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Quasi-harmonic computations of thermodynamic parameters of olivines at high-pressure and high-temperature. A comparison with experiment data
Abstract
Specific volumes of forsterite and san Carlos olivine have been measured by in-situ X-ray diffraction at simultaneously high pressure and high temperature up to 7 GPa and 1300 K in a cubic-anvil press coupled to synchrotron radiation. No difference could be evidenced between the thermal pressure of the two compounds. It has been shown that the parameter αKT is volume-independent within the volume range investigated in this study for both forsterite and San Carlos olivine. A quasi-harmonic calculation of the high-pressure high-temperature specific volumes of forsterite has been performed and shown to be in good agreement with the experimental results. Then, a self consistent model of all the thermodynamic functions of forsterite was constructed, the input consisting of the experimental ultrasonic and vibrational spectroscopic data exclusively. The model accurately reproduces not only the experimental P-V-T data measured in this study, but also the high-temperature, 1 bar thermal expansion, the adiabatic incompressibility, the constant pressure specific heat and the entropy measurements, without using other a priori information.
... Par ailleurs, dans tous les cas, les effets anharmoniques deviennent importants à plus haute température. On peut traiter ce type d'effets par l'approximation dite quasi-harmonique (Fiquet et al., 1992,Guyot et al., 1996. Dans cette étude, on n'a pas considéré ce deuxième effet, qui demande une approche théorique particulière. ...
... Dans ce cas, le phénomène de dilatation thermique, ainsi que la variation des fréquences de vibration, témoignent tous deux de l'anharmonicité du potentiel : plus la température augmente, plus les atomes se déplacent, plus ils "sondent" une grande partie du potentiel, au-delà des limites de validité de l'approximation harmonique. L'importance de cet effet sur les propriétés thermodynamiques a été évaluée sur quelques structures minéralogiques (Gillet et al., 1991, Fiquet et al., 1992, Guyot et al., 1996, à partir de l'évolution des fréquences vibrationnelles avec la température. Cet effet anharmonique n'a pas été évalué ici. ...
... On peut supposer que les fréquences des deux matériaux sont des fréquences harmoniques, du moins à température ambiante, et on s'attend à ce qu'une simple renormalisation de fréquences améliore l'accord avec l'expérience. Cependant, les données sur ces systèmes sont souvent disponibles à haute température seulement, et dans ce domaine de températures, des corrections liées à la dilatation, et à la variation des fréquences de vibration avec la température sont susceptibles de rentrer en ligne de compte (cf Guyot et al., 1996). On considérera ici que ces dernières sont négligeables. ...
In this work, we applied electronic structure calculations based on density functional theory to compute oxygen, hydrogen and silicon fractionation properties between several materials of interest in Earth Sciences. Understanding of such mechanisms is a main objective in geochemistry, since the isotopic content of minerals is a witness of geochemical events. Electronic structure calculations enables the determination of the complete vibrational properties of one material at the harmonic level, from which its fractionation properties can be deduced. Our goal is to validate an ab initio approach, i.e. free from any parameter coming from experiment, enaling the prediction of the fractionation between two minerals. We studied the materials: quartz, gas water, ice, kaolinite, lizardite, enstatite, forsterite, brucite and gibbsite. Some of them have been the subject of precise experimental studies of their various physical properties, which permits us to validate our approach, and all are of main interest in Earth Sciences, which permits us to propose new fractionation laws as a function of temperature, and to check the validity of laws existing in the litterature. This is the first time that such a systematic study, based on ab initio approaches, is realized on so many different systems. That is why we realized the possibly most complete study of the possible sources of error in our calculation. Moreover, this approach enables detailed analysis of the physical processes driving isotopic fractionation.
... The thermodynamics of Fo, Wds, Rwd, and Prv was studied by computational methods in many studies (Dorogokupets et al., 1999;Fabrichnaya, 1995;Fabrichnaya et al., 2004;Fei and Saxena, 1986;Fei et al., 1992;Guyot et al., 1996;Ita and Stixrude, 1992;Jacobs et al., 2013;Kuskov and Panferov, 1989;Pankov et al., 1996;Polyakov and Kuskov, 1994;Stixrude and Lithgow-Bertelloni, 2005), which were used to calculate the lines of the Fo-Wds, Wds-Rwd, and Rwd-(Prv + Per) transitions (Akaogi et al., 1989(Akaogi et al., , 2007Jacobs and de Jong, 2005;Kalachnikov et al., 1991;Liu, 1979) and the perovskitepostperovskite transition (PPrv) (Hirose, 2006;Tateno et al., 2009). At ambient pressure, the heat capacity, thermal expansion, and adiabatic bulk modulus of different olivine polymorphs, as well as perovskite, are studied in detail (Dachs et al., 2007;Gillet et al., 1991;Jacobs and de Jong, 2005;Jacobs and Oonk, 2001;Kojitani et al., 2012;Trots et al., 2012). ...
... At ambient pressure, the heat capacity, thermal expansion, and adiabatic bulk modulus of different olivine polymorphs, as well as perovskite, are studied in detail (Dachs et al., 2007;Gillet et al., 1991;Jacobs and de Jong, 2005;Jacobs and Oonk, 2001;Kojitani et al., 2012;Trots et al., 2012). The development of multianvil systems and diamond anvil cell have permitted the measurement of the P-V-T properties of Fo, Wds, Rwd, and Prv up to 30 GPa and 2100 K (Guyot et al., 1996;Katsura et al., 2004bKatsura et al., , 2009aMeng et al., 1993Meng et al., , 1994. The measured P-V-T properties were usually reduced and fitted with the use of simple thermodynamic relationships; as a result, bulk moduli, thermal expansion, and other characteristics were calculated depending on temperature and pressure with different degrees of certainty (Katsura et al., 2004b(Katsura et al., , 2009c. ...
... The equations of state for Fo, Wds, Rwd, Akm, Prv, and PPrv were set up with the use of the following data: C P (Akaogi and Ito, 1993;Akaogi et al., 2007Akaogi et al., , 2008Dachs et al., 2007;Gillet et al., 1991;Kojitani et al., 2012;Robie et al., 1982), α (Isaak et al., 1989;Matsui and Manghnani, 1985;Suzuki et al., 1979Suzuki et al., , 1980Suzuki et al., , 1983Trots et al., 2012;Ye et al., 2009), K S (T) (Isaak et al., 1989(Isaak et al., , 2007Jackson et al., 2000;Sumino et al., 1977;Suzuki et al., 1983), K S (P) (Duffy et al., 1995;Li et al., 1996;Yoneda and Morioka, 1992;Zha et al., 1996Zha et al., , 1997Zhou et al., 2014). The P-V-T relationships of these minerals were measured by many authors (Couvy et al., 2010;Fiquet et al., 2000;Funamori et al., 1996;Guignot et al., 2007;Guyot et al., 1996;Katsura et al., 2004bKatsura et al., , 2009aKomabayashi et al., 2008;Ono and Oganov, 2005;Ono et al., 2006;Saxena et al., 1999;Tange et al., 2012;Utsumi et al., 1995;Vanpeteghem et al., 2006;Wang et al., 1994). ...
The equations of state of forsterite, wadsleyite, ringwoodite, MgSiO3-perovskite, akimotoite, and postperovskite are set up by joint analysis of experimentally measured isobaric heat capacity, bulk moduli, thermal expansion depending on temperature at ambient pressure, and volume at room and higher temperatures. Modified equations of state based on the Helmholtz free energy are used to construct a thermodynamic model. The derived equations of state permit calculation of all thermodynamic functions for the minerals depending on temperature and volume or temperature and pressure. A phase diagram of the system MgSiO3-MgO is constructed based on the Gibbs energy calibrated using the referred experimental points. The seismic boundaries at depths of 410 and 520 km and in the zone D" are interpreted on the basis of the phase transitions. The global upper/lower mantle discontinuity at a depth of 660 km remains debatable; it is in poor agreement with experimental and computational data on the dissociation of ringwoodite to perovskite and periclase.
... (c) Comparison between measured entropy and that inferred from vibrational calculations including anharmonic behaviour of the vibrational modes. (d) Differences between the molar volumes calculated with the EK model of Figure 23 and values measured up to 8 GPa and 1300 K by x-ray diffraction (after Guyot et al. 1996). The agreement between both calculated and measured values is excellent and the difference never exceeds ±O.S%. ...
... All the parameters needed to calculate frequencies as a function of volume are thus available (Fig. 25). As discussed by Guyot et al. (1996) a Debye model is not nearly sufficient to describe the VDOS: a Kieffer-type model or the measured VDOS itself are required for thermodynamic calculations (Fig. 23) (Rao et al. 1988 em-I Figure 25. (a) Raman spectra of forsterite at high-pressure and room temperature (from Durben et al. 1993) and at high temperature and room pressure (from Gillet et al. 1997). ...
... The relative temperature-induced Shifts,~(~; )p' are likewise smaller for the internal modes than for the lattice modes, indicating a very small expansion of the Si-O bond relative to the Mg-O bond upon heating (Fig. 25). Chopelas (1990), Gillet et al. (1991), and Guyot et al. (1996) have used highpressure IR and Raman data to predict the thermodynamic and thermoelastic properties of forsterite over a wide P-Trange, in excellent agreement with available experiment data. ...
... These data show that the thermal pressure of hydrous pyrope varies linearly with temperature and is almost independent of volume. Therefore, we assume that (@K T /@T) V = 0 so the thermal pressures in Eq. (5) are independent of volume, an approximation that has been derived or assumed in many previous studies for mantle phases (e.g., Guyot et al., 1996;Wang et al., 1996Wang et al., , 1998Anderson, 1999;Shim et al., 2000;Nishihara et al., 2004;Liu and Li, 2006;Liu et al., 2008;Fan et al., 2015). With this assumption, fitting the data in this study yielded K 0 = 160 ± 2 GPa, K' 0 = 4.8 ± 0.4, and a 0 = (2.9 ± 0.2)Â10 À5 K À1 , which is reasonably consistent with those derived by the HTBM EoS fitting (Table 4). ...
... Furthermore, the density profiles of olivine, orthopyroxene, and clinopyroxene were also modeled using existing results on their thermoelasticity parameters (Suzuki and Anderson, 1983;Guyot et al., 1996;Zhao et al., 1995Zhao et al., , 1998Shinmei et al., 1999;Nishihara et al., 2003;Liu and Li, 2006;Lu et al., 2013;Huang and Chen, 2014). The den-sity profile of hydrous pure pyrope was $4% higher than the PREM density profile, but closer to the PREM than the anhydrous pure pyrope ($5% higher) and Fe-bearing pyrope ($6.5% higher) density profile. ...
High-pressure single-crystal / powder synchrotron X-ray diffraction was carried out on a hydrous pure magnesium pyrope (Mg3Al2Si3O12) containing 900 ppmw H2O, synthesized at 4.0 GPa and 1300 K. The pressure-volume (P-V) single-crystal data from room pressure to 9.81 GPa at ambient temperature were fitted by a third-order Birch-Murnaghan equation of state (BM-EoS) yielding a unit-cell volume of V0= 1505.14±0.38 ų, an isothermal bulk modulus of K0= 160±3GPa and its pressure derivative K′0= 5.2±0.4. When fixing K'0=4.0, the data yielded V0= 1504.58±0.32 ų and K0= 166±2GPa. The pressure-volume-temperature (P-V-T) EoS of the synthetic hydrous pyrope was also measured at temperatures up to 900 K and pressures up to 16.75 GPa, using a diamond anvil cell in conjunction with in situ synchrotron angle-dispersive powder X-ray diffraction. The P-V data at room temperature and in a pressure range of 0.0001-14.81 GPa were then analyzed by a third-order BM-EoS and yielded V0= 1505.35±0.25 ų, K0= 161±2 GPa, K′0= 5.0±0.3. With K'0 fixed to 4.0, we also obtained V0= 1505.04±0.29 ų and K0= 167±1 GPa. Consequently, we fitted the P-V-T data with the high-temperature third-order BM-EoS approach and obtained the thermoelastic parameters of V0= 1505.4±0.3 ų, K0=162±1 GPa, K′0= 4.9±0.2, the temperature derivative of the bulk modulus (∂K0/∂T)P= -0.018±0.004 GPaK⁻¹, and the thermal expansion coefficient at ambient conditions α0= (3.2±0.1)×10⁻⁵ K⁻¹. These properties were consistent with the thermal pressure EoS analysis. These new results on hydrous pyrope were also compared with previous studies of anhydrous pyrope. The main effect of hydration on pyrope is to decrease K0 and increase K'0 by increasing the vacancies or unoccupied volume in the structure. The entire dataset enabled us to examine the thermoelastic properties of important mantle garnets and this data has further applications for modeling the P-T conditions in the upper mantle of the Earth’s interior using deep mineral assemblages.
... The experimental dependency of volume on pressure for various isotherms were taken after Downs et al. (1996) (298 K and 0-17.2 GPa), Zhang (1998) (298 K and 0-9.72 GPa), and Guyot et al. (1996) (298-1272 K and 0-7.05 GPa). ...
