Finite-strain Equation Of State Research Articles

High-pressure single-crystal elasticity of corundum: Implication for multiple seismic structure of 660-km discontinuity

The complex multi-discontinuity structure at 660‐800 km depth is likely attributable to lateral mantle composition heterogeneities, which are closely related to the mid-ocean ridge basalts (MORB). To decipher the impact of varying composition on the seismic properties of MORB, detailed knowledge of the elasticity of candidate minerals is thus important. Here we employed Brillouin scattering coupled with diamond anvil cells to determine the single-crystal elasticity of corundum up to 14 GPa and 300 K. Using third-order finite strain equation of state, we calculate the pressure derivatives of adiabatic bulk modulus and shear modulus of corundum, which yields KS0’ = 3.8(1), G0’ = 1.8(1) with KS0 = 256(1) GPa and G0 = 163(1) GPa and ρ0 = 3.987(1) g/cm3. Combined with previous high-temperature data, the velocity and anisotropy of corundum have been calculated at 300 K or along normal geotherm. Our results are applied to model the density and velocity profiles of normal and alkali-depleted MORB. Our modeling demonstrates that varying the alkali content and temperature of MORB can significantly impact the discontinuity depth and velocity jump at 660–800 km depth. For normal MORB, reducing the temperature by 300 K from normal mantle geotherm results in a shift of the velocity jump from 670–710 to 650–710 km depth but hardly affects the magnitude of the velocity jump (5.8–6.8(3)% for VP and 10.0–10.8(5)% for VS). By contrast, in alkali-depleted MORB, the discontinuity will occur at a greater depth from 705–730 to 720–745 km depending on temperature with a VP jump of 3.8–4.6(2)% and VS jump of 6.3–7.4(4)%.

The seismically fastest chemical heterogeneity in the Earth's deep upper mantle—implications from the single-crystal thermoelastic properties of jadeite

Jadeite is a major mineral phase (up to 50 vol%) in the subducted sediments/crust with continental origin, which are one of the major heterogeneities and important enriched geochemical reservoirs (such as EM-1 and EM-2) for incompatible elements in the Earth's interior. Identifying and locating the enriched geochemical heterogeneities requires knowledge of the elastic properties of relevant mineral phases at high pressure-temperature conditions. Unfortunately, the single-crystal elastic properties of jadeite have never been measured at high-pressure conditions, partially due to its low crystal symmetry. In this study, we have experimentally determined the single-crystal elastic moduli of jadeite at high pressures for the first time up to 18 GPa at the ambient temperature condition using Brillouin spectroscopy. Fitting the third-order finite strain equation of state to the velocity-pressure data yields KS0′=3.9(1), G0′=1.09(4) with ρ0=3.302(5) g/cm3, KS0=138(3) GPa, and G0=84(2) GPa. In addition, we have also conducted synchrotron single-crystal X-ray diffraction experiments up to 25 GPa and 700 K. The fitting of a Holland-Powell type thermal-pressure Birch-Murnaghan equation of state yields KT0′=3.8(2) and α0=3.4(5) ×10−5 K−1. Based on the obtained thermoelastic parameters of jadeite, the density and seismic velocities of continent-derived sediments/crust are modeled at the depth range from 200 to 500 km. The seismic velocities of the subducted continental sediments/crust become extremely fast at depths greater than ∼300 km, up to 11.8% and 14.7% faster than the Vp and Vs of the ambient mantle, and 5.6% and 7.3% faster than the Vp and Vs of the subducted oceanic crust. The existence of even a small amount of the subducted continental sediments/crust can result in strong seismic anomalies in the Earth's interior.

