Geophysical Journal International
Geophys. J. Int. (2012) 190, 310–322
doi: 10.1111/j.1365-246X.2012.05511.x
Thermodynamic properties of (Mg,Fe2+ )SiO3 perovskite at the
lower-mantle pressures and temperatures: an internally consistent
LSDA+U study
Arnaud Metsue and Taku Tsuchiya
GJI Mineral physics, rheology, heat flow and volcanology
Geodynamics Research Center, Ehime University, 2–5 Bunkyo-cho, 790–8577 Matsuyama, Japan. E-mail: [email protected]
Accepted 2012 April 17. Received 2012 April 6; in original form 2011 October 17
SUMMARY
The thermodynamic properties of (Mg0.9375 Fe2+ 0.0625 )SiO3 perovskite have been investigated
at the pressure and temperature conditions of the lower mantle by first-principles calculations
where iron is incorporated in the high and low-spin states for the first time. The electronic
structure of ferrous Fe-bearing perovskite is modelled within the internally consistent local
spin density approximation with a Hubbard correction U. The thermodynamic properties
are derived from the calculation of the Helmholtz free energy within the quasi-harmonic
approximation, which requires the phonon frequencies determined by direct calculations of the
dynamic matrices. Incorporation of iron, irrespective of its spin states, decreases the acoustic
phonon mode frequencies, but less affects high-energy optic modes, leading to decreasing of
the acoustic wave velocities in Fe-bearing MgSiO3 perovskite, consistent with previous studies
on the elasticity of this phase. This study suggests that the thermodynamic properties of silicate
perovskite, such as the equation of state and isothermal bulk modulus, are not largely modified
by the incorporation of 6.25 per cent of ferrous iron. Calculations of the static enthalpy of the
iron-bearing perovskite in the 0–150-GPa-pressure range demonstrate that low-spin ferrous
iron is unstable at the pressure conditions of the lower mantle. Finally, we clarify the perovskiteto-post-perovskite phase transition boundary in an (Mg0.9375 Fe0.0625 )SiO3 composition. Ferrous
iron is found to decrease the transition pressure between the two phases with a small binary
phase loop of 3–4 GPa at the lowermost mantle conditions from 111 to 115 GPa at 2500 K
and from 116 to 119 GPa at 3000 K.
Key words: Equations of state; High-pressure behaviour; Phase transitions.
1 I N T RO D U C T I O N
Al-(Mg,Fe)SiO3 perovskite (Pv) is thought to be the most abundant mineral in the Earth’s lower mantle. Silicate Pv represents 68, 75 and
35 per cent of the mineral proportions in the representative pyrolitic, harzburgitic and MORB model compositions, respectively (Irifune &
Tsuchiya 2007). Therefore, the thermodynamic properties of the silicate Pv in a representative chemical composition in particular including
ferrous iron, such as the thermal equation of state (EOS) or the heat capacities, are of great interest to constrain seismological and thermal
models for the lower mantle (Poirier 1991).
The pressure–volume EOS and the thermodynamic properties of pure MgSiO3 Pv were intensively studied, both experimentally (Mao
et al. 1991; Wang et al. 1994; Funamori et al. 1996; Fiquet et al. 1998; Fiquet et al. 2000; Shim & Duffy 2000; Katsura et al. 2009; Mosenfelder
et al. 2009) and theoretically (Karki et al. 2000, 2001; Caracas & Cohen 2005; Tsuchiya et al. 2005). Experimentally, the zero-pressure
bulk modulus K 0 and its pressure derivative K 0 are estimated at 253–264 and 3.9–4.0 GPa, respectively at ambient temperature. These same
quantities are evaluated to 247 and 3.97 GPa at 300 K from ab initio calculations within the local density approximation (LDA; Karki et al.
2001). However, the determination of the thermodynamic quantities remains challenging for Pv with a composition representative of the
Earth’s lower mantle. Several experimental studies were performed to constrain the effects of the incorporation of Fe on the EOS and other
thermodynamic parameters of MgSiO3 Pv (Knittle & Jeanloz 1987; Mao et al. 1991; Wang et al. 1994; Andrault et al. 2001; Lundin et al.
2008). These experimental studies suggest that the incorporation of a small amount of Fe in the Pv phase increases slightly the volume while
the effects on the bulk modulus are unclear. Andrault et al. (2001) reported an increment of +0.3–0.4 per cent of the volume and +1.1–2.7 per
cent of K 0 in (Mg0.95 Fe0.05 )SiO3 compared to pure MgSiO3 Pv in the 0–60-GPa-pressure range. The results of Lundin et al. (2008) showed
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that the incorporation of 9 per cent of Fe in the Pv phase increases the specific volume at 0 GPa by 0.54 per cent. The value of the bulk
modulus at 0 GPa determined from the EOS depends on the used pressure scale and still not fully clarified.
First-principles calculations are an alternative way to investigate the effects of cationic substitution on the thermodynamic properties
of lower mantle minerals. The incorporation of ferrous iron on the pressure–volume EOS of Pv was previously investigated with ab initio
calculations employing conventional LDA or generalized gradient approximation (GGA) exchange–correlation functional (Kiefer et al. 2002;
Caracas & Cohen 2005, 2007, 2008; Tsuchiya & Tsuchiya 2006; Bengtson et al. 2008; Umemoto et al. 2010) or with an addition of a
correction on the local d electron–electron interactions of iron as the LSDA+U technique (Hsu et al. 2010a,b). The LSDA+U technique
gives a better description of the non-metallic nature of Fe-bearing MgSiO3 minerals (Hsu et al. 2011a). Bengtson et al. (2008) reported an
increasing of the bulk modulus K 0 with the increasing of ferrous iron concentration, irrespective of the spin state of Fe. This same study
indicates also an increasing of the specific volume V 0 with increasing high-spin (HS) Fe concentration while this quantity decreases with the
incorporation of low-spin (LS) Fe. The effects of the incorporation of ferric iron, with or without Al, on the elastic properties of MgSiO3
Pv were also investigated from first-principles calculations (Li et al. 2005; Hsu et al. 2011b). Hsu et al. (2011b) reported the bulk modulus
up to 150 GPa at 0, 300 and 2000 K for a (Mg0.875 Fe0.125 )(Si0.875 Fe0.125 )O3 composition from a combination of LSDA+U calculations and
the virtual crystal model approximation to model the solid solution. This study however suggests no clear temperature dependence of the
elastic properties of Fe2 O3 -bearing MgSiO3 Pv. Therefore, this mean-field-like treatment for the thermal effect is clearly too simple and all
of the previous theoretical investigations on the effects of the incorporation of Fe are performed only at static conditions. Instead, direct
lattice dynamic calculations in Fe-bearing MgSiO3 Pv, which have never been performed so far, are suitable to determine the thermodynamic
properties at finite temperature.
