Thermodynamic and Elastic Properties of Pyrope at High Pressure And

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Journal of Geophysical Research: Solid Earth

RESEARCH ARTICLE Thermodynamic and elastic properties of pyrope 10.1002/2016JB013026 at high pressure and high temperature Key Points: by ﬁrst-principles calculations • First-principles calculations of thermodynamic properties and Yi Hu1,2, Zhongqing Wu1,3, Przemyslaw K. Dera2,4, and Craig R. Bina5 elasticity of pyrope are done over a wide pressure and temperature range 1Laboratory of Seismology and Physics of Earth's Interior, School of Earth and Space Sciences, University of Science and • If pyrope can be preserved in cold 2 subducted slab, both low temperature Technology of China, Hefei, China, Department of Geology and Geophysics, School of Ocean and Earth Science and 3 and dense mineral assemblage Technology, University of Hawaii at Manoa, Honolulu, Hawaii, USA, National Geophysical Observatory, Mengcheng, contribute to the negative buoyancy China, 4Hawaii Institute of Geophysics and Planetology, School of Ocean and Earth Science and Technology, University of • Contrast in the seismic velocities Hawaii at Manoa, Honolulu, Hawaii, USA, 5Department of Earth and Planetary Sciences, Northwestern University, of the subducted slab and the Evanston, Illinois, USA surrounding mantle results from the lower temperature

Abstract Pyrope is an Mg end-member garnet, which constitutes up to ~15% of the upper mantle. Supporting Information: Thermodynamic and thermoelastic properties of garnet at high pressure and high temperature are • Supporting Information S1 important for understanding the composition and structure of the Earth. Here we report the ﬁrst-principles Correspondence to: calculations of vibrational properties, thermodynamic properties, and elasticity of pyrope over a wide Z. Wu, pressure and temperature range. The calculated results exhibit good consistency with the available [email protected] experimental results and provide values up to the pressure and temperature range that are challenging for experiments to achieve and measure. Pyrope is almost isotropic at high pressure. The elastic moduli, Citation: especially the shear modulus of pyrope, exhibit nonlinear pressure and temperature dependences. Density Hu, Y., Z. Wu, P. K. Dera, and C. R. Bina and seismic velocities of pyrope along with piclogite, pyrolite, and eclogite are compared with seismic (2016), Thermodynamic and elastic properties of pyrope at high pressure models along the normal upper mantle and the cold subducting lithospheric plate geotherms. Pyrope has and high temperature by ﬁrst-principles the highest density and seismic velocities among the major minerals of the upper mantle along these calculations, J. Geophys. Res. Solid Earth, geotherms. Eclogite with 30% pyrope and 70% clinopyroxene is denser than the surrounding mantle even 121, doi:10.1002/2016JB013026. along the normal upper mantle geotherm and becomes still denser along the cold slab geotherm. Seismic Received 29 MAR 2016 velocities of eclogite with 30% pyrope along the normal upper mantle geotherm are slower than the Accepted 21 AUG 2016 surrounding mantle but along the cold slab geotherm are faster than the surrounding mantle. Accepted article online 23 AUG 2016

1. Introduction

Garnet with chemical composition (Mg,Fe)3Al2Si3O12 is an important rock-forming mineral in both Earth's upper mantle and subducted lithospheric plates. According to the pyrolitic compositional model, Earth's upper mantle is composed of ~65% olivine, ~ 20% pyroxene, and ~15% garnet by volume [Ringwood, 1975; Frost, 2008; Ita and Stixrude, 1992]. Upper mantle garnet is mainly made of its Mg end-member pyrope,

with the formula Mg3Al2Si3O12 and a space group Ia3d [Rickwood et al., 1968]. With increasing pressure and temperature, pyrope garnet gradually incorporates clinopyroxene and orthopyroxene and transforms into majoritic garnet near the base of the upper mantle [Ringwood, 1967]. The detailed volume proportions of these minerals are still under debate because of limited knowledge on wave velocities and density proﬁles of these minerals at temperature and pressure conditions of the upper mantle [Ringwood, 1979; Ita and Stixrude, 1992]. The detailed elastic properties of these minerals are also critical to understanding the anisotropic property of the upper mantle, since lattice-preferred orientation and anisotropic elasticity of major mantle minerals are main reasons for the seismic anisotropy observed in the upper mantle [Kawasaki and Kon'no, 1984; Mainprice et al., 2000]. Pyrope is also an important mineral in the subducted lithospheric plate. Eclogite—a rock formed by high- pressure metamorphism of basalt or gabbro at subduction zones—is mainly composed of garnet and clinopyroxene [Poli, 1993; Ringwood, 1982]. Recent studies show that at temperatures well below the mantle geotherm, pyroxene-garnet transformation is kinetically inhibited at the conditions of subducting slab, so that pyropic garnet may be well preserved down to the transition zone in the subducting slab [Nishi et al., 2008; Van Mierlo et al., 2013; Bina, 2013]. Vibrational and thermodynamic properties (heat capacity, entropy, ©2016. American Geophysical Union. All Rights Reserved. enthalpy etc.) of the minerals are important for determining the stable and metastable regions of these

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minerals in the slab. Detailed elastic properties at the conditions of subducting slab are important for comparing petrological models with seismic data. Wave velocities and elastic moduli of pyrope have been extensively studied. However, due to the challenge of measuring elastic properties at simultaneous high pressure and temperature, most of these studies are limited to high pressure but ambient temperature [Chen et al., 1997; Leitner et al., 1980; O'Neill et al., 1991; Sinogeikin and Bass, 2002a] or high temperature but ambient pressure [Sinogeikin and Bass, 2002b]. Ultrasonic measurements in multianvil press provide bulk and shear moduli, instead of individual elastic constants at high pressure and temperature [Gwanmesia et al., 2006; Zou et al., 2012a; Chantel et al., 2016]. The Brillouin scattering measurement by Lu et al. [2013] provides detailed elastic constants of iron-bearing pyrope up to 20 GPa and 750 K, which reaches the pressure of the upper mantle but does not reach the temperature of the normal upper mantle. Erba et al. [2014] and Mahmoud et al. [2014] investigated the elasticity of pyrope up to 60 GPa at static condition by using first-principles calculations. The vibrational data for pyrope are also limited; only some Raman [Kolesov and Geiger, 1998, 2000; Hofmeister and Chopelas, 1991] and infrared [Hofmeister et al., 1996] data are available. Several studies reported the heat capacity of pyrope [Richet and Fiquet, 1991; Anderson et al., 1991] but are limited to low pressure. The vibrational properties and thermodynamic properties of pyrope at 0 GPa have also been investigated by using a classical potential [Chaplin et al., 1998] or the first-principles calculations with hybrid DFT functional [Baima et al., 2016; Dovesi et al., 2011; Maschio et al., 2013; Pascale et al., 2005]. Here we report the vibrational properties, thermodynamic properties, and elasticity of pyrope at high pressure and high temperature by using first-principles calculations. Our results show a good agreement with experimental and calculated data. Nonlinear pressure and temperature dependence of the elastic moduli are observed. Density, compressional, and shear velocities along the normal upper mantle and cold subducting lithospheric plate geotherms are calculated and compared with seismic models.