The Orson Anderson Era of Mineral Physics at Lamont in the 1960s

From 1964 to the early 1970s, Orson Anderson led a research program at the Lamont Geological Observatory in the newly-emerging field of “mineral physics”. In collaboration with colleagues Edward Schreiber and Naohiro Soga, Orson exploited the techniques of physical acoustics to study the behavior of the sound velocities of minerals at elevated pressures and temperatures. This research program also included semi-empirical studies of the relationships between the bulk modulus and the molar volume of solids, the use of lattice dynamics to calculate the elastic moduli of cubic structures as a function of pressure to predict instabilities, and theoretical investigations of the Lagrangian and Eulerian formulations of finite strain equations of state.

High‐Pressure Single‐Crystal Elasticity and Thermal Equation of State of Omphacite and Their Implications for the Seismic Properties of Eclogite in the Earth's Interior

AbstractOmphacite is a major mineral phase of eclogite, which provides the main driving force for the slab subduction into the Earth's interior. We have measured the single‐crystal elastic moduli of omphacite at high pressures for the first time up to 18 GPa at ambient temperature using Brillouin spectroscopy. A least squares fit of the velocity‐pressure data to the third‐order finite strain equation of state yields KS0′ = 4.5 (3), G0′ = 1.6 (1) with ρ0 = 3.34 (1) g/cm3, KS0 = 123 (3) GPa, and G0 = 74 (2) GPa. In addition, the synchrotron single‐crystal X‐ray diffraction data have been collected up to 18 GPa and 700 K. The fitting to Holland‐Powell thermal‐pressure equation of state yields KT0′ = 4.6 (5) and α0 = 2.7 (8) × 10−5 K−1. Based on the obtained thermoelastic parameters of omphacite, the anisotropic seismic velocities of eclogite are modeled and compared with pyrolite between 200 and 500 km. The largest contrast between the eclogite and pyrolite in terms of seismic properties is observed between ~310 and 410 km.

Equation of state, phase stability, and phase transformations of uranium-6 wt. % niobium under high pressure and temperature

In-situ time-of-flight neutron diffraction experiments were conducted on the uranium-niobium alloy with 6 wt. % Nb (U–6Nb) at pressures up to 4.7 GPa and temperatures up to 1073 K. Upon static compression at room temperature, the monoclinic structure of U–6Nb (α″ U–6Nb) remains stable up to the highest experimental pressure. Based on the pressure-volume measurements at room temperature, the least-squares fit using the finite-strain equation of state (EOS) yields an isothermal bulk modulus of B0 = 127 ± 2 GPa for the α″-phase of U–6Nb. The calculated zero-pressure bulk sound speed from this EOS is 2.706 ± 0.022 km/s, which is in good agreement with the linear extrapolation of the previous Hugoniot data above 12 GPa for α″ U–6Nb, indicating that the dynamic response under those shock-loading conditions is consistent with the stabilization of the initial monoclinic phase of U-6Nb. Upon heating at ambient and high pressures, the metastable α″ U–6Nb exhibits complex transformation paths leading to the diffusional phase decomposition, which are sensitive to applied pressure, stress state, and temperature-time path. These findings provide new insight into the behavior of atypical systems such as U-Nb and suggest that the different U-Nb phases are separated by rather small energies and hence highly sensitive to compositional, thermal, and mechanical perturbations.

The Elastic Anisotropy Change Near the 410‐km Discontinuity: Predictions From Single‐Crystal Elasticity Measurements of Olivine and Wadsleyite

AbstractWe present a model of the single‐crystal elasticity of olivine with ~9–10 mol% Fe based on all available single‐crystal velocity measurements. A set of finite strain equations of state that account for thermoelastic effects is used for fitting the individual elastic moduli at various pressure‐temperature conditions. The same analysis is applied to reanalyze the experimentally determined single‐crystal elastic moduli of hydrous and anhydrous Fe‐bearing wadsleyite. Based on the obtained thermoelastic parameters of individual elastic moduli, we then calculated various elastic anisotropy indices of olivine and wadsleyite and extrapolated them to the pressure‐temperature conditions expected in the Earth's mantle. The results suggest that the elastic anisotropy in the Earth's upper transition zone (410–520 km depth) is likely below the seismic detection limit if it originates primarily from the lattice preferred orientation of wadsleyite crystals. On the other hand, the elastic anisotropy of olivine is still relatively high (~19–25%) at 410 km depth. Thus, if a mantle flow field induces lattice preferred orientation of olivine and wadsleyite near 410 km depth, a reduction of elastic anisotropy is expected across the 410‐km discontinuity.

