P-V-T equation of state of magnesiowüstite (Mg0.6Fe0.4)O

The lattice parameter of magnesiowüstite (Mg_0.6Fe_0.4)O has been measured up to a pressure of 30 GPa and a temperature of 800 K, using an external heated diamond anvil cell and diffraction using X-rays from a synchrotron source. The experiments were conducted under quasi-hydrostatic condition, using neon as a pressure transmitting medium. The experimental P-V-T data were fitted to a thermal-pressure model with the isothermal bulk modulus at room temperature K _T0 = 157 GPa, (?K _TO /?P) _T =4, (?K _T /?T) _P =-2.7(3) × 10^-2 GPa/K, (?K _T /?T) _v =-0.2(2) × 10^-2 GPa/K and the Anderson-Grüneisen parameter δ _T =4.3(5) above the Debye temperature. The data were also fitted to the Mie-Grüneisen thermal equation of state. The least-squares fit yields the Debye temperature θ _DO = 500(20) K, the Grüneisen parameter γ ₀=1.50(5), and the volume dependence q=1.1(5). Both thermal-pressure models give consistent P-V-T relations for magnesiowüstite to 140 GPa and 4000 K. The P-V-T relations for magnesiowüstite were also calculate by using a modified high-temperature Birch-Murnaghan equation of state with a δ _t of 4.3. The results are consistent with those calculated by using the thermal-pressure model and the Mie-Grüneisen relation to 140 GPa and 3000 K.     

Density variations of Cr-rich garnets in the upper mantle inferred from the elasticity of uvarovite garnet

The thermoelastic parameters of Ca₃Cr₂Si₃O₁₂ uvarovite garnet were examined in situ at high pressure up to 13 GPa and high temperature up to 1100 K by synchrotron radiation energy-dispersive X-ray diffraction within a 6-6-type multi-anvil press apparatus. A least-square fitting of room T data to a third-order Birch–Murnaghan (BM3) EoS yielded K₀ = 164.2 ± 0.7 GPa, V₀ = 1735.9 ± 0.3 ų (K’₀ fixed to 4.0). P–V–T data were fitted simultaneously by a modified HT-BM3 EoS, which gave the isothermal bulk modulus K₀ = 163.6 ± 2.6 GPa, K’₀ = 4.1 ± 0.5, its temperature derivative (?K_0,T/?T)_P = –0.014 ± 0.002 GPa K^?1, and the thermal expansion coefficients a₀ = 2.32 ± 0.13 ×10^?5 K^?1 and b₀ = 2.13 ± 2.18 ×10^?9 K^?2 (K’₀ fixed to 4.0). Our results showed that the Cr³⁺ enrichment in natural systems likely increases the density of ugrandite garnets, resulting in a substantial increase of mantle garnet densities in regions where Cr-rich spinel releases chromium through a metasomatic reaction.     

Thermal equation of state of natural tourmaline at high pressure and temperature

Synchrotron-based in situ angle-dispersive X-ray diffraction experiments were conducted on a natural uvite-dominated tourmaline sample by using an external-heating diamond anvil cell at simultaneously high pressures and temperatures up to 18 GPa and 723 K, respectively. The angle-dispersive X-ray diffraction data reveal no indication of a structural phase transition over the P–T range of the current experiment in this study. The pressure–volume–temperature data were fitted by the high-temperature Birch–Murnaghan equation of state. Isothermal bulk modulus of K ₀ = 96.6 (9) GPa, pressure derivative of the bulk modulus of \(K_{0}^{\prime } = 12.5 \;(4)\), thermal expansion coefficient of α ₀ = 4.39 (27) × 10^?5 K^?1 and temperature derivative of the bulk modulus (?K/?T) _P  = ?0.009 (6) GPa K^?1 were obtained. The axial thermoelastic properties were also obtained with K _a0 = 139 (2) GPa, \(K_{a0}^{\prime }\) = 11.5 (7) and α _a0 = 1.00 (11) × 10^?5 K^?1 for the a-axis, and K _c0 = 59 (1) GPa, \(K_{c0}^{\prime }\) = 11.4 (5) and α _c0 = 2.41 (24) × 10^?5 K^?1 for the c-axis. Both of axial compression and thermal expansion exhibit large anisotropic behavior. Thermoelastic parameters of tourmaline in this study were also compared with that of the other two ring silicates of beryl and cordierite.     