Important and controversial issues remain on the spin state of Fe in Fe2+ -bearing MgSiO3 Pv and on the effect of iron on the phase
transition between Pv and post-Pv (PPv) occurring at the P–T conditions of the bottom of the lower mantle. Experimentally, Badro et al.
(2004) reported a spin transition of iron in a (Mg0.9 Fe0.1 )SiO3 composition, from HS to an intermediate spin state at 70 GPa and to the LS
state at 120 GPa. Recently, McCammon et al. (2010) reported the LS state of ferrous iron at 120 GPa and 1000 K in an (Mg0.82 Fe0.18 )SiO3
composition. On the other hand, Jackson et al. (2005) did not observe any spin transition of ferrous iron which remains in HS in the pressure
condition of the lower mantle. From a theoretical point of view, the evidence of the spin transition of ferrous iron from HS to LS is strongly
dependent of the exchange–correlation functional, iron concentration and configuration (see Hsu et al. 2010b, for a recent review). On the
effects of Fe incorporation on the Pv-to-PPv phase transition, Mao et al. (2004) suggested that the presence of iron decreases the transition
pressure between Pv and PPv while Hirose (2006) argued that the effects of incorporated iron are small. Catalli et al. (2009) determined
experimentally the phase diagram of (Mg0.9 Fe0.1 )SiO3 at P = 105–140 GPa and T = 1500–3000 K and reported a decreasing of the transition
pressure between Pv and PPv with a wide Pv–PPv binary phase loop thickness of 20 GPa. Sinmyo et al. (2011) reported that iron affects to
increase this transition pressure. As well as the transition pressure, its transition width still remains quite unclear. Very recently, Grocholski
et al. (2012) determined the Pv-to-PPv phase boundaries in pyrolitic, MORB and San Carlos olivine from laser-heated diamond-anvil cell
(DAC) experiments with Pv–PPv coexisting domains of about 30, 15 and 3 GPa for these representative compositions, respectively. Andrault
et al. (2010) reported a Pv-to-PPv phase transition in the MgSiO3 –FeAlO3 system in the whole pressure range of the D layer for a pyrolitic
composition. These experimental results largely scatter possible due to difficulty in accurate controlling of kinetic effects in DAC (Catalli
et al. 2009; Grocholski et al. 2012). From a theoretical point of view, Caracas & Cohen (2007, 2008) calculated the phase diagram of the
MgSiO3 –FeSiO3 solid solution at the P–T conditions of the bottom of the lower mantle and pointed a decreasing of the transition pressure
of the Pv–PPv phase transformation with the incorporation of iron in the phases. This latter theoretical study neglected the vibrational and
magnetic contribution to the entropy that could influence the P–T conditions of the Pv–PPv phase transition.
In this paper, we calculate the thermodynamic parameters, such as the P–V equations of state, thermal expansion coefficient and heat
capacities, of (Mg0.9375 Fe0.0625 )SiO3 Pv at pressure and temperature of the lower mantle from ab initio calculations. We focus on ferrous Fe in
this study because it is believed that this oxidation state of Fe represents 85 and 50 per cent of total Fe in Al-free and Al-bearing (Mg,Fe)SiO3
Pv, respectively (McCammon 1997). The internally consistent LSDA+U formalism is used to model the electronic structure of the ferrous
iron-bearing systems with partially occupied d shells (Anisimov et al. 1991). Iron is incorporated in HS and LS state since the spin state
of ferrous iron in the P–T condition of the lower mantle is still under significant debate in Fe-bearing silicate Pv. The high-temperature
thermodynamic quantities are determined within the quasi-harmonic approximation, which involved the phonon frequencies calculated with
a direct method as our recent study on the thermodynamic properties of (Mg0.9375 Fe0.0625 )SiO3 PPv (Metsue & Tsuchiya 2011). Finally, we
extend the calculation of the Helmholtz free energy to the Gibbs free enthalpy to determine the spin state of iron and the phase transition
between ferrous Fe-bearing Pv and PPv.
2 METHODS
We performed first-principles calculations within the density functional theory (DFT; Hohenberg & Kohn 1964; Kohn & Sham 1965)
implemented in the PWSCF code (Giannozzi et al. 2009). We investigate an (Mg0.9375 Fe0.0625 )SiO3 composition for the Pv phase, close to
the amount of iron observed experimentally in Pv in a pyrolitic composition (∼7 molar per cent; Kesson et al. 1998). Iron is incorporated in
the HS and LS state as Mg substitution defect in an 80-atom supercell, similarly to our previous study of the incorporation of Fe in the PPv
phase (Metsue & Tsuchiya 2011). Hsu et al. (2010a) reported two stable atomic sites for HS Fe with different quadrupole splitting values.
The first one is the eightfold coordinated position of HS Fe and is associated with a quadrupole splitting value of 3.3 mm s–1 . The other one
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Table 1. Definition of the models used in this study for incorporated HS (a) and LS (b). The Fe–Fe
interatomic distance, volume, bulk modulus and energy at 0 GPa and 0 K are provided for each
investigated models. K is fixed at 3.94. We give the values of the volume and the bulk modulus at
0 GPa for pure MgSiO3 Pv in (c).
(a)
Supercell
d (Fe–Fe) (Å)
V 0 (cm3 mol−1 )
K 0 (GPa)
E0 (Ry)
Model 1
Model 2
Model 3
2×2×1
2×1×2
1×2×2
6.8794
4.9246
4.7723
24.3325
24.3326
24.3317
259.3412
259.2916
259.4608
−1974.55147
−1974.55165
−1974.55121
(b)
Supercell
d (Fe–Fe) (Å)
V 0 (cm3 mol−1 )
K 0 (GPa)
E0 (Ry)
Model 1
Model 2
Model 3
2×2×1
2×1×2
1×2×2
6.8752
4.9197
4.7644
24.2583
24.2605
24.2596
260.7987
260.7980
260.7441
−1974.45387
−1974.45044
−1974.44879
(c)
V 0 (cm3 mol−1 )
K 0 (GPa)
MgSiO3
24.2717
258.7857
for HS Fe comes from a translation of the former position along the (1 0 0) and (0 1 0) directions with a quadrupole splitting of 2.3 mm s–1 .