Elasticity and phase transformation at high pressure in coesite from experiments and first-principles calculations

The crystal structure and equation of state of coesite (space group C2/c) and its high-pressure polymorph coesite-II (space group P21/n) under pressure have been studied using X-ray powder diffraction in a diamond-anvil cell (DAC) up to 31 GPa at room-temperature and first-principles calculations at 0 K up to 45 GPa. New diffraction peaks appear above 20 GPa, indicating the formation of coesite-II structure. The calculated enthalpies provide theoretical support for the pressure-induced phase transformation from coesite to coesite-II at ~21.4 GPa. Compared with coesite, the coesite-II structure is characterized by a “doubled” b-axis and the breakdown of the linear Si1-O1-Si1 angle in coesite into two distinct angles—one is ~176°, close to linear, whereas the other decreases by 22 to 158°. Coesite is very anisotropic with the a-axis the shortest and twice more compressible than the b- and c-axis. By comparison, coesite-II is not so anisotropic with similar compressibilities in its a-, b-, and c-axis. As analyzed by a third-order Eulerian finite strain equation of state, the bulk modulus of coesite at 21.4 GPa is 182.3 GPa, and that of coesite-II is 140.8 GPa, indicating that coesite-II is much more compressible than coesite. The existence of coesite-II in the coldest subduction zone will change the elasticity and anisotropic properties of the subducting materials dramatically.

High-pressure elasticity of sodium majorite garnet, Na2MgSi5O12

Garnet is the second most abundant mineral phase in the upper mantle and transition zone settings. The crystal structure of garnet is quite flexible and hence it is able to accommodate various cations, including large incompatible cation such as sodium. We used first-principles simulation based on density functional theory and two widely used approximations—local density approximation (LDA) and generalized gradient approximation (GGA)—to explore the crystal structure, equation of state, and elasticity of sodium bearing majorite garnet with Na 2 MgSi 5 O 12 stoichiometry at pressures relevant to the upper mantle and transition zone. We find that the pressure-volume results based on LDA can be explained by a Birch Murnaghan finite strain equation of state with V 0 = 1447.6 (± 0.1) A 3 , K 0 = 177.4 (± 0.4) GPa, and K ′ 0 = 3.93 (± 0.02). The results based on GGA can be explained by a Birch Murnaghan finite strain equation of state with V 0 = 1525.8 (± 0.2) A 3 , K 0 = 160.2 (± 0.4) GPa, and K ′ 0 = 3.96 (± 0.02). The full elastic moduli tensor for Na-majorite with tetragonal symmetry exhibits slight deviation from the cubic symmetry with C 11 C 33 , C 12 ~ C 13 , and C 44 ~ C 66 . The magnitude of the tetragonal strain also captures the slight deviation from the cubic symmetry. At pressures corresponding to the upper mantle and mantle transition zone, the compressional wave velocity, v P , and shear wave velocity, v S , for the Na-majorite garnet are fast compared to a wide variety of garnets such as pyrope, grossular, almandine, and majorite garnet. Although single-crystal anisotropy of Na-majorite is greater than pyrope, it is still low compared to the major mantle phases.

