A general moment NRIXS approach to the determination of

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Geochimica et Cosmochimica Acta 94 (2012) 254–275

A general moment NRIXS approach to the determinationof equilibrium Fe isotopic fractionation factors: Application

to goethite and jarosite

N. Dauphas a,⇑, M. Roskosz b, E.E. Alp c, D.C. Golden d, C.K. Sio a, F.L.H. Tissot a,M.Y. Hu c, J. Zhao c, L. Gao c, R.V. Morris e

a Origins Laboratory, Department of the Geophysical Sciences and Enrico Fermi Institute, The University of Chicago,

5734 South Ellis Avenue, Chicago, IL 60637, USAb Unite Materiaux et Transformations, Universite de Lille 1, CNRS UMR 8207, 69655 Villeneuve d’Ascq, France

c Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USAd Engineering and Science Contract Group-Hamilton Sundstrand, Mail Code JE23, Houston, TX 77058, USA

e NASA Johnson Space Center, Houston, TX, USA

Received 14 November 2011; accepted in revised form 21 June 2012

Abstract

The equilibrium Fe isotopic fractionation factors of goethite and jarosite have considerable importance for interpreting Feisotope variations in low temperature aqueous systems on Earth and possibly Mars in the context of future sample returnmissions. We measured the b-factors of goethite FeO(OH), potassium-jarosite KFe3(SO4)2(OH)6, and hydronium-jarosite(H3O)Fe3(SO4)2(OH)6, by Nuclear Resonant Inelastic X-ray Scattering (NRIXS, also known as Nuclear Resonance Vibra-tional Spectroscopy – NRVS or Nuclear Inelastic Scattering – NIS) at the Advanced Photon Source. These measurementswere made on synthetic minerals enriched in 57Fe. A new method (i.e., the general moment approach) is presented to calculateb-factors from the moments of the NRIXS spectrum S(E). The first term in the moment expansion controls iron isotopic frac-tionation at high temperature and corresponds to the mean force constant of the iron bonds, a quantity that is readily mea-sured and often reported in NRIXS studies. The mean force constants of goethite, potassium-jarosite, and hydronium-jarositeare 314 ± 14, 264 ± 12, and 310 ± 14 N/m, respectively (uncertainties include statistical and systematic errors). The general

moment approach gives 56Fe/54Fe b-factors of 9.7, 8.3, and 9.5& at 22 �C for these minerals. The b-factor of goethite mea-sured by NRIXS is larger than that estimated by combining results from laboratory exchange experiments and calculationsbased on electronic structure theory. Similar issues have been identified previously for other pairs of mineral–aqueous species,which could reflect inadequacies of approaches based on electronic structure theory to calculate absolute b-factors (differencesin b-factors between aqueous species may be more accurate) or failure of laboratory experiments to measure mineral–fluidequilibrium isotopic fractionation at low temperature. We apply the force constant approach to published NRIXS dataand report 1000 � ln b for important Fe-bearing phases of geological and biochemical relevance such as myoglobin, cyto-chrome f, pyroxene, metal, troilite, chalcopyrite, hematite, and magnetite.� 2012 Elsevier Ltd. All rights reserved.

1. INTRODUCTION

In near-neutral oxic conditions, oxidation of ferrousiron (Fe2+) can take place very rapidly to produce ferric

0016-7037/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.gca.2012.06.013

⇑ Corresponding author.E-mail address: [email protected] (N. Dauphas).

iron (Fe3+) that rapidly precipitates as insoluble forms suchas ferric oxyhydroxide, explaining the widespread occur-rence of these minerals in low temperature aqueous systems(Stumm and Morgan, 1996). Goethite and hematite are themost abundant Fe-bearing oxides present at Earth’s sur-face, where they are ubiquitous in soils as well as lacustrineand marine sediments. At low temperature (<100 �C),


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N. Dauphas et al. / Geochimica et Cosmochimica Acta 94 (2012) 254–275 255

goethite is thermodynamically favored over most iron oxi-des but other parameters such as reaction kinetics, grainsize, water activity, and element substitution can influencethe nature of the minerals that form (Cornell and Schwert-mann, 2003).

On Mars, acidic conditions resulted in precipitation ofthe sulfate mineral jarosite at Meridiani Planum (Kling-elhofer et al., 2004) and goethite has been detected in out-crop rocks of the Columbia Hills at Gusev Crater (Morriset al., 2006, 2008). Goethite may also be a precursor tohematite associated with jarosite at Meridiani Planum(Tosca et al., 2005; Zolotov and Shock, 2005). Similarmineral assemblages to Martian outcrops have been foundon Earth in acid mine drainage environments, in naturalpyritic ore bodies (Nordstrom Darrell et al., 1979; Fernan-dez-Remolar et al., 2005) and in volcanic edifices nearfumaroles (Morris et al., 2005).

Biogeochemical transformations of iron bearing miner-als and their relationships to dissolved species can be inves-tigated by measuring iron isotope variations in rocks,minerals and fluids (Dauphas and Rouxel, 2006; Johnsonet al., 2008). For example, coupled carbon and iron isotopestudies of banded-iron formations have revealed the antiq-uity of microbial iron respiration in sediments (Heimannet al., 2010; Craddock and Dauphas, 2011). However, a ma-jor difficulty for interpreting iron isotope variations is thatmost equilibrium fractionation factors have not beenrigorously determined. Several approaches can be used toestimate equilibrium Fe isotope fractionation factors:

(1) Products of laboratory experiments can be directlymeasured (Skulan et al., 2002; Welch et al., 2003;Schuessler et al., 2007; Shahar et al., 2008; Poitrassonet al., 2009; Beard et al., 2010; Saunier et al., 2011;Wu et al., 2011). The main difficulties in thisapproach are to assess the degree of equilibrationand to monitor and correct for all kinetic isotopeeffects that could affect the results.

(2) Calculations based on electronic structure theory andother computational methods provide estimates ofreduced partition function ratios or b-factors (Schau-ble et al., 2001; Anbar et al., 2005; Domagal-Gold-man and Kubicki, 2008; Hill and Schauble, 2008;Ottonello and Vetuschi Zuccolini, 2008, 2009; Blan-chard et al., 2009; Domagal-Goldman et al., 2009;Hill et al., 2009; Rustad and Yin, 2009). The accuracyof these methods can be difficult to assess and theresults are variable depending on the assumptionsthat are made and the parameters that are used.

(3) The vibrational properties of iron in solids can beinvestigated by using nuclear resonance inelastic X-ray scattering (NRIXS), from which b-factors canbe derived (Polyakov et al., 2007; Polyakov, 2009;Polyakov and Soultanov, 2011).

This last method is relatively new in isotope geochemistry(Polyakov et al., 2005a,b). Initially, Polyakov and cowork-ers used Mossbauer spectroscopy to calculate iron b-factorsand to identify the parameters that could affect iron isotopicfractionation such as oxidation state (Polyakov, 1997;

Polyakov and Mineev, 2000). While this method is theoret-ically sound, the determination of second-order Dopplershifts is difficult using conventional Mossbauer spectroscopymeasurements. The positions of the absorption lines shift asa function of temperature for a variety of reasons, which areadditive. These include the second-order Doppler shift,which is determined by kinetic energy and (chemical) isomershift. Additional complications arise if there is a distributionof hyperfine field parameters caused by changes of chemicalor magnetic nature. Carefully crafted experiments areneeded to disentangle these effects reliably and reproducibly.For example, Polyakov and Mineev (2000) predicted theequilibrium fractionation factor of magnetite, based on con-ventional Mossbauer data from De Grave et al. (1993), thatwas later proven to be inaccurate when mineral equilibrationexperiments were performed (Shahar et al., 2008) and newconventional Mossbauer and NRIXS data became available(Polyakov et al., 2007 and references therein). NRIXS is afar superior method than Mossbauer spectroscopy that usesa synchrotron X-ray source to probe the vibrational proper-ties of iron atoms in the solid lattice. Polyakov and cowork-ers used the kinetic energy derived from the partial phonondensity of states (PDOS) of the Fe sub-lattice to estimate b-factors. In this paper, we outline a superior approach basedon the determination of the moments of the NRIXS spec-trum S(E). At high temperature, this approach reduces todetermining the iron force constant, which is given by thethird moment of the NRIXS spectrum. NRIXS data for syn-thetic powders of 57Fe-enriched goethite, K-jarosite, andH3O-jarosite are reported and the geochemical implicationsof these measurements are presented.

2. MATERIALS AND METHODS

2.1. Synthesis of 57Fe-enriched minerals

Because NRIXS is only sensitive to 57Fe among all Feisotopes (its natural abundance is 2.119%) and synchrotronbeam-time is valuable, samples enriched in 57Fe were pre-pared in order to minimize beam-time. The starting mate-rial consists of 57Fe-rich metal (95%) purchased fromCambridge Isotope Laboratories, Inc. Details of the min-eral synthesis have been published previously (Goldenet al., 2008) and are briefly described below. For the synthe-sis of hydronium-jarosite, 100 mg of 57Fe powder wastransferred into a clean 23-mL Teflon-lined Parr reactionvessel. Approximately 145 lL of 18 M H2SO4, 40 lL ofHClO4 (oxidant), and 2 mL of H2O were added to this ves-sel, which was closed airtight and heated for 17 h in an ovenat 140 �C. The reaction vessel was cooled to room temper-ature in a freezer before opening. The pH was recorded(�0.6) and it was brought up to �1 (actual pH = 1.08) byadding appropriate amounts of Mg-hydroxycarbonate(�1 mg). The vessel was closed airtight and was heated asecond time at 140 �C for 17 h. The run-product waswashed in distilled water by centrifugation and decantationin a high-speed centrifuge. The final product was freeze-dried and stored until further use. Potassium-jarosite wasprepared in a similar fashion except that before the secondheating step, 54 mg of K2SO4 was added and the pH was

256 N. Dauphas et al. / Geochimica et Cosmochimica Acta 94 (2012) 254–275

adjusted to �2.1–2.75 (actual value = 2.72) by adding Mg-hydroxycarbonate. Goethite was precipitated by the sameprotocol as that used for the synthesis of hydronium-jaro-site except that the pH was adjusted to 2.5 instead of 1. Itis worth noting that the addition of Mg-hydroxycarbonatefor pH adjustment does not affect jarosite formation be-cause Mg only forms soluble sulfates that were efficientlyremoved during the washing step. The nature of all the min-erals synthesized was checked by X-ray powder diffraction.The particle sizes (equivalent sphere diameters of coherentdomain sizes) were determined using the peak broadeningof Rietveld refined XRD (JADE software package, Materi-als Data Inc.). The particle sizes are 19.8 nm for goethiteand 223.0 nm for K-jarosite.