... Figure 14 shows the temperature dependencies of the Grüneisen parameter and the product αK T . The calculated volume at 298 K is consistent with the experimental data ( Fig. 15) and results of Guyot et al. (1996) at higher temperature and dK'/dT = 1.8 × 10 -3 K -1 . Gillet et al. (1991) was calculated from enthalpy measurements at 788-1847 K, and isochoric heat capacity after the same authors is shown for the anharmonic model. ...
A numeric model was constructed for the joint optimization of data on the isobaric heat capacity, volume, thermal expansion coefficient, and bulk moduli of minerals. The model is based on the classic thermodynamic relationship between isobaric and isochoric heat capacities, volume, thermal expansion coefficient, isothermal bulk modulus, and Grüneisen parameter. The internal energy and isochoric heat capacity were approximated by the Nernst-Lindeman function with additional terms accounting for the deviation of the internal energy of a real crystal from the Debye model (or the Nernst-Lindeman in our case). Volume and thermal expansion coefficient were described through the internal energy in the Suzuki approximation and bulk modulus, through volume and the Anderson-Grüneisen parameter, which was taken to be constant. As a result, a system of equations was obtained, which allowed us to optimize the experimental data on heat capacity, thermal expansion coefficient, and bulk moduli under standard-state pressure. The adjusted parameters of the model are volume, Debye temperature, isothermal bulk modulus, its pressure derivative, Grüneisen and Anderson-Grüneisen parameters at zero temperature, and three empirical parameters accounting for the effects of anharmonicity, premelting, etc. The equations approximate experimental data within experimental uncertainties from 50-100 K (depending on the value of the Debye temperature) to melting temperature. The model was tested on the example of tungsten, periclase, corundum, and forsterite.
... Most first principles calculations of mantle minerals do not evaluate the effects of intrinsic phonon anharmonicity explicitly but rely on a QH model that can successfully reproduce many of the thermoelastic properties under the range of high-P,T conditions encountered within the lower mantle and transition zone (Gillet et al., 2000b;Guyot et al., 1996;Hamann et al., 2005;Karki and Wentzcovich, 2002;Wentzcovich et al., 2004b). However, some phase equilibrium calculations for phase transitions involving major structural transformations, such as that between MgSiO 3 ilmenite and perovskite, have shown significant differences from the experimental P,T transition line that could be assigned to intrinsic anharmonic phonon effects (Wentzcovich et al., 2004b). ...
... where the integration runs over the VDOS (g(n)) to the maximum frequency n max . Experimental or theoretical determinations of g(n) thus lead to prediction and rationalization of important thermodynamic quantities derived from F th , including the heat capacity (C v or C p ), the 'third law' or vibrational entropy (S vib ), and the thermal pressure (P th ), using the methods of statistical thermodynamics (Chopelas, 1991;Chopelas et al., 1994;Fiquet et al., 1992;Gillet et al., 1991Gillet et al., , 1997Gillet et al., , 1998Gillet et al., , 2000bGuyot et al., 1996;Hofmeister, 1996;Hofmeister and Ito, 1992;Hofmeister et al., 1989;Kieffer, 1979aKieffer, ,b,c, 1985Wallace, 1972). The contribution of a given vibrational mode (n i (q)) to the specific heat at constant volume (C Vi ) is ...
Abstract: Vibrational spectroscopic techniques, including Raman, infrared (IR), and inelastic neutron scattering (INS) or x-ray scattering (IXS) spectroscopies, are coupled with ab initio or first principles calculations to study the vibrational lattice dynamics and to predict and elucidate the thermodynamic behavior of minerals under high-pressure–high-temperature conditions relevant to the Earth’s mantle and other deep planetary interiors. Principal lower mantle and transition zone phases are (Mg,Fe)O magnesiowu¨ stite, silicate perovskite, majorite, wadsleyite, ringwoodite, and SiO2 stishovite. They result from high-pressure transformation of olivine, pyroxene, and garnet phases that occur mainly within the upper mantle. Phase transformations in aluminous and hydrous mineral phases are also discussed.
... P was calculated using a thermal equation of state for San Carlos olivine (Guyot et al., 1996) in the reference sample and in the two-phase sample. Both were fairly consistent (Table 1). ...
Weak serpentine minerals affect the mechanical behavior of serpentinized peridotites at depth, and may play a significant role in deformation localization within subduction zones, at local or regional scale. Mixtures of olivine with 5, 10, 20 and 50 vol. % fraction of antigorite, proxies for serpentinized peridotites, were deformed in axial shortening geometry under high pressures (ca. 2–5 GPa) and moderate temperatures (ca. 350°C), with in situ stress and strain measurements using synchrotron X‐rays. We evaluate the average partitioning of stresses at the grains scale within each phase (mineral) of the aggregate and compare with pure olivine aggregates in the same conditions. The in situ stress balance is different between low antigorite contents up to 10 vol. %, and higher contents above 20 vol. %. Microstructure and stress levels suggest the deformation mechanisms under these experimental conditions are akin to (semi)brittle and frictional processes. Unlike when close to dehydration temperatures, hardening of the aggregate is observed at low serpentine fractions, due to an increase in local stress concentrations. Below and above the 10–20 vol. % threshold, the stress state in the aggregate corresponds to friction laws already measured for pure olivine aggregates and pure antigorite aggregates respectively. As expected, the behavior of the two‐phase aggregate does not evolve as calculated from simple iso‐stress or iso‐strain bounds, and calls for more advanced physical models of two‐phase mixtures.
... The free energy is the thermodynamic potential, which is minimized by the atomic configuration at T > 0 K. Structures showing spectra with low phonon frequency energies minimize i and thus also A. One implication of this dependence is research papers thermal expansion, where larger lattice parameters lead to weaker interactions of atoms and thus also affect the energy spectrum of atomic vibrations (red shift). Therefore, the increasing cell parameters minimize A when T increases (Guyot et al., 1996). On the other hand, it has been shown using the example of cryolite that anharmonic effects can lead to a quasi-degeneracy of numerous low-energy states, resulting in a decrease of free energy and structural stabilization. ...
Perovskites ABX 3 with delocalized positions of the X atoms represent a distinct class of dynamically distorted structures with peculiar structural relations and physical properties. The delocalization originates from atoms crossing shallow barriers of the potential energy surface. Quantum mechanically, they can be treated similar to light atoms in diffusive states. Many of these perovskite structures are widely used functional materials thanks to their particular physical properties, such as superconductivity, ferroelectricity and photo-activity. A number of these properties are related to static or dynamic motion of octahedral units. Yet, a full understanding of the relationships between perovskite crystal structure, chemical bonding and physical properties is currently missing. Several studies indicate the existence of dynamic disorder generated by anharmonic motion of octahedral units, e.g. in halide perovskite structures. To simplify structural analysis of such systems we derive a set of space groups for simple perovskites ABX 3 with dynamical octahedral tilting. The derived space groups extend the well established space group tables for static tiltings by Glazer [ Acta Cryst. B (1972). 28 , 3384–3392], Aleksandrov [ Ferroelectrics (1976). 24 , 801–805] and Howard & Stokes [ Acta Cryst. B (1998). 54 , 782–789]. Ubiquity of dynamical tilting is demonstrated by an analysis of the structural data for perovskites reported in recent scientific publications and the signature of dynamic tilting in the corresponding structures is discussed, which can be summarized as follows: ( a ) volume increase upon a lowering of temperature, ( b ) apparent distortion of octahedra (where Jahn–Teller distortions can be ruled out), ( c ) mismatch between observed instantaneous symmetry and average symmetry, ( d ) deviation of the experimental space group from the theoretically predicted structures for static tilting, ( e ) inconsistency of lattice parameters with those suggested by the theory of static tilts, and ( f ) large displacement parameters for atoms at the X and B sites. Finally, the possible influence of dynamic disorder on the physical properties of halide perovskites is discussed.
... Later, Yanbin utilized transmission electron microscopy to study dislocation dissociation and twinning in minerals in collaboration with colleagues from France, including Jean-Paul Poirier, François Guyot Powder X-ray diffraction and thermal equations-of-state Laboratory X-ray facilities, and later synchrotron X-radiation facilities enabled investigations of thermal equations of state and phase transitions in minerals by Xing Liu, Yanbin Wang, Yue Meng, and Jun Liu [3] [19]- [28]. ...
... The pressure is about 5 GPa. The calculated volumes of forsterite at elevated temperatures were compared with previous experimental measurements (Anderson & Suzuki, 1983;Downs et al., 1996;Guyot et al., 1996) as shown in Figure S2 and Table S1. The differences are less than 1%. ...
Plain Language Summary
Previous experimental studies show that hydrous olivine has higher conductivity than dry olivine, and the conductivity is highly anisotropic. However, the conduction mechanism in hydrous olivine is still under controversy. Density functional theory (DFT) calculation based on quantum mechanics is able to investigate the mechanism in atomic‐scale. Therefore, we conducted DFT calculations on several hydrous olivine models with different hydrogen content, iron content, and kinds of hydrogen defects. We observed the ionization of hydrogen to form free protons at high temperature, which is able to diffuse fast in the olivine crystalline leading to high and anisotropic proton conductivity. The calculated conductivity is consistent with experimental studies at high temperature and gives a mechanism for the observed anisotropic conductivity. In addition, we found hydrogen associated with Mg vacancy site is more mobile than hydrogen associated with Si vacancy site. This atomic‐scale investigation is beneficial in understanding the experimental results as well as the high conductivity anomalies at the asthenosphere.
... The results obtained through this method have been surprisingly close to the experimental values reported for some specific materials [11]. The growing interest for the double perovskite materials based on Co and Mo has to do with the tuning of the physical characteristics from its composition and the type of hybridizations of the electronic orbitals 3d and 4d, from which they can be generated properties that suggest its applicability in spintronics [12] and solid oxide fuel cells [13], among other modern technology devices. ...
Perovskite-like materials which include magnetic elements have relevance due to the technological perspectives in the spintronics industry. In this work, the magnetic, structural and electronic properties of the Ba2CoMoO6double perovskite are investigated. Calculations are carried out through the Full-Potential Linear Augmented Plane Wave method within the framework of the Density Functional Theory with exchange and correlation effects in the Generalized Gradient and Local Density approximations, including spin polarization. From the minimization of energy as a function of volume using the Murnaghan’s state equation the equilibrium lattice parameter and cohesive properties of this compound were obtained. The study of the electronic structure was based in the analysis of the electronic density of states, and the band structure, showing that this compoundevidences a conductive character for a spin channel and insulation for the other, and presents an integer value for the effective magnetic moment (3.0 μB), which allows it to be classified as a half-metallic material. The effects of pressure and temperature on thermophysical properties such as specific heat, Debye temperature, coefficient of thermal expansion and the Grüneisen parameter were calculated and analyzed from the state equation of the system. The obtained results reveal that, in the low temperature regime, the specific heat at constant volume and pressure presents an analogous behavior to each other, with a tendency to the limit of Dulong-Petit typical of the structures of cubic perovskite type, showing a value of 246.3 J/mol.Kat constant volumeand slightly higher values at constant pressure. The dependence of the coefficient of thermal expansion, the temperature of Debye and the Grüneisen parameter with the increase in temperature is discussed in relation to other perovskite-like materials.
... For instance, when the Ca/(Ca ? Mg) of forsterite decreases from 2/16 to 1/64, the 10 3 lna of 44 Ca/ 40 Ca between forsterite and diopside increase from 0.43% to 0.92% at 1000 K. Combining previous studies with our results, the enrichment sequence of heavier Ca isotope in Guyot et al. (1996), ambient conditions; Exp. 2, Downs et al. (1996), ambient conditions; Exp. 3, Kirfel et al. (2005), ambient conditions. The three experimental data were done on pure forsterite Fig. 2 Calculated frequencies of pure forsterite compared with the measured frequencies. ...
Previous theoretical studies have found that the concentration variations within a certain range have a prominent effect on inter-mineral equilibrium isotope fractionation (10³lnα). Based on the density functional theory, we investigated how the average Ca–O bond length and the reduced partition function ratios (10³lnβ) and 10³lnα of ⁴⁴Ca/⁴⁰Ca in forsterite (Fo) are affected by its Ca concentration. Our results show that Ca–O bond length in forsterite ranges from 2.327 to 2.267 Å with the Ca/(Ca + Mg) varying between a narrow range limited by an upper limit of 1/8 and a lower limit of 1/64. However, outside this narrow range, i.e., Ca/(Ca + Mg) is lower than 1/64 or higher than 1/8, Ca–O bond length becomes insensitive to Ca concentration and maintains to be a constant. Because the 10³lnβ is negatively correlated with Ca–O bond length, the 10³lnβ significantly increases with decreasing Ca/(Ca + Mg) when 1/64 < Ca/(Ca + Mg) < 2/16. As a consequence, the 10³lnα between forsterite and other minerals also strongly depend on the Ca content in forsterite. Combining previous studies with our results, the heavier Ca isotopes enrichment sequence in minerals is: forsterite > orthopyroxene > clinopyroxene > calcite ≈ diopside > dolomite > aragonite. Olivine and pyroxenes are enriched in heavier Ca isotope compared to carbonates. The 10³lnα between forsterite with a Ca/(Ca + Mg) of 1/64 and clinopyroxene (Ca/Mg = 1/1, i.e., diopside) is up to ~ 0.64‰ at 1200 K. The large 10³lnαFo-diopside relative to the current analytical precision for Ca isotope measurements suggests that the dependence of 10³lnαFo-diopside on temperature can be used as a thermometer, similar to the one based on the 10³lnα of ⁴⁴Ca/⁴⁰Ca between orthopyroxene and diopside. These two Ca isotope thermometers both have a precision approximate to that of elemental thermometers and provide independent constraints on temperature.