2. Method 2.1. Computational Details The calculations were performed based on density functional theory (DFT) [Hohenberg and Kohn, 1964; Kohn and Sham, 1965] with local density approximation (LDA) [Ceperley and Alder, 1980] by using the Quantum Espresso package [Giannozzi et al., 2009]. The pseudopotential for magnesium was generated by the method of von Barth and Car [Karki et al., 2000]. The weights of five valence configurations with a cutoff radii 2.5 Bohr, 3s23p0,3s13p1,3s13p0.53d0.5, and 3s13p0.5 3s13d1, are 1.5, 0.6, 0.3, 0.3, and 0.2, respectively. Those for oxygen and silicon were generated by the method of Troullier and Martins [1991] with valence configuration 2s22p4 and cutoff radii 1.45 Bohr for oxygen and valence configuration 3s23p43d0 and cutoff radii 1.45 Bohr for silicon. The pseudopotential for aluminum was generated by the Vanderbilt method [Vanderbilt, 1990] with valence configuration 3s23p1 and cutoff radii 1.77 Bohr. Plane- wave kinetic energy cutoff applied was 70 Ry. Brillouin zone summations over the electronic states were performed at gamma point. Structural optimizations were achieved by using damped variable cell shape molecular dynamics [Wentzcovitch, 1991]. For each optimized structure, dynamical matrix was calculated by using density functional perturbation theory [Baroni et al., 2001] to obtain vibrational frequencies at gamma point. 2.2. Quasi-Harmonic Approximation With static internal energy and phonon frequencies known, Helmholtz free energy at high temperature can be calculated by using the quasi-harmonic approximation (QHA) [Wallace, 1998], that is, X X 1 ℏωq;jðÞe; V FeðÞ¼; V; T U ðÞþe; V ℏω ; ðÞþe; V K T ln 1 exp ; (1) st q;j q j B q;j 2 KBT

where subscripts j represent normal mode index, q is phonon wave vector, T is the temperature and e is strain tensor. The first, second, and third terms are the static internal, zero point, and vibrational energies, respectively. We repeated the above-mentioned calculation for seven different volumes as shown in Table S1. The largest volume explored is 757.1109 Å3, which corresponds to the volume at 700 K and 0 GPa, and the smallest volume explored is 666.7611 Å3, which corresponds to the condition of 2000 K and 34 GPa. The calculated Helmholtz free energies versus volume were fitted by the isothermal third-order finite strain

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equation of state, and then the pressure and volume relationship are described by the third-order Birch- Murnaghan equation of state as shown below: "#()"# 7 5 2 3 3 3 3K V V 3 ′ V PVðÞ¼ T0 0 0 1 þ K 4 0 1 ; (2) 2 V V 4 T0 V

where P is pressure. V and V0 represent the volume at high pressure and ambient pressure, respectively. KT0 and KT0′ are isothermal bulk modulus and its pressure derivative at 0 GPa [Prewitt and Downs, 1998]. All thermodynamic properties can be derived from the calculated free energies. Although the thermodynamic properties at 0 GPa and above 700 K are obtained by extrapolation, the data at pressure and temperature conditions relevant to the Earth's interior are from interpolation. 2.3. Elastic Constants Isothermal elastic constants are given by [Barron and Klein, 1965] ∂2 T 1 F 1 δ δ δ δ δ δ : cijkl ¼ þ P 2 ij kl il kj ik jl (3) V ∂eij∂ekl 2

Here eij (i, j =1–3) is inﬁnitesimal strains and F is the Helmholtz free energy, which is expressed in the quasiharmonic approximation as equation (1). Adiabatic elastic constants are obtained by using the relation ∂ ∂ S T T S S δ δ ; cijkl ¼ cijkl þ ij kl (4) VCV ∂eij ∂ekl

where CV, V, and S are heat capacity, volume, and entropy, respectively. In general, calculations of the elastic tensors require vibrational density of states for many strained conﬁgurations. The computational workload for vibrational density of states is 2 or 3 orders larger than that for static internal energy. Therefore, the computation of the elastic tensors at high temperature and high pressure is considerably expensive. This is especially noticeable for a crystal like pyrope, with 80 atoms in the primitive cell, whose static internal-energy calculations are already expensive. Wu and Wentzcovitch [2011] developed a new method to reduce signiﬁ- cantly the computational workload. Using the relation between volume dependence and strain dependence of vibrational frequencies and introducing an approximation, Wu and Wentzcovitch [2011] deduced an analytical formula for the second derivative of the free energy with respect to strain. The volume dependence of the frequency, ω d q; j γ dV ¼ q; j (5) ωq; j V and the strain dependence of the frequency, ω d q; j γ11 γ22 γ33 ; ¼ q; je11 þ q; je22 þ q; je33 (6) ωq; j

should produce the same frequency change for the same case. Here γq, j is the mode Grüneisen parameter γii θ and q; j is the strain Grüneisen parameter. For each mode, we can introduce the azimuthal parameters q,j and ϕq,j with γ11 e11 þ e22 þ e33 γ 2θ 2ϕ ; q; j ¼ q; jcos q; jsin q; j (7a) e11 γ22 e11 þ e22 þ e33 γ 2θ 2ϕ ; q; j ¼ q; jsin q; jsin q; j (7b) e22 γ33 e11 þ e22 þ e33 γ 2θ : q; j ¼ q; jcos q; j (7c) e33

The distribution function f(θ, ϕ), which deﬁnes the numbers of the azimuthal parameters between (θ, ϕ) and (θ +dθ, ϕ +dϕ), is a constant for an isotropic material. By assuming an isotropic distribution function γii for a crystal, we can obtain all kinds of the average values of q; j, required for the calculation of the elasticity,

from the Grüneisen parameter γq, j. Thus, the method developed by Wu and Wentzcovitch [2011] only requires knowledge of volume dependence of vibrational frequencies. The vibrational modes are calculated at seven different volumes, which correspond to the pressure from negative pressure to the pressure of the

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Table 1. Frequencies and Symmetries of Raman and Infrared Modes of Pyropea Raman IR

This Study γ Mode Exp1 Exp2 Exp3 Calc1 Calc2 This Study γ Mode Exp4 Cal2 Cal3

141.9 3.3 T2g 133.0 135 - 97.9 106.5 134.0 2.4 T1u 134.4 121.1 114.6 176.0 2.4 - - - 170.1 171.9 145.1 4.2 134.4 139.8 137.7 208.3 2.0 209.8 222 208 203.7 204.2 201.5 1.5 194.6 189.3 185.5 - - - - 230 - - 221.3 1.9 220.8 215.8 213.5 275.3 1.3 285.0 - 272 266.9 268.3 260.0 1.3 258.5 259.9 259.1 - - - - 285 - - 340.0 1.1 336.0 334.2 333.4 316.8 1.1 319.0 322 318 319.0 320.0 346.6 1.3 - 348.6 346.5 353.5 1.3 351.4 353 350 350.6 353.3 379.7 1.0 383.0 282.9 380.2 384.0 1.4 381.2 383 379 381.9 383.8 418.1 0.8 422.0 423.0 421.3 486.1 0.9 490.8 492 490 492.6 495.3 456.4 1.0 455.0 459.0 457.0 502.0 0.9 510.4 512 510 513.5 514.1 470.9 1.0 478.0 483.6 480.9 588.9 1.0 - 598 598 605.9 607.2 540.1 1.1 535.0 532.8 530.9 635.1 1.0 647.5 650 648 655.3 655.2 583.3 1.0 581.0 582.9 580.6 876.7 0.9 867.6 871 866 861.0 860.6 660.6 0.9 650.0 674.1 673.3 909.7 1.0 - 902 899 896.7 895.9 878.8 1.0 871.0 864.8 865.1 1066.2 0.8 1063.1 1066 1062 1068.4 1067.2 909.2 1.0 902.0 895.9 896.2 208.1 1.7 Eg - 211 203 209.2 209.2 979.4 0.9 972.0 969.7 970.5 309.6 1.3 - 284 - 308.5 309.3 342.7 1.2 - 344 342 336.5 336.7 381.4 1.2 - 375 365 376.9 378.9 - - - - 439 - - 515.1 1.0 - 525 524 526.6 530.7 616.7 0.6 - 626 626 636.0 635.8 858.9 0.8 - - - 864.4 863.2 - - - - 911 - - 947.8 0.9 - 945 938 937.4 936.3 355.6 1.6 A1g 363.6 364 362 352.5 356.5 553.5 1.0 561.4 563 562 564.8 567.5 920.0 0.9 926.0 928 925 926.0 923.8 aExp1, Kolesov and Geiger [2000]; Exp2, Kolesov and Geiger [1998]; Exp3, Hofmeister and Chopelas [1991]; Exp4, Hofmeister et al. [1996]. Cal1, Maschio et al. [2013]. Cal2, Pascale et al. [2005]. Cal3, Dovesi et al. [2011].

top of the lower mantle. Then these vibrational frequencies are ﬁtted and interpolated to other volumes to get the volume dependence of vibrational frequencies. Therefore, the method avoids the computation of vibrational density of states for the strained conﬁgurations and reduces the computational workload to less than a tenth of the usual approach. The method has been applied successfully to MgO [Wu and Wentzcovitch, 2011], diamond [Núñez-Valdez et al., 2012], olivine and its polymorph [Núñez-Valdez et al.,

