique” from the University of Toulouse to F.P. This work is IPGP contribution n°2546.

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