Elastic Properties of Tricalcium Aluminate from High‐Pressure Experiments and First‐Principles Calculations

The structure and elasticity of tricalcium aluminate (C3A) have been experimentally and theoretically studied. From high‐pressure X‐ray diffraction experiments, the bulk modulus of 102(6) and 110(3) GPa were obtained by fitting second‐ and third‐order finite strain equation of state, respectively. First‐principles calculations with a generalized gradient approximation gave an isotropic bulk modulus of 102.1 GPa and an isothermal bulk modulus of 106.0 GPa. The static calculations using the exchange‐correlation functional show an excellent agreement with the experimental measurements. Based on the agreement, accurate elastic constants and other elastic moduli were computed. The slight difference of behavior at high pressure can be explained by the infiltration of pressure‐transmitting silicone oil into structural holes in C3A. The computed elastic and mechanical properties will be useful in understanding structural and mechanical properties of cementitious materials, particularly with the increasing interest in the advanced applications at the nanoscale.

Shock compression of liquid silicates to 125 GPa: The anorthite‐diopside join

We determined the equation of state (EOS) of three silicate liquid compositions by shock compression of preheated samples up to 127 GPa. Diopside (Di; Ca2Mg2SiO6) at 1773 K, anorthite (An; CaAl2Si2O8) at 1923 K and the eutectic composition Di64An36 at 1673 K were previously studied by shock compression to 38 GPa. The new data extend the EOS of each composition nearly to the Earth's core‐mantle boundary. The previously reported anomaly at 25 GPa for Di64An36 eutectic was not reproduced; rather all data for this composition fit within error a straight line Hugoniot in particle velocity vs. shock velocity. Di also displays a linear Hugoniot consistent with ultrasonic data and a third‐order finite strain EOS. The full anorthite data set is complex; we examine a model with a transition between two structural states and a fourth‐order finite strain model excluding two points that may not display relaxed behavior. We also report an experiment on room‐temperature solid Di64An36 aggregate that clearly demonstrates increase upon compression of the Grüneisen parameter of this liquid, much as experiment and theory have shown for forsterite and enstatite liquids. We construct isentropes and isotherms from our Hugoniots using Mie‐Grüneisen thermal pressure and evaluate the model of ideal mixing of volumes. Volume may mix almost linearly at high temperature, but deviates strongly when calculated along an isotherm; it remains difficult to reach a firm conclusion. We compare the densities of liquids to lower mantle solids. Our results suggest that basaltic liquids rich in CaO and Al2O3 are notably denser than liquids in the MgO‐SiO2 binary and, subject to uncertainties in the behavior of FeO and in corrections for thermal pressure, such liquids may be the most likely candidates for achieving negative buoyancy in the lowermost mantle.

Equation of state and phase stability of mantle perovskite up to 140 GPa shock pressure and its geophysical implications

We performed shock wave experiments on a natural pyroxene with chemical composition close to (Mg0.92, Fe0.08)SiO3 and initial density of 3.06 g/cm3, between 48 and 140 GPa. The relationship of shock wave velocity us and particle velocity up can been described linearly by us = 3.76(±0.24) + 1.48(±0.07) up (km/s). The model Hugoniot for the assemblage of (Mg0.92, Fe0.08)O (Mw) + SiO2(St) is significantly different from the experimental data, excluding the possibility of chemical decomposition of perovskite to oxides during the shock compression. The Grüneisen parameter γ obtained by fitting the experimental data can be expressed by γ = γ0 (ρ0/ρ)q, where γ0 = 1.84(2) and q = 1.69(3). Using the third‐order Birch‐Murnaghan finite strain equation of state, the shock experimental data yield a zero‐pressure bulk modulus K0s = 260.1(9) GPa and its pressure derivative K′0s = 4.18(4), with ρ0 = 4.19 g/cm3. A comparison of the experimental Hugoniot densities of perovskite with the PREM density profile prefers a perovskite‐dominant lower mantle model.