2013], ferropericlase [Wu et al., 2013], stishovite, the CaCl2-type silica [Yang and Wu, 2014], and bridgmanite [Shukla et al., 2015]. 2.4. Geophysical Modeling In order to better understand the contribution of pyrope to the bulk upper mantle and subducting lithospheric plate, a geophysical model is built and compared to seismic observations. The geotherm of the bulk upper mantle is adopted from Brown and Shankland [1981]. The geotherm of the plate top boundary is estimated with TEMSPOL [Negredo et al., 2004] with a lithospheric age of 130 Myr, a dip angle of 30°, subduction rate of 9.1 cm/yr, and a lithospheric thickness of 95 km. These conditions are equivalent to the subducting plate under northeast Japan [Zhao et al., 1997; Peacock and Wang, 1999]. Mineral data including olivine [Isaak, 1992; Zha et al., 1998; Liu and Li, 2006], orthopyroxenes [Chai et al., 1997; Jackson et al., 2003, 2007], and clinopyroxene [Zhao et al., 1998; Isaak et al., 2006; Duffy and Anderson, 1989] are used. Both piclogite and pyrolite models are adopted as representative bulk mantle mineral assemblages. The piclogite model has ~43% olivine, ~ 36% clinopyroxene, ~ 6% orthopyroxene, and ~15% pyrope by volume, and pyrolite has ~63% olivine, ~ 16% clino-pyroxene, ~ 6% ortho-pyroxene, and ~15% pyrope by volume [Ita and Stixrude, 1992]. The uppermost 6 km layer of the subducting plate is modeled by eclogite, a high pressure, and high temperature metamorphic product of mid-ocean ridge basalt, with ~30 vol % garnet and

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~70 vol % clinopyroxene [Ringwood, 1982; Perrillat et al., 2006]. Density and seismic velocities of the aggregate mineral assemblage are calculated by using a Voigt-Reuss-Hill average [Hill, 1952] and are compared with the preliminary reference Earth model (PREM) [Dziewonski and Anderson, 1981] and ak135 [Kennett et al., 1995] models.

3. Results 3.1. Vibrational and Thermodynamic Properties The 240 modes of pyrope at the Brillouin zone center for a primitive Figure 1. Volume-pressure-temperature relationship of pyrope. The solid cell with 80 atoms are divided by sym- lines represent our calculation results. Scattered points are from experi- metry as mental data with circles from Zou et al. [2012b], crosses from Zhang et al. Γ 3A 5A 16E [1998], and down-pointing triangles from Milani et al. [2015]. The dashed total ¼ 1g þ 2g þ g lines correspond to the conditions where the validity of QHA may be þ 42T 1g þ 42T 2g þ 5A1u questionable. þ 5A2u þ 20Eu þ 54T 1u þ 48T 2u: (8) Mulliken notation for vibrational modes is used here [Mulliken, 1955, 1956]. A, E, and T refer to singly, doubly,

and triply degenerate states, respectively. Among them, T1u are infrared active modes, and T2g, Eg, and A1g are Raman active modes. The remaining modes are optically inactive. The calculated results using LDA show good consistency with experimental results and previous ﬁrst-principles calculation results using B3LYP functional (Table 1). The relative differences between experimentally observed and our calculated frequencies for ﬁ most of the vibrational modes are less than 3%. Only ve modes differ from observations by more than 3%. γ ∂lnω Mode Grüneisen parameter ¼∂lnV is also calculated and listed in Table 1. The pressure-temperature-volume relation of pyrope up to 2000 K and 40 GPa is shown in Figure 1. The volume given by static energy calculation is ~2% smaller than experiments at ambient temperature. Zero

point motion and room temperature effects increase V0 by ~1.6%. The calculated V0 at ambient condition agrees well with experimental data [Zou et al., 2012b; Zhang et al., 1998; Milani et al., 2015] with a difference within 0.5% (Table 2). The good agreements with experiment also hold at higher pressures with the difference within 0.8% at ~33 GPa. The calculated volumes at high temperature are also in good agreement with Zou et al. [2012b], which suggests that the temperature effect on volume and thermal pressure are well included in this calculation. α 1 ∂V Figure 2a shows the thermal expansion ¼ V ∂T P of pyrope at high pressure and temperature. At low temperature the calculations agree well with experimental results [Anderson et al., 1991; Milani et al., 2015; Zou et al., 2012a]. The calculation results start to noticeably deviate the experimental results at ~500 K. The discrepancy increases with increasing the temperature. This pattern is typical for QHA calculations and was also observed by previous studies using the same method [e.g., Wu and Wentzcovitch, 2007; Karki et al., 2000]. QHA, which assumes that phonon frequencies are independent of temperature, omits intrinsic phonon-phonon interaction (anharmonicity). The intrinsic anharmonicity, which increases with increasing the temperature, becomes noticeable at high temperature and results in deviation of the thermal expansion from experiment at high temperature [Wu, 2010; Wu and Wentzcovitch, 2009]. Among thermodynamic properties, thermal expansion is most sensitive to anharmonicity. The effect of anharmonicity at high pressure is less pronounced than at low pressure and can be negligible up to 1500 K at 10 GPa. As shown in Figure 2a, the difference between the calculated and experimental thermal expansion at 10 and 20 GPa is signiﬁcantly smaller than the one at 0 GPa. ∂E ∂E Heat capacity at constant volume (isochoric) CV ¼ ∂T V and constant pressure (isobaric) CP ¼ ∂T P, which are related by CP = CV(1 + αγthT), are shown in Figures 2b and 2c. CV agrees with experimental [Richet and

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Table 2. Thermodynamic Parameters of Pyrope at Ambient Conditions Calculated Experimental 3 a b V0,Å 747.09 750.215(40) , 751.45(15) Thermal expansion, 105 K1 2.724 2.97(45)a, 2.36c 1 1 h d CV, J mol K 324.63, 325.79 323.5 , 1 1 h d CP, J mol K 329.08, 330.70 326.4 Grüneisen parameter 1.68 1.19e, 1.15f Entropy, J mol1 K1 264.46, 273.69h 266.27(70)g aZou et al. [2012b]. bZhang et al. [1998]. cAnderson et al. [1991]. dRichet and Fiquet [1991]. eZou et al. [2012b], q = 1.0 ﬁxed. fZou et al. [2012b], q = 1.5 ﬁxed. gHaselton and Westrum [1980]. hBaima et al. [2016].

Fiquet, 1991] and previous DFT calculation data [Baima et al., 2016] and shows a typical QHA pattern, increasing from ~324.63 J/mol K at 300 K to ~487.2 J/mol K at 2000 K, a value very close to the Dulong-

Petit limit (Table 2). CP is calculated to be 329.08 J/mol K at ambient condition (Table 2). At low temperature, the calculated CP is consistent with experimental measurements. Similar to the thermal expansion, the calculated CP begins to deviate from the experimental data at ~1000 K and ambient pressure; the anharmonicity omitted by QHA becomes important above this temperature. On the other hand, the anharmonicity becomes less visible with increasing pressures. In general, QHA works well along a mantle geotherm [e.g., Karki et al., 2000; Wentzcovitch et al., 2010]. The thermal Grüneisen parameter is an important thermodynamic parameter used to quantify the relationship between the thermal and elastic properties of a solid (Figure 2d). The thermal Grüneisen parameter is

(a) (b)

Figure 2. (a) Thermal expansion, (b) constant volume heat capacity, (c) constant pressure heat capacity, and (d) thermal Grüneisen parameter of pyrope at various temperatures. The solid lines represent the results of this work. The dashed lines along with the shaded area are from Zou et al. [2012b]. The circles are from Anderson et al. [1991]. The plus-sign lines are from Richet and Fiquet [1991]. The dotted lines are from Baima et al. [2016].

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(a) (b)

Figure 3. (a) Elastic constants of pyrope at various pressures. The solid lines represent the calculated results of this study. The dashed lines represent the calculated static results by Erba et al. [2014]. Scattered points are from experimental results: Sinogeikin and Bass [2000] (right-pointing triangles), Conrad et al. [1999] (circles), Leitner et al. [1980] (rectangles), O'Neill et al. [1991] (downward-pointing triangles), and Chen et al. [1997] (six-pointed stars). (b) Elastic constants of pyrope at various temperatures. The circles are from Sinogeikin and Bass [2002b].

