Temperature Derivative Of Bulk Modulus Research Articles

Compressibility and thermoelasticity of CrN

ABSTRACT We report experimental determination of compressibility and thermoelasticity for CrN based on in situ high pressure/high temperature synchrotron x-ray diffraction measurements. The pressure-induced cubic-to-orthorhombic phase transition is observed at ∼ 5 GPa in high-quality CrN samples synthesized using a high pressure reaction route. Equation of state, bulk modulus and axial compressibility of this material are obtained by analysis of high pressure x-ray diffraction data collected at room temperature. Both cubic and orthorhombic phases have a similar bulk modulus value of 257 (5) and 262 (6) GPa, respectively. For orthorhombic CrN, the b-axis has the most axial incompressibility, which should be associated with magnetic properties. In addition, the isothermal equation of states is also determined at different temperatures of 300, 400, 600, 800, 1000, and 1200 K, respectively. Some of the important thermoelastic properties of cubic CrN are thus derived, including temperature derivative of bulk modulus and volumetric thermal expansivity where a 0 = 1.95 (16) × 10−5 K−1 and a 1 = 4.39 (3) × 10−9 K−1.

Revisit of Pressure-Induced Phase Transition in PbSe: Crystal Structure, and Thermoelastic and Electrical Properties

Lead selenide, PbSe, an important lead chalcogenide semiconductor, has been investigated using in-situ high-pressure/high-temperature synchrotron X-ray diffraction and electrical resistivity measurements. For the first time, high-quality X-ray diffraction data were collected for the intermediate orthorhombic PbSe. Combined with ab initio calculations, we find a Cmcm, InI-type symmetry for the intermediate phase, which is structurally more favorable than the anti-GeS-type Pnma. At room temperature, the onset of the cubic-orthorhombic transition was observed at ∼3.5 GPa with a ∼3.4% volume reduction. At an elevated temperature of 1000 K, the reversed orthorhombic-to-cubic transition was observed at 6.12 GPa, indicating a positive Clapeyron slope for the phase boundary. Interestingly, phase-transition induced elastic softening in PbSe was also observed, which can be mainly attributed to the loosely bonded trigonal prisms along the b-axis in the Cmcm structure. In a comparison with the cubic phase, orthorhombic PbSe exhibits a large negative pressure dependence of electrical resistivity. In addition, thermoelastic properties of orthorhombic PbSe have been derived from isothermal compression data, such as the temperature derivative of bulk modulus and thermally induced pressure.

Temperature and pressure effects of multiferroic Bi2NiTiO6 compound

Bi2NiTiO6 compound which shows both magnetic (TM = 58 K) and ferroelectric properties (TC = 513 K) was synthesized under high pressure of 5 GPa and temperature of 1273 K. The crystal structure, as determined by X-ray powder diffraction and neutron powder diffraction, is a distorted A(B1B2)O3 type perovskite with space group Pn21a. Structural evolution of multiferroic Bi2NiTiO6 shows that there are two isostructural phase transitions at ∼2 GPa and ∼15 GPa under high pressure and at room temperature and indicates that isostructural phase transitions occurred with temperature higher than 823 K under ambient condition. All the isostructural phase transitions come from the Bi ion discontinuous shift, which identifies the phase transition at ∼15 GPa and at temperature higher than 823 K are the same. Using a modified high-T Birch-Murnaghan equation of state and a thermal-pressure approach, we have derived the thermoelastic parameters of high pressure phase Bi2NiTiO6, including the ambient bulk modulus K0, temperature derivative of bulk modulus at constant pressure, volumetric thermal expansivity, pressure derivative of thermal expansion, and temperature derivative of bulk modulus at constant volume.

Thermal equations of state and phase relation of PbTiO3: A high P-T synchrotron x-ray diffraction study