2.2. Principles of NRIXS spectroscopy and experimental

setup

Nuclear Resonant Inelastic X-ray Scattering (NRIXS) isa recently established spectroscopic technique that allowsone to probe the vibrational properties of certain elementsin solids (Seto et al., 1995; Sturhahn et al., 1995; Chumakovand Sturhahn, 1999). Its use in Earth sciences has focusedon high-pressure applications to determine seismic veloci-ties and phonon density of states of minerals relevant toEarth’s deep mantle and core (Sturhahn and Jackson,2007). More recently, Polyakov and coworkers have shownthat NRIXS could be used in stable isotope geochemistry todetermine equilibrium fractionation between minerals(Polyakov et al., 2005a,b, 2007; Polyakov, 2009; Polyakovand Soultanov, 2011). The NRIXS method, like conven-tional Mossbauer spectroscopy, relies on the fact that57Fe possesses a low-lying nuclear excited state at14.4125 keV that can be populated by X-ray photons ofthe appropriate energy. Conventional Mossbauer Spec-trometers are not sufficiently tunable to cover the phononspectral range, which is typically within ±80 meV. TypicalMossbauer drives operate at ±100 mm/s, correspondingto ±500 neV (for 57Fe transition). Sophisticated apparatushave extended this range to 2.6 meV, but for very limitedapplications (Rohlsberger et al., 1997). In NRIXS, the pho-ton source is a bright X-ray beam produced by a synchro-tron. The primary beam is pulsed and has a broad energyspectrum, from 1 to 1000 keV. It passes through two mono-chomators that reduce this energy spread to 1 meV (Fig. 1).These high-resolution monochromators use several crystalsin Bragg diffraction positions (Toellner, 2000). This mono-chromatic beam is then focused using X-ray mirrors to�10 lm. The signal resulting from interaction of the inci-dent beam with 57Fe in the sample is measured at an angleon the same side as the incident beam. The detector is anAvalanche Photo-Diode (APD). Extensive details onNRIXS and its applications can be found in several contri-butions (Sturhahn et al., 1995; Chumakov and Sturhahn,1999; Alp et al., 2002; Sturhahn and Jackson, 2007). Inter-actions of the incident beam with the sample are describedin more detail below.

Most of the incident X-rays are scattered by electronicshells without interacting with the Fe nuclei. The signal

from these photons is almost instantaneous (<10�12 s). Asmall fraction of the incident X-rays can excite the 57Fe nu-cleus. The lifetime of this excited nucleus is 141 ns, meaningthat the signal from nuclear scattering will be received withsome delay. Because the primary beam is pulsed (153 ns be-tween pulse, each of 70 ps duration), it is possible to apply atime discriminator on the incoming signal, remove the elec-tron contribution and retain only the nuclear scattering sig-nal. The monochromator is tunable by physically rotatingthe crystals to change the Bragg angle. A typical energyscan would be from �80 to +80 meV around the elasticpeak at 14.4125 keV.

Recoilless elastic scattering, as in Mossbauer spectros-copy, produces part of the signal. The rest of the nuclearscattering signal is influenced by the vibrational propertiesof the iron lattice. Atoms in solids are in oscillating move-ments. These collective movements can be decomposed intonormal modes of vibration that have particle-like propertiesknown as phonons. When a photon of higher energy thanthe resonance energy impacts a 57Fe nucleus, part of thatenergy can go into exciting the nucleus while the rest cango into exciting certain modes of vibration of the solid lat-tice. This process is known as phonon creation. Conversely,the incident photon can have lower energy than what is re-quired to excite 57Fe but part of the vibrational energy inthe crystal lattice can fill this energy gap and allow forthe transition to happen. This is known as phonon annihi-lation. Thus, the nuclear resonant transitions that happenoff the nominal resonance energy are influenced by the pho-non energy spectrum. After data processing and assumingthat the interatomic potential is quadratic in the atomic dis-placement (harmonic approximation), it is possible to re-trieve the partial phonon density of states (PDOS)(Sturhahn et al., 1995; Chumakov and Sturhahn, 1999).The term “partial” refers to the fact that the phonon den-sity of state is only relevant to the iron sub-lattice. Whenworking on a large monocrystal, the PDOS is also projectedin the sense that it only probes the solid in the direction ofthe incident beam. When working with fine powder orglass, this is not an issue as the material is isotropic at thescale of the incident beam.

The powdered samples of jarosite and goethite weremounted as compressed pellets into aluminum holders.The NRIXS measurements were done at beamline 3-ID-Bof the Advanced Photon Source (APS) at Argonne Na-tional Laboratory. The storage ring was operated in top-up mode with 24 bunches separated by 153 ns. The sampletilt relative to the incident beam was 5� and an APD detec-tor was positioned a few millimeters away from the sample.A second APD was positioned downstream to measure theaverage energy resolution (full width at half maximum) of1.33 meV. The monochromator was tuned from �120 to+130 meV with a step size of 0.25 meV and a collectiontime of 5 s per step. The minerals were rich in Fe, so 1 or2 scans were sufficient to yield high-quality data. All datawere acquired at room pressure and room temperature(299 ± 1 K). Data reduction was done with the PHOENIXsoftware (Sturhahn, 2000) and a Mathematica script writ-ten by the authors (Appendix A).

Si

Si

Si

ΔE=1.2 eV ΔE=1 meV

Pre-monochromator

High-resolution monochromator

APD

Sample

Detector

Si1-1,000 keV

Synchrotron X-ray source

153 ns

70 ps

14.4 keV14.4 keV

Time

Log(intensity)Electronic scattering

Nuclear scattering

0

Data acquisition

Fig. 1. Experimental set-up used for NRIXS spectroscopy (modified from Sturhahn and Jackson, 2007). The pulsed X-ray beam passesthrough two monochromators to achieve an energy dispersion of ±1 meV at 14.4 keV. This energy can induce a nuclear transition to anexcited state of 57Fe, which decays to the ground state with a lifetime of 141 ns. The signal from electronic scattering is separated from thatfrom nuclear scattering using the fact that the latter signal is produced with some delay. The energy spectrum is produced by scanning theenergy (e.g., �120 to +130 meV) with the high-resolution monochromator, and recording the cummulative signal from 15 to 140 ns after eachpulse of the synchrotron beam.


3. APPLICATION OF NRIXS SPECTROSCOPY TO

ISOTOPE GEOCHEMISTRY

Iron possesses an isotope with a low-lying nuclear excitedstate, making it possible to directly probe its vibrationalproperties using synchrotron radiation and to calculateequilibrium isotopic fractionation factors (Polyakov et al.,2007; Polyakov, 2009; Polyakov and Soultanov, 2011). Tobe more precise, NRIXS allows one to derive reduced parti-tion function ratios or b-factors, from which equilibriumisotopic fractionation between two phases A and B canbe calculated by using, dA � dB ’ 1000 � (ln bA � ln bB).Below, we present a new approach to calculate b-factorsfrom the moments of the NRIXS spectrum S(E). The firstterm in the moment expansion controls iron isotopicfractionation at high temperature and corresponds to themean force constant of the iron bonds, a quantity that isreadily measured and often reported in NRIXS studies.

3.1. General moment approach

b-factors can be calculated based on series expansions inmoments of the PDOS of the iron kinetic energy (Polyakovet al., 2005b, 2007) or reduced partition function ratio(Appendix B). The expansion is well approximated by thethree lowest order terms,

1000� ln bI=I� ’1000MM� �1

� �mg

2

8k2T 2� mg

4

480k4T 4þ mg

6

20;160k6T 6

� �;

ð1Þ

where I and I* are two isotopes of iron of masses M andM*, and the jth moment of the PDOS g(E) is given by,

mgj ¼

Z þ1

0

EjgðEÞdE: ð2Þ

Polyakov et al. (2005a,b, 2007), Polyakov (2009) and Polya-kov and Soultanov (2011) used this equation to calculateb-factors in minerals from NRIXS data. In Appendix C,we show for the first time that the even moments of g can

be obtained directly from the moments of S and Eq. (1)can be rewritten as,

1000� ln bI=I� ¼1000MM� �1

� �� 1

Er

RS3

8k2T 2�RS

5�10RS2RS

3

480k4T 4þRS

7þ210ðRS2Þ

2RS3�35RS

3RS4�21RS

2RS5

20;160k6T 6

" #; ð3Þ

where the jth moment of S centered on ER is,

RSj ¼

Z þ1

�1SðEÞðE � ERÞjdE; ð4Þ

with ER ¼ E20=2c2 the free recoil energy (1.956 meV for the

E0 = 14.4125 keV nuclear transition of 57Fe). When rawNRIXS data are available, all the moments of S(E) canbe calculated and 1000 � ln b can be written as A1/T2 + A2/T4 + A3/T6. As further discussed below, it is rec-ommended that Eq. (3) be used over Eq. (1) in future appli-cations to Fe isotope geochemistry because the forceconstant estimated from S(E) is less sensitive to the as-sumed background than that derived from g(E) (providedthat the background is constant and the energy range issymmetric), S(E) can be extrapolated beyond the energyacquisition range to account for missing n-phonon contri-butions (but not missing normal mode frequencies), andcoefficients obtained using Eq. (3) have smaller statisticaluncertainties than those obtained using Eq. (1) (Fig. 2).

At high temperature, the first terms in Eqs. (1) and (3)provide a sufficient approximation (Fig. 3 shows underwhich conditions this formula is valid),

1000� ln bI=I� ’ 1000MM� � 1

� �mg

2

8k2T 2

¼ 1000MM� � 1

� �1

Er

RS3

8k2T 2: ð5Þ

We recognize in these expressions a term that correspondsto the average restoring force constant of the harmonicoscillators holding the element in position (Lipkin, 1995,1999; Kohn and Chumakov, 2000),

2×105

4×105

6×105

8×105

10×105

12×105

2×105 4×105 6×105 8×105 10×105 12×105

A1

fro

m g

(E)

A1 from S(E)

-11×109

-8×109

-5×109

-2×109

1×109

-11×109 -8×109 -5×109 -2×109 1×109

A2

fro

m g

(E)

A2 from S(E)

-5×1013

5×1013

15×1013

25×1013

-5×1013 5×1013 15×1013 25×1013

A3

fro

m g

(E)

A3 from S(E)

Fig. 2. Comparison between the coefficients in the polynomialexpansion 1000 � ln b = A1/T2 + A2/T4 + A3/T6 obtained fromg(E) (Eq. (1)) and S(E) (Eq. (3); Appendix C). The samples wereall measured at the APS using the same acquisition protocol as thatused in the present study. There is good agreement between thevarious coefficients but those obtained from S(E) have uncertain-ties that are on average �2 times (and up to 5 times) smaller thanthose obtained from g(E). For this, and other reasons discussed inthe text, Eq. (3) is preferred over Eq. (1).