∂ α deﬁned as γ ¼ V P and can be expressed as γ ¼ V KT V. Starting from 273 K, the thermal Grüneisen para- th ∂U V th CV meter decreases as pressure increases and has a nonmonotonic behavior with temperature. It has a minimum at ~600 K and then increases slightly with temperature. Such a pattern is also obtained in wadsleyite by Wu and Wentzcovitch [2007].

3.2. Thermal Elasticity

The calculated elastic constants c11, c12, and c44 along with experimental and previous calculated data are shown in Figure 3 and Table 3. The static elastic constants at 0 GPa using LDA in this study are larger than those reported by Erba et al. [2014] using hybrid functional B3LYP (Table 3). This is normal because LDA predict a smaller equilibrium volume than B3LYP. Both results show similar pressure dependence (Figure 3 and Table 3). The vibrational term of the elastic constants is important. The static elastic constants at 0 GPa in this study are about 5%–6% higher than experimental data at ambient condition. Including the phonon contributions signiﬁcantly improves the consistency between the calculations and experiments at room temperature, as has been found in previous calculations [e.g., Wu and Wentzcovitch, 2011]. This is because the vibrational contribution expands the volume at 0 GPa ~ 1.6%, and elastic constants are sensitive to volume change. The good agreement between the calculations and experiments depends critically on the predictive power of LDA calculations on the equation of state of pyrope (Figure 1). The temperature dependence of the elastic constants is also well described by the Wu and Wentzcovitch [2011] method (Figure 3b). This is to be expected, because the only approximation introduced by Wu and Wentzcovitch [2011] works best for crystals with high symmetry, and pyrope has cubic symmetry. The elastic anisotropy of materials expresses the difference of material stiffness in different directions. It can provide insights into seismic anisotropy and can be an indicator of the mechanical stability of the structure. For cubic crystals, an anisotropy factor can be expressed as A ¼ 2c44 þ c12 1[Karki et al., c11 1997], which indicates the deviation from elastic isotropy. Elastically isotropic material has an anisotropy factor of zero. Our calculated anisotropy of pyrope at high pressures and temperatures is presented in Figure S1 in the supporting information. The absolute values of the anisotropy factor decrease as pressure increases and temperature decreases. As shown in Figure S1, the anisotropy of pyrope is low, especially at high pressure and low temperature. This result suggests that the anisotropy in upper mantle and subducting lithospheric plate is unlikely to be caused by pyrope [Sinogeikin and Bass, 2000; Lu et al., 2013].

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Table 3. Thermal Elastic Constants c11, c12, and c44 and Their First and Second Derivatives With Respect to Pressure and Temperature of Pyrope at Ambient Pressure

Composition Pmax (GPa) M (GPa) dM/dP

c11 c12 c44 c11 c12 c44 This study, 300 K Py100 25 292.66 107.3 88.09 6.29 3.32 1.84 This study, static Py100 25 307.63 112.83 93.64 5.23 3.65 0.89 Erba et al. [2014], static Py100 60 296 109 89 5.9 3.2 1.4 Lu et al. [2013] Py68Alm24Gr5Sp1 24.6 290(2) 106(2) 92(1) 6.0(1) 3.5(1) 1.2(1) Sinogeikin and Bass [2000] Py100 20 297(3) 108(2) 93(2) 5.8(4) 3.2(4) 1.3(3) Conrad et al. [1999] Py100 8.75 297.6(5) 109.8(6) 92.7(7) O'Neill et al. [1991] Py100 ambient 296.2(5) 111.1(6) 91.6(3) Leitner et al. [1980] Py100 ambient 295(2) 117(1) 90(3) d2M/dP2 (×103 GPa1) c11 c12 c44 This study, 300 K Py100 25 57.3 10.8 46 This study, static Py100 25 4.84 0.63 3.73 Erba et al. [2014], static Py100 60 33.0 1.9 17.0 2 2 6 2 Tmax (K) dM/dT (MPa/K) d M/dT (×10 GPa/K ) c11 c12 c44 c11 c12 c44 This study, 300 K Py100 2000 35.5 12.9 13.4 7.08 2.06 3.06 Lu et al. [2013] Py68Alm24Gr5Sp1 750 21(4) 16.3(5) 3(1) Sinogeikin and Bass [2002b] Py100 1073 31(3) 6(2) 7(2) d2M/dTdP (×103 K1) c11 c12 c44 This study 300 K Py100 2000 0.54 0.03 0.33

For cubic crystals, the adiabatic bulk modulus KS and shear modulus G can be obtained by using the Voigt-Reuss-Hill averages [Hill, 1952]

2c K ¼ c ; (9) S 11 3 1 c 3c 5c c G ¼ þ 44 þ 44 ; (10) 2 5 5 4c44 þ 3c

c ¼ c11 c12: (11) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 4 =ρ =ρ The calculated elastic modulus and the velocities expressed as VP ¼ KS þ 3 G and VS ¼ G are shown in Figure 4 and Table 4. Again, the static results on the elastic modulus and velocities in this study are slightly larger than those reported by Erba et al. [2014]; including vibration contribution significantly reduces the elastic modulus and sound velocities. The calculated results show good consistency with experimental data at various pressures and temperatures. The calculated data are fit with the equation: ∂ ∂ ∂2 ∂2 M M 1 M 2 1 M 2 M ¼ M0 þ P þ ðÞþT 300 P þ ðÞT 300 ∂P ∂T 2 ∂P2 2 ∂T 2 0 0 0 0 ∂2 M ; þ ∂ ∂ PðÞT 300 (12) T P 0

where M refers to the elastic moduli (c11, c12, and c44), bulk modulusKS or shear modulusG. M0 refers to the ∂M ∂M 1 ∂2M 1 ∂2M ∂2M fi values of M at ambient condition (0 GPa and 300 K). ∂ , ∂ , ∂ 2 , ∂ 2 , and ∂ ∂ are the rst and P 0 T 0 2 P 0 2 T 0 T P 0 second derivatives of M at ambient condition. Data in 1300–2000 K and 0–7 GPa are excluded from fitting, since the anharmonic effect may not be negligible at these conditions. The calculated first and second deri-

vatives of elastic moduli (c11, c12, and c44), bulk modulus K, and shear modulus G are shown in Tables 3 and 4. The pressure and temperature derivatives ofc11, c12, and c44 are comparable to experimental data [Lu et al., ∂KS ∂G 2013; Sinogeikin and Bass, 2002b]. ∂P 0 and ∂P 0 are determined to be 4.31 and 1.71 at ambient conditions. ∂KS ∂G ∂T 0 and ∂T 0 are determined to be 20.40 and 12.60 MPa/K at ambient conditions, which agrees well with results from ultrasonic measurements [Gwanmesia et al., 2006, 2007]. Nonlinear pressure and

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(a) (b)

Figure 4. (a) Bulk modulus K and shear modulus G, (b) VS and VP of pyrope at various pressures. The dash-dotted lines represent the calculated static result by Erba et al. [2014]. Scattered points are from experimental results: Chen et al. [1997] (six-pointed stars), Leitner et al. [1980] (rectangles), O'Neill et al. [1991] (downward-pointing triangles), Sinogeikin and Bass [2000] (right-pointing triangles), Zou et al. [2012a] (crosses), Gwanmesia et al. [2006] (asterisk), and Chantel et al. [2016] (left-pointing triangles). (c) Bulk modulus K and shear modulus G, (d) VS and VP of pyrope at various temperatures. The circles are from Gwanmesia et al. [2007], and the stars are from Sinogeikin and Bass [2002b].

temperature dependence is noticeable in all elastic moduli (Tables 3 and 4). The dimensionless logarithmic ∂lnKS 1 ∂KS anharmonic Anderson-Grüneisen parameter, deﬁned as δS ¼ fgKS ¼ ¼ andfg G P ∂lnρ αKS ∂T P P ∂lnG 1 ∂G P ¼ ∂ ρ ¼α ∂ , can be a good indicator of anharmoncity and are calculated to be 4.4 and 5.1, ln P G T P respectively in this study.