The phase relation of tetragonal and cubic PbTiO3 and their unit-cell parameters have been determined by synchrotron x-ray diffraction at pressures up to 7.8 GPa and temperatures up to 1074 K with a cubic anvil apparatus. From these measurements, a pressure-temperature phase boundary between the tetragonal and cubic phases has been established. With increasing temperature or pressure, the c/a ratio of the ferroelectric, tetragonal PbTiO3 becomes closer to unity, suggesting that both heating and compression favor the paraelectric, cubic structure. Using a modified high-T Birch-Murnaghan equation of state and a thermal-pressure approach, we have derived the thermoelastic parameters of tetragonal and cubic PbTiO3, including the ambient bulk modulus K0, temperature derivative of bulk modulus at constant pressure, volumetric thermal expansivity, pressure derivative of thermal expansion, and temperature derivative of bulk modulus at constant volume. Our obtained K0 value for tetragonal PbTiO3 is consistent with previously reported results, while that for cubic PbTiO3 is smaller than earlier results probably due to differences in the experimental techniques used (cubic anvil apparatus versus diamond anvil cell) and related stress conditions of the samples. All other thermoelastic parameters for both tetragonal and cubic PbTiO3 have been determined for the first time. Compared with previous high-temperature data at atmospheric pressure, our P-V-T dataset for tetragonal PbTiO3 infers a pressure-induced crossover in volumetric thermal expansion from negative to positive between 0 and 1 GPa, an phenomenon that is of fundamentally interest and practically important.

Incompressibility of osmium metal at ultrahigh pressures and temperatures

Osmium is one of the most incompressible elemental metals, and is used as a matrix material for synthesis of ultrahard materials. To examine the behavior of osmium metal under extreme conditions of high pressure and temperature, we measured the thermal equation of state of osmium metal at pressures up to 50 GPa and temperatures up to 3000 K. X-ray diffraction measurements were conducted in the laser heated diamond anvil cell at GeoSoilEnviroCARS and the High Pressure at the Advanced Photon Source and beamline 12.2.2 at the advanced light source. Ambient temperature data give a zero pressure bulk modulus of 421 (3) GPa with a first pressure derivative fixed at 4. Fitting to a high temperature Birch–Murnaghan equation of state gives a room pressure thermal expansion of 1.51(0.06)×10−5 K−1 with a first temperature derivative of 4.9(0.7)×10−9 K−2 and the first temperature derivative of bulk modulus of be dK0/dT=−0.055 (0.004). Fitting to a Mie–Grüneisen–Debye equation of state gives a Grüneisen parameter of 2.32 (0.08) with a q of 7.2 (1.4). A comparison of the high pressure, temperature behavior among Re, Pt, Os, shows that Os has the highest bulk modulus and lowest thermal expansion of the three, suggesting that Os-based ultrahard materials may be especially mechanically stable under extreme conditions.

Phase Transition and EOS of Marmatite (Zn0.76Fe0.23S) upto 623 K and 17 GPa

In situ energy dispersive x-ray diffraction for natural marmatite(Zn0.76Fe0.23S) is performed up to 17.7 GPa and 623 K. Itis fitted by the Birch–Murnaghan equation of state (EOS) that K0 andα0 for marmatite are 85(3)GPa and 0.79(16)×10−4 K−1, respectively. Fe2+ isomorphic replacing toZn2+ in natural crystal is responsible for high bulk modulus andthermal expansivity of marmatite. Temperature derivative of bulk modulus(∂K/∂T)P for marmatite is fitted to be-0.044(23) GPaK−1. The unambiguous B3–B1 phase boundaries formarmatite are determined to be Pupper(GPa) = 15.50−0.016T(°C) and Plower(GPa)=9.94–0.012T(°C) at 300–623 K.

P– V– T equation of state of MgSiO 3 perovskite and MgO periclase: Implication for lower mantle composition

The P– V– T equation of state (EOS) for magnesium silicate perovskite (MgSiO 3) and periclase (MgO) was obtained by fitting measured P, V, T data. The EOS of MgSiO 3 perovskite was determined by applying the Mie–Grüneisen–Debye approach, based on three alternative bulk modulus values ( K T0 = 240, 250 and 261 GPa), yielding substantially different values for high-order thermoelastic parameters such as the Grüneisen parameter ( γ) and the temperature derivative of bulk modulus ((∂ K T /∂ T) P ). Combined with the EOS of magnesiowüstite, we calculated the density ( ρ), bulk modulus ( K S) and seismic parameter ( ϕ) for simplified pyrolitic lower mantle along a plausible adiabat (temperature at 670 km, T a = 1900 K), and compared these values to those of a seismological model (PREM). The results based on K T0 of 261 GPa, which has been commonly used, showed good agreement with PREM, within 0.5% for ρ, and 2–3% for K S and ϕ, throughout the lower mantle. The EOS obtained using the recent K T0 data (240–250 GPa), substantially overestimated K S and ϕ by 5–10%, requiring a progressive enrichment in (Mg,Fe)O with increasing depth. In addition, a more realistic vibrational density of states (VDOS) model (Kieffer-type model) was applied to calculate the EOS. The effect of VDOS on thermoelastic parameters was not large enough to modify those conclusions.