0

50

100

150

200

250

300

350

400

450

500

0 100 200 300

F (

N/m

)

Fig. 3. Relative departure from Eq. (7) introduced by the B2

term in Eq. (12) for the 56Fe/54Fe ratio. The relative departure isequal to (B1hFi/T2 � B2hFi2/T4)/B1hFi/T2 = 1 � (B2/B1)hFi/T2.With B1 = 2904 and B2 � 52,000, the correction factor becomes1–17.9hFi/T2 (gray continuous lines). For high-temperature phases,to a good approximation we have 1000 � ln b ’ 2904hFi/T2.


hF i ¼ M

�h2

Z þ1

0

E2gðEÞdE ¼ M

ER�h2

Z þ1

�1ðE � ERÞ3SðEÞdE:

ð6Þ

Therefore, Eq. (5) can be rewritten as,

1000� ln bI=I� ¼ 10001

M� �1

M

� ��h2

8k2T 2hF i; ð7Þ

which is a familiar formula in isotope geochemistry (Herz-feld and Teller, 1938; Bigeleisen and Goeppert-Mayer,1947). The force constant in Eq. (6) corresponds to the sec-ond-order derivative of the interaction potential, whichshould be constant for a harmonic oscillator (Lipkin,1995, 1999). Measuring the force constant at different tem-peratures offers a means of testing the possible anharmonic-ity of lattice vibrations (Sturhahn, 2004). The forceconstant calculated using the excitation function S(E) is lesssensitive to background subtraction than that derived fromg(E) because positive and negative terms annihilate in Eq.(6) for symmetric energy scans;

Rþx�x bE3dE ¼ 0 with b con-

stant. Note that in reality, b may change with time andthe scan may not be perfectly centered and symmetric,which can affect the force constant calculated from S(E).Most papers reporting NRIXS data give the mean forceconstant computed in this manner (Table 1 and referencestherein). The force constant can also be calculated fromthe PDOS g(E) but this method is less reliable as thebackground can affect it and the statistical uncertaintiesare larger (Fig. 2). Nonetheless, in properly acquired mea-surements, the force constant calculated using either meth-od should agree. The energy cube or square (Eq. (6)) arefactors in the integrands that give the force constant. Thismeans that even small bumps in the high-energy tails ofthe excitation function S(E) or in the PDOS g(E) can havesufficient weight to affect the force constant and hence the

0

20,000

40,000

60,000

80,000

100,000

B2

Biomolecule MetalOxide/Hydroxide Sulphate

SilicateCarbonate

Debye

Fig. 4. Determination of B2 used to calculate equilibrium isotopicfractionation factors (56Fe/54Fe) from the force constant;1000 � ln b = 2904hFi/T2 � B2hFi2/T4. B2 is a parameter thatrelates the 4th moment of the PDOS to the 2nd moment. For aDebye solid, this value is 37,538. For other solids with moreextended PDOS, the B2 values can be higher and reach �70,000. Inmost cases, adopting a constant value of 52,000 provides sufficientaccuracy on calculated equilibrium isotopic fractionation factors asthis is only a minor second-order correction that affects phases withhigh hFi values at low temperature.


equilibrium isotopic fractionation factors. Because thesetails are characterized by low counting statistics, determin-ing mean force constants is particularly challenging, as it re-quires broad energy scans and long acquisition times. Apotential difficulty arises when the energy scan has insuffi-cient width and some NRIXS signal from multiple phononsat high energy is truncated. The PHOENIX software allowsone to assess and correct for this possible shortcoming bycalculating the multiple phonon-contribution after singlephonon decomposition and using this to extrapolate S(E)in a physically sound manner (“force constant after refine-ment” in Table 2). However, this extrapolation methodcannot account for normal vibration modes that would bepresent outside of the scanned energy range and those bur-ied under the background.

3.2. Application to published force constant data

Below, we provide an alternative approximate approachthat is particularly useful for calculating equilibrium iso-tope fractionation from published force constants derivedfrom NRIXS data. The 4th moment of the PDOS (secondterm in Eq. (1)) can be estimated from the 2nd moment ifwe approximate the PDOS with a Debye model (a Debyemodel assumes linear dispersions, i.e., phonon energy is alinear function of its momentum).

gðEÞ ¼ 3E2=E2D; E 6 ED

gðEÞ ¼ 0; E > ED;ð8Þ

where ED is the Debye energy cutoff. In this framework wehave,

hF i ¼ M

�h2

Z ED

0

gðEÞE2dE ¼ 3

5

M

�h2E2

D; ð9Þ

so,Z ED

0

gðEÞE4dE ¼ 3

7E4

D ¼25

21

�h4

M2hF i2: ð10Þ

We can therefore rewrite the second term of Eq. (1) in theframework of a Debye model as,

1000� ln bI=I� ¼ 1000�h2

8k2

hF iT 2� 5�h4

2016k4M

hF i2

T 4Þ 1

M� �1

M

� �:

ð11Þ

Replacing the constants with the relevant numerical values,we have for iron isotopic ratio 56Fe/54Fe,1000 � ln b = 2904hFi/T2 � 37,538hFi2/T4, with hFi in N/m and T in K. This formula, which is only valid for a Debyesolid, can be generalized for any PDOS by writing,

1000� ln b ¼ B1hF iT 2� B2

hF i2

T 4ð12Þ

where B1 = 2904 and B2 is a numerical value that dependson the shape of g(E) or S(E). Values of B2 were evaluatedfor 78 phases ranging from metals with Debye-like behav-iors to hydroxides or high-pressure phases with extendedPDOS profiles. The exact values of 1000 � ln b were calcu-lated in the temperature range 0–500 �C using Eq. (B6)when only the PDOS was available or using Eq. (3) when

raw NRIXS data were available. The first term (Eq. (11);2904hFi/T2) was then subtracted and the residuals were fit-ted with a function in 1/T4. The values of B2 range from�40,000 to 80,000 and depend on the nature of the phase(Fig. 4; B2 � 39,000 for biomolecules, 36,000 for metals,51,000 for oxides and hydroxides, 64,000 for sulfates,52,000 for sulfides, 68,000 for silicates, 80,000 for one car-bonate). To a good approximation one can take B2 con-stant and write,

1000� ln b ¼ 2904hF iT 2� 52; 000

hF i2

T 4; ð13Þ

with hFi in N/m and T in K. The relative correction intro-duced by the second term is shown in Fig. 3. For mineralsrelevant to low temperature geochemistry, using a constantB2 value of 52,000 rather than measured ones introduces aninaccuracy of less than �0.2& at 22 �C (Table 1), which issmaller than the overall uncertainty of the method. The ap-proach outlined here is particularly useful for calculating b-factors from published data as the mean force constant isoften reported. Another significant advantage of the pro-posed method is that error propagation is straightforwardif the uncertainties on the force constant and B2 are knownand not correlated,

r21000�ln b ¼

B1

T 2� 2B2

hF iT 4

� �2

r2F þ

hF i2

T 4

!2

r2B2: ð14Þ

This approach is easier to implement than the one used byPolyakov and coworkers to treat published NRIXS data.For example, Polyakov and Soultanov (2011) used previ-ously published NRIXS data (Kobayashi et al., 2007) tocalculate the b-factor of chalcopyrite and obtained valuesof 5.4 and 0.81& for the 56Fe/54Fe ratio at 22 and500 �C, respectively. Kobayashi et al. (2007) reported a

Table 156Fe/54Fe b-factors calculated from force constants, 1000 � ln b = 2904hFi/T2 � B2hFi2/T4 with hFi in N/m and T in K. B2 values are estimates based on g(E) or S(E) data available (actual ordigitized values). In some previous studies, the energy scan during NRIXS measurements may have been too narrow and some force constants may need to be reevaluated when new data becomesavailable.

Phase P, T, size Forceconstant(N/m)

B2 B2 from g or S,actual ordigitized

103 � ln b22 �C

103 � ln b22 �C,B2 = 52,000

103 � ln b500 �C

References Litteraturedata, 22 �C

103 � ln b500 �C

References

Goethite 314.1 ± 13.9 61,951 S + a 9.67 ± 0.39 9.8 1.51 ± 0.07 This studyPotassium-jarosite 264.5 ± 11.6 58,394 S + a 8.28 ± 0.34 8.3 1.27 ± 0.06 This studyHydronium-jarosite 309.7 ± 13.6 64,918 S + a 9.50 ± 0.38 9.7 1.49 ± 0.06 This studyDeoxymyoglobin 174 ± 16 42,068 g + d 5.63 ± 0.50 5.6 0.84 ± 0.08 Adams et al.

(2006)Metmyoglobin 245 ± 9 39,704 g + d 7.85 ± 0.28 7.8 1.18 ± 0.04 Adams et al.

(2006)Cytochrome f, oxidized 313 ± 34 39,280 g + d 9.93 ± 1.02 9.8 1.51 ± 0.16 Adams et al.

(2006)Cytochrome f, reduced 342 ± 18 36,353 g + d 10.84 ± 0.54 10.6 1.65 ± 0.09 Adams et al.

(2006)Orthoenstatite (Mg0.93,Fe0.07)SiO3

195 ± 5 66,123 g + d 6.17 ± 0.15 6.2 0.94 ± 0.02 Jackson et al.(2009)

Orthoenstatite (Mg0.87,Fe0.13)SiO3

170 ± 3 65,493 g + d 5.42 ± 0.09 5.5 0.82 ± 0.01 Jackson et al.(2009)

Orthoenstatite (Mg0.80,Fe0.20)SiO3

165 ± 5 65,442 g + d 5.27 ± 0.15 5.3 0.80 ± 0.02 Jackson et al.(2009)

Hematite 5 K 252 ± 4 54,915 g + a 7.94 ± 0.12 8.0 1.21 ± 0.02 Sturhahn et al.(1999)








Magnetite 230.48 ± 5.5 57,163 g + a 7.28 ± 0.16 7.3 1.11 ± 0.03 Alp, pers. comm.FeS, troilite 1.5 Gpa 102 61,060 g + d 3.32 3.3 0.49 Kobayashi et al.

(2004)3.33 0.50 Polyakov and

Soultanov (2011)FeS, troilite 2.5 Gpa 102 61,060 g + d 3.32 3.3 0.49 Kobayashi et al.

(2004)FeS, troilite 3 Gpa 109 ± 8 61,060 g + d 3.54 ± 0.25 3.6 0.53 ± 0.04 Kobayashi et al.