Compression mechanisms and equations of state

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Angel R. J. and Ross N. L. 1996Compression mechanisms and equations of statePhil. Trans. R. Soc. A.3541449–1459http://doi.org/10.1098/rsta.1996.0057SectionRestricted accessArticleCompression mechanisms and equations of state R. J. Angel Google Scholar Find this author on PubMed Search for more papers by this author and N. L. Ross Google Scholar Find this author on PubMed Search for more papers by this author R. J. Angel Google Scholar Find this author on PubMed Search for more papers by this author and N. L. Ross Google Scholar Find this author on PubMed Search for more papers by this author Published:15 June 1996https://doi.org/10.1098/rsta.1996.0057AbstractThe derivations of equations of state to describe the volume-pressure variation of a solid are based upon certain assumptions about the properties of the solid. For finite strain equations of state, these assumptions include homogeneity and isotropy of the strain distribution in the sample, the continuous differentiability of the equation of state parameters with respect to extensive variables, and the assumption that terms involving higher-order powers of the finite strain do not contribute significantly to the free energy of the material. We examine these assumptions and demonstrate that, within the experimental uncertainties, crystalline solids with no or limited degrees of internal structural freedom compress in the manner predicted by finite strain equations of state, even though in some cases the assumptions involved in the derivation of the equation of state are demonstrably violated. In more complex structures with a larger number of degrees of structural freedom, a variety of behaviour is observed; most undergo continuous structural change with increasing pressure and the evolution of the volume with pressure again follows that predicted by the finite strain equations of state. However, a significant number of complex structures undergo changes in compression mechanism which, in some cases, result in significant deviations from the behaviour predicted by the equations of state.FootnotesThis text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Zhou X and Zhang Z (2020) Strain induced Rashba splitting in CH 3 NH 3 PbBr 3 organic–inorganic lead halide perovskite , AIP Advances, 10.1063/5.0020236, 10:8, (085210), Online publication date: 1-Aug-2020. Clarke S, Steele B, Kroonblawd M, Zhang D, Kuo I and Stavrou E (2019) An Isosymmetric High-Pressure Phase Transition in α-Glycylglycine: A Combined Experimental and Theoretical Study, The Journal of Physical Chemistry B, 10.1021/acs.jpcb.9b07313, 124:1, (1-10), Online publication date: 9-Jan-2020. Baxter S, Hester B, Wright B and Wilkinson A (2019) Controlling the Negative Thermal Expansion and Response to Pressure in ReO 3 -type Fluorides by the Deliberate Introduction of Excess Fluoride: Mg 1– x Zr 1+ x F 6+2 x , x = 0.15, 0.30, 0.40, and 0.50 , Chemistry of Materials, 10.1021/acs.chemmater.9b00592, 31:9, (3440-3448), Online publication date: 14-May-2019. Ballaran T (2010) Equations of State and Their Applications in Geosciences High-Pressure Crystallography, 10.1007/978-90-481-9258-8_12, (135-145), . Amulele G, Manghnani M, Marriappan S, Hong X, Li F, Qin X and Liermann H (2008) Compression behavior of WC and WC-6%Co up to 50 GPa determined by synchrotron x-ray diffraction and ultrasonic techniques, Journal of Applied Physics, 10.1063/1.2938024, 103:11, (113522), Online publication date: 1-Jun-2008. Mueller H, Lathe C and Schilling F (2005) Simultaneous determination of elastic and structural properties under simulated mantle conditions using multi-anvil device MAX80 Advances in High-Pressure Technology for Geophysical Applications, 10.1016/B978-044451979-5.50006-5, (67-94), . Oganov A, Brodholt J and Price G Ab initio theory of phase transitions and thermoelasticity of minerals Energy Modelling in Minerals, 10.1180/EMU-notes.4.4, (83-170) Knoche R, Webb S and Rubie D (1998) Measurements of acoustic wave velocities at P-T conditions of the Earth's mantle Properties of Earth and Planetary Materials at High Pressure and Temperature, 10.1029/GM101p0119, (119-128) Hugh-Jones D and Angel R (1997) Effect of Ca 2+ and Fe 2+ on the equation of state of MgSiO 3 orthopyroxene , Journal of Geophysical Research: Solid Earth, 10.1029/96JB03485, 102:B6, (12333-12340), Online publication date: 10-Jun-1997. Knoche R, Webb S and Rubie D (1997) Experimental determination of acoustic wave velocities at Earth mantle conditions using a multianvil press, Physics and Chemistry of the Earth, 10.1016/S0079-1946(97)00088-8, 22:1-2, (125-130), Online publication date: 1-Jan-1997. Angel R and Ross N (1997) Equations of state of mantle minerals from high-pressure diffraction, Physics and Chemistry of the Earth, 10.1016/S0079-1946(97)00087-6, 22:1-2, (119-123), Online publication date: 1-Jan-1997. This Issue15 June 1996Volume 354Issue 1711 Article InformationDOI:https://doi.org/10.1098/rsta.1996.0057Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online01/01/1997Published in print15/06/1996 License:Scanned images copyright © 2017, Royal Society Citations and impact