Thermal expansion and crystal structure of FeSi between 4 and 1173 K determined by time-of-flight neutron powder diffraction

The thermal expansion and crystal structure of FeSi has been determined by neutron powder diffraction between 4 and 1173?K. No evidence was seen of any structural or magnetic transitions at low temperatures. The average volumetric thermal expansion coefficient above room temperature was found to be 4.85(5)?×?10^?5?K^?1. The cell volume was fitted over the complete temperature range using Grüneisen approximations to the zero pressure equation of state, with the internal energy calculated via a Debye model; a Grüneisen second-order approximation gave the following parameters: θ_D=445(11)?K, V ₀=89.596(8)?ų, K ₀′=4.4(4) and γ′=2.33(3), where θ_D is the Debye temperature, V ₀ is V at T=0?K, K ₀′ is the first derivative with respect to pressure of the incompressibility and γ′ is a Grüneisen parameter. The thermodynamic Grüneisen parameter, γ_th, has been calculated from experimental data in the range 4–400?K. The crystal structure was found to be almost invariant with temperature. The thermal vibrations of the Fe atoms are almost isotropic at all temperatures; those of the Si atoms become more anisotropic as the temperature increases.     

Hydrostatic compression of γ-Mg2SiO4 to mantle pressures and 700 K: Thermal equation of state and related thermoelastic properties

P-V-T equations of state for the γ phase of Mg₂SiO₄ have been fitted to unit cell volumes measured under simultaneous high pressure (up 30 GPa) and high temperature (up to 700 K) conditions. The measurements were conducted in an externally heated diamond anvil cell using synchrotron x-ray diffraction. Neon was used as a pressure medium to provide a more hydrostatic pressure environment. The P-V-T data include 300 K-isothermal compression to 30 GPa, 700 K-compression to 25 GPa and some additional data in P-T space in the region 15 to 30 GPa and 300 to 700 K. The isothermal bulk modulus and its pressure derivative, determined from the isothermal compression data, are 182(3) GPa and 4.2(0.3) at T=300 K, and 171(4) GPa and 4.4(0.5) at T=700 K. Fitting all the P-V-T data to a high-temperature Murnaghan equation of state yields: K _TO=182(3.0) GPa, K _TO=4.0(0.3), ?K _T /?T)₀=?2.7(0.5)×10^?2 GPa/K and (?² K _T /?P?T)₀=5.5(5.2)×10^?4/K at the ambient condition.     

Thermoelastic properties of chromium oxide Cr<Subscript>2</Subscript>O<Subscript>3</Subscript> (eskolaite) at high pressures and temperatures

A new synchrotron X-ray diffraction study of chromium oxide Cr₂O₃ (eskolaite) with the corundum-type structure has been carried out in a Kawai-type multi-anvil apparatus to pressure of 15 GPa and temperatures of 1873 K. Fitting the Birch–Murnaghan equation of state (EoS) with the present data up to 15 GPa yielded: bulk modulus (K _0,T0), 206 ± 4 GPa; its pressure derivative K′_0,T , 4.4 ± 0.8; (?K _0,T /?T) = ?0.037 ± 0.006 GPa K^?1; a = 2.98 ± 0.14 × 10^?5 K^?1 and b = 0.47 ± 0.28 × 10^?8 K^?2, where α _0,T  = a + bT is the volumetric thermal expansion coefficient. The thermal expansion of Cr₂O₃ was additionally measured at the high-temperature powder diffraction experiment at ambient pressure and α _0,T0 was determined to be 2.95 × 10^?5 K^?1. The results indicate that coefficient of the thermal expansion calculated from the EoS appeared to be high-precision because it is consistent with the data obtained at 1 atm. However, our results contradict α ₀ value suggested by Rigby et al. (Brit Ceram Trans J 45:137–148, 1946) widely used in many physical and geological databases. Fitting the Mie–Grüneisen–Debye EoS with the present ambient and high-pressure data yielded the following parameters: K _0,T0 = 205 ± 3 GPa, K′_0,T  = 4.0, Grüneisen parameter (γ ₀) = 1.42 ± 0.80, q = 1.82 ± 0.56. The thermoelastic parameters indicate that Cr₂O₃ undergoes near isotropic compression at room and high temperatures up to 15 GPa. Cr₂O₃ is shown to be stable in this pressure range and adopts the corundum-type structure. Using obtained thermoelastic parameters, we calculated the reaction boundary of knorringite formation from enstatite and eskolaite. The Clapeyron slope (with \({\text{d}}P/{\text{d}}T = - 0.014\) GPa/K) was found to be consistent with experimental data.     