The study of Hsu et al. (2010a) suggests a transition of the most stable position of HS Fe from the site associated with the value of the
quadrupole splitting of 2.3 mm s–1 to the site associated with the value of 3.3 mm s–1 at 7 GPa, out of the thermodynamic stability field of
the Pv phase. Consequently, only the atomic position for HS Fe associated with a quadrupole splitting value of 3.3 mm s–1 is investigated
in our study. The calculations are performed for three different supercell shapes based on the repetition of the primitive Pv cell to take into
accounts the effect of the iron distribution. The details of the three investigated models are given in Table 1. The supercells are relaxed at 0,
30, 60, 90, 120 and 150 GPa with the variable-cell-shape molecular dynamics algorithm (Wentzcovitch et al. 1993). The convergence of the
calculations is achieved when the total forces acting on the atoms are less than 10−5 Ry a.u–1 and the energy differences on the electronic
charge density less than 10−9 Ry.
In this study, the energy of the system is calculated with the internally consistent LSDA+U formalism, since conventional DFT
techniques involving the LDA and the GGA to approximate the exchange–correlation energy both fail to describe the correct electronic
structure in transition metal oxides. This technique was already used for Fe-bearing MgO and MgSiO3 Pv and PPv and the reader is invited
to refer these papers and reference therein for further details (Tsuchiya et al. 2006a,b; Hsu et al. 2010a,b; Metsue & Tsuchiya 2011). The
on-site screened Coulomb interaction U is optimized for each volume and spin state based on the linear response theory with the constrained
total energy variational principles (Cococcioni & de Gironcoli 2005). The values of the correction U used in this study are given Table 2.
These values are similar when Fe is incorporated in LS state and two times higher at low pressures when Fe is incorporated in HS state than
those calculated by Hsu et al. (2010a) for an (Mg0.875 Fe0.125 )SiO3 composition. We check that our system remains non-metallic within the
LSDA+U formalism, with energy band gaps equal to 1.4 and 2.5 eV at 60 GPa when iron is incorporated in HS and LS, respectively. We
used pseudo-potentials built within the Troullier–Martins method (Troullier & Martins 1991) for Mg, Si and O atoms and with the ultrasoft
method (Vanderbilt 1990) for Fe atoms to model the ion–electrons interactions. These pseudo-potentials were already tested and used in our
previous studies (Tsuchiya et al. 2004; Tsuchiya et al. 2006a,b; Metsue & Tsuchiya 2011). The plane wave cut-off was set to 70 Ry and the
irreducible Brillouin zone (IBZ) was sampled with a Monkhorst–Pack grid (Monkhorst & Pack 1976) adapted for each supercell geometry:
2 × 2 × 2, 2 × 3 × 1, 3 × 2 × 1 for the model 1, 2 and 3, respectively.
The thermodynamic properties of iron-bearing phases are determined by applying the quasi-harmonic approximation (QHA) to the
lattice free energy, where the Helmholtz free energy F(V , T) adopts this expression:
1
hω j (q, V ) + k B T
ln[1 − exp(−hω j (q, V )/k B T )] − T (Smag + Sconf ),
(1)
F(V, T ) = E(V ) +
2 q, j
q, j
where the first, second and third terms are the static energy, the zero-point and thermal contributions, respectively. S mag and S conf are the
magnetic and configurational contributions of iron atoms to the total entropy, respectively. The zero-point and thermal contributions include
Table 2. On-site Hubbard correction U parameters
(in eV) for HS/LS (Mg0.9375 Fe0.0625 )SiO3 perovskite
used in this study.
P (GPa)
0
30
60
90
120
150
U (eV) for HS
U (eV) for LS
6.74
4.44
3.85
3.65
3.64
3.64
5.05
5.14
5.23
5.31
5.40
5.49
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the phonon frequencies ωj . Those are determined by calculating the phonon dispersion relations by diagonalizing the dynamic matrix here
calculated with a direct method. We usually apply the density functional perturbation theory (DFPT) for calculating the phonon relations,
but it is less efficient for the Fe-bearing supercells due to a significant number of displacements to be considered. The same technique was
recently used in the cases for Fe-bearing MgSiO3 PPv (Metsue & Tsuchiya 2011) and for the Si–Fe solid solution at the Earth’s inner core
conditions (Côté et al. 2010). For the 80-atom supercell of (Mg0.9375 Fe0.0625 )SiO3 Pv, in total 240 displacements are required to build a single
force constant matrix. The applied displacements are equal to 0.01 Å and the dynamic matrices (or phonon dispersions) are calculated with
the PHON code by the Fourier transformation of the force constant matrix (Alfè 2009). The magnetic contribution to the entropy S mag is given
by (Tsuchiya et al. 2006b; Wentzcovitch et al. 2009)
Smag = −kB X Fe ln[m(2S + 1)].
(2)
In this expression, X Fe is the iron concentration, S, the total spin quantum number and m, the orbital degeneracy: S = 2 (S = 0) and m =
5 (m = 1) for HS (LS) Fe, respectively. The configurational contribution to the entropy, S conf , is approximated with that for an ideal mixing
where the different Fe configurations are energetically equivalent
Sconf = −kB [X Fe ln(X Fe ) + (1 − X Fe ) ln(1 − X Fe )].