(2004)(continued on next page)

260N

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Table 1 (continued)

Phase P, T, size Forceconstant(N/m)

B2 B2 from g or S,actual ordigitized

103 � ln b22 �C

103 � ln b22 �C,B2 = 52,000

103 � ln b500 �C

References Litteraturedata, 22 �C

103 � ln b500 �C

References

FeS, MnP-type 4 GPa 98 ± 10 55,909 g + d 3.20 ± 0.32 3.2 0.47 ± 0.05 Kobayashi et al.(2004)



FeS, monoclinic 9.5 GPa 173 ± 13 44,277 g + d 5.59 ± 0.41 5.6 0.84 ± 0.06 Kobayashi et al.(2004)

FeS, monoclinic 12 GPa 173 ± 21 44,277 g + d 5.59 ± 0.66 5.6 0.84 ± 0.10 Kobayashi et al.(2004)

CuFeS2 Chalcopyrite 0 Gpa 146 ± 2 56,616 g + d 4.71 ± 0.06 4.7 0.71 ± 0.01 Kobayashi et al.(2007)

5.4 0.8 Polyakov andSoultanov (2011)



CuFeS2 Chalcopyrite 4.5 Gpa 149 ± 2 58,621 g + d 4.80 ± 0.06 4.8 0.72 ± 0.01 Kobayashi et al.(2007)


CuFeS2 Amorphous 8 Gpa 141 ± 2 58,621 g + d 4.55 ± 0.06 4.6 0.68 ± 0.01 Kobayashi et al.(2007)

CuFeS2 Amorphous 14 Gpa 139 ± 3 56,255 g + d 4.49 ± 0.09 4.5 0.67 ± 0.01 Kobayashi et al.(2007)

Fe3S 0 119.7 ± 6.6 53,756 g + d 3.89 ± 0.21 3.9 0.58 ± 0.03 Lin et al. (2004) 3.6 0.5 Polyakov andSoultanov (2011)

Fe3S 6.1 GPa 175.9 ± 6 49,944 g + d 5.66 ± 0.19 5.7 0.85 ± 0.03 Lin et al. (2004)Fe3S 13.1 GPa 198.9 ± 5 47,057 g + d 6.39 ± 0.15 6.4 0.96 ± 0.02 Lin et al. (2004)Fe3S 21.3 GPa 213.1 ± 4.9 44,568 g + d 6.84 ± 0.15 6.8 1.03 ± 0.02 Lin et al. (2004)Fe3S 28 GPa 237 ± 12 44,344 g + d 7.57 ± 0.37 7.5 1.14 ± 0.06 Lin et al. (2004)Fe3S 37.7 GPa 258.2 ± 5 43,560 g + d 8.22 ± 0.15 8.2 1.25 ± 0.02 Lin et al. (2004)Fe3S 45.2 GPa 291.3 ± 6.2 43,034 g + d 9.23 ± 0.19 9.1 1.40 ± 0.03 Lin et al. (2004)Fe3S 57.1 GPa 323.2 ± 7.8 41,812 g + d 10.20 ± 0.23 10.1 1.56 ± 0.04 Lin et al. (2004)(Mg0.75, Fe0.25)O 0 GPa 180 53,934 g + d 5.77 5.8 0.87 Lin et al. (2006) 5.3 0.8 Polyakov (2009)(Mg0.75, Fe0.25)O 8 Gpa 203 51,097 g + d 6.49 6.5 0.98 Lin et al. (2006) 6.1 1.0 Polyakov (2009)(Mg0.75, Fe0.25)O 23 Gpa 211 51,097 g + d 6.75 6.7 1.02 Lin et al. (2006)(Mg0.75, Fe0.25)O 23 Gpa 234 51,097 g + d 7.44 7.4 1.13 Lin et al. (2006)(Mg0.75, Fe0.25)O 31 Gpa 245 52,392 g + d 7.75 7.8 1.18 Lin et al. (2006) 6.1 1.0 Polyakov (2009)(Mg0.75, Fe0.25)O 42 Gpa 245 48,188 g + d 7.78 7.8 1.18 Lin et al. (2006) 7.3 1.2 Polyakov (2009)(Mg0.75, Fe0.25)O 52 Gpa 306 50,378 g + d 9.57 9.5 1.47 Lin et al. (2006) 8.8 1.4 Polyakov (2009)(Mg0.75, Fe0.25)O 58 Gpa 360 50,378 g + d 11.14 11.1 1.73 Lin et al. (2006)(Mg0.75, Fe0.25)O 62 Gpa 368 49,932 g + d 11.39 11.4 1.77 Lin et al. (2006) 10.6 1.7 Polyakov (2009)(Mg0.75, Fe0.25)O 74 Gpa 400 49,932 g + d 12.28 12.2 1.92 Lin et al. (2006)

N.

Dau

ph

aset

al./G

eoch

imica

etC

osm

och

imica

Acta

94(2012)

254–275261

(Mg0.75, Fe0.25)O 92 Gpa 477 41,983 g + d 14.65 14.3 2.29 Lin et al. (2006) 12.8 2.0 Polyakov (2009)(Mg0.75, Fe0.25)O 109 Gpa 666 38,115 g + d 19.96 19.2 3.19 Lin et al. (2006) 13.7 2.1 Polyakov (2009)a-Fe (bcc) 175.15 ± 2.1 37,089 g + a 5.69 ± 0.07 5.6 0.85 ± 0.01 Alp et al. (2001) 5.6 0.8 Polyakov et al.

(2007)c-Fe (fcc) precipitate in Cu 3 nm 132 ± 9 37,639 g + d 4.31 ± 0.29 4.3 0.64 ± 0.04 Tsunoda et al.






(2002)e-Fe (hcp) 36 Gpa, 300 K 291.2 ± 8.7 35,773 g + d 9.31 ± 0.27 9.1 1.41 ± 0.04 Lin et al. (2005)e-Fe (hcp) 43.3 Gpa, 300 K 263.2 ± 5.2 35,773 g + d 8.45 ± 0.16 8.3 1.27 ± 0.02 Lin et al. (2005)e-Fe (hcp) 44 Gpa, 300 K 276.6 ± 6 35,773 g + d 8.86 ± 0.18 8.7 1.34 ± 0.03 Lin et al. (2005)e-Fe (hcp) 50.5 Gpa, 300 K 296 ± 7.6 35,773 g + d 9.45 ± 0.23 9.3 1.43 ± 0.04 Lin et al. (2005)e-Fe (hcp) 54.7 Gpa, 300 K 308.6 ± 9.9 35,773 g + d 9.84 ± 0.30 9.6 1.49 ± 0.05 Lin et al. (2005)e-Fe (hcp) 51.1 Gpa, 300 K 312.8 ± 8.7 35,773 g + d 9.97 ± 0.26 9.8 1.51 ± 0.04 Lin et al. (2005)e-Fe (hcp) 54.6 Gpa, 300 K 325.6 ± 5.9 35,773 g + d 10.35 ± 0.18 10.1 1.57 ± 0.03 Lin et al. (2005)e-Fe (hcp) 60 Gpa, 300 K 329.5 ± 9.9 35,773 g + d 10.47 ± 0.30 10.2 1.59 ± 0.05 Lin et al. (2005)e-Fe (hcp) 71 Gpa, 300 K 328.6 ± 4.9 35,773 g + d 10.45 ± 0.15 10.2 1.59 ± 0.02 Lin et al. (2005)e-Fe (hcp) 72.8 Gpa, 700 K 299.3 ± 14 35,773 g + d 9.56 ± 0.43 9.4 1.45 ± 0.07 Lin et al. (2005)e-Fe (hcp) 73.3 Gpa, 880 K 299.5 ± 21.5 35,773 g + d 9.56 ± 0.66 9.4 1.45 ± 0.10 Lin et al. (2005)e-Fe (hcp) 39 Gpa, 1000 K 267.3 ± 7.8 35,773 g + d 8.57 ± 0.24 8.4 1.29 ± 0.04 Lin et al. (2005)e-Fe (hcp) 57.5 Gpa, 1000 K 292.8 ± 7.7 35,773 g + d 9.36 ± 0.24 9.2 1.41 ± 0.04 Lin et al. (2005)e-Fe (hcp) 46.5 Gpa, 1100 K 257 ± 10 38,686 g + d 8.23 ± 0.31 8.1 1.24 ± 0.05 Lin et al. (2005)e-Fe (hcp) 47.2 Gpa, 1100 K 277.8 ± 18.5 38,686 g + d 8.87 ± 0.56 8.7 1.34 ± 0.09 Lin et al. (2005)e-Fe (hcp) 57.9 Gpa, 1200 K 294.9 ± 13.1 38,686 g + d 9.39 ± 0.40 9.2 1.42 ± 0.06 Lin et al. (2005)e-Fe (hcp) 54.6 Gpa, 1300 K 266.8 ± 22.9 38,686 g + d 8.53 ± 0.70 8.4 1.29 ± 0.11 Lin et al. (2005)e-Fe (hcp) 54.1 Gpa, 1500 K 166.5 ± 8.2 38,686 g + d 5.41 ± 0.26 5.4 0.81 ± 0.04 Lin et al. (2005)e-Fe (hcp) 58.1 Gpa, 1700 K 196.9 ± 15 38,686 g + d 6.37 ± 0.47 6.3 0.95 ± 0.07 Lin et al. (2005)e-Fe (hcp) 158 Gpa, c axis 511.7 ± 5.6 33,820 g + d 15.89 ± 0.16 15.3 2.46 ± 0.03 Lin et al. (2010)e-Fe (hcp) 158 Gpa, 30 � to a

axis508.1 ± 3.5 33,820 g + d 15.79 ± 0.10 15.2 2.44 ± 0.02 Lin et al. (2010)

e-Fe (hcp) 158 Gpa, a axis 491.1 ± 4.5 33,820 g + d 15.30 ± 0.13 14.7 2.36 ± 0.02 Lin et al. (2010)e-Fe (hcp) 172 Gpa, c axis 531.4 ± 8.8 33,820 g + d 16.46 ± 0.25 15.8 2.55 ± 0.04 Lin et al. (2010)e-Fe (hcp) 172 Gpa, 28 �

to a axis534.5 ± 7.7 33,820 g + d 16.54 ± 0.22 15.9 2.57 ± 0.04 Lin et al. (2010)

e-Fe (hcp) 172 Gpa, a axis 526.8 ± 8.8 33,820 g + d 16.32 ± 0.25 15.7 2.53 ± 0.04 Lin et al. (2010)a-Fe (hcp) 0 Gpa 185 ± 12 36,246 g + d 6.00 ± 0.38 5.9 0.90 ± 0.06 Lubbers et al.

(2000)e-Fe (hcp) 20 Gpa 320 ± 15 34,950 g + d 10.20 ± 0.46 10.0 1.54 ± 0.07 Lubbers et al.

(2000)(continued on next page)

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(2012)254–275

Tab

le1

(co

nti

nu

ed)

Ph

ase

P,

T,

size

Fo

rce

con

stan

t(N

/m)

B2

B2

fro

mg

or

S,

actu

alo

rd

igit

ized

103�

lnb

22�C

103�

lnb

22�C

,B

2=

52,0

00

103�

lnb

500

�CR

efer

ence

sL

itte

ratu

red

ata,

22�C

103�

lnb

500

�CR

efer

ence

s

e-F

e(h

cp)

32G

pa

365

±15

34,9

50g

+d

11.5

5±

0.45

11.3

1.76

±0.

07L

ub

ber

set

al.

(200

0)e-

Fe

(hcp

)42

Gp

a38

8±

1534

,950

g+

d12

.24

±0.

4511

.91.

87±

0.07

Lu

bb

ers

etal

.(2

000)

e-F

e(h

cp)

29G

pa,

300

K25

3±

1738

,678

g+

d8.

11±

0.52

8.0

1.22

±0.

08S

hen

etal

.(2

004)

e-F

e(h

cp)

29G

pa,

430

K25

0±

1935

,552

g+

d8.

04±

0.59

7.9

1.21

±0.

09S

hen

etal

.(2

004)

e-F

e(h

cp)

29G

pa,

720

K23

6±

1533

,720

g+

d7.

62±

0.47

7.5

1.14

±0.

07S

hen

etal

.(2

004)

a-F

e 52.5

Cr 4

7.5

298

K15

6±

136

,329

g+

d5.