Thermal equations of state of dioctahedral micas on the join muscovite–paragonite

 Powder diffraction measurements at simultaneous high pressure and temperature on samples of 2M1 polytype of muscovite (Ms) and paragonite (Pg) were performed at the beamline ID30 of ESRF (Grenoble), using the Paris-Edinburgh cell. The bulk moduli of Ms, calculated from the least-squares fitting of V–P data on each isotherm using a second-order Birch–Murnaghan EoS, were: 57.0(6), 55.1(7), 51.1(7) and 48.9(5) GPa on the isotherms at 298, 573, 723 and 873 K, respectively. The value of (∂K _T /∂T) was −0.0146(2) GPa K⁻¹. The thermal expansion coefficient α varied from 35.7(3) × 10⁻⁶ K⁻¹ at P ambient to 20.1(3) × 10⁻⁶ K⁻¹ at P = 4 GPa [(∂α/∂P) _T = −3.9(1) × 10⁻⁶ GPa⁻¹ K⁻¹]. The corresponding values for Pg on the isotherms at 298, 723 and 823 K were: bulk moduli 59.9(5), 55.7(6) and 53.8(7) GPa, (∂K _T /∂T) −0.0109(1) GPa K⁻¹. The thermal expansion coefficient α varied from 44.1(2) × 10⁻⁶ K⁻¹ at P ambient to 32.5(2) × 10⁻⁶ K⁻¹ at P = 4 GPa [(∂α/∂P) _T = −2.9(1) × 10⁻⁶ GPa⁻¹ K⁻¹]. Thermoelastic coefficients showed that Pg is stiffer than Ms; Ms softens more rapidly than Pg upon heating; thermal expansion is greater and its variation with pressure is smaller in Pg than in Ms. Received: 28 January 2002 / Accepted: 5 April 2002     

Single crystal X-ray diffraction study of MgSiO3 perovskite from 77 to 400 K

Single crystal X-ray diffraction study of MgSiO₃ perovskite has been completed from 77 to 400 K. The thermal expansion coefficient between 298 and 381 K is 2.2(8) × 10^-5 K^-1. Above 400 K, the single crystal becomes so multiply twinned that the cell parameters can no longer be determined.From 77 to 298 K, MgSiO₃ perovskite has an average thermal expansion coefficient of 1.45(9) × 10^-5 K^-1, which is consistent with theoretical models and perovskite systematics. The thermal expansion is anisotropic; the a axis shows the most expansion in this temperature range (_a = 8.4(9) × 10^-6 K^-1) followed by c(_c = 5.9(5) × 10^-6 K^-1) and then by b, which shows no significant change in this temperature range. In addition, the distortion (i.e., the tilting of the [SiO₆] octahedra) decreases with increasing temperature. We conclude that the behavior of MgSiO₃ perovskite with temperature mirrors its behavior under compression.     

Effect of incorporation of iron and aluminum on the thermoelastic properties of magnesium silicate perovskite

In situ X-ray diffraction measurements of Fe- and Al-bearing MgSiO₃-rich perovskite (FeAl-Pv), which was synthesized from a natural orthopyroxene, were performed at pressures of 19–32 GPa and temperatures of 300–1,500 K using a combination of a Kawai-type apparatus with eight sintered-diamond anvils and synchrotron radiation. Two runs were performed using a high-pressure cell with two sample chambers, and both MgSiO₃ perovskite (Mg-Pv) and FeAl-Pv were synthesized simultaneously in the same cell. Thus we were able to measure specific volumes (V/V ₀) of Mg-Pv and FeAl-Pv at the same P−T conditions. At all the measurement conditions, values of the specific volume of FeAl-Pv are consistent with those of Mg-Pv within 2 Standard Deviation, strongly suggesting that effect of incorporation of iron and aluminum on the thermoelastic properties of magnesium silicate perovskite is undetectable in this composition, pressure, and temperature range. Two additional runs were performed using a high-pressure cell that has one sample chamber and unit-cell volumes of FeAl-Pv were measured at pressures and temperatures up to 32 GPa and 1,500 K, respectively. All the unit-cell volume data of FeAl-Pv perovskite were fitted to the high temperature Birch–Murnaghan equation of state and a complete set of thermoelastic parameters of this perovskite was determined with an assumption of K′ _300,0 = 4. The determined parameters are K _300,0 = 243(3) GPa, (∂K _T,0/∂T) _P = −0.030(8) GPa/K, a ₀ = 2.78(18) × 10⁻⁵ K⁻¹, and b ₀ = 0.88(28) × 10⁻⁸ K⁻², where a ₀ and b ₀ are the coefficients of the following expression describing the zero-pressure thermal expansion: α _T,0 = a ₀ + b ₀ T. The equation-of-state parameters of FeAl-Pv are in good agreement with those of MgSiO₃ perovskite at the conditions corresponding to the uppermost part of the lower mantle.     