(3)
We compare the static enthalpy of our three investigated configurations to validate this approximation. We conduct additional static calculations
at 30 and 120 GPa on a supercell built on a 2 × 2 × 2 repetition of the Pbnm primitive cell (160 atoms) where two atoms of Fe in HS and
LS spin states substitute two atoms of Mg so that the Fe–Fe is maximized or minimized. We found that the enthalpy differences between
the three configurations for the 80-atom supercells are less than 1.3 × 10−2 eV per supercell in the case of incorporated HS iron and 7.8 ×
10−2 eV per supercell in the case of incorporated LS iron in the 30–90-GPa-pressure range. The enthalpy differences between the two
investigated configurations of Fe in the 160-atom supercells are equal to 6.9 × 10−2 eV per supercell at 30 GPa and 3.2 × 10−2 eV per
supercell at 120 GPa when Fe atoms are incorporated in HS state. These enthalpy differences are equal to 1.4 × 10−2 eV per supercell at 30
GPa and 3.0 × 10−3 eV per supercell at 120 GPa when LS Fe are substituted to Mg atoms. We note that the 160-atom supercell with the shorter
Fe–Fe distance has lower enthalpy, similarly to the previous study of Stackhouse et al. (2007). These small enthalpy differences between each
model in HS and LS Fe-bearing systems are however compensated by a thermal energy k B T with T ∼ 800 and 900 K, clearly lower than the
estimated temperatures in the lower mantle, that is, T > 1900 K (Ito & Katsura 1989) and suggest that the investigated configurations have
the same probability at the P–T conditions of the lower mantle.
3 R E S U LT S A N D D I S C U S S I O N
3.1 Phonon dispersion relations
The vibrational properties of Fe-bearing MgSiO3 Pv are investigated through the determination of the phonon dispersion curves up to
150 GPa with the direct method. Pure MgSiO3 Pv has 20 atoms per Pbnm primitive cell and leads to 60 vibration modes at any point in
the Brillouin zone (BZ). The incorporation of iron in the Pv structure breaks the symmetries and 240 vibration modes are generated for the
80-atom supercell. The phonon dispersion curves are computed for in total 500 q-vectors between each high symmetry point in the BZ.
The phonon dispersion curves at 60 GPa and the vibrational density of states (VDoS) for pure and incorporated HS and LS Fe in MgSiO3
Pv calculated with the model 1 are displayed in Fig. 1. The phonon dispersion relation calculated for pure MgSiO3 Pv is in good agreement
with the previous results of Karki et al. (2000) obtained by DFPT, except some optic mode frequencies close to the -point due to the LO–TO
splitting which was not considered in this study. The analysis of the partial VDoS of Fe indicates that the iron participation on the phonon
frequencies is limited to low frequencies. At 60 GPa, Fe contributes less than 1 per cent to the total VDoS for frequencies higher than 802.24
and 882.29 cm−1 when Fe is incorporated in HS and LS, respectively. In addition, the frequency where Fe mostly participates to the VDoS
is shifted to higher frequencies for Pv with LS iron compared to Pv with HS iron. The most intensive peaks stand at 235.2 and 315.2 cm−1
in incorporated HS and LS Fe systems, respectively. This frequency shift of the iron partial VDOS can be attributed to the position of LS
iron, which is slightly displaced along the (001) axis from eightfold to sixfold coordinated position and leads to small hardening in the force
constants. In consequence, Fe affects mainly the low phonon frequencies and, in particular, the acoustic phonon frequencies. These latter are
clearly softer compared to pure MgSiO3 . For example, at X -point, the longitudinal and transverse acoustic frequencies are equal to 238.64,
227.63 and 227.63 cm−1 for pure MgSiO3 at 60 GPa. For the same point in the BZ and the same pressure, these frequencies decrease to 122.36,
115.38 and 121.03 cm−1 in incorporated HS Fe Pv and 124.80, 114.89 and 120.58 cm−1 in incorporated LS Fe Pv. A decreasing of the acoustic
mode phonon frequencies with the incorporation of Fe implies a decreasing of the acoustic wave velocities in MgSiO3 Pv. This is consistent
with previous theoretical studies that showed a decreasing of the acoustic wave velocities with the incorporation of ferrous Fe (Kiefer et al.
2002; Tsuchiya & Tsuchiya 2006; Stackhouse et al. 2007). As a consequence of the limited participation to the low frequency part of the
phonon dispersion relation, the presence of Fe has no significant influence on the highest optic mode frequencies. At -point, the highest
optic mode frequency is 1032.65 cm−1 for pure MgSiO3 and 1095.66 and 1096.75 cm−1 when Fe is incorporated in HS and LS, respectively,
at 60 GPa. The effects of the incorporation of Fe on the phonon dispersion relations in the Pv phase are similar to those observed in Fe-bearing
MgSiO3 PPv (Metsue & Tsuchiya 2011), while these behaviours are not correctly captured by mean-field type simplification such as the
virtual crystal approximation (Wentzcovitch et al. 2009). This limited contribution of iron on the vibrational properties of (Mg,Fe)SiO3 Pv is
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Figure 1. Phonon dispersion curves for MgSiO3 (a) and for (Mg0.9375 Fe0.0625 )SiO3 Pv with iron in the HS (b) and LS (c) state at 60 GPa. We add the
vibrational density of states on the right side for the Fe-bearing phases. The partial density of states of Fe is filled in pink and is not normalized according to
the total density of states for clarity.
also consistent with the results of infrared spectroscopy of (Mg0.9 Fe0.1 )SiO3 and pure MgSiO3 Pv, which show no significant differences in
the infrared spectra between the Fe-bearing and Fe-pure phases (Lu et al. 1994).
We investigate the effect of the iron distribution on the vibrational properties of Fe2+ -bearing MgSiO3 Pv. In Fig. 2, we compare the
VDoS for the three investigated models at 60 GPa for both incorporated HS and LS Fe. According to this figure, the iron distribution seems
to have a minor effect on the vibrational properties. The highest phonon frequencies are marginally affected by the spin state of Fe and are
equal to 1119.1, 1112.5 and 1122.5 cm−1 for model 1, 2 and 3, respectively and for both spin states. This limited effect of the iron distribution
suggests a similar thermal contribution to the Helmholtz free energy, irrespective of the iron distribution.
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Figure 2. Vibrational density of states of (Mg0.9375 Fe0.0625 )SiO3 Pv with Fe in the HS state (a) and in the LS state (b) at 60 GPa for the three investigated
models defined in Table 1. The partial density of states of Fe is not normalized according to the total density of states for clarity.
The phonon dispersion curves are calculated for several pressures, from 0 to 150 GPa for both spin states and investigated models.