Shock wave equation of state and finite strain theory

The linear shock‐velocity (Us), particle‐velocity (up) relation, which is known to be exceptionally successful in describing a wide variety of Hugoniot equation of state measurements, is shown to be virtually indistinguishable from the Birch‐Murnaghan (third‐order Eulerian finite strain) equation of state. For typical values of zero‐pressure parameters, specifically for the pressure derivative of the adiabatic bulk modulus K′0S ≃ 3 to 6, the Grüneisen parameter γ0 ≃ 1.5 (±1.0), and the logarithmic volume derivative of the Grüneisen parameter q ≃ 1.0 (±1.0), the Eulerian finite strain formulation yields the appropriate compressional moduli at infinitesimal strains, and it closely reproduces the Hugoniot at large compressions when compared with the linear Us‐up relation. Thus shock wave measurements provide strong empirical support for the success of the Eulerian finite strain equation of state in describing the compression of condensed matter to high pressures.

Compressibility and stability of icosahedral Al-Mn up to 28 GPa.

The equation of state of icosahedral ${\mathrm{Al}}_{86}$${\mathrm{Mn}}_{14}$ (i-Al-Mn) at room temperature has been determined up to \ensuremath{\sim}28 GPa using diamond-anvil x-ray powder-diffraction techniques. The i-Al-Mn remained icosahedral over this pressure range; no phase transition was observed. The measured compression of the two strongest observed i-Al-Mn diffraction peaks (100000) and (110001), was isotropic within the measurement error: d(110001)/d(100000)=1.776\ifmmode\pm\else\textpm\fi{}0.006. The pressure-volume data were least-squares fit to a finite-strain equation of state to determine the zero-pressure bulk modulus, ${K}_{0}$=117.6\ifmmode\pm\else\textpm\fi{}16.8 GPa, and its pressure derivative, ${K}_{0}^{\mathcal{'}}$=6.0\ifmmode\pm\else\textpm\fi{}2.4. .AE

Static compression of iron to 78 GPa with rare gas solids as pressure‐transmitting media

New data are presented for the room temperature, static compression of iron to 78 GPa with solid neon and argon as pressure‐transmitting media. X ray diffraction studies have been performed on a geophysically relevant material, for the first time to such pressures under quasihydrostatic conditions, in a diamond anvil cell. The hydrostatic technique leads to increased precision in the measurement of high pressures and has placed closer constraints on the equation of state of ε iron. From a linear least squares fit of a finite strain equation of state to the present data combined with earlier, low‐pressure data for metastable ε iron, the preferred values for the zero‐pressure isothermal bulk modulus, K0, and first pressure derivative, K0′, are 192.7 (±9.0) GPa and 4.29 (±0.36), respectively. The zero‐pressure volume for the ε phase is 6.687 (±0.018) cm3/mol. On the basis of the pressure‐volume curve calculated from fits of the finite strain equation of state, ε iron appears to be less compressible under nonhydrostatic conditions, but the differences are within the error of the nonhydrostatic experiment. The results also confirm that the absence of a soft medium in static compression experiments with the diamond anvil cell results in an overestimate of the unit cell volume (measured with the incident X ray beam parallel to the load axis) for pressures calculated with the nonhydrostatic ruby calibration scale. It is found that for ε iron, substantial compensation for this nonhydrostatic effect is implicit in the nonhydrostatic ruby pressure scale up to intermediate strains. The hydrostatic data and the ε iron isotherm derived from shock wave experiments on iron samples are in very close agreement.