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Table 4. Bulk Modulus K and Shear Modulus G and Their First and Second Derivatives With Respect to Pressure and Temperature of Pyrope 2 2 3 1 Composition Pmax (GPa) M (GPa) dM/dP d M/dP (×10 GPa )

KS GKS GKS G This study, 300 K Py100 25 169.09 89.9 4.31 1.71 11.9 41.5 This study, static Py100 25 177.76 95.14 4.18 0.85 1.19 3.34 Brillouin Scattering Lu et al. [2013] Py68Alm24Gr5Sp1 24.6 168(2) 92(1) 4.4(1) 1.2(1) Sinogeikin and Bass [2000] Py100 20 171(2) 94(2) 4.1(3) 1.3(2) Sinogeikin and Bass [2002b] Py100 ambient 170(2) 94(2) Conrad et al. [1999] Py100 8.75 172.7 92 3.2 1.4 O'Neill et al. [1991] Py100 ambient 172.8(3) 92.0(2) Leitner et al. [1980] Py100 ambient 177(1) 89(1) Ultrasonic interference Gwanmesia et al. [2007] Py100 0.3 166.0(2) 92(1) Gwanmesia et al. [2006] Py100 8.7 175(2) 91(2) 3.9(3) 1.7(2) Chen et al. [1999] Py100 10 171(2) 92(1) 5.3(4) Chen et al. [1997] Py98Alm2 ambient 173 91.7 1.6(2) Chantel et al. [2016] Py100 23.8 172(2) 89.1(5) 4.38(8) 1.66(5) 2 2 6 2 2 3 1 Tmax (K) dM/dT (MPa/K) d M/dT (×10 GPa/K ) d M/dTdP (×10 K ) KS GKS GKG This study Py100 2000 20.4 12.6 3.73 2.84 0.2 0.3 Brillouin scattering Lu et al. [2013] Py68Alm24Gr5Sp1 750 17(1) 5(1) Sinogeikin and Bass [2002b] Py100 1073 14(2) 9(1) Ultrasonic interference Gwanmesia et al. [2007] Py100 1000 19.3(4) 10.4(2) 0.2(3) 0.2(1) Gwanmesia et al. [2006] Py100 700 18(2) 10(1) Chantel et al. [2016] Py100 1273 18(2) 8(1)

4. Discussion 4.1. Nonlinear Dependence of Elastic Constants

The c11, c12, and c44 for static calculations show almost linear pressure dependence (Figure 3). However, the nonlinear components in elastic constants become noticeable after considering the vibrational contributions

and increase with temperatures. The second pressure derivatives of elastic constants c11 and c44 are negative, while those of c12 are positive. At 20 GPa and ambient temperature, the contribution from the second pressure derivative is about 5.6%, 2.5%, and 15.9% for c11, c12, and c44, respectively. The second pressure derivative of the bulk modulus is small because the contributions of c11 and c12 cancel each other partly. In contrast, the second pressure derivative of shear modulus is significant because both c44 and c = c11 c12 have large nonlinear dependence on pressure. At 20 GPa, the second pressure derivatives have contribution of 1.9% and 14.4% to K and G (Figure 4a). The high nonlinear pressure dependence of G also leads to significant nonlinear pressure dependences of velocities, especially shear velocity. Although velocities are usually fit with linear functions in most studies, we illustrate that velocities are not necessarily a linear function of pressure. This nonlinear behavior is implied by the discrepancy in temperature dependence of elastic modulus of grossular at ambient pressure between Isaak et al. [1992] and Kono et al. [2010]. Isaak et al. [1992] investigated the elasticity of grossular at ambient pressure up to 1350 K by using ultrasonic measurements. Kono et al. [2010] obtained the elasticity by linearly extrapolating the measured elasticity of grossular at high pressure to ambient pressure. As shown in Figure 4a, the linear extrapolation will overestimate the elastic modulus at ambient pressure because of nonlinear behavior of the elastic moduli. In contrast to Isaak et al. [1992], Kono et al. [2010] overestimate the elasticity at ambient pressure, and the overestimation grows with increasing temperature, which is consistent with nonlinear behavior of the elastic moduli (Figures 3 and 4). Furthermore, the overestimation in the shear modulus is much larger than that in the bulk modulus. This is also consistent with the difference in nonlinear behavior between the shear modulus and bulk modulus (Figure 4). The elastic constants also exhibit nonlinear temperature dependence. The second temperature derivatives of

c11, c12, c44, K, and G (Table 4) have a negative contribution of 7.97%, 5.41%, 12.25%, 6.79%, and

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(a) 11.3% to each term at 2000 K and 20 GPa. The concave relationship between temperature and each elastic modulus or velocity can be seen in Figures 3b, 4c, and 4d. Such nonlinear behavior was reported for majorite garnet but not for pyrope and grossular, so the nonlinear behavior is attributed to chemical composition rather than structure by previous studies [Irifune et al., 2008; Zou et al., 2012a; Kono et al., 2010]. However, the available experimental data cannot rule out nonlinear beha- (b) vior in pyrope. Sinogeikin and Bass [2002b] measured the velocity of pyrope up to 800 K. As shown in Figure 4d the nonlinear temperature dependence of velocities is negligible below 1000 K at 0 GPa. Instead, the elastic modulus of grossular up to 1350 K at ambient pressure measured by Isaak et al. [1992] exhi- bits clearly nonlinear temperature dependence. In contrast, the linear extrapolation of elasticity at high pressure to ambient pressure by Kono et al. [2010] produces the linear (c) temperature of the elastic moduli of grossular at ambient pressure. It can be expected that Zou et al. [2012a] obtained the linear temperature dependence of elastic modulus of pyrope at ambient pressure since they used the same procedure as Kono et al. [2010]. As suggested by this study, however, the elastic modulus of pyrope shows the nonlinear temperature dependence (Figure 4 c). Therefore, ﬁtting the data with linear pressure dependence may not be a good choice, especially at high temperatures (Figures 4a and 4b). Nonlinear pressure and temperature Figure 5. (a) Density of upper mantle minerals (olivine, orthopyroxene, clin- dependence of elastic moduli and opyroxene, and pyrope) and aggregate mineral models (piclogite, pyrolite, and eclogite with 30% pyrope) along with seismic models PREM [Dziewonski sound velocities should be a com- and Anderson, 1981] and ak135 [Kennett et al., 1995] along the normal mantle mon feature for pyrope, grossular, geotherm. (b). Seismic velocities of upper mantle minerals along the normal and majorite garnet no matter the mantle geotherm (c). Seismic velocities of aggregate mineral models composition. (piclogite, pyrolite, and eclogite with 30% pyrope) and seismic models (PREM and ak135) along the normal upper mantle geotherm. Olivine data are from The nonlinear behavior observed in Isaak et al. [1992], Zha et al. [1998], and Liu and Li [2006]. Orthopyroxene data this study is analyzed along the man- are from Chai et al. [1997], Jackson et al. [2007], and Jackson et al. [2003]. Clinopyroxene data are from Zhao et al. [1998], Isaak et al. [2006], and Duffy tle and slab geotherms (Figure S2). and Anderson [1989]. Pyrope data are from this study. The solid lines show the results from

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(a) a second-order ﬁtting, and the dash lines take only the linear term in the second-order ﬁtting. The nonlinear contribution has an uneven negative effect on velocity of pyrope along the bulk mantle and cold slab geotherms. This nonlinear effect is more pronounced at

greater depths. It lowers VP by ~0.3% and VS by ~0.5% at 400 km under the normal upper mantle conditions. For the cold slab geotherm, the nonlinear effect has a positive inﬂuence and raises

VP by ~0.3% and VS by ~0.8% at 400 km. A linear ﬁt of low pressure and low tem- (b) perature elastic moduli data will result in higher error under normal upper mantle conditions.