... where the integration runs over the vibrational density of states (g(#)). Experimental or theoretical determinations of g(#) thus lead to prediction and rationalization of important thermodynamic quantities derived from F th , including the heat capacity (C v or C p ), the 'third law' or vibrational entropy (S vib ), and the thermal pressure (P th ), using the methods of statistical thermodynamics (Wallace, 1972;Kieffer, 1979aKieffer, , 1979bKieffer, , 1979cKieffer, , 1985Hofmeister et al., 1989;Chopelas, 1991;Gillet et al., 1991Gillet et al., , 1997Gillet et al., , 1998Gillet et al., , 2000Fiquet et al., 1992;Hofmeister and Ito, 1992;Chopelas et al., 1994;Guyot et al., 1996;Hofmeister, 1996). The contribution of a given vibrational mode (# i (q)) to the specific heat at constant volume (C vi ) is ...
... Thus, the thermal-pressure equation of state consists of two separate functions: static pressure that is a function of volume only and thermal pressure which is a function of temperature only. A number of silicates exhibit this property over the experimental P-T range, including (Mg,Fe)2SiO 4 olivine (Anderson et al., 1992;Guyot et al., 1996), CaSiO 3 perovskite , and garnet (this study). As pointed out by Anderson et al. (1992), this implies that the Anderson-Griineisen parameter ~r = -(aKr/aT)r/(aKr) = K'r over the entire P-T range of our experiment. ...
... Within the uncertainty of the fit, (∂K T /∂T) V value is roughly zero, indicating that the thermal pressure is independent of volume, which is the same with the conclusions ofWang et al. (1998)who also found that the thermal pressure in garnet shows linear variation with T up to 1200 K independent of compression (V/ V 0 ), concluding that (∂K T /∂T) V is very close to zero. The same conclusions have been drawn for some other silicate minerals, such as uvarovite (Fan et al. 2015b), olivine (Guyot et al. 1996), CaSiO 3 perovskite (Wang et al. 1996), and ringwoodite (Nishihara et al. 2004). By using the thermodynamic relation: ...
The pressure–volume–temperature (P–V–T) equation of state (EoS) of synthetic grossular (Grs)–andradite (And) solid-solution garnet sample have been measured at high temperature up to 900 K and high pressures up to 22.75 GPa for Grs50And50, by using in situ angle-dispersive X-ray diffraction and diamond anvil cell. Analysis of room-temperature P–V data to a third-order Birch–Murnaghan (BM) EoS yields: V0 = 1706.8 ± 0.2 ų, K0 = 164 ± 2 GPa and K′0 = 4.7 ± 0.5. Fitting of our P–V–T data by means of the high-temperature third-order BM EoS gives the thermoelastic parameters: V0 = 1706.9 ± 0.2 ų, K0 = 164 ± 2 GPa, K′0 = 4.7 ± 0.2, (∂K/∂T)P = −0.018 ± 0.002 GPa K⁻¹, and α0 = (2.94 ± 0.07) × 10⁻⁵ K⁻¹. The results also confirm that grossular content increases the bulk modulus of the Grs-And join following a nearly ideal mixing model. The relation between bulk modulus and Grs mole fraction (XGrs) in this garnet join is derived to be K0 (GPa) = (163.7 ± 0.7) + (0.14 ± 0.02) XGrs (R² = 0.985). Present results are also compared to previously studies determined the thermoelastic properties of Grs-And garnets.
... The elastic properties of olivine have been measured under various pressure (P)-temperature (T) conditions. A list of previous studies can be found in the supporting information Table S1 [Abramson et al., 1997;Aizawa et al., 2001;Darling et al., 2004;Downs et al., 1996;Duffy et al., 1995;Guyot et al., 1996;Isaak et al., 1994;Isaak, 1992;Isaak et al., 1989;Jacobsen et al., 2009;Liu et al., 2005;Liu and Li, 2006;Li et al., 2004;Mao et al., 2010Mao et al., , 2015Manghnani et al., 2008;Speziale et al., 2004;Suzuki et al., 1983;Webb, 1989;Wang, 2008;Zha et al., 1996Zha et al., , 1998Zaug et al., 1993]. However, the few existing simultaneous high P-T measurements cover a very limited P-T range. ...
We present the elastic properties of San Carlos olivine up to P = 12.8(8) GPa and T = 1300(200) K using Brillouin spectroscopy with CO2 laser-heating. A comparison of our results with the global seismic model AK135 yields average olivine content near 410 km depth of about 37% and 43% in a dry and wet (1.9%wt H2O) upper mantle, respectively. These olivine contents are far less than in the pyrolite model. However, comparisons of our results with regional seismic models lead to very different conclusions. High olivine contents of up to 87% are implied by seismic models of the western U.S. and eastern Pacific regions. In contrast, we infer less than 35% olivine under the Central Pacific. Strong variations of olivine content and upper mantle lithologies near the 410 km discontinuity are suggested by regional seismic models.
... The model based on equations (3) and (7) or (9) and (7) [18][19][20], the underestimation of CV resulting from neglecting the intrinsic anharmonicityexplicit dependence of frequencies of thermal vibrations on temperature. To express this dependence the following anharmonic parameter was suggested [18]: ...
New adiabatic calorimeter measurements of heat capacity of sphalerite from 13.23 to 312.85 K at ambient pressure agree well with previous adiabatic calorimeter results but overestimate PPMS-derived data (Cardona et al., 2010) at T > 150 K. The adiabatic calorimeter low-temperature heat capacity data and drop-calorimetric high-temperature data on H0(Т) − H0(298.15) are rationalized in the framework of the Kieffer-type vibrational spectrum model accounting for dependence of vibrational frequencies on volume and temperature. Thermodynamic properties of sphalerite are calculated by the elaborated model from 0 to 1300 K.
... Over the last decade, quasi-harmonic approximation (QHA) [22] has emerged as a powerful tool in extending the commonly obtained 0 K DFTderived properties to any set of operational temperatures and pressures. The QHA formalism [23,24] has been widely deployed to assess the thermal stability of naturally occurring materials at elevated temperatures and pressures encountered in geological reservoirs. Molybdenum nitrides adopt various phases among which are the cubic and hexagonal forms of fully saturated MoN; with no vacant nitrogen lattice. ...
This contribution aims to investigate volume-dependent thermal and mechanical properties of the two most studied phases of molybdenum nitride (c-MoN and h-MoN) by means of the quasi-harmonic approximation approach (QHA) via first-principles calculations up to their melting point and a pressure of 12 GPa. Lattice constants, band gaps, and bulk modulus at 0 K match corresponding experimental measurements well. Calculated Bader's charges indicate that Mo-N bonds exhibit a more ionic nature in the cubic MoN phase. Based on estimated Gibbs free energies, the cubic phase presents thermodynamic stability higher than that detected for hexagonl, with no phase transition observed in the selected T-P conditions as detected experimentally. The elastic stiffness coefficients of MoN in hexagonal structure revealed that it is stable elastically; in contrast to the cubic structure. The temperature dependence on the bulk modulus is more profound on the dense cubic phase than on the hexagonal phase. Overall, the two considered structures of molybdenum nitride display very minimal harmonic effects, evidenced by the slight variation of thermal and mechanical properties with the increase of pressure and temperature. The optical conductivity of both phases near a zero photon energy coincides well with their metallic character inferred by their corresponding DOS curves. It is expected that the thermo-elastic properties of saturated molybdenum nitrides reported in this study will aid in the continuous pursuit to enhance their catalytic and mechanical utilizations.
... Yusa and Inoue, 1997;Yusa et al., 2000;Ross and Crichton, 2001;Crichton and Ross, 2002;Kudoh et al., 2002;Kuribayashi et al., 2003Kuribayashi et al., , 2004Kuribayashi et al., , 2008Smyth et al., 2004;Manghnani et al., 2005Manghnani et al., , 2013Holl et al., 2006Holl et al., , 2008Ganskow et al., 2010; al., 2010, 2012). On the other hand, some studies have been conducted for the P-V-T EoS of anhydrous forsterite, wadsleyite, and ringwoodite (Meng et al., 1993(Meng et al., , 1994Guyot et al., 1996;Li et al., 2001;Katsura et al., 2004Katsura et al., , 2009aKatsura et al., , 2009bCouvy et al., 2010). Some high-temperature measurements (at ambient pressure) have also been carried out for these DHMS and hydrous NAMs phases (Pawley et al., 1995: Inoue et al., 2004Ye et al., 2009Ye et al., , 2011Ye et al., , 2012Ye et al., , 2013Ye et al., , 2015, which are useful for establishing simultaneous P-V-T EoSs of these hydrous silicate minerals in the future. ...
There are potentially huge amounts of water stored in Earth’s mantle, and the water solubilities in the silicate minerals range from tens to thousands of part per minion (ppm, part per million). Exploring water in the mantle has attracted much attention from the societies of mineralogy and geophysics in recent years. In the subducting slab, serpentine breaks down at high temperature, generating a series of dense hydrous magnesium silicate (DHMS) phases, such as phase A, chondrodite, clinohumite, etc. These phases may serve as carriers of water as hydroxyl into the upper mantle and the mantle transition zone (MTZ). On the other hand, wadsleyite and ringwoodite, polymorphs of olivine, are most the abundant minerals in the MTZ, and able to absorb significant amount of water (up to about 3 wt.% H2O). Hence, the MTZ becomes a very important layer for water storage in the mantle, and hydration plays important roles in physics and chemistry of the MTZ. In this paper, we will discuss two aspects of hydrous silicate minerals: (1) crystal structures and (2) equations of state (EoSs).
... Ces résultats sont en accord avec les résultats publiés par Chakraborty et al. (1994). Les énergies de formation des différents types de défauts présents dans la forstérite sont données en annexe 1. La dépendance en température de propriétés thermodynamiques de la forstérite a été étudiée par Gillet et al. (1991), Guyot et al. (1996), Cynn et al. (1996), Andersson et al. (1989) et Bouhfid et al. (1996. Toutes ces études ont montré une augmentation de l'anharmonicité de la forstérite à haute température. ...
With increasing temperature, some oxide compounds that are transparent in the near infrared range become progressively opaque when approaching the liquid phase. Such a behavior is unusual and deeply impacts their thermal radiative properties. To understand this phenomenon, infrared emittance spectra were acquired from room temperature up to the liquid state on several crystalline oxides (Mg2SiO4, LiAlO2, LiGaO2, ZnO, YAlO3, LaAlO3, LiNbO3, MgO). These data have been selectively completed by electrical conductivity measurements, NMR and X-ray diffraction experiments versus temperature. The analysis of the experimental emittance data with a semi-quantum dielectric function model including an extended Drude term, allowed to characterize finely the material responses and to suggest a physical origin for the opacification mechanism. The phenomenon is thermally activated and can be explained by the formation and the mobility of polarons. This work also showed the existence of a close link between the material microstructure and the characteristics of the opacification.
... Within the uncertainty of the fit (∂K T /∂T) V value is roughly zero, indicating that the thermal pressure is independent of volume, which is the same with the conclusions of Wang et al. (1998) who also found that the thermal pressure in garnet shows linear variation with T up to 1200 K independent of compression (V/ V 0 ), concluding that (∂K T /∂T) V is very close to zero. The same conclusions have been drawn for some other silicate minerals, such as olivine (Guyot et al. 1996), CaSiO 3 perovskite (Wang et al. 1996), and ringwoodite (Nishihara et al. 2004 ). By using the thermodynamic relation: (∂K T /∂T) P = (∂K T /∂T) V – aK T K∂ T , we can obtain (∂K T /∂T) P = –0.020(3) ...