An evaluation of finite strain equations of state using a lattice model

Analytic expressions have been derived for ρ ρ 0 , V p V b0 and V s V b0 as functions of p K 0 for rigid ion models of the NaCl, CsCl and ZnS structures. In this dimensionless form, the only independent parameter in the lattice model is the repulsive parameter n (or equivalently K′ 0). Analytic first and second pressure derivatives of the bulk and shear moduli calculated from the lattice model were used as parameters in Murnaghan and Birch equations of state (EOS) of first and second order. Comparison between the various finite strain equations and the lattice model for the three crystal structures leads to the following general conclusions. (1) For each finite strain EOS, V p is in better agreement with the lattice model than V s; (2) the Birch EOS is in better agreement with the lattice model than is the Murnaghan EOS of comparable order; (3) no first-order EOS is capable of extrapolating V s to p K 0 > 0.2 without deviating by more than 5% from the lattice model, and (4) the second-order Murnaghan EOS gives very little improvement over the first-order equations, while the second-order Birch EOS allows V s to be extrapolated to p K 0 ≈ 0.8 without deviating from the lattice model by more than 5%. For close-packed oxides and silicates this covers the pressure range throughout the lower mantle.

Quasi-harmonic finite strain equations of state of solids

Thomsen's ‘fourth-order anharmonic’ theory, which explicitly evaluates thermal effects in finite strain equations of elasticity according to the fourth-order approximation in lattice dynamics, is reconsidered for the special case of isotropic stresses and strains. It is shown that the approximations made in the finite strain theory are independent from those made in the lattice dynamics theory, with the result that strain dependence may be described in terms of any frame-indifferent strain tensor, not just the ‘Lagrangian’ strain tensor, η, and that the finite strain expansions may be taken to any order, not just the fourth. This result is valid for general stresses and strains. Illustrative pressure-volume equations are derived in terms of three strain measures, including η and the frame-indifferent analogue, E, of the ‘Eulerian’ strain tensor, ε. The reference state is here left arbitrary, rather than identifying it with the ‘rest’ state. This results in greater convenience in applying the equations. Not being restricted to fourth order, the present equations do not depend for their application on knowing the second pressure derivatives of the bulk modulus. Expressions are obtained for isentropes and Hugoniots in terms of the same parameters as enter the original equations, which have the form of isotherm. Ultrasonic, thermal expansion and calorimetric data for MgO are used to evaluate the parameters of third-order equations of state of MgO. The equations of state are tested and refined with Hugoniot data. The third-order ‘ E’ Hugoniot is much closer to the data than the third-order ‘η’ Hugoniot. Inclusion of fourth-order terms allows both ‘ E’ and ‘ρ’ Hugoniots to fit the data within their scatter. The separation of Hugoniots corresponding to different initial densities is predicted within the accuracy of the data by the thermal part of this theory.

Quasi-harmonic finite strain equations of state of solids

Quasi-harmonic finite strain equations of state of solids

Approximate Compressibility of Elements and Compounds

The zero-pressure incompressibilities and volumes per atom for metals and solid nonmetals are found to fit the relation $({K}_{0}{Z}^{\frac{\ensuremath{-}10}{3}}){(Z{v}_{0})}^{\frac{7}{3}}=(\frac{{\ensuremath{\pi}}^{\frac{10}{3}}{6}^{\frac{2}{3}}}{15})a_{0}^{}{}_{}{}^{3}{e}^{2}$ to good accuracy. The relation is derived by an empirical attempt to interpolate between the finite-strain equation of state and the Thomas-Fermi equation of state. The strain energy is quadratic in the finite strain for the heavier alkali metals but higher order terms are important for the silicates, at least to twofold compression.