High-pressure behaviour and equation of state of calcite, CaCO3

The high-pressure response of the cell parameters of calcite, CaCO₃, has been investigated by single crystal X-ray diffraction. The unit cell parameters have been refined from 0 to 1.435?GPa, and the linear and volume compressibilities have been measured as β _a =2.62(2)?×?10^?3?GPa^?1,β _c =7.94(7)?×?10^?3?GPa^?1, β _v =13.12?×?10^?3?GPa^?1. The bulk modulus has been obtained from a fit to the Birch-Murnaghan equation of state, giving K ₀=73.46?±?0.27?GPa and V ₀=367.789 ±?0.004?ų with K′=4. Combined with earlier data for magnesite, ankerite and dolomite, these data suggest that K ₀ V ₀ is a constant for the Ca-Mg rhombohedral carbonates.     

The high-pressure and temperature equation of state of a majorite solid solution in the system of Mg4Si4O12-Mg3Al2Si3O12

The high-pressure and temperature equation of state of majorite solid solution, Mj_0.8Py_0.2, was determined up to 23 GPa and 773 K with energy-dispersive synchrotron X-ray diffraction at high pressure and high temperature using the single- and double-stage configurations of the multianvil apparatuses, MAX80 and 90. The X-ray diffraction data of the majorite sample were analyzed using the WPPD (whole-powder-pattern decomposition) method to obtain the lattice parameters. A least-squares fitting using the third-order Birch-Murnaghan equation of state yields the isothermal bulk modulus, K _T0  = 156 GPa, its pressure derivative, K′ = 4.4(±0.3), and temperature derivative (∂K _T /∂T) _P = −1.9(±0.3)× 10⁻² GPa/K, assuming that the thermal expansion coefficient is similar to that of pyrope-almandine solid solution. Received: 5 October 1998 / Revised, accepted: 24 June 1999     

Thermal equation of state of CaIrO<Subscript>3</Subscript> post-perovskite

The pressure–volume–temperature (P–V–T) relation of CaIrO₃ post-perovskite (ppv) was measured at pressures and temperatures up to 8.6 GPa and 1,273 K, respectively, with energy-dispersive synchrotron X-ray diffraction using a DIA-type, cubic-anvil apparatus (SAM85). Unit-cell dimensions were derived from the Le Bail full profile refinement technique, and the results were fitted using the third-order Birth-Murnaghan equation of state. The derived bulk modulus \( K_{T0} \) at ambient pressure and temperature is 168.3 ± 7.1 GPa with a pressure derivative \( K_{T0}^{\prime } \) = 5.4 ± 0.7. All of the high temperature data, combined with previous experimental data, are fitted using the high-temperature Birch-Murnaghan equation of state, the thermal pressure approach, and the Mie-Grüneisen-Debye formalism. The refined thermoelastic parameters for CaIrO₃ ppv are: temperature derivative of bulk modulus \( (\partial K_{T} /\partial T)_{P} \) = ?0.038 ± 0.011 GPa K^?1, \( \alpha K_{T} \) = 0.0039 ± 0.0001 GPa K^?1, \( \left( {\partial K_{T} /\partial T} \right)_{V} \) = ?0.012 ± 0.002 GPa K^?1, and \( \left( {\partial^{2} P/\partial T^{2} } \right)_{V} \) = 1.9 ± 0.3 × 10^?6 GPa² K^?2. Using the Mie-Grüneisen-Debye formalism, we obtain Grüneisen parameter \( \gamma_{0} \) = 0.92 ± 0.01 and its volume dependence q = 3.4 ± 0.6. The systematic variation of bulk moduli for several oxide post-perovskites can be described approximately by the relationship K _T0  = 5406.0/V(molar) + 5.9 GPa.     