08±

0.03

5.0

0.76

±0.

00D

ub

iel

etal

.(2

010)

r-F

e 52.5

Cr 4

7.5

298

K15

7±

239

,792

g+

d5.

10±

0.06

5.1

0.76

±0.

01D

ub

iel

etal

.(2

010)

r-F

e 52.5

Cr 4

7.5

20K

155.

1±

0.7

39,7

92g

+d

5.04

±0.

025.

00.

75±

0.00

Du

bie

let

al.

(201

0)


mean force constant of 146 ± 2 N/m for this mineral.Based on Eq. (12), we calculate for the 56Fe/54Fe ratio,1000 � ln b = 4.71 and 0.71& at 22 and 500 �C, respec-tively. The discrepancy between Polyakov and Soultanov(2011) and the present study (5.4–4.7 = 0.7&) reflects adifference in the force constant estimate as it persists athigh temperature, where the second terms in Eqs. (1)and (3) become negligible. The reason for this discrepancyis unknown but it could reflect a problem of backgroundsubtraction. Kobayashi et al. (2007) acquired NRIXSdata over a symmetric energy interval (form �80 to+80 meV) and calculated the force constant from thethird-order moment of S(E), so the force constant thatthey report should be relatively insensitive to backgroundsubtraction, as long as the background is constant. On theother hand, Polyakov and Soultanov (2011) used thePDOS g(E), which is sensitive to the assumed backgroundand can lead to erroneous results.

In Table 1, we summarize our estimates of b-factors forseveral molecules and minerals based on published forceconstants and Eq. (12). For several of these phases,including deoxymyoglobin, metmyoglobin, cytochrome f

(oxidized and reduced), and orthoenstatite, it is the firsttime that b-factors are calculated. We warn the reader thatprevious NRIXS studies were not done for the purpose ofestimating force constants and some b-factors may needto be reevaluated when better quality measurements be-come available.

There is overall good agreement between the resultspresented here and those reported by Polyakov andcoworkers (Table 1; Fig. 5). The discrepancies concernmostly high-pressure phases with strong bonds. For exam-ple, the b-factors calculated for ferropericlase by Polyakov(2009) were too low, sometimes by 30% (e.g., 2.1 vs. 3.2&

at 109 Gpa). The reason for this discrepancy is unknownbut it could be a smoothing or digitization artifact. In gen-eral, one should be particularly cautious with high-pres-sure phases as they have extended high-energy tails andas such, they are particularly prone to biases.

0

1

2

3

0 1 2 3

1000

ln

(fr

om

<F

>, t

his

stu

dy)

1000 ln (Polyakov and coworkers)

Fig. 5. Comparison between 1000 � ln b (56Fe/54Fe) at 500 �Creported by Polyakov and coworkers and values derived from forceconstants (Table 1). Overall, there is good agreement between thetwo approaches except for a few high-pressure phases characterizedby PDOS with extended high-energy tails.


4. RESULTS

The NRIXS spectra of 57Fe-enriched goethite, potas-sium-jarosite, and hydronium-jarosite were measured.Fig. 6 shows these spectra after removal of the elastic peakfollowing the methodology of Sturhahn (2004). The spectrahave been normalized, so the integral

R Emax

EminSðEÞdE is equal

to the recoil fraction during resonant nuclear scattering(1 � flm, where flm is the Lamb-Mossbauer factor). Table 2summarizes all the thermo-elastic properties that can be re-trieved from such spectrum, namely the Lamb-Mossbauerfactor, the mean kinetic energy/atom, the mean force con-stant, the Lamb-Mossbauer factor at T = 0 K, the mean ki-netic energy at T = 0 K, the vibrational specific heat, thevibrational entropy, and the critical temperature. For Feisotope geochemistry, the critical number is the mean forceconstant hFi. As discussed in Section 3, this value is obtain-able either directly from the NRIXS data or indirectly fromthe PDOS (Eq. (6)). The force constants obtained usingNRIXS data S(E) (the preferred method) are

Fig. 6. NRIXS derivation of the average force constant for goethitehydronium-jarosite (bottom panels). The force constant is calculahF i ¼ M

ER�h2

Rþ1�1 ðE � ERÞ3SðEÞdE (Eq. (6)). While S(E) reaches near-backg

may still be present above that value, which gets amplified in the integrapanel). The right panels show the values of the force constahF i ¼ M

ER�h2

RþX�X ðE � ERÞ3SðEÞdE with x = 0 to 130 meV. While the force

show that little signal is expected above 130 meV and the force constants tthe force constants used for calculation of b-factors are obtained from S(Econtribution of multiple phonons outside of the acquisition range. The forand 309.7 ± 13.6 N/m for goethite, potassium-jarosite, and hydronium-ja

314.10 ± 9.66, 264.48 ± 5.87, and 309.71 ± 9.24 N/m forgoethite, potassium-jarosite, and hydronium-jarosite,respectively. The force constants obtained using the PDOSg(E) are 316.57 ± 10.83, 270.83 ± 7.04, and 307.26.7 ±12.67 N/m for goethite, potassium-jarosite, and hydro-nium-jarosite, respectively.

While visual inspection of the NRIXS spectra suggeststhat there is no signal beyond �70 to +80 meV, the forceconstant is calculated by integrating an integrand that mul-tiplies S(E) by E3, thus dramatically amplifying the contri-bution of the high-energy tails. The relevant quantity toassess whether the background has been reached is there-fore S(E) � E3, which extends to lower and higher energies(middle vertical panels of Fig. 6). Ideally, the force constantintegral should plateau as the integration limits broaden,meaning that no signal is present above background. Thevalue of the force constant integral is plotted in the rightpanels of Fig. 6. In the case of potassium-jarosite, a plateauis clearly reached. Despite the wide energy scans performedin this study, goethite and hydronium-jarosite do not show

(top panels), potassium-jarosite (middle horizontal panels), andted from the NRIXS spectrum (left) by using the formularound values above 80 meV for all three phases, significant signal

l when S(E) is multiplied by the cube of the energy (middle verticalnt integrals for different values of the integration limits;

constant integrals never reach a plateau, the vertical middle panelshat we derive are not affected by truncation in energy. The values of) after refinement (see text for details), which takes into account thece constants derived from these data are 314.1 ± 13.9, 264.5 ± 11.6,rosite, respectively (Table 2).


well-defined plateaus. As discussed in Sect. 3, the PHOE-NIX software can correct for truncated multiple phononcontributions by extrapolating the raw data beyond the ac-quired energy range. Comparison of the force constants be-fore and after refinement shows that the correction is small(306.9 vs. 314.1 N/m for goethite, 302.4 vs. 309.7 N/m forH-jarosite, and 262.0 vs. 264.5 N/m for K-jarosite). Thedata thus appear reliable. Note that this extrapolationmethod cannot account for the presence of normal vibra-tion modes outside of the scanned energy range. Raman/IR and theoretical calculations can provide clues on the en-ergy range required to capture all normal vibration fre-quencies. If present, high-energy modes could be belowbackground, which would make them difficult to detectby NRIXS even if the energy range is expanded.

The same analysis was performed on the determinationof the force constant from the PDOS (Fig. 7). At first sight,the PDOS of all three phases have reached background val-ues at �100 meV. However, the relevant integrand for cal-culation of the force constant is g(E) � E2, shown in themiddle vertical panels of Fig. 7. For potassium-jarosite,no significant signal is present above �100 meV, which is

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120

PD

OS

(1/

eV)

Energy (meV)

Goethite

-0.05

0

0.05

0.1

0.15

0 20 40 60

En

erg

y2 P

DO

S (

eV)

Energy

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120

PD

OS

(1/

eV)

Energy (meV)

Potassium-Jarosite

-0.05

0

0.05

0.1

0.15

0 20 40 60

En

erg

y2 P

DO

S (

eV)

Energy

0

20

40

60

80

100

120

0 20 40 60 80 100 120

PD

OS

(1/

eV)

Energy (meV)

Hydronium-Jarosite

-0.05

0

0.05

0.1

0.15

0 20 40 60

En

erg

y2 P

DO

S (

eV)

Energy

Fig. 7. PDOS derivation of the average force constant for goethitehydronium-jarosite (bottom panels). The force constant is calculated from(6)). While the PDOS reaches near-background values above 80 meV fovalue, which gets amplified in the integral when the PDOS is multiplied bshow the values of the force constant integrals for different values of the iPDOS reaches background level, the integral should reach a plateau correthis is the case for the potassium-jarosite sample, goethite and hydroniumlittle signal is expected above 130 meV and the force constants that we d

confirmed by examination of the force constant integral(right panels of Fig. 7). However, high-energy phononsseem to be present in hydronium-jarosite and goethite, allthe way to 110 meV.

Background subtraction can affect the force constant de-rived from the PDOS and, to a lesser degree, that derivedfrom S(E). The background counts per channel were esti-mated by averaging the counts measured in the lowest10 meV of the scans (i.e., from �120 to �110 meV) andthe 2r uncertainty on the average was calculated. The back-ground was then tuned within the limits allowed by theuncertainties to minimize the detailed balance (a balancebetween phonon annihilation and creation that dependson temperature). As expected, the force constant derivedfrom the PDOS g(E) is more sensitive to the backgroundthan that derived from S(E).

In order to take into account the possibility that (1)some NRIXS signal may still be present at 130 meV and(2) background subtraction may not be optimum for thehigh-energy tail of the PDOS, we have added quadratically±10 N/m of systematic error to the statistical uncertaintygiven by the PHOENIX software. This was assessed by

80 100 120 (meV)

Goethite

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

Fo

rce

con

stan

t in

teg

ral (

N/m

)

Upper limit of the integral (meV)

Goethite

80 100 120 (meV)

Potassium-Jarosite

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

Fo

rce

con

stan

t in

teg

ral (

N/m

)


Potassium-Jarosite

80 100 120 (meV)

Hydronium-Jarosite

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

Fo

rce

con

stan

t in

teg

ral (

N/m

)


Hydronium-Jarosite

(top panels), potassium-jarosite (middle horizontal panels), andthe PDOS (left) by using the formula hF i ¼ M

�h2

Rþ10

gðEÞE2dE (Eq.r all three phases, significant signal may still be present above thaty the square of the energy (middle vertical panel). The right panels

ntegration upper-limit; M�h2

R x0

gðEÞE2dE with x = 0 to 130 meV. If thesponding to the average force constant of the chemical bond. While-jarosite never reach a plateau. The vertical middle panels show thaterive are not too affected by truncation in energy.

Table 2Thermoelastic properties derived from NRIXS data. Uncertainties in parentheses take into account systematic errors on the mean force constant from S(E).