The phonon dispersion curves do not exhibit any imaginary frequencies in the case of incorporated HS Fe in the 0–150-GPa-pressure range
for all the investigated models. The situation is different for incorporated LS Fe, where the phonon dispersion curves display imaginary
frequencies in the Z–T–R–U pathway in the BZ at 0 GPa for the three investigated models. Dynamic instability is in principle resolved by
atomic rearrangements relevant to the soft phonon modes. Hsu et al. (2010a) proposed several structures of LS Fe-bearing MgSiO3 Pv at low
pressure, involving the rotation of the Si–O6 octahedra around the Fe atom. However, LS Fe-bearing silicate Pv is not stable at 0 GPa and
further investigation on the stable Pv phase at such low pressure is out of the purpose of this study.
3.2 Finite temperature thermodynamic properties
The pressure–volume and the energy–volume equations of states at static conditions for the three investigated models are determined first.
We use the third-order Birch–Murnaghan EOSs (Poirier 1991) to fit the pressure–volume and energy–volume relations where the pressure
derivative of the bulk modulus at 0 GPa, K 0 , is close to 3.94 for the three investigated models and the two spin states. In the following,
K 0 is fixed to 3.94 to clearly describe the effects of the incorporation of Fe on the volume and the bulk modulus at static conditions. The
volume, the bulk modulus and the energy at 0 GPa for the three investigated HS/LS Fe configurations are presented in Table 1. The specific
volume V 0 and the bulk modulus K 0 at static pressure for pure MgSiO3 are calculated from the EOS and are equal to 24.272 cm3 mol−1 and
258.79 GPa, respectively. Previous studies showed that incorporation of Fe expands the volume of the Pv phase with a trend of ∂∂ lnX FeV around
0.030–0.034 (Fei et al. 1996; Kiefer et al. 2002; Tsuchiya & Tsuchiya 2006). We found values of 0.038 and −0.006 for this trend when Fe
is incorporated in HS and LS, respectively, from our LSDA+U calculations. The slightly larger value for HS Fe-bearing phase compared to
previous studies comes from the use of the Hubbard correction on the d electrons of Fe, which is thought to emphasize the effect of iron. The
bulk modulus of the Pv phase at 0 GPa increases with the inclusion of iron, with a more pronounced effect when LS Fe is incorporated. We
adopt a linear relationship to describe the Fe-content dependence on the bulk modulus as K 0 (X Fe ) = K 0 (MgSiO3 )(1+bX Fe ). We found values
of b equal to 0.035 and 0.123 when Fe is incorporated in HS and LS, respectively, slightly lower to the previous value of 0.039 calculated by
Tsuchiya & Tsuchiya (2006) for HS Fe-bearing Pv. This difference comes from the larger volume calculated within the LSDA+U formalism.
The comparison of our results with previous theoretical studies dealing with the incorporation of ferric iron can constrain the effects of the
oxidation state of Fe on the elastic properties. LSDA+U calculations by Hsu et al. (2011b) and GGA calculations by Li et al. (2005) reported
values of K 0 about 260 GPa at 0 K for a 12.5 per cent Fe2 O3 -bearing phase and about 230 GPa at 0 K for a (Mg0.9375 Fe0.0625 )(Si0.9375 Al0.0625 )O3
composition, respectively. This comparison between the values of K 0 indicates that the oxidation state of Fe seems have a small effect on
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the elastic properties of the Pv phase. The significantly smaller K 0 found in the latter study might be related to the use GGA and not due to
the incorporation of Al. The ferrous iron configuration has minor effects on the equations of state at static condition, both in energy and in
pressure. The EOS parameters differ less than 7 × 10−2 per cent in the models with a same spin state. This result, combined with the limited
effect of the iron configuration on the VDoS described in the previous section, allows us to use only one iron configuration to determine the
thermodynamic properties at least for this small iron concentration. We choose the model 1 with the most homogeneous iron distribution.
Based on the phonon dispersion relations presented in Section 3.1, the Helmholtz free energy was calculated within the QHA for
every volume as a function of temperature (eq. 1). The thermodynamic properties are derived from the Helmholtz free energy with standard
thermodynamic relations (e.g. Tsuchiya 2003). The P–V –T EOS, thermal expansion coefficient α, isothermal bulk modulus K T , isochoric
C V and isobaric heat capacities C P and the vibrational contribution to the entropy S vib , are shown in Fig. 3.
We present the compression curves at various temperatures of 300, 1000, 2000, 3000 and 4000 K up to 150 GPa in Fig. 3(a). Comparing
the present results with the compression curves of pure MgSiO3 Pv calculated by Tsuchiya et al. (2005) and the room temperature experimental
results of Andrault et al. (2001) with 5 per cent Fe and Lundin et al. (2008) with 9 per cent Fe, it is found that the incorporation of HS Fe in
the Pv phase tends to increase the volume, similarly to the static case. The volumes of HS Fe and LS Fe-bearing Pv are +0.27 and +0.06 per
cent larger than that of pure MgSiO3 at 60 GPa and 300 K, respectively. At 2000 K and 60 GPa, the differences in the volumes between pure
MgSiO3 and the Fe-bearing phases increase to +0.35 and +0.10 per cent for HS Fe and LS Fe-bearing Pv, respectively. Pressure seems to
have a limited effect on the volume increases associated with the incorporation of iron (+0.29–0.26 per cent for HS Fe and +0.09–0.01 per
cent for LS Fe in 25–125 GPa at 300 K).
The thermal expansion coefficient α at 30, 60 and 100 GPa of (Mg0.9375 Fe0.0625 )SiO3 Pv with HS and LS Fe is exposed in Fig. 3(b)
as a function of temperature. The differences of α between HS Fe and LS Fe-bearing Pv decreases with increasing pressure and become
undistinguishable at 100 GPa. The incorporation of HS Fe tends to increase the thermal expansion coefficient in the investigated pressure
and temperature range. The variation of the thermal expansion coefficient, α, between HS Fe-bearing and Fe-pure phases decreases with
pressure. At 2000 K, α decreases from +4.6 per cent at 30 GPa to +1.9 per cent at 100 GPa. This distinctive increase in α at the uppermost
lower-mantle P–T condition is related to the volume expansion by the incorporation of HS Fe, which is more marked at low pressure. The
effects of the incorporation of LS Fe on the thermal expansion coefficient are smaller, with values of α equal to −0.9 per cent at 30 GPa
and 2000 K and +2.1 per cent at 100 GPa and 2000 K.