X-ray diffraction study of magnesite at high pressure and high temperature

 P–V–T measurements on magnesite MgCO₃ have been carried out at high pressure and high temperature up to 8.6 GPa and 1285 K, using a DIA-type, cubic-anvil apparatus (SAM-85) in conjunction with in situ synchrotron X-ray powder diffraction. Precise volumes are obtained by the use of data collected above 873 K on heating and in the entire cooling cycle to minimize non-hydrostatic stress. From these data, the equation-of-state parameters are derived from various approaches based on the Birch-Murnaghan equation of state and on the relevant thermodynamic relations. With K′₀ fixed at 4, we obtain K₀=103(1) GPa, α(K⁻¹)=3.15(17)×10⁻⁵ +2.32(28)×10⁻⁸ T, (∂K_T/∂T)_P=−0.021(2) GPaK⁻¹, (dα/∂P)_T=−1.81×10⁻⁶ GPa⁻¹K⁻¹ and (∂K_T/∂T)_V= −0.007(1) GPaK⁻¹; whereas the third-order Birch-Murnaghan equation of state with K′₀ as an adjustable parameter yields the following values: K₀=108(3) GPa, K′₀=2.33(94), α(K⁻¹)=3.08(16)×10⁻⁵+2.05(27) ×10⁻⁸ T, (∂K_T/∂T)_P=−0.017(1) GPaK⁻¹, (dα/∂P)_T= −1.41×10⁻⁶ GPa⁻¹K⁻¹ and (∂K_T/∂T)_V=−0.008(1) GPaK⁻¹. Within the investigated P–T range, thermal pressure for magnesite increases linearly with temperature and is pressure (or volume) dependent. The present measurements of room-temperature bulk modulus, of its pressure derivative, and of the extrapolated zero-pressure volumes at high temperatures, are in agreement with previous single-crystal study and ultrasonic measurements, whereas (∂K_T/∂T)_P, (∂α/∂P)_T and (∂K_T/∂T)_V are determined for the first time in this compound. Using this new equation of state, thermodynamic calculations for the reactions (1) magnesite=periclase+CO₂ and (2) magnesite+enstatite=forsterite+CO₂ are consistent with existing experimental phase equilibrium data. Received September 28, 1995/Revised, accepted May 22, 1996     

Effect of pressure on the thermal expansion of MgO up to 8.2 GPa

Isobaric volume measurements for MgO were carried out at 2.6, 5.4, and 8.2 GPa in the temperature range 300–1073 K using a DIA-type, large-volume apparatus in conjunction with synchrotron X-ray powder diffraction. Linear fit of the thermal expansion data over the experimental pressure range yields the pressure derivative, (∂α/∂P) _T , of −1.04(8) × 10⁻⁶ GPa⁻¹ K⁻¹ and the mean zero-pressure thermal expansion α_{0, T}  = 4.09(6) × 10⁻⁵ K⁻¹. The α_{0, T} value is in good agreement with results of Suzuki (1975) and Utsumi et al. (1998) over the same temperature range, whereas (∂α/∂P) _T is determined for the first time on MgO by direct measurements. The cross-derivative (∂α²/∂P∂T) cannot be resolved because of large uncertainties associated with the temperature derivative of α at all pressures. The temperature derivative of the bulk modulus, (∂K _T/∂T) _P , of −0.025(3) GPa K⁻¹, obtained from the measured (∂α/∂P) _T value, is in accord with previous findings. Received: 2 April 1999 / Revised, accepted: 22 June 1999     

Crystal structure, Raman and FTIR spectroscopy, and equations of state of OH-bearing MgSiO3 akimotoite