Thermal equation-of-state of osmium: a synchrotron X-ray diffraction study

The pressure–volume–temperature behavior of osmium was studied at pressures and temperatures up to 15 GPa and 1273 K. In situ measurements were conducted using energy-dispersive synchrotron X-ray diffraction in a T-cup 6–8 high pressure apparatus. A fit of room-temperature data by the third-order Birch–Murnaghan equation-of-state yielded isothermal bulk modulus K 0=435(19) GPa and its pressure derivative K′ 0=3.5(0.8) GPa. High-temperature data were analyzed using Birch–Murnaghan equation of state and thermal pressure approach. The temperature derivative of bulk modulus was found to be −0.061(9) GPa K −1. Significant anisotropy of osmium compressibility was observed.

Thermal equation of state of (Mg0.91Fe0.09)2SiO4 ringwoodite

In situ synchrotron X-ray diffraction (XRD) experiments were conducted using the SPEED-1500 multi-anvil press of SPring-8 on (Mg0.91Fe0.09)2SiO4 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) Å3 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 (∂KT/∂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 (∂KT/∂T)P are consistent with previous studies on Mg2SiO4 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 Mg2SiO4 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.

Thermal equation of state of (Mg 0.91Fe 0.09) 2SiO 4 ringwoodite

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 V 0=527.83(7) Å 3 and isothermal bulk modulus K T0 =187 GPa yielded a pressure derivative of isothermal bulk modulus K T ′=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 K T ′ 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.

Thermal equation of state of garnets along the pyrope-majorite join

P-V-T relations of two garnet samples along the pyrope (Py)-majorite (Mj) join (Py n Mj 1− n , where subscripts indicate molar percent) were measured at pressure and temperature conditions up to 11 GPa and 1163 K, respectively, with energy-dispersive synchrotron X-ray diffraction in a cubic-anvil, DIA-6 type apparatus (SAM-85). For each volume measurement, non-hydrostatic stress was determined from relative shifts of the diffraction lines of NaCl, within which the sample was embedded. Heating to ∼ 1100 K reduced the strength of NaCl below 0.1 GPa, making the measurements nearly hydrostatic. The recovered samples were examined by transmission electron microscopy (TEM) to detect irreversible changes that may affect data quality. For the cubic garnet Py 62Mj 38, no irreversible changes took place during the experiment. A fit using the third-order Birch-Murnaghan equation of state yielded ambient cell volume V 0 = 1509.2(3) Å 3 (113.62 cc/mol), isothermal bulk modulus K T0 = 160(3) GPa, and its pressure derivative K′ T0 = 4.9(5). Various thermal equations-of-state analyses gave consistent results for the temperature derivative of bulk modulus ( ∂K T ∂T ) P = −0.020(1) GPa K −1 pressure derivative for thermal expansion ( ∂α ∂P ) T = −7.8 × 10 −7 GPa −1 K −1 , with the ambient volumetric thermal expansion α 0 = 2.5 × 10 −5 K −1. For the tetragonal sample Py 21Mj 79, TEM indicated that micro-twin domains were significantly coarsened after the P-V-T experiment. Stress relaxation may have occurred during the coarsening, thereby compromising the P-V-T data. In addition, the tetragonal distortion in this sample was too small to be resolved by the energy-dispersive technique. Therefore, data for the latter sample were not suitable for accurate equation-of-state analysis. Previous data on pyrope were compiled, analyzed, and compared with the new data on the majoritic garnet. The entire data set enabled us to examine systematics of the thermoelastic properties of garnets along the majorite-pyrope join. These data will have important applications for modeling the transition zone of the mantle. Published by Elsevier Science B.V.

Thermoelastic Equation of State of Monoclinic Pyroxene: CaMgSi2O6 Diopside.

We have conducted an in situ synchrotron x-ray diffraction study on a monoclinic pyroxene: CaMgSi2O6 diopside at simultaneous high pressures and high temperatures up to 8. 2 GPa and 1280 K. A modified Rietveld profile refinement program has been applied to refine the diffraction spectra of low symmetry and multiple phases observed in energy dispersive mode. Thermoelastic parameters for diopside CaMgSi2O6 are derived by fitting the P-V-T data to the high-T Birch-Murnaghan equation of state. We obtained isothermal bulk modulus KTO =109. 1 GPa with a pressure derivative of bulk modulus K'=∂K/∂P=4. 84, temperature derivative of bulk modulus K=∂K/∂T=-2. 05×10-2 GPa/K, volumetric thermal expansivity α=a+ bT with values of a=2. 32×10-5K-1 and b=1. 88×10-8K-2.