We estimated a pressure and temperature domain of the QHA validity based on the temperature dependence of α. At high temperature,
this thermodynamic quantity calculated with the QHA can deviate from the usual linear behaviour and put in forward the upper limit of the
QHA validity (Tsuchiya et al. 2005). We found that the inflexion points of the temperature dependence of α of HS/LS Fe-bearing Pv are close
to those observed in pure MgSiO3 Pv (Tsuchiya et al. 2005). Therefore, the QHA validity domain is the same in pure and Fe-bearing MgSiO3
and covers the P–T conditions of the lower mantle except maybe at the top of this layer (P ∼ 24 GPa and T ∼ 1900 K) (Ito & Katsura 1989).
Dashed lines in Fig. 3 indicate the out-of-validity range of the QHA.
The isothermal bulk modulus K T at 30, 60 and 100 GPa up to 4000 K are displayed in Fig. 3(c). The presence of HS and LS Fe in silicate
Pv has marginal effects on the isothermal bulk modulus at low temperature. The temperature derivative of K T is larger in the HS Fe-bearing
phase, leading to K T slightly lower than that of pure MgSiO3 at high temperature. This behaviour is related to the larger thermal expansion
coefficient in the HS Fe-bearing Pv than in pure MgSiO3 Pv. The temperature derivatives of K T at constant pressure are −3.32 × 10−2
and −2.88 × 10−2 GPa K–1 at 30 GPa for (Mg0.9375 Fe0.0625 )SiO3 Pv with HS and LS Fe, respectively. These are comparable to an experimental
value for pure MgSiO3 , −2.3 × 10−2 GPa K–1 (Wang et al. 1994), but smaller than −6.3×10−2 GPa K–1 for the (Mg0.9 Fe0.1 )SiO3 composition
(Mao et al. 1991). Wang et al. (1994) explained that the large value of the temperature derivative of the bulk modulus of Mao et al. (1991)
could be related to possible non-hydrostaticity or heterogeneous iron distribution under room temperature compression.
We present the vibrational contribution to the entropy, the isobaric and isochoric heat capacities in Figs 3(d)–(f), respectively. The
vibrational contribution to the entropy is marginally affected by the incorporation of Fe. This thermodynamic parameter increases less than
1.0 and 0.5 per cent when HS and LS Fe is incorporated in MgSiO3 Pv. The configuration and magnetic entropies, S conf and S mag , expected
at high temperatures (eqs 2 and 3) are also plotted. These latter contributions to the entropy are clearly lower than S vib in the whole pressure
and temperature conditions of the lower mantle and indicate that the vibrational entropy becomes predominant compared to the entropy due
to the atomic disorder. The isobaric and isochoric heat capacities, C P and C V , displayed in Figs 3(e) and (f) are not modified by the presence
of iron. A detailed analysis shows that theses quantities increase slightly with the incorporation of Fe, ∼+0.6 per cent for C P and ∼+0.1 per
cent for C V in the P–T conditions of the lower mantle.
3.3 PPv phase relation in (Mg0.9375 Fe0.0625 )SiO3 in the lowermost mantle P–T condition
The determination of the phase relations in (Mg0.9375 Fe0.0625 )SiO3 between Pv and PPv at the lowermost mantle P–T conditions requires
information on the spin state of Fe at these conditions in these phases. Recently, we found that Fe remains in the HS state in (Mg0.9375 Fe0.0625 )SiO3
PPv (Metsue & Tsuchiya 2011). The stable spin state of ferrous iron in Pv is investigated by comparing the static enthalpies of incorporated HS
and LS Fe in MgSiO3 Pv. The relative enthalpies of LS Fe-bearing Pv with respect to the enthalpy of HS Fe-bearing Pv are shown in Fig. 4 up
to 150 GPa for the three investigated models. The comparison between the enthalpies show that LS Fe is unlikely also in (Mg0.9375 Fe0.0625 )SiO3
Pv in the pressure range of the lower mantle. This stability of HS Fe in an (Mg0.9375 Fe0.0625 )SiO3 composition is consistent with the previous
ab initio calculations performed with the GGA approximation of Stackhouse et al. (2007) who suggest a HS-to-LS transition at 146
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Figure 3. Thermodynamic properties of MgSiO3 and HS/LS (Mg0.9375 Fe0.0625 )SiO3 Pv. (a) Pressure–volume equation of state at 300, 1000, 2000, 3000 and
4000 K isotherms. A comparison with the experimental data of Andrault et al. (2001) (1) and Lundin et al. (2008) (2) is also provided. Temperature dependence
of the thermal expansion coefficient α (b), the isothermal bulk modulus K T (c), the vibrational entropy (d) and the heat capacities C P (e) and C V (f) at 30, 60
and 100 GPa. For the heat capacities, the curves are given from the top to bottom as the pressure increases. The configuration and magnetic entropies expected
at sufficiently high temperatures, S conf and S mag , are shown in (d). The data for pure MgSiO3 are taken from Tsuchiya et al. (2005). The dashed lines represent
the pressure and temperature domains where the validity of the QHA is questionable.
GPa. The differences of the enthalpies between the phases with Fe in the LS state and in the HS state, H, vary between approximately
+0.12 Ry/Fe at 40 GPa and approximately +0.08 to +0.09 Ry/Fe at 150 GPa for all the three models. Our values of H are slightly higher
than those calculated by Hsu et al. (2010a) for a (Mg0.875 Fe0.125 )SiO3 composition with LSDA+U calculations, indicating that the increase
of iron concentration in the Pv phase decreases the spin transition pressure. This tendency is similar to the results of Bengtson et al. (2008)
with conventional LDA and GGA calculations where a significant drop of the spin transition pressure was reported for a concentration of
Fe larger than 0.25 per cent in the Pv phase. Umemoto et al. (2008, 2010) reported a possible spin transition at lower-mantle pressures
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Figure 4. Relative enthalpy of LS (Mg0.9375 Fe0.0625 )SiO3 Pv with respect to HS (Mg0.9375 Fe0.0625 )SiO3 Pv up to 150 GPa calculated for the three investigated
models of iron configuration presented in Table 1.
with the LDA approximation for specific iron distributions, in particular, when all irons are in the (1 1 0) plane. The possible HS-to-LS
transition pressure found from our static calculations is far higher than the lower-mantle pressure range and is not investigated at finite
temperature. Experimentally, the HS to LS ferrous iron was reported at the lower-mantle pressures (Badro et al. 2004; Li et al. 2004, 2006;
McCammon et al. 2010) or HS to intermediate spin state (Lin et al. 2008; Narygina et al. 2010). The observed spin transition is sensitive to
the concentration of iron (Bengtson et al. 2008; Umemoto et al. 2008) and maybe to the contents of aluminium and ferric iron in the sample.