MgSiO₃ akimotoite is stable relative to majorite-garnet under low-temperature geotherms within steeply or rapidly subducting slabs. Two compositions of Mg–akimotoite were synthesized under similar conditions: Z674 (containing about 550 ppm wt H₂O) was synthesized at 22 GPa and 1,500 °C and SH1101 (nominally anhydrous) was synthesized at 22 GPa and 1,250 °C. Crystal structures of both samples differ significantly from previous studies to give slightly smaller Si sites and larger Mg sites. The bulk thermal expansion coefficients of Z674 are (153–839 K) of a ₁ = 20(3) × 10^?9 K^?2 and a ₀ = 17(2) × 10^?6 K^?1, with an average of α ₀ = 27.1(6) × 10^?6 K^?1. Compressibility at ambient temperature of Z674 was measured up to 34.6 GPa at Sector 13 (GSECARS) at Advanced Photon Source Argonne National Laboratory. The second-order Birch–Murnaghan equation of state (BM2 EoS) fitting yields: V ₀ = 263.7(2) Å³, K _T0 = 217(3) GPa (K′ fixed at 4). The anisotropies of axial thermal expansivities and compressibilities are similar: α _a  = 8.2(3) and α _c  = 10.68(9) (10^?6 K^?1); β _a  = 11.4(3) and β _c  = 15.9(3) (10^?4 GPa). Hydration increases both the bulk thermal expansivity and compressibility, but decreases the anisotropy of structural expansion and compression. Complementary Raman and Fourier transform infrared (FTIR) spectroscopy shows multiple structural hydration sites. Low-temperature and high-pressure FTIR spectroscopy (15–300 K and 0–28 GPa) confirms that the multiple sites are structurally unique, with zero-pressure intrinsic anharmonic mode parameters between ?1.02 × 10^?5 and +1.7 × 10^?5 K^?1, indicating both weak hydrogen bonds (O–H···O) and strong OH bonding due to long O···O distances.     

High pressure study of ScAlO3 perovskite

The crystal structure of orthorhombic (Pbnm) ScAlO₃ perovskite has been refined to 5 GPa using single-crystal X-ray diffraction. The compression of the structure if anisotropic with β _a =1.39(3)×10⁻³ GPa⁻¹, β _b =1.14(3)×10⁻³ GPa⁻¹ and β _c =1.84(3)×10⁻³ GPa⁻¹. The isothermal bulk modulus of ScAlO₃, K _T , determined from fitting a Birch-Murnaghan equation of state (K ^′ _T =4) to the volume compression data is 218(1) GPa. The interoctahedral angles to not vary significantly with pressure, and the compression of the structure is entirely attributable to compression of the AlO₆ octahedra. The compressibilities of the constituent AlO₆ and ScO₁₂ are well matched: β_Al−O=1.6×10⁻³ GPa⁻¹ and β_Sc−O=1.5×10⁻³ GPa⁻¹. Therefore the distortion of the structure shows no significant change with increasing pressure. Received: 18 August 1997 / Revised, accepted: 11 November 1997     

Compressibility of strontium orthophosphate Sr<Subscript>3</Subscript>(PO<Subscript>4</Subscript>)<Subscript>2</Subscript> at high pressure

High pressure in situ synchrotron X-ray diffraction experiment of strontium orthophosphate Sr₃(PO₄)₂ has been carried out to 20.0 GPa at room temperature using multianvil apparatus. Fitting a third-order Birch–Murnaghan equation of state to the P–V data yields a volume of V ₀ = 498.0 ± 0.1 Å³, an isothermal bulk modulus of K _T  = 89.5 ± 1.7 GPa, and first pressure derivative of K _T ′ = 6.57 ± 0.34. If K _T ′ is fixed at 4, K _T is obtained as 104.4 ± 1.2 GPa. Analysis of axial compressible modulus shows that the a-axis (K _a  = 79.6 ± 3.2 GPa) is more compressible than the c-axis (K _c  = 116.4 ± 4.3 GPa). Based on the high pressure Raman spectroscopic results, the mode Grüneisen parameters are determined and the average mode Grüneisen parameter of PO₄ vibrations of Sr₃(PO₄)₂ is calculated to be 0.30(2).     

High-temperature and high-pressure equation of state for the hexagonal phase in the system NaAlSiO4 – MgAl2O4