4.2. Implications for the Upper Mantle and Subducting Lithospheric Plates The density and seismic velocities of pyrope and other relevant mantle minerals and seismic models are shown in Figure 5. Pyrope shows the highest density among all the minerals under (c) the P-T conditions equivalent to the normal upper mantle (Figure 5). The density of pyrope increases from ~3.75 g/cm3 at 220 km to ~3.95 g/cm3 at 400 km along the normal upper mantle geotherm. Pyrope is ~9% denser than PREM and ak135 models and is the only mineral that shows higher density than the seismic models. Both piclogite and pyrolite models show densities ~3% lower than the seismic models. Pyrope has the highest seismic velocities among all the minerals, with P wave velocity of ~8.74 km/s at 220 km and ~9.03 km/s at 400 km and with S wave velocity of ~4.77 km/s at 220 km and ~4.88 km/s Figure 6. (a) Density of upper mantle minerals (olivine, orthopyroxene, clin- at 400 km (Figure 5b). The calculated P opyroxene, and pyrope) and aggregate mineral models (eclogite with 10%, wave velocities of both piclogite and 30%, and 50% pyrope) along with seismic models PREM [Dziewonski and pyrolite models are ~2% smaller than Anderson, 1981] and ak135 [Kennett et al., 1995] along the cold subducted the seismic observations. S wave veloci- lithospheric plate geotherm. (b) Seismic velocities of upper mantle minerals along the cold subducted lithospheric plate geotherm. (c) Seismic velocities ties of aggregate mineral assemblages of aggregate mineral models (piclogite, pyrolite, and eclogite with 30% show a good match with seismic obser- pyrope) and seismic models (PREM and ak135) along the cold subducted vations; especially, the pyrolite model lithospheric plate geotherm. Olivine data are from Isaak et al. [1992], Zha shows a perfect match with PREM. et al. [1998], and Liu and Li [2006]. Orthopyroxene data are from Chai et al. Incorporating more pyrope should [1997], Jackson et al. [2007], and Jackson et al. [2003]. Clinopyroxene data are from Zhao et al. [1998], Isaak et al. [2006], and Duffy and Anderson [1989]. increase the density and P wave velocity Pyrope data are from this study. of the aggregate mineral models, thus

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making it closer to seismic observations; however, this would yield more Al in the mantle chemical model. Previous research also found similar results [Lu et al., 2013]. Element partitioning data suggest 25% Fe2+ and 20% Ca2+ in substitution of Mg2+ in pyrope [Irifune et al., 1998; Fumagalli and Poli, 2005], which can affect both density and seismic velocities. Almandine is ~22% den-

ser than pyrope [Arimoto et al., 2015] and has ~10% and 9% lower VS and VP than pyrope. Grossular is ~1.4% denser than pyrope and has 2% and 5.4% higher VP and VS than pyrope. Assuming a linear effect, incorporating 25% Fe2+ and 20% Ca2+ into garnet in the model can increase the density of pyrope by ~6% and

decrease the VP and VS of pyrope by 2% and 1%, thus increasing the density of pyrolite and piclogite by ~1% and decreasing the VP and VS of the two models by ~0.3% and ~0.2%, respectively. In the cold subducting slab, the density and seismic velocities of pyrope along with other minerals and seismic models are shown in Figure 6. The density of pyrope increases from ~3.84 g/cm3 at 220 km to ~4.01 g/ cm3 at 400 km, which is ~2% more than the density along the bulk mantle geotherm. Pyrope also shows the highest density among all the minerals in the cold subducting slab geotherm. Eclogites with 10% to 50% pyrope show higher density than the seismic models, and the density of the slab increases as the pyrope proportion increases. Eclogite with 30% pyrope shows higher density than seismic models even along the bulk mantle geotherm (Figure 5a). This implies that both low temperature and dense mineral phase contribute to the driving force of lithospheric plate subduction. Along the slab geotherm, pyrope has the highest seismic velocity among all the minerals, with P wave velocity of 9.14 km/s at 220 km and 9.31 km/s at 400 km and S wave velocity of 5.02 km/s at 220 km and 5.04 km/s at 400 km. Eclogite with 30% pyrope shows ~1.5% higher P wave velocities and ~1.5% higher S wave velocities than PREM and shows ~2.4% higher P wave velocities and ~2.9% higher S wave velocities than ak135 model. On the other hand, along the normal mantle geotherm, eclogite with 30% pyrope has the lowest seismic velocity compared to pyrolite and piclogite model, due to the very low seismic velocities of clinopyrox- enes (Figure 5c). Thus, if the pyroxene-garnet transformation is kinetically inhibited so that both pyrope garnet and pyroxene are preserved in the condition of subducting slab as shown by Nishi et al. [2008], then the higher seismic velocities of the slab relative to the surrounding mantle should arise from the lower temperatures rather than from the differences in mineral assemblage. Equilibrium dissolution of pyroxene into garnet-majorite solid solution would, of course, further increase velocities in the subducting slab by up to a few percent.

5. Conclusions

The thermodynamic and elastic properties of pyrope (Mg3Al2Si3O12) are calculated at pressure and temperature conditions equivalent to the Earth's upper mantle under the scheme of density functional theory and quasi-harmonic approximation. Among the major minerals in the upper mantle, pyrope has the highest density and sound velocities. Pyrope shows isotropic elasticity at high pressure. Nonlinear pressure and temperature dependence of bulk and shear moduli are observed. Eclogite with 30% pyrope and 70% clinopyroxene is denser than the surrounding mantle even along the normal upper mantle geotherm and becomes yet denser along the cold slab geotherm, while the seismic velocities of eclogite with 30% pyrope along the normal upper mantle geotherm are slower than the surrounding mantle but along the cold slab geotherm are faster than the surrounding mantle. If the pyroxene-garnet transformation is kinetically inhibited at the conditions Acknowledgments of subducting slab so that both pyroxene and pyropic garnet are preserved, both low temperature and dense This work is supported by the State Key Development Program of Basic mineral assemblage contribute to the negative buoyancy, while the contrast in the seismic velocities of the Research of China (2014CB845905), the slab and the surrounding mantle results from the lower temperature, instead of the mineral assemblage. Natural Science Foundation of China (41590621, 41274087, and 41473011), and the National Science Foundation Division of Earth Sciences Geophysics References grant 1344942. The calculations were Anderson, O. L., D. L. Isaak, and H. Oda (1991), Thermoelastic parameters for 6 minerals at high-temperature, J. Geophys. Res., 96, conducted at the supercomputing 18,037–18,046, doi:10.1029/91JB01579. center of USTC. We would like to thank Arimoto, T., et al. (2015), Sound velocities of Fe3Al2Si3O12 almandine up to 19GPa and 1700K, Phys. Earth Planet. Inter., 246,1–8. Chao Yao and Rui Yang from USTC for Baima, J., M. Ferrabone, R. Orlando, A. Erba, and R. Dovesi (2016), Thermodynamics and phonon dispersion of pyrope and grossular silicate their helpful discussion. The data for this garnets from ab initio simulations, Phys. Chem. Miner., 43(2), 137–149. paper are available by contacting the Baroni, S., S. de Gironcoli, A. Dal Corso, and P. Giannozzi (2001), Phonons and related crystal properties from density-functional perturbation corresponding author at the University theory, Rev. Mod. Phys., 73(2), 515. of Science and Technology of China. Barron, T. H. K., and M. L. Klein (1965), Second-order elastic constants of a solid under stress, Proc. Phys. Soc., 85(3), 523.

HU ET AL. HIGH-PT ELASTICITY OF PYROPE 13 Journal of Geophysical Research: Solid Earth 10.1002/2016JB013026