Further theoretical investigations on the stability of intermediate spin state are required to clarify this issue.
We determine the effects of 6.25 per cent HS ferrous iron on the phase transition between Pv and PPv in the P–T condition of the base
of lower mantle. We use the same approach as developed in Tsuchiya & Tsuchiya (2008) in the determination of the PPv equilibrium in the
MgSiO3 –Al2 O3 system. We compute the total Gibbs free energy Gss (P,T,x) of the (Mg1−x Fex )SiO3 Pv and PPv solid solutions which is given
by the general expression for a binary system (Cemič 2005)
FeSiO3
3 − TS
G ss
+ (1 − x)G MgSiO
conf ,
α (P, T, x) = x G α
α
(4)
where α denotes Pv or PPv, x the concentration of Fe in the phase and S conf the configuration entropy given in eq. (3). We limit our investigation
to Fe-poor compositions in a first time. The calculation of the Fe-rich part of the MgSiO3 –FeSiO3 solid solution is ongoing and may improve
the phase diagram. Consequently, we approximate the Gibbs free energy of the low concentration Fe-bearing phases from a linear interpolation
of the results obtained for a (Mg0.9375 Fe0.0625 )SiO3 composition from this study for the Pv phase and from our previous theoretical study
performed with the LSDA+U formalism for the PPv phase (Metsue & Tsuchiya 2011). Therefore, the Gibbs free energy of the solid solution
for a Fe-poor (Mg1−x Fex )SiO3 solid solution is given by
(Mg
Fe
)SiO
(Mg
Fe
)SiO
MgSiO3
[Hα 0.9375 0.0625 3 (P, T ) + Fvib−α0.9375 0.0625 3 (P, T )] − 0.9375G α
(P, T )
ss
∗
− T Smag
G α (P, T, x) = x
0.0625
3 (P, T ) − T S
+ (1 − x)G MgSiO
conf ,
α
where
(Mg
Fe
)SiO
Hα 0.9375 0.0625 3 (P, T )
Hα(Mg0.9375 Fe0.0625 )SiO3 (P, T )
(Mg
Fvib−α0.9375
Fe0.0625 )SiO3
=
(V, T ) =
and
E α(Mg0.9375 Fe0.0625 )SiO3 (V )
1
2
q, j
(5)
(Mg
Fe
)SiO
Fvib−α0.9375 0.0625 3 (V, T )
+
hω j (q, V ) + kB T
have the following expressions
P Vα(Mg0.9375 Fe0.0625 )SiO3 (P, T )
ln[1 − exp(−hω j (q, V )/kB T )].
(6)
(7)
q, j
The quantities involved in eqs (6) and (7) are the same as defined for eq. (1) and are calculated from this study for the Pv phase and from
Metsue & Tsuchiya (2011) for the PPv phase.
∗
is the magnetic contribution to the total entropy, with the same quantities described for eq. (2) and where the contribution of the
Smag
iron concentration x is taken into account on the left side of eq. (5)
∗
Smag
= −kB ln[m(2S + 1)].
(8)
We use the results of Tsuchiya et al. (2005) to calculate the Gibbs free of the pure MgSiO3 phases. The eq. (5) is computed for several
pressures and temperatures and the fraction x of Fe in Pv and PPv during the phase transition is determined from the calculated cotangent of
ss
ss
the curves G ss
Pv (P, T, x) and G PPv (P, T, x) when the curves cross each other. We show an example of computed G (x) at 3000 K at 110 and
115 GPa in Fig. 5. For a given temperature, we search the two pressures where x = 0.0625 in the Pv and in the PPv phases, which give the
Pv/PPv phase boundary and the pressure width of the binary phase loop for an (Mg0.9375 Fe0.0625 )SiO3 composition.
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Figure 5. Total Gibbs free energy of solid solution (Gss ) of Pv and PPv at 110 GPa (a) and 115 GPa (b) at 3000 K relative to the end-members of Pv phase.
We show the case of Pv and PPv coexistence in (b). The cotangent of the two curves (green line) gives the fraction of Fe in Pv and PPv (dashed green lines).
Figure 6. Proposed phase diagram for a (Mg0.9375 Fe0.0625 )SiO3 composition at the pressure and temperature of the lowermost mantle. Green line is the normal
adiabatic geotherm of Brown & Shankland (1981). The blue line is the geotherm with thermal boundary layer thickness of 400 km proposed by Kawai &
Tsuchiya (2009). The filled red P–T domain represents the coexisting domain between Pv and PPv phases. Orange line represents the boundary for pure
MgSiO3 calculated by Tsuchiya et al. (2005). The black dashed line represents the expected pressure of the core–mantle boundary (135 GPa). Comparison
with experimental studies for (Mg0.91 Fe0.09 )SiO3 and San Carlos olivine compositions from Catalli et al. (2009) and Grocholski et al. (2012) are provided.