The pressure-volume-temperature (P-V-T) equation of state (EoS) of synthetic uvarovite has been measured at high temperatures up to 900 K and high pressures up
to 16.20 GPa, by using in situ angle-dispersive X-ray diffraction and diamond-anvil cell. Analysis of room-temperature P-V data to a third-order Birch-Murnaghan EoS yielded: V0 = 1736.9 ± 0.5 Å3, K0 = 162 ± 2 GPa, and K′0 = 4.5 ± 0.3. With K′0 fixed to 4.0, we obtained: V0 = 1736.5 ± 0.3 Å3 and K0 = 164 ± 1 GPa. Fitting of our P-V-T data by means of the high-temperature third-order Birch-Murnaghan equations of state, given the thermoelastic parameters:
V0 = 1736.8 ± 0.8 Å3, K0 = 162 ± 3 GPa, K′0 = 4.3 ± 0.4, (∂K/∂T)P = −0.021 ± 0.004 GPa/K, and α0 = (2.72 ± 0.14)×10−5 K−1. We compared our elastic parameters to the results from the previous studies for uvarovite. From the comparison of these
fittings, we propose to constrain the bulk modulus and its pressure derivative to K0 = 162 GPa and K′0 = 4.0–4.5 for uvarovite. Present results were also compared with previous studies for other ugrandite garnets, grossular
and andradite, which indicated that the compression mechanism of uvarovite might be similar with grossular and andradite.
Furthermore, a systematic relationship, K0 (GPa) = 398.1(7)−0.136(8) V0 (Å3) with a correlation coefficient R2 of 0.9999, has been established based on these isostructural analogs. Combining these results with previous studies for pyralspite
garnets—pyrope, almandine, and spessartine—the compositional dependence of the thermoelastic parameters (bulk modulus, thermal
expansion, and the temperature derivative of the bulk modulus) were discussed.
... Considering the upper basalt layer, however, the overall subducted oceanic lithosphere is denser than the surrounding mantle ( Figure 8) and slabs will be driven to the bottom of the MTZ. Guyot et al., 1996;g , Li et al., 1998;h , Li et al., 1996;i , Li et al., 2001;j , Li, 2003;k , Sinogeikin et al., 2003;l , Meng et al., 1994;m , Kung et al., 2005;n , Wang et al., 1998;o , Sinogeikin and Bass, 2002;p , Gwanmesia et al., 2006;q , Liu et al., 2000;r , Wang et al., 2004;s , Weidner and Ito, 1985;t , Nishihara et al., 2005;u , Li and Liebermann, 2007;v , Zhou et al., 2011;w , Kung et al., 2002;x , Fei et al., 1992;y , Li and Zhang, 2005;z , Wang et al., 1994;aa , Woodland and Angel, 1997;ab , Xu et al., 2008. [19] Mantle dynamics is a complex process and cannot be determined by simple gravitation buoyancy alone. ...
relations in harzburgite have been determined between 14 and 24 GPa and
1473 and 1673 K. At 1673 K, harzburgite transformed to wadsleyite +
garnet + clinopyroxene below 19 GPa and decomposed into an assemblage of
ringwoodite + garnet + stishovite above 20 GPa. Certain amounts of
akimotoite were produced at still higher pressures (22-23 GPa). Finally,
perovskite and magnesiowüstite were found to coexist with garnet at
24.2 GPa. Compositions of all the phases were analyzed and elemental
partitioning coefficients were determined among coexisting phases.
Combining our experimental data with available thermoelastic properties
of major minerals in the earth's mantle, we modeled the velocity and
density signatures of the stagnated oceanic slab in the mantle
transition zone (MTZ) under eastern China, based on kinematic slab
thermal structure analysis. We examined two end-member slab models: a
conventional straight slab with deformation thickening and an undulate
slab with an oscillating wavelength of 200 km. We found that an
undulated (buckled) slab model yields velocity anomalies (about 1-2% for
Vp) that are consistent with seismic tomography models, taking into
account low-pass filtering effects in seismic tomography studies. On the
other hand, straight slab models yield velocity anomalies that are too
high compared with seismic tomography models. Our models provide
important constraints on the thermal structure, mineralogy, composition,
density, and velocities of slab materials in the MTZ.
... A lot of papers deal with high pressure conditions (Presnall and Walter 1993; Wang et al. 1993; Durben et al. 1993; Li et al. 2006; Ottonello et al. 2007; Koker et al. 2008; Liu et al. 2008; Yu et al. 2008), and some experimental studies were dedicated to high temperature. The temperature dependences of thermodynamic properties of forsterite were investigated in the following studies (Hofmeister 1987; Chopelas 1990; Gillet et al. 1991; Guyot et al. 1996; Cynn et al. 1996; Anderson 1996; Bouhifd et al. 1996). Hofmeister and Chopelas were able to fit thermodynamic properties with vibrational models but their data are limited to 1,000 K. Higher temperature measurements performed by the other authors indicated the existence of a sudden enhancement of the anharmonic behavior of forsterite above 1,200 K. Synthetic forsterite and 26 Mg-enriched forsterite were also investigated by using Raman spectroscopy from 293 to 1,473 K (Kolesov and Geiger 2004). ...
Polarized emittance measurements were acquired for synthetic forsterite, the pure magnesium end member of the olivines group, on the whole infrared spectral range and up to the melting point by using CO2 laser heating. The experimental data, fitted with a semi-quantum dielectric function model, allowed the retrieval of the temperature dependence of the absorption coefficient of forsterite both in the opaque and semi-transparent regions. The analysis of the phonon parameters indicates that the lattice dynamics evolve drastically with increasing temperature. The normal modes involving motions of the magnesium cations located in site 1 are the more impacted, and some of them vanish around 1,200 K. The results confirm that the enhancement of the lattice anharmonicity and the increasing mobility of the magnesium cations are closely linked and are at the origin of the anomalies observed in the evolution of the thermophysical properties. This complete set of spectroscopic data may be a step toward a more precise evaluation of the impact of thermal radiation heat transfer inside systems involving forsterite and quantification of their heat budget.
... Olivine is a near Debye-like solid. Guyot et al. (1996) showed that although Debye theory gives values close to experiment for the properties of C V , S, and ⌬P Th in olivine, the Kieffer approximation to the density of states gives values even closer to experiment. ...
MgSiO 3 perovskite is shown to be a Debye-like mineral by the determination of specific heat, C v , entropy, S, and thermal pressure, Delta P Th , using the Debye theory up to 1800 K. Sound velocities and bulk moduli at ambient conditions published by Yeganeh-Haeri
were used to find the ambient acoustic Debye temperature, Theta ac D . The variation of Theta ac D with T was assumed to be a curve parallel to the Theta ac D vs. T curves previously found for Al 2 O 3 , MgO, and MgSiO 3 , enabling Theta ac D (T) to be given up to 1800 K. To determine C p , the thermal expansivity, alpha , and the isothermal bulk modulus, K T , are needed. After considering several sets of alpha (T), the alpha (T) data of Funamori and his colleagues were chosen.
Using the ambient K T and the values of (theta K T /theta T) P vs. T reported by Jackson and Rigden, K T (T) up to 1800 K was found. Then C P (T) up to 1800 K was found assuming quasiharmonicity in C v . The data behind the C P (T) calculation are also sufficient to find the Gruneisen parameter, gamma (T), and the Anderson-Gruneisen parameters, delta
T and delta S , up to 1800 K. The value of q = (theta ln gamma /theta ln V) T was found, and with gamma and rho , Delta P Th vs. V and T was determined. The three sound velocities, v s , v p , and v b = K (sub s/rho ) , were then determined to 1800 K. From v s and v p , Poisson's ratio and the isotropic shear modulus, G, were found to 1800 K. MgSiO 3 perovskite is one of a small, select group of Debye-like minerals for which thermoelastic properties and the equation of
state (EOS) are calculable from acoustic data.
... Olivine is a near Debye-like solid. Guyot et al. (1996) showed that although Debye theory gives values close to experiment for the properties of Cu, S, and APrn in olivine, the Kieffer approximation to the density of states gives values even closer to experiment. ...
MgSiO. perovskite is shown to be a Debye-like mineral by the determination of specific heat, Cu, entropy, S, and thermal pressure, APrn, using the Debye theory up to 1800 K. Sound velocities and bulk moduli at ambient conditions published by Yeganeh-Haeri were used to find the ambient acoustic Debye temperature, Oij. The variation of @5 with Z was assumed to be a curve parallel to the @3.^ vs. Z curves previously found for AlrO., MgO, and MgSiO., enabling OS(7") to be given up to 1800 K. To determine C", the thermal expansivity, a, and the isothermal bulk modulus , K, are needed. After considering several sets of a(Z), the a(Q data of Funamori and his colleagues were chosen. Using the ambient K,and the values of (aK,lAT), vs. Zreported by Jackson and Rigden, K,(Dup to 1800 K was found. Then C"(7-) up to 1800 K was found assuming quasiharmonicity in C". The data behind the Cr(T) calculation are also sufficient to find the Grtineisen parameter, 1(7-), and the Anderson-Griineisen parameters, D, and 6r, up to 1800 K. The value of q : @ ln 1i/0ln VS, was found, and with ^y and p, APrn vs. V and Z was determined. The three sound velocities, v", vo, and ,o: t/KJp, were then determined to 1800 K. From v" and vo, Poisson's ratio and the isotropic shear modulus, G were found to 1800 K. MgSiO. perov- skite is one of a small, select group of Debye-like minerals for which thermoelastic prop- erties and the equation of state (EOS) are calculable from acoustic data.
... Owing to its abundance, Mg 2 SiO 4 forsterite has been thoroughly studied experimentally, e.g., its vibrational spectroscopy and thermal elasticity have been measured (Chopelas 1990), and its equation of state parameters determined from experiments (e.g. Guyot et al. 1996, Bouhifd et al. 1996 and calculated by various methods , Li et al. 2006. The thermodynamic properties of forsterite calculated by LDA and GGA are compared with experiments and shown in Table 1. ...
Quasiharmonic theory combined with first-principles phonon density of states gives accurate thermodynamics properties of minerals at the high pressures and temperatures of the Earth interior. Care must be exercised in using this method within its regime of validity. A simple criterion to establish the approximate upper temperature limit of validity of the QHA versus pressure was introduced based on a posteriori inspection of the thermal expansivity. This criterion shows that the QHA is a good approximation for minerals at mantle conditions, except for truly anharmonic crystals like CaSiO3- perovskite, and perhaps for minerals at conditions of the core-mantle boundary. In general, the temperature range of applicability of the QHA increases with pressure. The systems analyzed here, were all investigated consistently and systematically, using the same pseudopotentials, convergence criteria, plane-wave cut-offs, and k- and q-point samplings in LDA and PBE-GGA calculations. The trends extracted from these calculations are therefore reliable. The importance of zero-point-motion effects on structural properties cannot be overemphasized. The LDA exchange-correlation functional gives considerably superior results for 0 K properties after the zero-point-motion energy is included in the calculation. GGA results consistently overestimate volume and underestimate the bulk modulus by several percent. LDA thermodynamic properties at ambient conditions are in excellent agreement with experimental values. The performance of these exchange-correlation functionals was also investigated for predictions of thermodynamic phase boundaries. In general, the LDA phase boundaries are "shifted" by 5-10 GPa to lower pressures compared to the experimental ones, while GGA boundary are closer to experimental ones but still "shifted" by 2-5 GPa to higher pressures. LDA and GGA Clapeyron slopes are very similar and in fairly good agreement with experimental slopes. Phase boundaries, however, may be more affected by anharmonicity since bond-breaking and bond-reconstruction, phonon softening, diffusion, etc may take place at these pressures and temperatures. Discontinuities in density and bulk sound velocity for important phase transformations in the mantle transition zone were systematically investigated. Only the magnesium endmembers, Mg 2SiO4, were studied. Predicted discontinuities in density, bulk modulus, and bulk sound velocity are sharp and have useful accuracy for analysis of mantle discontinuities, despite uncertainties in the predicted phase boundary. With the rough premise that the olivine system does not interact with other minerals such as pyroxene/garnet/Ca-perovskite, we were able estimate density discontinuities at 410-km, 520-km, and 660-km depth and compare them with those inferred from seismic data. We conclude that seismic density discontinuities observed at 410-km (0.2-4%) and 660-km (∼5.2%) depth can be produced by phase transition in the olivine system alone in a mantle with pyrolite composition (~60 vol% olivine). However, the 520-km discontinuity (2.1±0.8%) cannot. It requires contributions from other mantle transformations, e.g., the Ca-perovskite ex-solution from majorite garnet. We also discuss the post-perovskite phase boundary. Our predicted Clapeyron slope, ∼7.5±0.3 MPa/K, differs somewhat from the preferred experimental slopes, > 9.7 MPa/K. This suggests that this phase boundary might be sensitive to anharmonic effects. We have introduced a semi-empirical ansatz to compute anharmonic contributions to the free energy. This method utilizes experimental data at low pressures and high temperatures on one property, preferably thermal expansivity, to compute all thermodynamic properties at high pressures and high temperatures. It offered excellent results for MgO, and the P-V-T relation in this mineral was offered for pressure calibration in diamond-anvil-cells experiments. It was also applied to forsterite (α), wadsleyite (β), and to the α-to-β transformation. The thermodynamic properties of the α- and β-phases improve and are further reconciled with experimental measurements beyond the QHA validity regime after correction for anharmonic effects. This study indicated that anharmonicity manifests differently in different systems, depending whether the "average" phonon frequency increases (β and MgO) or decreases (α) with temperature at constant volume. This difference in behavior affects the Clapeyron slope of the α-to-β transformation, raising it from 2.5-2.6 to 3.6 MPa/K and reconciling it with the latest experimental determinations. Anharmonic effects are most evident in the thermal expansivity, followed by the thermal Grüneisen parameter, constant pressure specific heat, and least evident in the bulk modulus.