Bina, C. R. (2013), Mineralogy: Garnet goes hungry, Nat. Geosci., 6(5), 335–336. Brown, J., and T. Shankland (1981), Thermodynamic parameters in the Earth as determined from seismic profiles, Geophys. J. Int., 66(3), 579–596. Ceperley, D., and B. Alder (1980), Exchange-correlation potential and energy for density-functional calculation, Phys. Rev. Lett., 45, 567–581. Chai, M., J. M. Brown, and L. J. Slutsky (1997), The elastic constants of an aluminous orthopyroxene to 12.5 GPa, J. Geophys. Res., 102, 14,779–14,785, doi:10.1029/97JB00893. Chantel, J., G. M. Manthilake, D. J. Frost, C. Beyer, T. B. Ballaran, Z. Jing, and Y. Wang (2016), Elastic wave velocities in polycrystalline Mg3Al2Si3O12-pyrope garnet to 24 GPa and 1300 K, Am. Mineral., 101(4), 991–997. Chaplin, T., G. D. Price, and N. L. Ross (1998), Computer simulation of the infrared and Raman activity of pyrope garnet, and assignment of calculated modes to specific atomic motions, Am. Mineral., 83(7-8), 841–847. Chen, G. L., R. Miletich, K. Mueller, and H. A. Spetzler (1997), Shear and compressional mode measurements with GHz ultrasonic interferometry and velocity-composition systematics for the pyrope-almandine solid solution series, Phys. Earth Planet. Inter., 99(3-4), 273–287. Chen, G., et al. (1999), Elastic wave velocities of Mg3Al2Si3O12-pyrope garnet to 10 GPa, Am. Mineral., 84(3), 384–388. Conrad, P. G., C. S. Zha, H. K. Mao, and R. J. Hemley (1999), The high-pressure, single-crystal elasticity of pyrope, grossular, and andradite, Am. Mineral., 84(3), 374–383. Dovesi, R., M. De La Pierre, A. M. Ferrari, F. Pascale, L. Maschio, and C. M. Zicovich-Wilson (2011), The IR vibrational properties of six members of the garnet family: A quantum mechanical ab initio study, Am. Mineral., 96(11–12), 1787–1798. Duffy, T. S., and D. L. Anderson (1989), Seismic velocities in mantle minerals and the mineralogy of the upper mantle, J. Geophys. Res., 94, 1895–1912, doi:10.1029/JB094iB02p01895. Dziewonski, A. M., and D. L. Anderson (1981), Preliminary reference Earth model, Phys. Earth Planet. Inter., 25(4), 297–356. Erba, A., A. Mahmoud, D. Belmonte, and R. Dovesi (2014), High pressure elastic properties of minerals from ab initio simulations: The case of pyrope, grossular and andradite silicate garnets, J. Chem. Phys., 140(12), 124703. Frost, D. J. (2008), The upper mantle and transition zone, Elements, 4(3), 171–176. Fumagalli, P., and S. Poli (2005), Experimentally determined phase relations in hydrous peridotites to 6.5 GPa and their consequences on the dynamics of subduction zones, J. Petrol., 46(3), 555–578. Giannozzi, P., et al. (2009), QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials, J. Phys. Condens. Matter, 21(39), 395502. Gwanmesia, G. D., J. Z. Zhang, K. Darling, J. Kung, B. S. Li, L. P. Wang, D. Neuville, and R. C. Liebermann (2006), Elasticity of polycrystalline pyrope (Mg3Al2Si3O12) to 9 GPa and 1000 degrees C, Phys. Earth Planet. Inter., 155(3–4), 179–190. Gwanmesia, G. D., I. Jackson, and R. C. Liebermann (2007), In search of the mixed derivative∂ 2M/∂ P∂ T (M = G, K): Joint analysis of ultrasonic data for polycrystalline pyrope from gas-and solid-medium apparatus, Phys. Chem. Miner., 34(2), 85–93. Haselton, H. T., and E. F. Westrum (1980), Low-temperature heat capacities of synthetic pyrope, grossular, and pyrope 60 grossular 40, Geochim. Cosmochim. Acta, 44(5), 701–709. Hill, R. (1952), The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc. Sect. A, 65(5), 349. Hofmeister, A. M., and A. Chopelas (1991), Vibrational spectroscopy of end-member silicate garnets, Phys. Chem. Miner., 17(6), 503–526. Hofmeister, A. M., T. J. Fagan, K. M. Campbell, and R. B. Schaal (1996), Single-crystal IR spectroscopy of pyrope-almandine garnets with minor amounts of Mn and Ca, Am. Mineral., 81(3–4), 418–428. Hohenberg, P., and W. Kohn (1964), Inhomogeneous electron gas, Phys. Rev., 136(3B), B864. Irifune, T., and M. Isshiki (1998), Iron partitioning in a pyrolite mantle and the nature of the 410-km seismic discontinuity, Nature, 392(6677), 702–705. Irifune, T., Y. Higo, T. Inoue, Y. Kono, H. Ohfuji, and K. Funakoshi (2008), Sound velocities of majorite garnet and the composition of the mantle transition region, Nature, 451(7180), 814–817. Isaak, D. G. (1992), High-temperature elasticity of iron-bearing olivines, J. Geophys. Res., 97, 1871–1885, doi:10.1029/91JB02675. Isaak, D. G., O. L. Anderson, and H. Oda (1992), High-temperature thermal expansion and elasticity of calcium-rich garnets, Phys. Chem. Miner., 19(2), 106–120. Isaak, D. G., I. Ohno, and P. Lee (2006), The elastic constants of monoclinic single-crystal chrome-diopside to 1,300 K, Phys. Chem. Miner., 32(10), 691–699. Ita, J., and L. Stixrude (1992), Petrology, elasticity, and composition of the mantle transition zone, J. Geophys. Res., 97, 6849–6866, doi:10.1029/ 92JB00068. Jackson, J. M., J. W. Palko, D. Andrault, S. V. Sinogeikin, D. L. Lakshtanov, W. Jingyun, and Z. Chang-Sheng (2003), Thermal expansion of natural orthoenstatite to 1473 K, Eur. J. Mineral., 15(3), 469–473. Jackson, J. M., S. V. Sinogeikin, and J. D. Bass (2007), Sound velocities and single-crystal elasticity of orthoenstatite to 1073 K at ambient pressure, Phys. Earth Planet. Inter., 161(1), 1–12. Karki, B., L. Stixrude, S. Clark, M. Warren, G. Ackland, and J. Crain (1997), Structure and elasticity of MgO at high pressure, Am. Mineral., 82(1), 51–60. Karki, B., R. Wentzcovitch, S. De Gironcoli, and S. Baroni (2000), High-pressure lattice dynamics and thermoelasticity of MgO, Phys. Rev. B, 61(13), 8793. Kawasaki, I., and F. Kon'no (1984), Azimuthal anisotropy of surface waves and the possible type of the seismic anisotropy due to preferred orientation of olivine in the uppermost mantle beneath the Pacific Ocean, J. Phys. Earth, 32(3), 229–244. Kennett, B., E. Engdahl, and R. Buland (1995), Constraints on seismic velocities in the Earth from traveltimes, Geophys. J. Int., 122(1), 108–124. Kohn, W., and L. J. Sham (1965), Self-consistent equations including exchange and correlation effects, Phys. Rev., 140(4A), A1133. Kolesov, B. A., and C. A. Geiger (1998), Raman spectra of silicate garnets, Phys. Chem. Miner., 25.2, 142–151. Kolesov, B. A., and C. A. Geiger (2000), Low-temperature single-crystal Raman spectrum of pyrope, Phys. Chem. Miner., 27(9), 645–649. Kono, Y., S. Gréaux, Y. Higo, H. Ohfuji, and T. Irifune (2010), Pressure and temperature dependences of elastic properties of grossular garnet up to 17 GPa and 1 650 K, J. Earth Sci., 21(5), 782–791. Leitner, B. J., D. J. Weidner, and R. C. Liebermann (1980), Elasticity of single-crystal pyrope and implications for garnet solid-solution series, Phys. Earth Planet. Inter., 22(2), 111–121. Liu, W., and B. Li (2006), Thermal equation of state of (Mg 0.9 Fe 0.1) 2 SiO 4 olivine, Phys. Earth Planet. Inter., 157(3), 188–195. Lu, C., Z. Mao, J. F. Lin, K. K. Zhuravlev, S. N. Tkachev, and V. B. Prakapenka (2013), Elasticity of single-crystal iron-bearing pyrope up to 20 GPa and 750 K, Earth Planet. Sci. Lett., 361, 134–142. Mahmoud, A., A. Erba, K. Doll, and R. Dovesi (2014), Pressure effect on elastic anisotropy of crystals from ab initio simulations: The case of silicate garnets, J. Chem. Phys., 140, 234703.