Thermoelastic equation of state of jadeite NaAlSi2O6: An energy‐dispersive Reitveld Refinement Study of low symmetry and multiple phases diffraction

We report the first measurement of a complete set of thermoelastic equation of state of a clinopyroxene mineral. We have conducted an in situ synchrotron x‐ray diffraction study of jadeite at simultaneous high pressures and high temperatures. A modified Rietveld profile refinement program has been applied to refine the diffraction spectra of low symmetry and multiple phases observed in energy dispersive mode. Unit cell volumes, measured up to 8.2 GPa and 1280 K, are fitted to a modified high‐temperature Birch‐Murnaghan equation of state. The derived thermoelastic parameters of the jadeite are: bulk modulus K=125 GPa with assumed pressure derivative of bulk modulus K′ = ∂K/∂P = 5.0, temperature derivative of bulk modulus , and volumetric thermal expansivity α = a + bT with values of a=2.56×10−5K−1 and b=0.26x10−8 K−2. We also derived thermal Grüneisen parameter γth=1.06 for ambient conditions; Anderson‐Grüneisen parameter δTo=5.02, and pressure derivative of thermal expansion ∂α/∂P = −1.06×10−6 K−1 GPa−1. From the P‐V‐T data and the thermoelastic equation of state, thermal expansions at five constant pressures of 1.0, 2.5, 4.0, 5.5, and 7.5 GPa are calculated. The derived pressure dependence of thermal expansion is: Δα/ΔP = −0.97×10−6K−1GPa−1, in good agreement with the thermodynamic relations.

A high P-T single-crystal X-ray diffraction study of thermoelasticity of MgSiO3 orthoenstatite

P-V-T data of MgSiO3 orthoenstatite have been measured by single-crystal X-ray diffraction at simultaneous high pressures (in excess of 4.5 GPa) and temperatures (up to 1000 K). The new P-V-T data of the orthoenstatite, together with previous compression data and thermal expansion data, are described by a modified Birch-Murnaghan equation of state for diverse temperatures. The fitted thermoelastic parameters for MgSiO3 orthoenstatite are: thermal expansion ∂α/∂P with values of a=2.86(29)×10-5 K-1 and b=0.72(16)×10-8 K-2; isothermal bulk modulus K T o =102.8(2) GPa; pressure derivative of bulk modulus K′=∂K/∂P=10.2(1.2); and temperature derivative of bulk modulus K=∂K/∂T=-0.037(5) GPa/K. The derived thermal Gruneisen parameter is γ th=1.05 for ambient conditions; Anderson-Gruneisen parameter is δ T o =11.6, and the pressure derivative of thermal expansion is ∂α/∂P=-3.5×10-6K-1 GPa-1. From the P-V-T data and the thermoelastic equation of state, thermal expansions at two constant pressures of 1.5 GPa and 4.0 GPa are calculated. The resulting pressure dependence of thermal expansion is Δα/ΔP=-3.2(1)× 10-6 K-1 GPa-1. The significantly large values of K′, K, δ T and ∂α/∂P indicate that compression/expansion of MgSiO3 orthoenstatite is very sensitive to changes of pressure and temperature.

In situ high P‐T X ray diffraction studies on three polymorphs (α, β, γ) of Mg2SiO4

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.

Temperature variation of elastic constants of pyrope-almandine garnets.

Elastic constants of three natural pyrope-almandine garnets (Fe/(Mg+Fe)=0.18, 0.48 and 0.58) were measured by an RPR method in the temperature range between -190° and 80°C. The temperature derivative of bulk modulus ∂Ks/∂T does not depend much on the composition, but those of shear constants ∂C44/∂T and∂Cs/∂T [Cs=(C11-C12)/2] increase considerably with increase in the Fe-content. The temperature derivatives of the elastic constants (kbar/deg) of pyrope and almandine at room temperature are determined by extrapolation: From the measured elastic parameters of garnet, the material of the high velocity layer (Vp=8.6km/s and Vs=4.9km/s) in the oceanic lithosphere is interpreted as eclogite or garnet peridotite constituted dominantly of garnet.