Goethite H-jarosite K-jarosite

From S(E) Lamb-Moessbauer factor 0.77 ± 0.00 0.65 ± 0.00 0.68 ± 0.00Kinetic energy/atom (meV) 45.43 ± 0.38 46.41 ± 0.38 45.98 ± 0.27Mean force constant (N/m) 306.87 ± 9.43 302.38 ± 9.02 262.04 ± 5.82

From S(E) after refinement Lamb-Moessbauer factor 0.77 ± 0.00 0.65 ± 0.00 0.68 ± 0.00Kinetic energy/atom (meV) 45.54 ± 0.38 46.51 ± 0.38 46.00 ± 0.27Mean force constant (N/m)* 314.10 ± 9.66 (13.90) 309.71 ± 9.24 (13.61) 264.48 ± 5.87 (11.60)

From PDOS Lamb-Moessbauer factor 0.77 ± 0.00 0.66 ± 0.00 0.69 ± 0.00Kinetic energy/atom (meV) 45.32 ± 0.46 45.12 ± 0.52 44.50 ± 0.31Mean force constant (N/m) 316.57 ± 10.83 307.26 ± 12.67 270.83 ± 7.04Lamb-Moessbauer factor at T = 0 0.92 ± 0.00 0.91 ± 0.00 0.91 ± 0.00Kinetic energy/atom at T = 0 (meV) 24.86 ± 0.40 23.78 ± 0.47 23.07 ± 0.28Vibrational specific heat (kB/atom) 2.57 ± 0.01 2.59 ± 0.02 2.62 ± 0.01Vibrational entropy (kB/atom) 2.92 ± 0.01 3.21 ± 0.01 3.18 ± 0.01Critical temperature (K) 1209.20 ± 5.80 745.00 ± 2.06 850.60 ± 1.87

Mathematica script Force constant from S(E) (N/m) 307 ± 9 303 ± 9 263 ± 6Force constant from g(E) (N/m) 316 ± 11 307 ± 13 271 ± 7

1000 � ln b coefficients from S(E) * A1 8.93E + 05 ± 2.73E + 04 8.80E + 05 ± 2.61E + 04 7.63E + 05 ± 1.69E + 04A1/T2 + A2/T4 + A3/T6 (Eq. (3)) A2 �7.35E + 09 ± 6.93E + 08 �7.45E + 09 ± 6.61E + 08 �6.64E + 09 ± 3.95E + 08with T in K A3 1.39E + 14 ± 2.97E + 13 1.38E + 14 ± 2.86E + 13 1.33E + 14 ± 1.65E + 131000 � ln b coefficients from g(E) A1 9.19E + 05 ± 3.16E + 04 8.92E + 05 ± 3.79E + 04 7.86E + 05 ± 2.11E + 04A1/T2 + A2/T4 + A3/T6 (Eq. (1)) A2 �8.15E + 09 ± 8.30E + 08 �8.06E + 09 ± 1.01E + 09 �4.88E + 09 ± 5.33E + 08with T in K A3 1.96E + 14 ± 3.59E + 13 1.93E + 14 ± 4.43E + 13 6.45E + 13 ± 2.29E + 13B1hFi/T2 � B2hFi2/T4 (Eq. (12)) * B1 2904.48 2904.48 2904.48with < F > in N/,m and T in K B2 61951.30 64917.80 58394.20

* Preferred methods.

266N

.D

aup

has

etal./

Geo

chim

icaet

Co

smo

chim

icaA

cta94

(2012)254–275


testing the sensitivity of force constant estimates to back-ground subtraction and refinement for missing multiplephonon contribution. The resulting equilibrium fraction-ation of the three minerals investigated for the 56Fe/54Feratio are given below (T is in K),

Goethite : 1000� ln b ¼ 9:12� 105=T 2 � 6:11� 109=T 4

K-jarosite : 1000� ln b ¼ 7:68� 105=T 2 � 4:08� 109=T 4

H3O-jarosite : 1000� ln b ¼ 9:00� 105=T 2 � 6:23

� 109=T 4

To obtain the b-factors for the 57Fe/54Fe ratio, one canmultiply the factors on the right by �1.475, correspondingto a high-temperature approximation (Matsuhisa et al.,1978; Young et al., 2002). The b-factors of K-jarosite andH3O-jarosite are the first ever published and no direct com-parison can be made with previous studies. Using previ-ously published Mossbauer data for goethite (De Graveand Van Alboom, 1991), Polyakov and Mineev (2000) cal-culated at 25 �C, 1000 � ln b = 5.9& while the NRIXSdata presented here give a b-factor of �9.49 ± 0.38&.The b-factor derived from Mossbauer data uses the sec-ond-order Doppler shift, which is difficult to determineexperimentally. As discussed in the introduction and byPolyakov et al. (2007), NRIXS b-factors are more accurateand should supersede Mossbauer data whenever available.For example, Polyakov and Mineev (2000) calculated ab-value for magnetite at 25 �C of 10.8& based on conven-tional Mossbauer data (De Grave et al., 1993), which wassubsequently revised by Polyakov et al. (2007) to 7.3&

based on new NRIXS data (Seto et al., 2003) and new con-ventional Mossbauer data. We calculate a value of 7.14&

for magnetite (Table 1). The NRIXS-value is in goodagreement with the experimentally measured equilibriumisotopic fractionation between fayalite and magnetite(Shahar et al., 2008). For hematite, Mossbauer gives1000 � ln b = 7.79& for the 56Fe/54Fe ratio at 25 �C(Polyakov et al., 2007) while the NRIXS data of Sturhahnet al. (1999) give a value of 7.33 ± 0.27& (Table 1). In thiscase, there is excellent agreement between the two ap-proaches. A DFT-based quantum chemical calculationgives a value of 7.1& for hematite (Blanchard et al., 2009).

5. DISCUSSION

5.1. Goethite

In the past decade, iron oxyhydroxides have been the fo-cus of extensive iron isotope studies (Dauphas and Rouxel,2006; Johnson et al., 2008). Iron isotope measurements offerric iron oxides have been used to understand subsurfacefluid transport (Teutsch et al., 2005; Chan et al., 2006;Busigny and Dauphas, 2007), the conditions prevailing dur-ing banded-iron formation (Johnson et al., 2003; Dauphaset al., 2004; Heimann et al., 2010; Craddock and Dauphas,2011), the nature of metamorphosed rocks and the pres-sure–temperature conditions that affected them (Dauphaset al., 2007a,b; Frost et al., 2007). Goethite was detectedon Mars in the Columbia Hills at Gusev Crater, providing

evidence of aqueous activity on the planet (Morris et al.,2006, 2008). To properly interpret Fe isotope variationsin the rock record, one must know equilibrium fraction-ation factors. Several laboratory experiments have studiedisotopic fractionation in systems involving goethite. Crosbyet al. (2005, 2007a) studied experimentally the isotopic frac-tionation associated with biological dissimilatory ironreduction (DIR) of goethite, a form of microbial respirationthat uses ferric iron as electron acceptor. Icopini et al.(2004) and Jang et al. (2008) examined Fe isotopic fraction-ation between Fe(II)aq and goethite. They argued that iso-topic fractionation during sorption of Fe(II)aq onto thesolid goethite substrate was the main factor controlling ironisotopic fractionation in their experiments. Mikutta et al.(2009) carried out an experiment where Fe(II)aq flowedthrough goethite-coated sandstone and they found that sig-nificant isotopic fractionation was present between Fe(II)aq

and adsorbed Fe(II). All these experiments illustrate howcomplex fluid–rock interactions can be in the presence ofa highly reactive mineral such as goethite.

Beard et al. (2010) recently estimated the equilibrium Feisotopic fractionation between Fe(II)aq and goethite. Theyused a three-isotope technique first pioneered for O isotopegeochemistry by Matsuhisa et al. (1978). Shahar et al.(2008) used the same technique to measure Fe isotopic frac-tionation between magnetite and fayalite. This approachuses an isotope spike (e.g., 57Fe) to track the extent of equil-ibration between the reactants (at equilibrium, the reactantsshould plot on a single mass fractionation line). However,application of this technique does not guarantee accuracyif equilibrium has not been reached and kinetic processessuch as diffusion or dissolution-re-precipitation reactionscontrol isotopic exchange between the reactants, or if morethan two chemical species are involved in the reaction.Beard et al. used a stepwise dissolution protocol to separatethe various iron components Fe(II)aq, sorbed Fe(II), sur-face Fe(III), and bulk goethite. They obtained an equilib-rium isotopic fractionation between Fe(II)aq and goethiteof �1.05 ± 0.08& at 22 �C for the 56Fe/54Fe ratio. Com-paring this value with the results presented here is notstraightforward as Beard et al. measured the difference1000 � ln bFe(II)aq � 1000 � ln bgoethite, while we calculated1000 � ln bgoethite. Therefore, one needs to know1000 � ln bFe(II)aq to compare the two studies.

Several studies based on electronic structure theory orother computational methods have reported the values ofbFe(II)aq and/or bFe(III)aq (Schauble et al., 2001; Anbaret al., 2005; Domagal-Goldman and Kubicki, 2008; Hilland Schauble, 2008; Hill et al., 2009, 2010; Ottonello andVetuschi Zuccolini, 2009; Rustad and Dixon, 2009; Rustadet al., 2010). These calculations match experimentally mea-sured values for the difference bFe(III)aq � bFe(II)aq (Welchet al., 2003; Hill et al., 2009). However, it remains uncertainat the present time whether the b-factors are accurate on anabsolute scale. Anbar et al. (2005) reported b-values for the56Fe/54Fe ratio of Fe(II)aq at 25 �C from 6.23 to 6.69&;Ottonello and Vetuschi Zuccolini (2009) reported valuesfrom 4.859 to 5.537&; Rustad et al. (2010) reported a valueof �4.67&; Domagal-Goldman and Kubicki (2008)reported values of �5.66–6.55&; Hill and Schauble


(2008) reported values of 5.42–6.70&. Overall, theoreticalcalculations give a b-value for Fe(II)aq that ranges over2& from 4.6 to 6.7&. Combining these predictions withthe b-factor that we calculated for goethite (9.67 ± 0.39&

at 22 �C and 9.49 ± 0.38& at 25 �C), we would predict anequilibrium isotopic fractionation factor between goethiteand Fe(II)aq of +3 to +5&; most of the uncertainty stemsfrom the large variability in predictions from electronicstructure theory. This is much larger than the equilibriumfractionation of +1.05& that was estimated by Beardet al. Clearly, some of these estimates must be in error.