Thermal equation of state of an Al-rich phase with Na_1.13Mg_1.51Al_4.47Si_1.62O₁₂ composition has been derived from in situ X-ray diffraction experiments using synchrotron radiation and a multianvil apparatus at pressures up to 24 GPa and temperatures up to 1,900 K. The Al-rich phase exhibited a hexagonal symmetry throughout the present pressure–temperature conditions and the refined unit-cell parameters at ambient condition were: a=8.729(1) Å, c=2.7695(5) Å, V ₀=182.77(6) Å³ (Z=1; formula weight=420.78 g/mol), yielding the zero-pressure density ρ₀=3.823(1) g/cm³ . A least-square fitting of the pressure-volume-temperature data based on Anderson’s pressure scale of gold (Anderson et al. in J Appl Phys 65:1534–543, 1989) to high-temperature Birch-Murnaghan equation of state yielded the isothermal bulk modulus K ₀=176(2) GPa, its pressure derivative K ₀ ^′ =4.9(3), temperature derivative (?K _T /?T) _P =?0.030(3) GPa K^?1 and thermal expansivity α(T)=3.36(6)×10^?5+7.2(1.9)×10^?9 T, while those values of K ₀=181.7(4) GPa, (?K _T /?T) _P =?0.020(2) GPa K^?1 and α(T)=3.28(7)×10^?5+3.0(9)×10^?9 T were obtained when K ₀ ^′ was assumed to be 4.0. The estimated bulk density of subducting MORB becomes denser with increasing depth as compared with earlier estimates (Ono et al. in Phys Chem Miner 29:527–531 2002; Vanpeteghem et al. in Phys Earth Planet Inter 138:223–230 2003; Guignot and Andrault in Phys Earth Planet Inter 143–44:107–128 2004), although the difference is insignificant (<0.6%) when the proportions of the hexagonal phase in the MORB compositions (～20%) are taken into account.     

The P–V–T equation of state of CaPtO3 post-perovskite

Orthorhombic post-perovskite CaPtO₃ is isostructural with post-perovskite MgSiO₃, a deep-Earth phase stable only above 100 GPa. Energy-dispersive X-ray diffraction data (to 9.4 GPa and 1,024 K) for CaPtO₃ have been combined with published isothermal and isobaric measurements to determine its P–V–T equation of state (EoS). A third-order Birch–Murnaghan EoS was used, with the volumetric thermal expansion coefficient (at atmospheric pressure) represented by α(T) = α₀ + α₁(T). The fitted parameters had values: isothermal incompressibility, $ K_{{T_{0} }} $  = 168.4(3) GPa; $ K_{{T_{0} }}^{\prime } $  = 4.48(3) (both at 298 K); $ \partial K_{{T_{0} }} /\partial T $  = ?0.032(3) GPa K^?1; α₀ = 2.32(2) × 10^?5 K^?1; α₁ = 5.7(4) × 10^?9 K^?2. The volumetric isothermal Anderson–Grüneisen parameter, δ _T , is 7.6(7) at 298 K. $ \partial K_{{T_{0} }} /\partial T $ for CaPtO₃ is similar to that recently reported for CaIrO₃, differing significantly from values found at high pressure for MgSiO₃ post-perovskite (?0.0085(11) to ?0.024 GPa K^?1). We also report axial P–V–T EoS of similar form, the first for any post-perovskite. Fitted to the cubes of the axes, these gave $ \partial K_{{aT_{0} }} /\partial T $  = ?0.038(4) GPa K^?1; $ \partial K_{{bT_{0} }} /\partial T $  = ?0.021(2) GPa K^?1; $ \partial K_{{cT_{0} }} /\partial T $  = ?0.026(5) GPa K^?1, with δ _T  = 8.9(9), 7.4(7) and 4.6(9) for a, b and c, respectively. Although $ K_{{T_{0} }} $ is lowest for the b-axis, its incompressibility is the least temperature dependent.     

A thermodynamic theory of the Grüneisen ratio at extreme conditions: MgO as an example

The Grüneisen ratio, γ, is defined as γy=αK _TV/C_v. The volume dependence of γ(V) is solved for a wide range in temperature. The volume dependence of αK _T is solved from the identity (? ln(αK _T)/? ln V)T ≡ δ _T-K′. α is the thermal expansivity; K _T is the bulk modulus; C _V is specific heat; and δ _Tand K′ are dimensionless thermoelastic constants. The approach is to find values of δ _T and K′, each as functions of T and V. We also solve for q=(? ln γ/? ln V) where q=δ _T -K′+ 1-(? ln C _V/? ln V)T. Calculations are taken down to a compression of 0.6, thus covering all possible values pertaining to the earth's mantle, q=? ln γ/? ln V; δ _T=? ln α/? ln V; and K′= (?K _T/?P)T. New experimental information related to the volume dependence of δ _T, q, K′ and C _V was used. For MgO, as the compression, η=V/V ₀, drops from 1.0 to 0.7 at 2000 K, the results show that q drops from 1.2 to about 0.8; δ _T drops from 5.0 to 3.2; δ _T becomes slightly less than K′; ? ln C _V/? In V→0; and γ drops from 1.5 to about 1. These observations are all in accord with recent laboratory data, seismic observations, and theoretical results.