The calculated phase diagram for an (Mg0.9375 Fe0.0625 )SiO3 composition is displayed in Fig. 6. The Pv/PPv phase boundary for pure
MgSiO3 of Tsuchiya et al. (2004) is also indicated in this figure. This previous first-principles calculations study of the phase diagram of
pure MgSiO3 at the D layer thermodynamic conditions showed a difference of 10 GPa between the Pv–PPv phase boundary calculated with
the LDA and GGA approximations (Tsuchiya et al. 2004). Consequently, a shift of +5 GPa is added to the boundaries to compensate the
use of the LDA approximation, which is expected to underestimate the transition pressure. The phase diagram determined clearly indicates
that 6.25 per cent of ferrous iron decreases the PPv phase boundary with a Pv + PPv binary phase loop, though it less affects the Clapeyron
slope itself. The pressure width of the divariant loop is estimated to be 5 GPa at 2000 K and 2 GPa at 4000 K. Along the adiabatic geotherm
proposed by Brown & Shankland (1981), our results predict that the PPv phase transition in the iron-bearing phases starts at a depth z of
2420 km (P = 109 GPa) and finish at z = 2495 km (P = 113 GPa), while the transition occurs at z = 2635 km (P = 121 GPa) along the
normal adiabatic geotherm for pure MgSiO3 (Tsuchiya et al. 2004). Our results suggest also that the PPv phase is enriched in Fe compared
to the Pv phase during the phase transition. At 2500 K, the first PPv phase appears at 111 GPa and has 8.75 per cent of Fe, when the bulk
iron concentration is 6.25 per cent. The amount of Fe in the first grain of PPv decreases to 8.25 per cent during the phase transition at
3000 K and 116 GPa. This preference of Fe for the PPv phase is consistent with the tendency of the partition coefficient of Mg–Fe between
Pv-ferropericlase and PPv-ferropericlase determined experimentally by Kobayashi et al. (2005) and Auzende et al. (2008) who showed that
the partition coefficient of Fe is larger in the PPv-ferropericlase system compared to the Pv-ferropericlase system. Seismological observations
that some superadiabatic temperature increases of ∼1300 K are expected in the thermal boundary D layer and deviations of the geotherm
from the mantle adiabat may start form ∼2500 km depth in the hotter regions like underneath central Pacific and Africa (e.g. Kawai &
Tsuchiya 2009). In these areas, the PPv transition in (Mg0.9375 Fe0.0625 )SiO3 occurs in the same pressure and temperature ranges than along
the normal adiabatic geotherm, indicating that unlike for the pure MgSiO3 composition, the transition condition is not significantly affected
by the geophysically relevant lateral temperature variation (Fig. 6). The depth of discontinuity can therefore vary with the iron concentration.
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Previous experimental determination of the Pv-to-PPv phase boundary in Fe2+ -bearing MgSiO3 in the D layer thermodynamic conditions
indicated that the incorporation of Fe2+ decreases the transition pressure between Pv and PPv (Mao et al. 2004; Catalli et al. 2009), similarly
to our and previous computational studies (Caracas & Cohen 2007, 2008). Sinmyo et al. (2011) in contrast reported a contradicting result
from chemical analyses in aluminium-bearing samples that iron increases the transition pressure. It is suggested that ferric iron and also
aluminium might have some effects to increase the Pv–PPv transition pressure (Akber-Knutson et al. 2005; Caracas & Cohen 2005; Tateno
et al. 2005; Zhang & Oganov 2006; Nishio-Hamane et al. 2007). Mao et al. (2004) estimated a Pv–PPv coexisting domain between 107 and
119 GPa at 2000–2500 K in a (Mg0.93 Fe0.07 )SiO3 composition. On the other hand, Catalli et al. (2009) reported a large Pv–PPv binary phase
loop thickness of 20 ± 5 GPa in a (Mg0.91 Fe0.09 )SiO3 composition. The linear interpolation of our calculated binary phase loop thickness of
5 GPa at X = 6.25 per cent gives a phase loop of 7 GPa at X = 9 per cent in (Mg1−x ,Fex )SiO3 at 2000 K, which is still substantially narrower
than the experimental results. Kinetic effects, deviatoric stresses and cation partitioning amongst different phases in binary, ternary and
multiphase systems are the main sources of uncertainties in laser-heated DAC experiments (Mao et al. 2004; Catalli et al. 2009; Grocholski
et al. 2012; Zhang et al. 2012) and could reasonably explain the discrepancies between computational and experimental studies. Along the
geotherm proposed by Brown & Shankland (1981), we found that the phase transition width is equal to 75 km. This is still shorter than the
typical seismic wavelengths (∼100 km) that are used to map the D seismic discontinuity (Wysession et al. 1998; Lay et al. 2006; Lay &
Garnero 2007; van der Hilst & Kárason 2007). It was reported that the binary phase loop of the olivine-to-wadsleyite phase transition in
the Mg2 SiO4 –Fe2 SiO4 solid solution is clearly reduced by the presence of additional garnet and pyroxene phases (Stixrude 1997). In the
MgSiO3 –FeSiO3 solid solution, the binary phase loop between Pv and PPv may be reduced with the presence of ferropericlase (Mg,Fe)O.
Further detailed effects including ferric iron and aluminium must be considered before discussing the real mantle mineralogy.
4 C O N C LU S I O N S
We have investigated the thermodynamic properties of (Mg0.9375 Fe0.0625 )SiO3 Pv at pressure and temperature conditions of the lower mantle
with a combination of internally consistent LSDA+U calculations and phonon relations determined with a direct method for the first time.
The phonon free energy is determined with the QHA and is extended to the calculation of the Gibbs free energy in order to determine the
Pv-to-PPv phase transition in a (Mg0.9375 Fe0.0625 )SiO3 composition at the P–T conditions of the lowermost part of the mantle. The main
conclusions are summed up here:
(1) The incorporation of iron affects the lowest frequencies part of the phonon dispersion relations for both HS and LS state. The acoustic
mode frequencies are lower in Fe-bearing phase and indicate a decreasing of the elastic wave velocities. Vibrational instabilities appear at
0 GPa in the case of incorporated LS Fe.
(2) The thermodynamic parameters are not clearly affected by a small amount of Fe in the Pv structure at the P–T conditions of the lower
mantle. The bulk modulus decreases slightly and the thermal expansion coefficient increases slightly in these conditions.
(3) Fe remains in the HS state in the pressure range of the lower mantle, like in Fe-bearing MgSiO3 PPv. The calculated phase diagram
between HS (Mg0.9375 Fe0.0625 )SiO3 Pv and PPv shows that Fe stabilizes the PPv structure at lower pressure, but the calculated binary phase
loop between Pv and PPv corresponding to a 75-km-depth range is smaller than the seismological observations. Further investigations,
including the effects of ferric iron and Al, are required to clarify the thermodynamic properties of MgSiO3 Pv in a relevant geophysical
mineral composition.
AC K N OW L E D G M E N T S
This research was supported by the Ehime University Global Center Of Excellence program ‘Deep Earth Mineralogy’ and, in part, the
KAKENHI 20001005 and 21740379. The authors gratefully acknowledge the two anonymous reviewers for their helpful comments.
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