HU ET AL. HIGH-PT ELASTICITY OF PYROPE 14 Journal of Geophysical Research: Solid Earth 10.1002/2016JB013026

Mainprice, D., G. Barruol, and W. B. Ismail (2000), The seismic anisotropy of the Earth's mantle: From single crystal to polycrystal, in Earth's Deep Interior: Mineral Physics and Tomography From the atomic to the Global Scale, pp. 237–264, AGU, Washington, D. C. Maschio, L., B. Kirtman, S. Salustro, C. M. Zicovich-Wilson, R. Orlando, and R. Dovesi (2013), Raman spectrum of pyrope garnet. A quantum mechanical simulation of frequencies, intensities, and isotope shifts, J. Phys. Chem. A, 117(45), 11,464–11,471. Milani, S., et al. (2015), Diamond–garnet geobarometry: The role of garnet compressibility and expansivity, Lithos 227, 140–147. Mulliken, R. (1955), Report on notation for the spectra of polyatomic molecules, J. Chem. Phys., 23(11), 1997. Mulliken, R. S. (1956), Erratum: Report on notation for the spectra of polyatomic molecules, J. Chem. Phys., 24(5), 1118. Negredo, A., J. Valera, and E. Carminati (2004), TEMSPOL: A MATLAB thermal model for deep subduction zones including major phase transformations, Comput. Geosci., 30(3), 249–258. Nishi, M., T. Kato, T. Kubo, and T. Kikegawa (2008), Survival of pyropic garnet in subducting plates, Phys. Earth Planet. Inter., 170(3), 274–280. Núñez-Valdez, M., K. Umemoto, and R. M. Wentzcovitch (2012), Elasticity of diamond at high pressures and temperatures, Appl. Phys. Lett., 101(17), 171902. Núñez-Valdez, M., Z. Wu, Y. G. Yu, and R. M. Wentzcovitch (2013), Thermal elasticity of (Fex,Mg1-x)2SiO4 olivine and wadsleyite, Geophys. Res. Lett, 40, 290–294, doi:10.1002/grl.50131. O'Neill, B., J. D. Bass, G. R. Rossman, C. A. Geiger, and K. Langer (1991), Elastic properties of pyrope, Phys. Chem. Miner., 17(7), 617–621. Pascale, F., C. M. Zicovich-Wilson, R. Orlando, C. Roetti, P. Ugliengo, and R. Dovesi (2005), Vibration frequencies of Mg3Al2Si3O12 pyrope. An ab initio study with the CRYSTAL code, J. Phys. Chem. B, 109(13), 6146–6152. Peacock, S. M., and K. Wang (1999), Seismic consequences of warm versus cool subduction metamorphism: Examples from southwest and northeast Japan, Science, 286(5441), 937–939. Perrillat, J.-P., A. Ricolleau, I. Daniel, G. Fiquet, M. Mezouar, N. Guignot, and H. Cardon (2006), Phase transformations of subducted basaltic crust in the upmost lower mantle, Phys. Earth Planet. Inter., 157(1), 139–149. Poli, S. (1993), The amphibolite-eclogite transformation; an experimental study on basalt, Am. J. Sci., 293(10), 1061–1107. Prewitt, C. T., and R. T. Downs (1998), High-pressure crystal chemistry, Rev. Mineral., 37, 284–318. Richet, P., and G. Fiquet (1991), High-temperature heat-capacity and premelting of minerals in the system MgO-CaO-Al2O3-SiO2, J. Geophys. Res., 96, 445–456, doi:10.1029/90JB02172. Rickwood, P., M. Mathias, and J. Siebert (1968), A study of garnets from eclogite and peridotite xenoliths found in kimberlite, Contrib. Mineral. Petrol., 19(4), 271–301. Ringwood, A. (1967), The pyroxene-garnet transformation in the Earth's mantle, Earth Planet. Sci. Lett., 2(3), 255–263. Ringwood, A. E. (1975), Composition and Petrology of the Earth's Mantle, McGraw-Hill, New York. Ringwood, A. E. (1979), Origin of the Earth and Moon, 295 pp., Spring, New York. Ringwood, A. E. (1982), Phase transformations and differentiation in subducted lithosphere: Implications for mantle dynamics, basalt petrogenesis, and crustal evolution, J. Geol., 611–643. 2+ Shukla, G., Z. Wu, H. Hsu, A. Floris, M. Cococcioni, and R. M. Wentzcovitch (2015), Thermoelasticity of Fe -bearing bridgmanite, Geophys. Res. Lett, 42, 1741–1749, doi:10.1002/2014GL062888. Sinogeikin, S. V., and J. D. Bass (2000), Single-crystal elasticity of pyrope and MgO to 20 GPa by Brillouin scattering in the diamond cell, Phys. Earth Planet. Inter., 120(1), 43–62. Sinogeikin, S. V., and J. D. Bass (2002a), Elasticity of majorite and a majorite-pyrope solid solution to high pressure: Implications for the Transition Zone, Geophys. Res. Lett, 29(2), 1017, doi:10.1029/2001GL013937. Sinogeikin, S. V., and J. D. Bass (2002b), Elasticity of pyrope and majorite-pyrope solid solutions to high temperatures, Earth Planet. Sci. Lett., 203(1), 549–555. Troullier, N., and J. L. Martins (1991), Efficient pseudopotentials for plane-wave calculations, Phys. Rev. B, 43(3), 1993. Vanderbilt, D. (1990), Soft self-consistent pseudopotentials in a generalized eigenvalue formalism, Phys. Rev. B, 41(11), 7892. Van Mierlo, W. L., F. Langenhorst, D. J. Frost, and D. C. Rubie (2013), Stagnation of subducting slabs in the transition zone due to slow diffusion in majoritic garnet, Nat. Geosci., 6(5), 400–403. Wallace, D. C. (1998), Thermodynamics of Crystals, Courier Dover, Mineola, New York. Wentzcovitch, R. M. (1991), Invariant molecular-dynamics approach to structural phase transitions, Phys. Rev. B, 44(5), 2358. Wentzcovitch, R. M., G. Y. Yonggang, and Z. Wu (2010), Thermodynamic properties and phase relations in mantle minerals investigated by first principles quasiharmonic theory, Rev. Mineral. Geochem., 71(1), 59–98. Wu, Z. (2010), Calculating the anharmonic free energy from first principles, Phys. Rev. B, 81(17), 172301. Wu, Z., and R. M. Wentzcovitch (2007), Vibrational and thermodynamic properties of wadsleyite: A density functional study, J. Geophys. Res., 112, B12202, doi:10.1029/2007JB005036. Wu, Z., and R. M. Wentzcovitch (2009), Effective semiempirical ansatz for computing anharmonic free energies, Phys. Rev. B, 79(10), 104304. Wu, Z., and R. M. Wentzcovitch (2011), Quasiharmonic thermal elasticity of crystals: An analytical approach, Phys. Rev. B, 83(18), 184115. Wu, Z. Q., J. F. Justo, and R. M. Wentzcovitch (2013), Elastic anomalies in a spin-crossover system: Ferropericlase at lower mantle conditions, Phys. Rev. Lett., 110(22), 5. Yang, R., and Z. Wu (2014), Elastic properties of stishovite and the CaCl2-type silica at the mantle temperature and pressure: An ab initio investigation, Earth Planet. Sci. Lett., 404,14–21. Zha, C.-s., T. S. Duffy, R. T. Downs, H.-k. Mao, and R. J. Hemley (1998), Brillouin scattering and X-ray diffraction of San Carlos olivine: Direct pressure determination to 32 GPa, Earth Planet. Sci. Lett., 159(1), 25–33. Zhang, L., H. Ahsbahs, and A. Kutoglu (1998), Hydrostatic compression and crystal structure of pyrope to 33 GPa, Phys. Chem. Miner., 25(4), 301–307. Zhao, D., T. Matsuzawa, and A. Hasegawa (1997), Morphology of the subducting slab boundary in the northeastern Japan arc, Phys. Earth Planet. Inter., 102(1), 89–104. Zhao, Y., R. V. Dreele, J. Zhang, and D. Weidner (1998), Thermoelastic equation of state of monoclinic pyroxene: CaMgSi2O6 diopside, Rev. High Pressure Sci. Technol., 7,25–27. Zou, Y., T. Irifune, S. Gréaux, M. L. Whitaker, T. Shinmei, H. Ohfuji, R. Negishi, and Y. Higo (2012a), Elasticity and sound velocities of polycrystalline Mg3Al2 (SiO4) 3 garnet up to 20 GPa and 1700 K, J. Appl. Phys., 112(1), 014910. Zou, Y., S. Greaux, T. Irifune, M. L. Whitaker, T. Shinmei, and Y. Higo (2012b), Thermal equation of state of Mg3Al2Si3O12 pyrope garnet up to 19 GPa and 1,700 K, Phys. Chem. Miner., 39(7), 589–598.

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