... With this data set, the calculated phase diagram is in good agreement with the experimental phase diagram determined by Katsura and Ito [1989]. For carbonates, all the thermodynamic data (except for standard enthalpies) were calculated using spectroscopic data (for more details about the technique, see Guyot et al., [1996]) and are given in Table 5. ...
Melting and subsolidus relations in the (Mg, Fe)SiO3-(Mg, Fe)CO3, (Mg, Fe)2SiO4-(Mg, Fe)CO3, and (Mg, Fe)O-(Mg, Fe)CO3 systems have been investigated at 14, 15, 16 and 25 GPa, 1973 K and 2173 K, using a 1000 t uniaxial multi anvil split sphere apparatus. The iron-magnesium partition coefficients between magnesite and silicates or oxides have been measured in subsolidus assemblages. Iron is always partitioned preferentially in the silicate and oxide phases, the order of increasing partitioning being pyroxene, olivine, silicate perovskite, wadsleyite and magnesiowüstite. A thermodynamic model of iron-magnesium distribution between magnesite and all these phases, based on Gibbs free energy minimization, is established. Melting of pyroxene-magnesite and olivine-magnesite pseudo binary systems is eutectic, with eutectic points close to 1973 K and 60 mol% carbonate at 15 GPa in both systems. In the more complex mantle system, it is likely that such melts would form in the transition zone by heating and homogenization of deep subducted carbonates. The melts formed in the olivine-carbonate system are characterized by high Mg+Fe/Si ratios and thus unlikely to be primary kimberlitic magmas, a conclusion in agreement with previous studies in the peridotite-CO2 system. On the other hand, the observed pyroxene-magnesite melts formed at transition zone conditions have Mg+Fe/Si ratios that are comparable to those of natural kimberlites, suggesting that melting of carbonated pyroxenites at high pressures could be a source of kimberlitic magmas.
Materials that exhibit zero thermal expansion have numerous applications, ranging from everyday ceramic hobs to telescope mirrors to devices in optics and micromechanics. These materials include glass ceramics containing crystal phases with negative thermal expansion in at least one crystallographic direction, such as Ba1-xSrxZn2-2yMg2ySi2O7 solid solutions. However, the volume increase associated with the martensitic phase transformation in these crystals often hinders their use as zero thermal expansion materials at operating temperatures near the transition temperature Tt. Here, an approach to rapidly predict Tt of such materials as a function of chemical composition based on a combination of density functional theory simulations and experiments has been developed and applied to Ba1-xSrxZn2-2yMg2ySi2O7. Its central element is the modeling of free energy as a function of temperature and chemical composition using a composition-dependent Debye model augmented by an empirical correction, which incorporates the effects of anharmonic lattice vibrations. This approach provides Tt predictions with an estimated uncertainty of about 100 K, which is similar to the accuracy of computationally much more demanding simulations of polymorphous phase transitions. In addition, this approach allows computationally efficient determination of the chemical compositions at which the Ba1-xSrxZn2-2yMg2ySi2O7 phase with the desired thermal properties will be formed during synthesis, facilitating the targeted design of zero thermal expansion materials.
Heat capacity of PbS (galena) has been measured in the temperature range from 12 to 338 K using adiabatic calorimetry. Results of the measurements are consistent with previous adiabatic calorimetric data measurements but differ notably from those obtained by relaxation calorimetry.
Using experimental data on adiabatic heat capacity measurements in the range of 12-338 K (including our data), results of EMF measurements, and literature data on H⁰(T) – H⁰(298.15 K) as inputs, we have optimized the thermodynamic properties of lead sulfide (PbS, galena) based on the quasi-harmonic model vibrational spectrum proposed by Kieffer and advanced to consider intrinsic anharmonicity. In this approach, the volume and temperature dependence of vibrational frequencies are considered, and the enthalpy of formation of PbS, ΔfH⁰(298.15 K), is treated as an optimized model parameter. The thermodynamic properties of PbS are described with a single set of model parameters over the entire range of its stability (0-1374 K) at a pressure of 1 bar. The model can be optimized based on either the reduced (Cp⁰ and ΔfG⁰(T)) or full (Cp⁰, H⁰(T) – H⁰(298.15 K), ΔfG⁰(T)) datasets. The both variants of the optimization predict the similar thermodynamic properties of PbS, in particular, values of the enthalpies of formation differ less than 0.1 kJ mol⁻¹ and are consistent with ΔfH⁰(298.15 K) obtained experimentally by oxide melt solution calorimetry. The low-temperature adiabatic calorimetry data and high-temperature results of EMF experiments (or other reliable experimental data on ΔfG⁰(T)) represent a minimal sufficient set of input data providing optimization of thermodynamic properties at ambient pressure by the technique established in present study.
Materials that exhibit zero thermal expansion have numerous applications, ranging from everyday ceramic hobs to telescope mirrors to devices in optics and micromechanics. These materials include glass ceramics containing crystal...
Polycrystalline samples of Sm2Bi2Fe4O12 were produced by the standard solid state reaction method. The structural characterization revealed that the material crystallizes in an orthorhombic structure (Pnma # 62 space group). Scanning electron microscopic images showed granular and densified characteristics on the surface and inside the samples, with two well-differentiated grain sizes: micrometric and submicrometric. From energy-dispersive X-ray spectra, it was established that the samples contained the expected elements in stoichiometric proportions suggested by the chemical formula of the compound. I–V curves and measurements of electrical permittivity as a function of temperature showed semiconductor-type behaviors, strongly dependent on the polycrystalline character of the material and a Maxwell–Wagner-type dielectric tendency. The optical spectral analysis corroborated the semiconductor behavior with the optical bandgap Eg = 2.62 eV. Temperature-dependent magnetization results have the expected form for ferromagnetic-type materials, with evidence of disorder that gives rise to magnetic irreversibility between the zero-field cooling and field cooling measurement procedures. Magnetization curves as a function of the applied field showed a hysterical response, even at room temperature, revealing the occurrence of ferromagnetic ordering in the material. The ab initio calculations of the electron density of states reveal the appearance of a mean semiconductor band gap whose value is in agreement with that measured experimentally. The effective magnetic moment is attributed to the majority contributions of the 3d-Fe orbitals. At low temperatures, the behaviors of the specific heat at constant volume and pressure are similar, with a tendency of the former towards the Dulong–Petit limit. Strong changes in all the thermophysical properties studied are evident due to the effects of pressure and temperature on the wave behavior of the crystal lattice. At higher temperatures, a strong variation in the Debye temperature induces divergence in thermophysical parameters for different applied pressures. The physical parameters related to the coexistence of the semiconductor and ferromagnetic properties in the material suggest possible technological implications in the spintronics industry.
Experiments of dynamical high pressure using shock waves are very effective to study physical properties of material under super-pressure. A shock wave experimental study on Damaping olivine with pressure from 10 to 45 GPa is presented in this paper. Combining previous work about isothermal equation of state for olivine, the temperature in experimental process is determined. The temperatures range from dozens of degree to 800℃ when the pressures of our experiments are between 10 and 30 GPa. The density variation with pressure is obtained from our experiments which ranges from 3.627 to 4.009 g·cm-3. According to the recover experiment of the sample and estimated temperatures under experimental pressures, it is derived that the phase transition during our experimental process under pressures of 30 GPa did not occur. Meanwhile, the parameters for equation of state are determined. Finally, the geodynamical implication of the experimental results to interior material movement in mantle is discussed, i.e. cold slab with metastable olivine is easily to sink into the mantle's transitional zone.
Large olivine samples were hot-pressed synthesized for shock wave experiments. The shock wave experiments were carried out at pressure range between 11 and 42 GPa. Shock data on olivine sample yielded a linear relationship between shock wave velocity D and particle velocity u described by D=3.56(±0.13)+2.57(±0.12)u. The shock temperature is determined by an energy relationship which is approximately 790°C at pressure 28 GPa. Due to low temperature and short experimental duration, we suggest that no phase change occurred in our sample below 30 GPa and olivine persisted well beyond its equilibrium boundary in metastable phase. The densities of metastable olivine are in agreement with the results of static compression. At the depth shallower than 410 km, the densities of metastable olivine are higher than those of the PREM model, facilitating cold slab to sink into the mantle transition zone. However, in entire mantle transition zone, the shock densities are lower than those of the PREM model, hampering cold slab to flow across the “660 km” phase boundary. © 2015 Science China Press and Springer-Verlag Berlin Heidelberg
This paper presents the characteristics of the properties of metallurgical molten slags based on the bond structure or the existence of certain ion clusters. Some experimental and theoretical approaches were adopted in the investigation. Two sets of high temperature Raman spectroscopy (HTRS) were established at Shanghai University; both can be successfully operated at 2000K or higher. One of them combines the accumulated time resolution together with the spatial resolution, and it is designed for both ultraviolet (UV) and visible (VIS) light. The SiOT model uses 5 kinds of Si-O tetrahedra (Qn) as microsturctural units. In this model, each sample contains 250000 or more tetrahedra. The calculation using a GF matrix and electro-optical parameter (EOP) methods were performed for every tetrahedron one by one to determine the partial Raman spectra of Qn. Then the spectra were integrated to obtain an envelope of the sample. In addition to the SiOT model, a CEMS model was also developed to link the ion cluster structure and the thermodynamic properties at equilibrium (as mixing free energy).
Simplified vibrational densities of states for five different carbonates are constructed using measured IR and Raman spectra. From the spectroscopic models we calculate thermodynamic and thermoelastic properties of magnesite, calcite, aragonite, dolomite, and siderite. The effects of temperature and pressure on the vibrational frequencies are explicitly introduced into the computations. These spectroscopic models provide high level agreement with the measured values of entropy and heat capacity (within +/- 2%), with the exception of aragonite (within +/- 5% above 600 K) due to its breakdown to calcite: For the molar volumes the agreement is within +/- 0.5 %. The Gibbs free energies of each mineral are then computed in order to obtain pressure and temperature equilibrium conditions for different chemical reactions involving carbonates. Comparing the predicted phase diagrams with those experimentally determined provides an additional constraint on the validity of spectroscopic models and in the values of formation enthalpies.
A simple numerical model for simultaneous optimization of heat capacity at constant pressure, C(P), heat capacity at constant volume, C(V), volume, V, thermal expansion coefficient, α, isothermal, K(T), and adiabatic, K(S), bulk moduli at zero pressure, and PVT data for minerals has been developed. The basic function is the Debye energy, expressed through the Nernst-Lindemann energy function for T > 0.2Θ. Three additional empirical parameters are included in the expression for energy, which take into account anharmonicity, premelting, and other effects for real minerals. The volume vs. energy dependence is calculated on the basis of either the Wachtman et al. (1962) or the Suzuki (1975) model or their linear combination. Volume, V298, Nernst-Lindemann characteristic temperature, Θ, isothermal bulk modulus, K(T)298, its pressure derivative, K', Gruneisen parameter, γ, isothermal Anderson-Gruneisen parameter, δ(T), and three empirical parameters, a, b, c, which can be equal to zero for Debye-like solids, are fitting parameters of the model. The proposed model enables one to calculate thermodynamic functions of simple substances, oxides, and minerals over a temperature range from 0.2Θ up to the melting temperature with a deviation within the scatter of experimental data. Correlation of the proposed model with PVT data is considered. It is shown that the isothermal equation of state results in an unsatisfactory extrapolation of volume in extreme regions. The Wachtman et al. (1962) and the Suzuki (1975) models of the volume vs. energy are extended to high pressure. The high-pressure Wachtman et al. (1962) and Suzuki (1975) models are versions of the Mie-Gruneisen equation of state and allow temperature dependencies of thermodynamic functions for any isobar to be easily calculated. The model described here and the classical Mie-Gruneisen model are found to be equivalent at q ~ 1. The model is tested using rock salt, corundum, and lawsonite.
Olivine is not only the dominant mineral in the upper mantle of the Earth, but also the major mineral in the subducted slab. It is generally accepted that the atom groups in olivine vibrate in an anharmonic manner at normal mantle temperatures. How they behave at the low temperatures typical in a subduction zone, however, is unclear. In this study, we systematically investigated the IR features of some doubly polished olivine thin sections with different thicknesses using infrared transmission spectroscopy, especially focusing on the overtone/combination bands in the 2100 ∼ 1500cm-1 region. With the IR data collected from room temperature to about 450°C, we obtained the relationships between the anharmonicity coefficient (χ) and temperature for two characteristic IR bands of the SiO4 tetrahedra: the equations are χ838 = 6.37(2) × 10-7 × T + 0.0014(1) and χ993 = 7.86(3) × 10-7 × T + 0.0015(1), with T in °C. When extrapolated to 600°C, these equations produce χ838 = 0.0018(1) and χ993 = 0.0020(2), which presumably indicate a negligible role of the anharmonic behavior for the SiO4 tetrahedra in olivine at the low temperatures in the subduction zone.