Polyakov and Soultanov (2011) recently reviewed allexisting constraints on mineral–fluid equilibrium isotopicfractionation and proposed that the b-value of Fe(II)aq at25 �C be revised to 4.9&, in agreement with some ab initio

calculations, in particular that of Rustad et al. (2010) whoexplicitly represented the second water solvation shell.Equilibrium isotopic fractionation between Fe(II)aq andFe(III)aq was measured for several ferric-iron bearing min-erals. Wu et al. (2011) determined a fractionation of+3.17 ± 0.08& between hydrous ferric oxide (ferrihydrite)and Fe(II)aq. Skulan et al. (2002) found no measurableequilibrium isotopic fractionation between hematite andFe(III)aq (i.e., +0.1 ± 0.2& at 98 �C, corresponding to+0.15 ± 0.30 at 25 �C assuming that 1000 � ln b scales as1/T2). Given that the equilibrium Fe isotopic fractionationbetween Fe(III)aq and Fe(II)aq is +2.9 ± 0.5& at 25 �C(Welch et al., 2003), the equilibrium isotopic fractionationbetween hematite and Fe(II)aq is +3.0 ± 0.6&. To summa-rize, the equilibrium isotopic fractionations for both fer-rihydrite-Fe(II)aq and hematite-Fe(II)aq are �+3&, whileBeard et al.’s experiment suggest that the isotopic fraction-ation goethite-Fe(II)aq is +1&. A +2& equilibrium isoto-pic fractionation between hematite and goethite at 25 �Cwould require a force constant for goethite of �180 N/m(assuming that the force constant of hematite is 244 N/m;Table 1). This is similar to values measured in Fe0 orFe2+-bearing phases and is very unlikely given that mostFe3+-bearing phases tend to have high force constant (i.e.,244 N/m for hematite, 264 N/m for potassium-jarosite,310 N/m for hydronium-jarosite, and 314 N/m for goethite;Fig. 8). The higher force constants for hydronium-jarosite

0

2

4

6

8

10

12

0 100 200 300 400

1,00

0 ln

Goethite

Potassium-Jarosite

Hydronium-Jarosite

Hematite

Fe3+aq

Fe2+aq

Fig. 8. 1000 � ln b (56Fe/54Fe) of Fe3+-bearing minerals (goethite,jarosite, and hematite; Table 1), Fe3+

aq, and Fe2+aq (Polyakov and

Soultanov, 2011).

and goethite relative to other Fe3+-bearing phases mightbe explained if OH– forms stiffer Fe–O bonds than arefound in anhydrous minerals.

Our results call for a reexamination of exchange experi-ments between goethite and Fe(II)aq. One way to assesswhether the measured isotopic fractionation reflects equi-librium would be to carry out experiments at higher tem-perature as equilibrium isotopic fractionation shoulddecrease as 1/T2 while kinetic isotope effects may still bepresent at higher temperature (i.e., one can measure thetemperature dependence of the fractionation factor and testwhether it conforms to expectations for equilibrium). Wenote that Crosby et al. (2007b) had measured a fraction-ation between Fe(II)aq and surface Fe(III) in goethite of+2.62 ± 0.57&, which would be in better agreement withthe value documented here.

5.2. Jarosite

The Mars rovers have revealed the occurrence of iron-rich rocks composed of hematite spherules embedded in ajarosite matrix (Klingelhofer et al., 2004). This detectionhas provided unambiguous evidence for the presence of li-quid water at the surface of Mars in the past. Considerableuncertainty remains, however, as to the chemical pathwaysthat permitted deposition of rocks composed of such anunusual mineral assemblage. Interpretation is complicatedby the fact that these rocks have no direct terrestrial ana-logues available for comparison. The field occurrence ofhematite spherules is reminiscent of the iron-oxide plusquartz concretions found in the Navajo sandstone in Utah,yet the sedimentology and details of the composition of theconcretions of the latter formation has no similarity withthe Martian rocks (Chan et al., 2004; Morris et al., 2005;Busigny and Dauphas, 2007). Some rocks near fumarolesin Hawaii contain jarosite and hematite spherules, repre-senting the best terrestrial analogues of the Martian out-crops (Morris et al., 2005). There are, however, majordifferences between these rocks and the Martian samples,starting with the size of the spherules. Iron isotopes havebeen used on Earth to understand how associated jaro-site-hematite mineral assemblages were formed and to traceFe mobilization during acid sulfate alteration (Dauphasand Morris, 2008). When returned samples are available,a similar approach can be used to study Martian surfacegeology. Such Fe isotope measurements will allow us to ad-dress the origin of Fe in hematite-jarosite and the chemicalpathways that led to the formation of these phases. It willbe possible to address two questions:

1. What is the provenance of the chemical sedimentarycomponent in these formations? It was suggested thatthe source of iron involved acid sulfate alteration of bas-alts (McLennan et al., 2005; Morris et al., 2005; Toscaet al., 2005). Studies of natural samples and laboratoryexperiments have shown that such alteration processeswould produce fluids that have light Fe isotopic compo-sition relative to the substrate (Rouxel et al., 2003;Brantley et al., 2004). The Fe isotopic composition ofMartian basaltic rocks is known from measurements of


SNC meteorites, which average a d56Fe value of�+0.02& relative to reference material IRMM-014(Poitrasson et al., 2004; Weyer et al., 2005; Anandet al., 2006; Wang et al., 2012). We thus expect the alter-ation fluid to have a d56Fe value of ��0.2&. OurNRIXS results show that at equilibrium, potassium-jarosite should have a d56Fe value higher than Fe(III)aq

by �+1& (Table 1 for K-jarosite; Polyakov and Soulta-nov, 2011 for Fe(III)aq). The isotopic composition ofjarosite will also depend on the extent of the conversionof Fe(III)aq into jarosite. The fluid–rock ratio may havebeen limited and water–rock interactions may have beenshort-lived. In this framework, one would expect jarositeto have a d56Fe value close to or slightly higher than thefluid.

2. Various geochemical pathways are possible for the for-mation of hematite spherules. Hematite spherules andjarosite can form by direct precipitation under hydrother-mal conditions (Golden et al., 2008). Other scenariosinvolve conversion of jarosite into goethite or hematiteupon fresh water recharge (Tosca et al., 2005; Zolotovand Shock, 2005), or oxidation of a late-stage evaporitemineral such as melanterite (Tosca et al., 2005). If theconversion is quantitative and is localized, one wouldexpect hematite to inherit the Fe isotopic compositionof the precursor mineral. On the other hand, the transfor-mation may reflect a displacement of equilibrium condi-tions in the whole system. Measuring the Fe isotopiccomposition of hematite and jarosite will provide a meansto assess the degree of disequilibrium between phases. Ifhematite has the exact same d56Fe value as jarosite, thiswill support the view that hematite was produced bydestabilization of jarosite during a recharge event.

Much of this discussion is speculative but it illustratesthe powerful constraints that Fe isotopes can provide tointerpret the record of iron-rich sediments observed onthe Martian surface. The fractionation factors calculatedhere for goethite, hematite, and jarosite will provide aframework to interpret Fe isotope variations in returnedMartian samples. Sooner, they will help interpret Fe iso-tope variations documented in terrestrial analogues of Mar-tian sediments and they will help understand iron-transportin acid mine drainage.

6. CONCLUSION

A new approach to determine equilibrium fractionationfactor from NRIXS data is presented. It relies on the factthat the reduced partition function ratio is related througha Bernoulli expansion to the moments of the NRIXS spec-trum S(E) (Eq. (3); Appendix C). At high temperature,1000 � ln b = B1hFi/T2, with B1 = 2,904 and hFi the meanforce constant of the bonds involving Fe atoms (Eq. (7)),which is easily obtained by taking the third-order momentof the NRIXS spectrum (Eq. (6)).

The force constants and the equilibrium fractionationfactors that are derived are highly sensitive to the high-en-ergy tails of the NRIXS spectrum. Such measurements aretherefore highly demanding as they require very long acqui-

sition times over broad energy intervals to accurately mea-sure the tails of the nuclear resonance scattering spectrum.This difficulty may explain the discrepancy found for high-pressure phases between the b-factors derived from theforce constant approach (this study) compared with thoseinferred by Polyakov (2009). For most low-pressure miner-als, there is good agreement between our approach and theresults reported by Polyakov et al. (2007), Polyakov (2009),and Polyakov and Soultanov (2011) based on the Thirringexpansion of the kinetic energy in moments of the PDOSg(E).

The approach that we propose based on the moments ofS(E) (Eq. (3); Appendix C) is more reliable than that basedon the moments of g(E) (Eq. (1); Polyakov et al., 2005b,2007) as it is less sensitive to background subtraction, it al-lows quantification of the contribution to the b-factor ofmultiple phonons outside the acquisition energy window,and it yields smaller statistical uncertainties.

We have applied the force constant approach to pub-lished NRIXS data and report 1000 � ln b for importantphases of geological and biochemical relevance such asmyoglobin, cytochrome f, orthoenstatite, metal, troilite,chalcopyrite, hematite, and magnetite (Table 1).

The NRIXS spectra of goethite, potassium-jarosite, andhydronium-jarosite are reported (Fig. 6). These are criticalphases for aqueous, low temperature iron isotope geochem-istry. The goethite fractionation factor derived fromNRIXS (Fig. 8) disagrees with values obtained by combin-ing laboratory exchange experiments (Beard et al., 2010)and calculations based on electronic structure theory (Rus-tad et al., 2010). The laboratory exchange experiments giveb-factors that differ markedly between hematite and hy-drous ferric oxide on the one hand (Skulan et al., 2002;Wu et al., 2011), and goethite on the other hand (Beardet al., 2010). Further experimental work will be neededbut we suspect that the equilibrium fractionation obtainedin laboratory exchange experiments is too low for goethite.

Our work shows that the relationship between iron equi-librium isotopic fractionation factor and force constantprovides a direct method of calculating b-factors in high-temperature/high-pressure systems using NRIXS data(Eqs. (6), (7)). High-pressure phases have extended NRIXSspectra, requiring wide energy scans to calculate iron forceconstants. Furthermore, the samples in diamond anvil cellsare small, requiring long acquisition times to get decent sig-nal to noise ratio. These two factors make the determina-tion of iron force constants in high-pressure phasesparticularly challenging, so published data should be con-sidered with a grain of salt for application to iron isotopegeochemistry.

The formalism developed here will be extremely usefulto analyze NRIXS data for application to the isotope geo-chemistry of elements with low-lying nuclear excited statessuch as Dy, Eu, Kr, and Sn.

ACKNOWLEDGMENTS

We thank C. Achilles for determination of particle sizes usingXRD data. Discussions with W. Sturhahn, R.N. Clayton, R. Cara-cas, and T. Fujii regarding data reduction, background


subtraction, and stable isotope fractionation were greatly appreci-ated. M. Meheut, V. Polyakov, A. Shahar, and Associate editorE. Schauble are thanked for their thoughtful reviews of the manu-script. This work was supported by NASA (NNX09AG59G), byNSF EAR Petrology and Geochemistry (EAR-1144429), and bya Packard Fellowship to N. Dauphas. L. Gao acknowledges thefinancial support from COMPRES under NSF Cooperative Agree-ment EAR 10-43050. Use of the Advanced Photon Source, anOffice of Science User Facility operated for the U.S. Departmentof Energy (DOE) Office of Science by Argonne NationalLaboratory, was supported by the US DOE under contract N�DE-AC02-06CH11367.

APPENDIX A. SUPPLEMENTARY DATA

Supplementary data associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.gca.2012.06.013.

APPENDIX B

Below, we demonstrate how equilibrium iron isotopefractionation factors at any temperature can be derivedfrom the moments of the partial phonon density of statesg(E). Bigeleisen (1958) carried out a Bernoulli expansionof the reduced partition function ratio and obtained aformula that is valid for dimensionless frequencies�hxmax=kT < 2p (also see Elcombe and Hulston, 1975),

ln bI=I� ¼1eN X1

j¼2

X3N

i

Bjduji

j!j; ðB1Þ

where eN is the number of isotopically substituted atoms, N

is the total number of atoms in a multi-atomic compound,duj

i ¼ u�ji � uji (u* and u are the dimensionless frequencies of

ith phonon mode for the two substituted iron isotopes;ui ¼ �hxi=kT ), and Bj are the Bernoulli numbers (i.e.,B0 = 1, B1 = �1/2, B2 = 1/6, B3 = 0, B4 = �1/30, B5 = 0,B6 = 1/42).