The successful parameterization of the volume thermal expansion in the Mg, Fe olivine solid-solution series stimulated us to carry the analysis further in view of the orthorhombic symmetry of the olivines and thus augment the available thermophysical data. To this purpose we studied the three axial expansivities, α a , b , c, on the basis of the same body of data and the same model as used for our parameterization of the volume expansion.
The results are applied to various thermophysical parameters that possess a significant dependence on α a , b , c. In particular, the temperature variations of the axial adiabatic and isothermal moduli (a), the axial thermodynamic Grüneisen parameters (b), and the axial isothermal Anderson-Grüneisen parameters (c) are obtained for both forsterite (Fo) and fayalite (Fa). This allowed us to also study the response of α a , b , c axial expansivities to pressure.
Since Fo and Fa are isomorphous, it is striking that the axial thermophysical properties of Fo are distinctly less anisotropic than those of Fa. Whereas in Fo, both the axial compressibilities and the axial expansivities follow the same sequence, i.e. β T, b > β T, c > β T, a and αb > αc > αa , respectively, this expectation is violated in Fa where αc > αa > αb above 400 K. It is shown that the different anisotropies of the expansivities and hence of the α-dependent thermophysical properties can be related to differing structural responses to heating.
In situ synchrotron X-ray diffraction (XRD) experiments were conducted
using the SPEED-1500 multi-anvil press of SPring-8 on (Mg
0.91Fe 0.09) 2SiO 4
ringwoodite (Rw), whose composition is similar to that expected in the
Earth's mantle transition zone. Pressure-volume-temperature data were
collected using a NaCl or MgO capsule up to 21 GPa and 1273 K. A fit to
high-temperature Birch-Murnaghan (HTBM) equation of state (EOS) with
fixed values of ambient cell volume V0=527.83(7) Å
3 and isothermal bulk modulus KT0 =187 GPa yielded
a pressure derivative of isothermal bulk modulus KT'=4.41(1),
a temperature derivative of bulk modulus (∂ K T/∂
T) P = -0.028(5) GPa K -1, and a volumetric
thermal expansivity α= a+ bT with values of a=1.9(2)×10
-5 K -1 and b=1.2(4)×10 -8 K
-2. These properties are consistent with the analysis using
the thermal pressure (Th-P) EOS. The derived KT' and (∂
K T/∂ T) P are consistent with previous
studies on Mg 2SiO 4 ringwoodite. However,
consistent with measurements at 0 GPa, the present equation of state
yields significantly higher thermal expansivity than derived by diamond
anvil experiments on Mg 2SiO 4 ringwoodite to 30
GPa and 700 K. On the basis of the equation of state, the density jump
around 660 km depth expected for pyrolitic homogeneous mantle (caused by
ringwoodite → Mg-perovskite (MgPv) + magnesiowüstite (Mw) and
garnet (GRT)→MgPv transitions) was estimated. The density jump
calculated for pyrolite (9.2%) is significantly larger than that across
the 660 km discontinuity derived by recent seismological studies (4-6%).
One possible explanation for this is that the recent seismic data
reflect only the sharp transition concerned with the Rw→MgPv+Mw
transition.
We examine the possibility of computing the high pressure P elastic
properties of forsterite using acoustic resonant frequencies measured at
elevated temperature T (but at P = 0). The transformation into P space
from T space requires three imposed conditions: (1) that the property
αKT be independent of volume V (α is the volume
coefficient of thermal expansivity, and KT is the isothermal
bulk modulus); (2) that CV, the heat capacity at constant
volume, be quasiharmonic at high T; and (3) that the acoustic resonant
mode frequencies be linear in P. We also compute the high-pressure
elastic properties of NaCl and MgO to provide comparisons with the
results for forsterite. We show that NaCl meets all three conditions,
but that for MgO, the first condition is not met. For forsterite the
second condition is not met.
In situ synchrotron X-ray diffraction measurements have been carried out
on San Carlos olivine (Mg 0.9Fe 0.1)
2SiO 4 up to 8 GPa and 1073 K. Data analysis using
the high-temperature Birch-Murnaghan (HTBM) equation of state (EoS)
yields the temperature derivative of the bulk modulus (∂
KT/∂ T) P = -0.019 ± 0.002 GPa K
-1. The thermal pressure (TH) approach gives
αKT = 4.08 ± 0.10 × 10 -3 GPa K
-1, from which (∂ KT/∂ T) P =
-0.019 ± 0.001 GPa K -1 is derived. Fitting the
present data to the Mie-Grüneisen-Debye (MGD) formalism, the
Grüneisen parameter at ambient conditions γ0 is
constrained to be 1.14 ± 0.02 with fixed volume dependence q = 1.
Combining the present data with previous results on iron-bearing olivine
and fitting to MGD EoS, we obtain γ0 = 1.11 ±
0.01 and q = 0.54 ± 0.36. In this study the thermoelastic
parameters obtained from various approaches are in good agreement with
one another and previous results.
As determined from powder X-ray diffraction experiments with synchrotron radiation, the thermal expansion coefficient of forsterite increases smoothly from 2.8 to 4.5 K-1 from 400 K to 2160 K. No anomalous increases of the cell parameters are observed near the melting point. The consistency between the observed and calculated value of the initial slope of the melting curve of forsterite suggests that defects do not make a large contribution to thermal expansion near the melting point. Along with previous results, the new data confirm the influence of anharmonicity on the high-temperature heat capacity of forsterite and indicate that both the Grüneisen parameter and αKT (α = thermal expansion coefficient, KT = bulk modulus) have nearly constant values at high temperatures.
Various models for determination of the temperature dependence of thermal pressure are critically examined in the light of experimental data. We have considered Mg2SiO4, Olivine, MgAl2O4, Pyrope rich garnet, Fe2SiO4, Grossular garnet, MnO, NaCl and KCl. The superiority of one model over others is discussed. The model is extended to study the compression behaviour at different temperatures, and the combined effect of pressure and temperature on thermal pressure of San Carlo Olivine. A good agreement obtained between theory and experiment demonstrates the validity of the present approach.
Phase equilibria of α, β, and γ (Mg,Fe)2SiO4 are important to understanding the mineralogy of the Earth’s upper mantle. Using the first principles approach, we studied thermodynamic properties and phase stability fields of Fe2SiO4. We show that the correct phase transition sequence in Fe2SiO4 (α → γ) can be obtained with the DFT + self-consistent Hubbard U method, while standard DFT methods (LSDA and σ-GGA) as well as the DFT + constant U method fail the task. The vibrational virtual crystal approximation was used to derive the phonon density of state of the Fe2SiO4 polymorphs. High-pressure thermodynamic properties of Fe2SiO4 are then derived with the aid of the quasi-harmonic approximation. They are in very good agreement with experiments. The phase diagram of the (Mg,Fe)2SiO4 system is calculated under the assumption of ideal mixing within α, β, and γ solid solutions. The model permits the investigation of the temperature and pressure effects on the phase boundaries. The widths of the divariant α–β and β-γ loops are barely sensitive to temperature between 1473 and 1873 K. This study shows the promise of applying the DFT + self-consistent Hubbard U method to study phase equilibria of iron-bearing Earth minerals.
Mineral physicists often refer to the quasi-harmonic approximation. This approximation accepts the intrinsically anharmonic effects of positive thermal expansion coefficient, alpha, and pressure-dependent bulk modulus, KT. However, for insulators at high temperatures, the classical specific heat, CV, and the product alphaKT are assumed to be independent of temperature at constant volume. Departures from this approximation are termed anharmonicity. We interpret this distinction using derivatives of the atomic potential function, phi(r), with atomic spacing, r. CV and KT are reasonably explained by harmonic bonds, that is, phi ~ (r - a)2, where a is the equilibrium value of r, so that phi'' = d2phi/dr2 is the only derivative considered. Thermal pressure or expansion and pressure dependence of KT depend on phi''' (referred to as first-order anharmonic effects). Temperature dependence of CV requires phiiv and is a second-order anharmonic effect. The temperature variation of alphaKT arises from phiv, a third-order effect. By calculating the Grüneisen parameter, gamma, as the ratio of thermal pressure to thermal energy, we relate its anharmonicity to a ratio of derivatives of phi. For all commonly used finite strain theories, this gives a temperature variation of gamma at high temperature and constant volume systematically less than that of CV. Anharmonicity of CV, which has been more comprehensively studied, may be a few percent at 2000 K, but decreases strongly with compression, so that anharmonicity of both gamma and CV is negligible (less than 1% in the case of gamma under deep-Earth conditions).
A recent paper [Bouhifd et al., 1996] has given the measured thermal expansivity, alpha, of forsterite up to 1900 K, which is 2.5 times the Debye temperature, Theta. Using the measured alpha, Bouhifd et al. [1996] found that the alphaKT curve (where KT is the isothermal bulk modulus) is parallel to the T axis, showing that (∂(alphaKT)/∂T)p=0. Using these results, I show that (∂CV/∂T)p>0 at high T, requiring CV to be anharmonic, is in agreement with other recent work. Further, I show that Bouhifd et al.'s [1996] work requires that all vestiges of anharmonicity be absent in the thermal pressure and the thermal expansivity of forsterite. Thus it is proven that forsterite is a solid that is both anharmonic in CV and quasiharmonic in the thermal pressure in the high-temperature region.
Chemical differentiation from pyrolite to harzburgite due to partial melting and melt extraction process causes the chemical heterogeneity in Earth's upper mantle that can be detected by seismological observations. The variation in major element chemistry in natural samples reflects complicated processes that include not only partial melting but also other various magmatic processes. On the basis of a comparison of chemical and mineralogical compositions of natural peridotites with those from melting experiment, density and seismic velocities of various peridotites are calculated for the range of pressure and temperature in the upper mantle using the latest data on mineral thermoelasticity. We conclude that the seismic velocities of shallow oceanic peridotites is characterized by a single parameter such as Mg # (molar ratio Mg/(Mg + Fe)), whereas the characterization of the deep continental peridotites requires two parameters, Mg # and Opx # (volume fraction of orthopyroxene). In agreement with previous studies, we find that in spinel stability field, the seismic velocities have positive correlation with Mg # from pyrolite to residual harzburgite, while in garnet stability field, seismic velocities of residual harzburgite are indistinguishable from those of pyrolite. The seismic velocities of the deep continental peridotites are lower than those of pyrolite and residual harzburgite because of the high concentration of orthopyroxene with low seismic velocities and have large pressure dependence. A jump of seismic velocity will occur at 300 km in orthopyroxene-rich continental harzburgite due to the orthorhombic to high-pressure monoclinic phase transition in (Mg, Fe)SiO3 pyroxene. This phase transition may correspond to the X discontinuity.
We have carried out an in situ synchrotron X-ray diffraction study on iron and an iron-silicon alloy Fe0.91Si0.09 at simultaneously high pressure and temperature. Unit-cell volumes, measured up to 8.9 GPa and 773 K on the bcc phases of
iron and Fe0.91Si0.09, are analyzed using the Birch-Murnaghan equation of state and thermal pressure approach of Anderson. Equation of state parameters
on iron are found to be in agreement with results of previous studies. For both iron and Fe0.91Si0.09, thermal pressures show strong dependence on volume; the (∂KT/∂T)V values are considerably larger than those previously reported for other solids. The present results, in combination with
our previous results on ɛ-FeSi, suggest a small dependency of the room-temperature bulk modulus upon the silicon content,
less than 0.3 GPa for 1 wt.% silicon. We also find that substitution of silicon in iron would not appreciably change the thermoelastic
properties of iron-rich Fe−Si alloys. If this behavior persists over large pressure and temperature ranges, the relative density
contrast between iron and iron-rich Fe−Si alloys at conditions of the outer core of the Earth could be close to that measured
at ambient conditions, i.e., 0.6% for 1 wt.% Si.
A large volume 2000-ton uniaxial split-sphere apparatus (USSA-2000) has been adapted to fabricate polycrystals of mantle minerals. By improving the gasket support and using NaC1 as the pressure medium, a 14/7.5 MgO cell assembly with a telescopic graphite furnace to provide a low temperature gradient (< 15øC/mm) and a working volume of 80 mm 3 can generate pressures up to 20 GPa at temperatures of 1200øC. Cylindrical specimens (3 mm long and 2-3 mm in diameter) of the beta (fi) (15 GPa, 1000øC) and spinel (y) (19 GPa, 900øC) phases ofMg2SiO 4 have been synthesized within their stability fields, and recovered at ambient conditions by simultaneously cooling and decompressing along computer-controlled pressure and temperature (P-T) paths designed to preserve the high-pressure phase and to relax intergranular stresses in the polycrystalline aggregates. These specimens were characterized by x-ray diffraction, optical, scanning and transmission electron microscopy, and by ultrasonic techniques, and found to be single-phased, fine-grained (<5 gm), free of microcracks and preferred orientation and to have bulk densities greater than 99% of x-ray density. Grains are uniform in size throughout the specimen and have boundaries that are well-sintered with no observable pores or cracks at the TEM scale. Ultrasonic measurements of the P-wave and S-wave velocities demonstrate that the polycrystals are isotropic and exhibit velocities within 2% of the Hashin-Shtrikman averages calculated from single-crystal elastic moduli. The successful fabrication of these high-quality polycrystalline specimens of the fi and y polymorphs of Mg2SiO 4 has made possible experiments to determine the pressure dependence of the acoustic velocities.
As determined from powder X-ray diffraction experiments with synchrotron radiation, the thermal expansion coefficient of forsterite increases smoothly from 2.8 to 4.5 K-1 from 400 K to 2160 K. No anomalous increases of the cell parameters are observed near the melting point. The consistency between the observed and calculated value of the initial slope of the melting curve of forsterite suggests that defects do not make a large contribution to thermal expansion near the melting point. Along with previous results, the new data confirm the influence of anharmonicity on the high-temperature heat capacity of forsterite and indicate that both the Grüneisen parameter and αKT (α = thermal expansion coefficient, KT = bulk modulus) have nearly constant values at high temperatures.
The predictions of compositional models of the mantle transition zone are compared to observed seismic properties by constructing phase diagrams in the MgO-FeO-CaO-Al2O3-SiO2 system and estimating the elasticity of the relevant materials. Mie-Grueneisen and Birch-Murnaghan finite strain theory is combined with an ideal solution to extrapolate experimental measurements of thermal and elastic properties to high pressures and temperatures. The resulting thermodynamic potentials are combined with the estimated phase diagrams to predict the density, seismic parameter, and mantle adiabats for a given compositional model. The properties of pyrolite are found to agree well with the observed density and bulk sound velocity of the upper mantle and transition zone. It is argued that within the transition zone, the dissolution of garnet to a Ca-perovskite near 18 GPa explains the proposed 520-km seismic discontinuity.
The high-temperature thermodynamic properties of forsterite were reviewed in the light of a new determination of the isobaric heat capacity (Cp), up to 1850 K, and Raman spectroscopic measurements, up to 1150 K and 10 GPa. The Cp measurements and available data on thermal expansion (alpha) and bulk modulus (K) show that the isochoric specific heat (Cnu) exceeds the harmonic limit of Dulong and Petit above 1300 K. This intrinsic anharmonic behavior of Cnu can be modeled by introducing anharmonic parameters ai=(∂lnnui/∂T)V which are calculated from the measured pressure and temperature shifts of the vibrational frequencies. These parameters are all negative, with absolute values lower for the stretching models of the SiO4 tetrahedra (ai~-1×10-5 K-1) than for the lattice modes (ai~-2×10-5 K-1). Through the relation Cp=Cnu+alpha2KTVT, the calculated anharmonic Cnu and the measured Cp are then used to determine the temperature dependence of the thermal expansion and bulk modulus of forsterite, up to 2000 k, in agreement with recent experimental results. Finally, all these data point to an inconsistency for the Grüneisen parameter of forsterite, whereby the macroscopic parameter gamma=alphaVKT/Cnu cannot be evaluated simply at high temperature by summation of the individual isothermal mode Grüneisen parameters gammatT=KT (∂lnnui/∂P).
Unit cell volumes of the beta (β) and spinel (γ) phases (Mg2SiO4) have been measured under simultaneous high pressure and high temperatures using synchrotron X-ray radiation, in a cubic anvil apparatus. The data on the temperature derivatives of the bulk modulus for the three phases are consistent with an upper mantle containing 60 to 65% olivine and the absence of a velocity signature for the β to γ phase transition near 520 km depth. -from Authors
The pressure dependence of the Raman spectrum of forsterite was measured over its entire frequency range to over 200 kbar. The shifts of the Raman modes were used to calculate the pressure dependence of the heat capacity, C
v, and entropy, S, by using statistical thermodynamics of the lattice vibrations. Using the pressure dependence of C
v and other previously measured thermodynamic parameters, the thermal expansion coefficient, , at room temperature was calculated from = K
S
(T/P)
S
C
V/TVK
T, which yields a constant value of ( ln / ln V)T= 6.1(5) for forsterite to 10% compression. This value is in agreement with ( ln / ln V)T for a large variety of materials.At 91 kbar, the compression mechanism of the forsterite lattice abruptly changes causing a strong decrease of the pressure derivative of 6 Raman modes accompanied by large reductions in the intensities of all of the modes. This observation is in agreement with single crystal x-ray diffraction studies to 150 kbar and is interpreted as a second order phase transition.
Written by a renowned expert in the field, this book is the most comprehensive treatment available on the applications of equations of state (EoS) in geophysics and materials science, a topic of fundamental importance to those studying the physics and chemistry of the Earth. Part one offers comprehensive treatments of thermal properties associated with EoS, thermodynamic and statistical mechanical backgrounds, and thermoelastic properties. Definitions of the physical properties needed for the EoS are provided as well. Part two discusses the isothermal pressure-volume relationship. The ab initio approach--EoS based upon quantum mechanics fundamentals using numerical methods--is utilized to clearly represent and analyze the measured data. Part three offers an advanced treatment of thermal properties at high temperature, and includes discussions of thermal pressure, shocked solids, and EoS applications to materials science topics such as melting and thermodynamic function. Advanced students, researchers, and professionals in geophysics, ceramics science, solid state physics, and geochemistry will want to read this book.
In view of the prime importance of silicate melting as a clue to the origin of chemical stratification in the mantle, we have developed an experimental technique to make static experiments on melting of silicates up to 20GPa. We have carried out melting experiments on several silicates in the pressure range from 10-20GPa. Melting behavior has been studied on fayalite to 16GPa, forsterite to 15GPa, pyrope to 10GPA, and the binary systems of fayalite-ferrosilite and pyrope-forsterite at 5GPa. Results are presented.-Authors
This is the first of a series of five papers in which the thermodynamic properties of minerals are interpreted in terms of lattice vibrational spectra. In this paper, measured heat capacities for minerals are examined in terms of the Debye theory of lattice vibrations, and it is demonstrated that heat capacities of silicates show large deviations from the behavior expected from Debye theory. The underlying assumptions of Debye theory are critically reviewed, and it is shown that the observed thermodynamic deviations in minerals probably arise from four effects not included in the Debye model: anisotropy of elastic parameters, dispersion of acoustic waves toward Brillouin zone boundaries, optic vibrations in excess of the Debye spectrum at low frequencies, and optic vibrations at frequencies much greater than the Debye cutoff frequency predicted by acoustic measurements. Each of the four effects influences the heat capacity in a particular temperature range: anisotropy, dispersion and low-frequency optic vibrations are important at low temperatures (0°K to ∼100°K); high-frequency vibrations are important at higher temperatures. It is necessary to include all four effects in a generalized lattice vibrational model for minerals; such a model is developed in papers 2-5 of this series. The minerals included in this study are halite, periclase, brucite, corundum, spinel, quartz, cristobalite, silica glass, coesite, stishovite, rutile, albite, microcline, jadeite, diopside, enstatite, tremolite, talc, muscovite, forsterite, zircon, kyanite, andalusite, sillimanite, pyrope, grossular, andradite, spessartine, almandine and calcite.
The first high-temperature data on the nine adiabatic elastic moduli for iron-bearing olivine are reported. These measurements are on two single-crystal specimens of natural olivine at ambient pressure and from room temperature to a maximum of 1500K. The data are high quality to extreme temperatures, with good agreement found when comparing the temperature derivatives of the elastic moduli of the two specimens. The temperature derivatives of the isotropic bulk modulus KS are (-1.69, -1.80) × 10-2GPa K-1 for the two olivine specimens, and the shear modulus G derivatives are (-1.38, -1.36) × 10-2GPa K-1. These derivatives are only slightly larger in magnitude than |(δKS/δT)P| = 1.56 × 10-2 and |(δG/δT)P| = 1.30 × 10-2GPa K-1 found previously for iron-bearing olivine over a very small temperature range. There are also no significant differences between the temperature derivatives found here and the average derivatives of end-member forsterite from data retrieved over a slightly larger temperature range. -from Author
Unit cell volumes of the beta (beta) and spinel (gamma) phases (Mg2SiO4) have been measured under simultaneous high pressure and high temperature using synchrotron X ray radiation, in a cubic anvil apparatus. With volume-temperature data at constant pressure, we determine the average volume thermal expansion coefficients of the beta phase from 724 to 872 K at 7.6 GPa to be 2.28(+/- 0.45) x 10(exp -5)/K and of the gamma phase from 759 to 962 K at 9.8 GPa to be 1.71(+/- 0.14) x 10(exp -5)/K. Thermodynamic relations are used to constrain the temperature derivative of the isothermal bulk modulus K(sub T) from the high-pressure thermal expansion data: (the partial derivative of K(sub T)/the partial derivative of T)(sub p)) is found to be -2.7(+/-0.5) x 10(exp -2) GPa/K for the beta phase and -2.8(+/-0.3) x 10(exp -2) GPa/K for the gamma phase. Unit cell volumes of the olivine (alpha) phase, back-transformed from the beta and gamma phases at high temperatures, have been measured under pressure at temperatures above the Debye temperature; using thermal pressure equations, we find (the partial derivative of K(sub T)/the partial derivative of T)(sub p)) for the alpha phase to be -2.1(+/-0.2) x 10(exp -2) GPa/K. These new data on the temperature derivatives of the bulk modulus for the three phases are consistent with an uppermantle containing 60 to 65% olivine and the absence of a velocity signature for the beta to gamma phase transition near 520 km depth.
The principles of lattice dynamics are briefly reviewed in this paper, and from these principles a simple, generally applicable lattice vibrational spectrum is proposed for minerals. The spectrum can be used to calculate the thermodynamic functions in the harmonic approximation. The model proposed is consistent with lattice dynamics and is sufficiently detailed in its assumptions about the distribution of modes that the thermodynamic functions are closely specified. The model is applied to give the temperature dependence of C//V over the range 0-1000 degree K of halite, periclase, brucite, corundum, spinel, quartz, cristobalite, silica glass, coesite, stishovite, rutile, albite, and microcline. The specific heat of the simple Debye-like substances, halite and periclase, is reproduced well by the model. The heat capacities of the relatively simple minerals, spinel and corundum, are given accurately by the model. The heat capacities of quartz, cristobalite, and coesite are accurately predicted from spectroscopic data through the model. The heat capacity of silica glass is discussed in terms of the classic continuous random network (CRN) model and a paracrystalline model. , the pentagonal dodecahedral (PD) model of Robinson (1965). The PD model appears to be more consistent with measured C//V data than the CRN model. The heat capacity data of rutile are reasonably reproduced by the model as are the data for stishovite at temperatures above 50 degree K. Measured data for stishovite below 50 degree K appear to contain an excess heat capacity relative to the model; this excess may arise from surface energy contributions, and it is suggested that the model provides a better estimate of the low-temperature vibrational heat capacity of stishovite than the measured data.
This paper is the fourth in a series relating the lattice vibrational properties to the thermodynamic properties of minerals. The temperature dependence of the harmonic lattice heat capacity is calculated from a model which uses only elastic, crystallographic, and spectroscopic data for the following minerals: calcite, zircon, forsterite, grossular, pyrope, almandine, spessartine, andradite, kyanite, andalusite, sillimanite, clinoenstatite, orthoenstatite, jadeite, diopside, tremolite, talc, and muscovite. The heat capacities of these minerals reflect structural and compositional differences. The ‘excess’ entropy of pyrope—compared with that of grossular—is shown to arise from low-frequency optic modes of vibration. The entropy differences between kyanite, andalusite, and sillimanite are well reproduced by the model, although the absolute values calculated are systematically about 3% high. Model values of the heat capacity and entropy are compared with experimental values at 298.15, 700, and 1000°K for the 32 minerals included in papers 1–4 of this series. The average deviation of the entropies at 298°K from well-determined calorimetric values is ±1.5%. A method is given for obtaining greater accuracy in the model thermodynamic functions by fitting one parameter to experimental data when partial calorimetric data (such as the heat capacity at a single temperature in the range 50–100°K) are available; such a method should permit accurate extrapolation of calorimetric data beyond the range of experiment.