Elcombe (1974) and Menendez et al. (1994) demon-strated using first-order perturbation theory that normalmode frequency shifts upon isotope substitution are relatedto the masses of the substituted isotopes,

u�2i � u2i

u2i¼ M �M�

M� � 1

N

Xm

jeimj2; ðB2Þ

where m denotes the eN isotopically substituted atoms. Onlyone isotope species is replaced, so the mass change is con-stant. |eim|

2 is the square modulus of the polarization vectorfor mode i and isotope m (eim is a vector of dimension 3, cor-responding to the 3 directions of space). The phonon polar-ization vectors are normalized to N. The polarization vectoris a mathematical construction that relates the displacementof a particular atom to the excitation of a particular vibra-tion mode. Given that x = u*/u is close to 1 andxj � 1 = (1 + x � 1)j � 1 ’ j(x � 1), it follows,

u�ji � uji

uji

� ju�i � ui

ui

� �� j

2

M �M�

M�

� �� 1

N

Xm

jeimj2; ðB3Þ

Using this formula, Eq. (B1) can be rewritten as (this is Eq.(13) of Elcombe and Hulston, 1975),

ln bI=I� ¼1

2

MM� � 1

� �X1j¼2

Bj

j!1eN Xm

1

N

X3N

i

jeimj2uji ; ðB4Þ

where the last two summations over all substituted isotopesand all phonon modes can be expressed in terms of the mo-ments of the projected partial phonon DOS (a definition isprovided in Hu et al., 2003; Eq. (2)),

ln bI=I� ¼1

2

MM� � 1

� �X1j¼2

Bj

j!3

mgj

ðkT Þj

¼ 3

2

MM� � 1

� �X1j¼2

Bj

j!ðkT Þjmg

j ; ðB5Þ

with mgj ¼

Rþ10

EjgðEÞdE the jth moment of iron partial

PDOS g(E) (note that the integral of the PDOS isRg(E)dE = 1). Here the sample is assumed to be isotropic

or powder. The factor of 3 comes from comparing thesquare modulus of the phonon polarization vector vs. itsprojection along a particular direction. We recognize the

sumP1j¼0

Bjxj=j! ¼ x=ðex � 1Þ, so Eq. (B5) equation can be

rewritten as,

ln bI=I� ¼3

2

MM� � 1

� �Z Emax

0

E2kTþ E=kT

eE=kT�1� 1

� �gðEÞdE:

ðB6Þ

Equations (B5) and (B6) had been derived by Polyakovet al. (2005b, 2007) using a different approach based onthe kinetic energy. Note that there is an error in Eq. (9)of Polyakov et al. (2005b); the factor should be h/k ratherthan (h/k)2i.

APPENDIX C

The derivation below establishes a relationship betweenthe moments of g and the moments of S. Let us denote,

mS12nþ1 ¼

Rþ1�1 E2nþ1S1ðEÞdE the 2n + 1 moment of S1(E),

mg2n ¼

Rþ10

E2ngðEÞdE the 2n moment of g(E).

In the harmonic approximation, the one phonon term inthe excitation probability is related to g(|E|) through,

S1ðEÞ ¼fERgðjEjÞ

Eð1� e�E=kT Þ ; ðC1Þ

where f is the Lamb-Mossbauer factor. The 2n + 1 (odd)moment of S1(E) with n an integer is,

mS12nþ1 ¼

Z 0

�1E2nþ1S1ðEÞdE þ

Z þ1

0

E2nþ1S1ðEÞdE: ðC2Þ

Using Eq. (C1), the first integral in Eq. (C2) can be rewrit-ten as,Z 0

�1E2nþ1S1ðEÞdE ¼

Z þ1

0

E2nþ1 fERgðEÞEð1� eE=kT Þ gðEÞdE; ðC3Þ

while the second integral takes the form,



Z þ1
0

E2nþ1S1ðEÞdE ¼Z þ1

0

E2nþ1 fERgðEÞEð1� e�E=kT Þ gðEÞdE:

ðC4Þ

We therefore have (C3 + C4),

mS12nþ1¼fER

Z þ1

0

E2nþ1 1�eE=kT� �

þE2nþ1 1�e�E=kT� �

E 2�e�E=kT �eE=kTð Þ gðEÞdE:

ðC5Þ

It follows,

mS12nþ1 ¼ fER

Z þ1

0

E2ngðEÞdE ¼ fERmg2n: ðC6Þ

For a 2n (even) moment, the temperature terms do not can-cel out and we have instead,

mS12n ¼ fER

Z þ1

0

E2n�1 1

1� e�E=kT� 1

1� eE=kT

� �gðEÞdE:

ðC7Þ

For energies that are most relevant to higher moment calcu-lations in phases with high force constants, the term inparenthesis is approximately 1,

mS12n � fERmg

2n�1: ðC8Þ

This is the same formula as the one that we had derived forodd moments of S (C6). This approximation can be refinedby recognizing that,

1

1� e�E=kT� 1

1� eE=kT� aþ b

E=kT: ðC9Þ

The figure below shows these two equations at T = 300 Kwith a = 0.63 and b = 1.48.

The even moments then take the form,

mS12n � afERmg

2n�1 þ bkTfERmg2n�2: ðC10Þ

Thus, the moments of S1 can be related to the moments ofg. The moments introduced here are centered on zero. Letus denote,

RSni ¼

Z þ1

�1ðE � ErÞiSnðEÞdE: ðC11Þ

the moment of the n-phonon excitation probabilitycentered on Er. Because Er� E over most of the PDOS,

RSni �

Z þ1

�1ðEi � iEi�1ErÞSnðEÞdE ¼ msn

i � iErmSni�1: ðC12Þ

We are now going to establish a relation between the mo-ments of the n-phonons and the moments of the 1 phonon.In the harmonic approximation, the n-phonon is calculatedusing,

SnðEÞ ¼1

nf

Z þ1

�1Sn�1ðxÞS1ðE � xÞdx: ðC13Þ

The ith moment centered on zero is then,

mSni ¼

1

nf

Z þ1

�1

Z þ1

�1Sn�1ðxÞS1ðE � xÞEidxdE; ðC14Þ

permuting the integration order and changing the variableto u = E � x, this equation takes the form,

mSni ¼

1

nf

Z þ1

�1Sn�1ðxÞ

Z þ1

�1S1ðuÞðuþ xÞidudx: ðC15Þ

We can now do a binomial expansion of (u + x)i,

mSni ¼

1

nf

Z þ1

�1Sn�1ðxÞ

Z þ1

�1S1ðuÞ

Xi

k¼0

Cki ui�kxkdudx; ðC16Þ

where Cki ¼ i!=½ði� kÞ!k! are the binomial coefficients.

Taking the summation out of the integral it follows,

mSni ¼

1

nf

Xi

k¼0

Cki

Z þ1

�1Sn�1ðxÞxk

Z þ1

�1S1ðuÞui�kdudx: ðC17Þ

We therefore have,

mSni ¼

1

nf

Xi

k¼0

Cki mSn�1

k mS1i�k : ðC18Þ

This relationship allows one to calculate the moments of allmultiple phonons using the moment of the single phonon.For example, the ith-moment of the 2-phonon is,

mS2i ¼

1

2f

Xi

k¼0

Cki mS1

k mS1i�k : ðC19Þ

Knowing the moments of the 2-phonon, it is straightfor-ward to calculate the moments of the 3-phonon and pro-ceed iteratively to calculate the ith moment of anymultiple phonon. The ith moment of S(E), the excitationprobability is the sum of all ith moments of Sn(E),

mSi ¼ mS0

i þ mS1i þ

1

f

X1n¼2

Xi

k¼0

Cki mSn�1

k mS1i�k

n; ðC20Þ

where S0 is a delta function and its moment is zero fori P 1. With Eqs. (C6) and (C10), we have shown that thereis a relationship between ms1

i and the moments of g. Eqs.(C18) and (C20) show that there is a relationship betweenmS1

i and mSi . We can thus express the moments of S as a

function of the moments of g and vice versa. We have cal-culated the second to tenth moments of S centered on Er byusing Eq. (C12) and expanding Eq. (C20) into moments ofS1, which were expressed as combinations of the momentsof g (Eqs. (C6) and (C10)). We used the fact that mg

0 ¼ 1and mS1

0 ¼ �f ln f . Furthermore, the series


f � ln f þ 1

2f ðln f Þ2 � 1

6f ðln f Þ3 þ 1

24f ðln f Þ4 þ . . . � 1

appeared in many coefficients, simplifying the final expres-sions. The terms that were found to be negligible are omit-ted in the following equation,

RS2 ¼ aErm

g1 þ bErkT e

RS3 ¼ Erm

g2

RS4 ¼ aErm

g3 þ 3a2E2

r ðmg1Þ

2 þ bErkT emg2 þ 6abE2

r kT emg1

þ 3b2E2r k2T 2

e

RS5 ¼ Erm

g4 þ 10aE2

r mg1mg

2 þ 10bE2r kT em

g2

RS6 ¼ aErm

g5 þ 15a2E2

r mg1mg

3 þ 10E2r ðm

g2Þ

2 þ 15b2E2r k2T 2

emg2

þ 15abE2r kT em

g3 þ 15abE2

r kT emg1mg

2 þ bErkT emg4þ

45a2bE3r kT eðmg

1Þ2 þ 45ab2E3

r k2T 2emg

1 þ 15b3E3r k3T 3

e

RS7 ¼ Erm

g6 þ 35aE2

r mg2mg

3 þ 21aE2r mg

1mg4 þ 105a2E3

r ðmg1Þ

2mg2

þ 35bE2r kT eðmg

2Þ2þ

21bE2r kT em

g4 þ 210abE3

r kT emg1mg

2 þ 105b2E3r k2T 2

emg2

ðC21Þ

where Te denotes the temperature at which the experimentswere performed (typically �300 K). The second equationcorresponds to Lipkin’s sum rule for the force constant.To our knowledge, this is the first time that this formulais established on mathematical grounds alone. There aresix unknowns (the moments of g) in 6 equations and thissystem can be solved,

mg2 ¼

RS3

Er

mg4 ¼

RS5 � 10RS

2RS3

Er

mg6 ¼

RS7 þ 210ðRS

2Þ2RS

3 � 35RS3RS

4 � 21RS2RS

5

Er:

ðC22Þ

Although the derivations were made in the most generalcase (for the even moment of S1, Eq. (C10) was used ratherthan Eq. (C8) and the relationship in Eq. (C21) has sometemperature dependence), the formulas that relate the evenmoments of g to the moments of S are temperature inde-pendent and should be valid for any phase.

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A general moment NRIXS approach to the determination of