Magnesium-rich olivine (Mg0.9Fe0.1)2SiO4 is considered to be a major constituent of the Earth's upper mantle. Because of its major geophysical importance, the temperature and pressure dependence of its crystal structure, elastic and dielectric constants, long-wavelength phonon modes and specific heat have been measured using a variety of experimental techniques. Theoretical study of lattice dynamics provides a means of analyzing and understanding a host of such experimental data in a unified manner. A detailed study of the lattice dynamics of forsterite, Mg2SiO4, has been made using a crystal potential function consisting of Coulombic and short-range terms. Quasiharmonic lattice dynamical calculations based on a rigid molecular-ion model have provided theoretical estimates of elastic constants, long-wavelength modes, phonon dispersion relation for external modes along the three high symmetry directions in the Brillouin zone, total and partial density of states and inelastic neutron scattering cross-sections. The neutron cross-sections were used as guides for the coherent inelastic neutron scattering experiment on a large single crystal using a triple axis spectrometer in the constant Q mode. The observed and predicted phonon dispersion relation show excellent agreement. The inelastically scattered neutron spectra from a powder sample have been analyzed on the basis of a phonon density of states calculated from a rigid-ion model, which includes both external and internal modes. The experimental data from a powder sample show good agreement with the calculated spectra, which include a multiphonon contribution in the incoherent approximation. The computed phonon densities of states are used to calculate the specific heat as a function of temperature using both the rigid molecular-ion and rigid ion models. These results are in very good agreement with the calorimetric measurement of the specific heat. The interatomic potential developed here can be used with some confidence to study physical properties of forsterite as a function of pressure and temperature.
Measurements of the nine adiabatic elastic constants of single-crystal
forsterite are reported for the temperature range 300-1700 K at ambient
pressure. The measured room temperature Cijs and
their first temperature derivatives at ambient conditions are generally
consistent with previous measurements, which were taken over a narrower
temperature range. Slight nonlinear behavior is observed in the
temperature dependence of the C11s,
C22s, and C33s moduli. The
resulting Hill-averaged bulk and rigidity moduli Ks and G are
lower at 1700 K than predicted by linear extrapolation of
low-temperature data. At 1700 K the new data give Ks = 103.8
± 1.1 GPa and G = 61.3 ± 0.5 GPa. Isotropic elastic wave
velocities have been calculated using the Hill averaging scheme over the
range of temperatures measured. At T = 1700 K and ambient pressure we
find vp = 7.79 ± 0.04 km s-1 and
vs = 4.48 ± 0.03 km s-1. The parameter
(∂ℓnvs/∂ℓnvp)P is
near to 1.2 and independent of temperature. Calculations are presented
for elastic properties and other parameters important for the equations
of state and anharmonicity of forsterite. Recent high-temperature
thermal expansion measurements on a forsterite specimen from the same
boule are used in this analysis. Effects of uncertainties in
high-temperature thermal expansion data on these parameters have been
evaluated and are discussed.
The high-temperature measurements of elastic constants and related temperature derivatives of nine minerals of interest to geophysical and geochemical theories of the Earth's interior are reviewed and discussed. A number of correlations between these parameters, which have application to geophysical problems, are also presented. Of especial interest is α, the volume coefficient of thermal expansion, and a section is devoted to this physical property. Here we show how α can be estimated at very high temperatures and how it varies with density. An estimate of α for Mg-perovskite at deep-mantle conditions is made. The formula for the Grüneisen ratio γ as a function of V and T is presented, including plots of the numerical values of γ over a wide T and V range. An example calculation of γ for MgO is made. The high-T-high-P values of γ calculated here agree well with results from the ab initio method of calculation for MgO. The use of the thermoelastic parameters is reviewed, showing application to the understanding of thermal pressure, thermal expansivity, enthalpy, and entropy. We review an extrapolation formula to determine Ks, the adiabatic bulk modulus, at very high T. We show that the thermal pressure is quite linear with T up to high temperatures (∼1800 K), and, as a consequence, the anharmonic contribution to the Helmholtz free energy is sufficiently small, so that it can and should be ignored in thermodynamic calculations for mantle conditions.
Ultrasonic wave velocities in single-crystal forsteritc (F) and single-crystal olivine (0) have been measured as a function of pressure and of temperature near ambient conditions. Shear and longitudinal velocities were measured in eighteen independent modes, so that each of the nine elastic constants could be calculated by at least two independent equations. The adiabatic stiffness constants c{j (in Mb), their temperature derivatives dc,j/dT (10 -4 Mb/ deg), and their pressure derivatives dc{ffdP, are ij 11 22 33 44 55 66 23 31 12 Cii (F) 3.284 1.998 2.353 0.6515 0.8120 0.8088 0.738 0.688 0.639 (O) 3.237 1.976 2.351 0.6462 0.7805 0.7904 0.756 0.716 0.664 dcii/dT (F) 3.31 2.81 2.83 1.30 1.32 1.51 0.46 0.82 1.04 (O) 3.40 2.85 2.86 1.28 1.30 1.57 0.51 0.94 1.05 dcii/dP (F) 8.47 6.56 6.57 2.12 1.66 2.37 4.11 4.84 4.67 (O) 7.98 6.37 6.38 2.17 1.64 2.31 3.76 4.48 4.74
Unit cell volumes of the beta (β) and spinel (γ) phases (Mg2SiO4) have been measured under simultaneous high pressure and high temperature using synchrotron X ray radiation, in a cubic anvil apparatus. With volume-temperature data at constant pressure, we determine the average volume thermal expansion coefficients of the β phase from 724 to 872 K at 7.6 GPa to be 2.28(±0.45)×10−5/K and of the γ phase from 759 to 962 K at 9.8 GPa to be 1.71(±0.14)×10−5/K. Thermodynamic relations are used to constrain the temperature derivative of the isothermal bulk modulus (KT) from the high-pressure thermal expansion data: (∂KT/∂T)P is found to be −2.7(±0.5)×10−2 GPa/K for the β phase and −2.8(±0.3)×10−2 GPs/K for the γ phase. Unit cell volumes of the olivine (α) phase, back-transformed from the β and γ phases at high temperatures, have been measured under pressure at temperatures above the Debye temperature; using thermal pressure equations, we find (∂KT/∂T)P for the α phase to be −2.1(±0.2)×10−2 GPa/K. These new data on the temperature derivatives of the bulk modulus for the three phases are consistent with an upper mantle containing 60 to 65% olivine and the absence of a velocity signature for the β to γ phase transition near 520 km depth.
Thermochemical characterization of high-pressure phases is essential to describe the phase relations and the thermal structure of the Earth's mantle. The first far-IR spectra of silicates at high pressure have been obtained by combining various state-of-the-art technologies. Three olivines (Fo{sub 100}, Fo{sub 87}, and Fo{sub 0}) were studied at pressures of up to 425 kbar. In addition, mid-IR data on Fo{sub 100} were obtained up to 240 kbar. The same pressure responses (i.e., changes in peak position, area, etc.) were seen in the hydrostatic and nonhydrostatic experiments. By augmenting these data with mid-IR studies at pressure by Xu et al. (1983), all IR modes in Mg-Fe olivines are characterized as a function of pressure. The data corroborate the band assignments for olivine and the correlation of the forsterite and fayalite 1-atm spectra by Hofmeister (1987), in that similar behavior with pressure is observed for analogous bands in these minerals, with each different type of vibrational motion having a distinct pressure response. The response to pressure of bands assigned to SiO{sub 4} vibrations differs fundamentally from that of bands involving M ions in a manner that is consistent with the relative incompressibility of the SiO{sub 4} tetrahedron. A gradual transformation from olivine to a more symmetric structure is supported by the generally continuous but individual responses of peak area and frequency to pressure, but the appearance of new bands at pressure, and by the occurrence of mode softening.
Thermodynamics of Crystals is a gold mine of a references bargain with more derivations of useful equations per dollar, or per page, than almost any other book I know. Useful to whom? To the solid state physicist, the solid state chemist working the geophysicist, the rock mechanic, the mineral physicist. Useful for what? For lattice dynamics, crystal potentials, band structure. For elegant, rigorous, and concise derivations of fundamental equations. For comparison of levels of approximation. For some data and physical insights, especially for metals and simple halides. This book is a reissue, with some changes and additions, of a 1970 treatise. It ages well, since the fundamentals do not change.
A calculation of the equation of state for NaCl from a Mie‐Grüneisen equation was repeated using more accurate values of the zero‐pressure compressibility. It was also extended to KCl and CsCl. An analysis of this approach to pressure calibration indicates that it will yield pressures with about the same accuracy as can be presently achieved by experimental measurements above 25 kbar, and thus furnishes a temporary practical pressure scale.
Evidence has steadily accumulated to show that at high temperatures (above the Debye temperature, θ) the thermal pressure, PTH, of solids, is linear with T to a close approximation. This empirical finding yields a simple relationship between P, V, and T quite useful for the computation of the equation-of-state (EOS). For geophysical applications, the empirical data is, so far, limited to a few minerals, all of which are important to our geophysical models of the Earth. The same results have been found for a variety of types of solids, including alkali metals, noble gas solids, alkali halides and metals in addition to minerals. It is argued that the linearity between PTH and T is a general high-temperature property of solids. This includes minerals. Thus it is proposed that there exists a common thermal EOS which transcends the chemical bonding type and crystallographic class.
Unit-cell volumes (V) of Mg1−xFexSiO3 perovskite (x = 0.0 and 0.1) have been measured along several isobaric paths up to P = 11 GPa and T = 1300 K using a DIA-type, cubic anvil high-pressure apparatus (SAM-85). With a combination of X-ray diffraction during heating cycles and Raman spectroscopy on recovered samples, pressure and temperature conditions were determined under which the P-V-T behavior of the perovskite remains reversible. At 1 bar, perovskites of both compositions begin to transform to amorphous phases at T ≈ 400 K, accompanied by an irreversible cell volume contraction. Electron microprobe and analytical electron microscopy studies revealed that the iron-rich perovskite decomposed into at least two phases, which were Fe and Si enriched, respectively. At pressures above 4 GPa, the P-V-T behavior of MgSiO3 perovskite remained reversible up to about 1200 K, whereas the Mg0.9Fe0.1SiO3 exhibited an irreversible behavior on heating. Such irreversible behavior makes equation-of-state data on Fe-rich samples dubious. Thermal expansivities (αV) of MgSiO3 perovskite were measured directly as a function of pressure. Overall, our results indicate a weak pressure dependence in αV for MgSiO3. Analyses on the P-V-T data using various thermal equations of state yielded consistent results on thermoelastic properties. The temperature derivative of the bulk modulus, , is −0.023(±0.011) GPa K−1 for MgSiO3 perovskite. Using these new results, we examine the constraints imposed by αV and on the and ratios for the lower mantle. For a temperature of 1800 K at the foot of an adiabat (zero depth), these results indicate an overall iron content of for a lower mantle composed of perovskite and magnesiowüstite. Although the ratio is very sensitive to the thermoelastic parameters of the perovskite and it is tentatively constrained between 1.4 and 2.0, these results indicate that it is unlikely for the lower mantle to have a perovskite stoichiometry.
Adiabatic elastic stiffness constants of single crystal forsterite measured as function of hydrostatic pressure and temperature, using pulse superposition technique
= 1.14), and the q parameter (q = 1), all taken from Ita and Stixrude (1992) No anharmonic term (a = 0) is taken into account, 2. An extended Kieffer's model (Kieffer, 1979, Ki-effer, 1980, see Fig. 4(b)) based on model 1 of A universal thermal equation of state
- Gillet
In this model, (hereafter designated ED-model) input parameters are the Debye temperature (0 d = 924 K), the Grfineisen parameter at V = V 0 (3' = 1.14), and the q parameter (q = 1), all taken from Ita and Stixrude (1992). No anharmonic term (a = 0) is taken into account, 2. An extended Kieffer's model (Kieffer, 1979, Ki-effer, 1980, see Fig. 4(b)) based on model 1 of Gillet et al. (1991), and hereafter designated EK-References Anderson, O.L., 1984. A universal thermal equation of state. J. Geodyn., 1: 185-214.
High-tem-perature thermodynamic properties of forsterite Elastic constants of single-crystal forsterite as a function of temperature and pres-sure
- Gillet
- Ph
- P Richet
- F Guyot
- G Fiquet
Gillet, Ph., Richet, P., Guyot, F. and Fiquet, G., 1991. High-tem-perature thermodynamic properties of forsterite. J. Geophys. Res., 96:11805-11 816. Graham, E.K. and Barsch, G.R., 1969. Elastic constants of single-crystal forsterite as a function of temperature and pres-sure. J. Geophys. Res., 74: 5949-5960.
High temperature equation of state for mantle minerals and their anharmonic properties
- Kajioshi
Kajioshi, K., 1986. High temperature equation of state for mantle
minerals and their anharmonic properties. M.S. Thesis,
Okayama University, Okayama, Japan.
High-Pressure Research, Application to Earth and Planetary Sciences
- Manghnani
Manghnani (Editors), High-Pressure Research, Application to
Earth and Planetary Sciences. Terra Scientific, Tokyo and
American Geophysical Union, Washington, pp. 13-17.