Donald G. Isaak

Anharmonicity and the equation of state for gold

The temperature dependence of the thermodynamic and the elastic properties of elemental gold are found from published data. It is shown that measurements for (∂KT/∂P)T near 5.5 are more thermodynamically consistent than are higher values of this parameter which have been reported earlier. Using 5.5 for (∂KT/∂P)T, we find that (∂KT/∂T)V is not zero but −11.5×10−3 GPa K−1 for high temperatures (T>θD, where θD is the Debye temperature). One consequence of this is that above θD the thermal pressure, PTH, along the room‐pressure isobar can be expressed as PTH(T)−PTH(300)=[7.14×10−3 +(∂K T/∂T)v ln(Va/V)] ×(T−300) GPa for T at absolute temperature and Va being the volume at ambient conditions. These results give thermal pressure values near those previously reported at small compressions, but give lower thermal pressures at large compressions. This study suggests that in order to ensure thermodynamic consistency, the value of d ln γ/d ln V is near 2.5–3.0 which is higher than values of 1.0 and 1.7 reported previ...

High‐temperature elastic constant data on minerals relevant to geophysics

The high-temperature measurements of elastic constants and related temperature derivatives of nine minerals of interest to geophysical and geochemical theories of the Earths 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 Gruneisen 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.

High-temperature elasticity of iron-bearing olivines

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 1500 K. The two specimens contain 8 and 9 modal percent fayalite, which required the oxygen fugacity be controlled at high temperature to preserve their chemical stability. The rectangular parallelepiped resonance apparatus was adapted to buffer the specimens from the atmosphere with a mixture of CO and CO2 gas. A small increase (∼1–2 GPa) in the adiabatic bulk modulus of each specimen, over that of end-member forsterite, was found. 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. Neither specimen exhibits measurable nonlinear temperature dependence in the computed isotropic bulk and shear moduli, which is in contrast to published forsterite data. The temperature derivatives of the isotropic bulk modulus KS are (−1.69, −1.80) × 10−2 GPa K−1 for the two olivine specimens, and the shear modulus G derivatives are (−1.38, −1.36) × 10−2 GPa K−1. These derivatives are only slightly larger in magnitude than |(∂KS/∂T)P| = 1.56 × 10−2 and |(∂G/∂T)P| = 1.30 × 10−2 GPa 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. Several dimensionless parameters have been calculated from these results and are discussed in view of systematics which bear on high-pressure phases in Earths transition zone. One result from these systematics related to the seismic velocities in the Earth, and especially the shear wave velocities, is that an olivine content of less than 50% is implied at the 400-km discontinuity if Earths upper mantle is isochemical. Furthermore, the substitution of almost 10% iron for magnesium at the forsterite end of the olivine solid solution series has little effect on the dimensionless parameters or on the temperature derivatives of the elastic moduli at high temperature.

Measured elastic moduli of single-crystal MgO up to 1800 K

Using the rectangular parallelepiped resonance method we measured the temperature dependence of the adiabatic elastic moduli of single-crystal MgO over the temperature range 300–1800 K. The high temperature limit of our measurements extends by 500 K the upper limit over which elasticity data on MgO are now available. Although our measured temperature dependence of Cijsare generally in good agreement with previous measurements over a more narrow range in temperature, we found that C44sdecreases more rapidly with temperature, for T > 1000 K, than previous studies suggest. We also found that each of the slopes (ϱC11s/ϱT)p, (ϱKs/ϱT)p, and (C44s/ϱT)p become less negative with increasing temperature for T > 1400 K. From our measurements on elasticity we are able to confirm that the Grüneisen parameter at zero pressure is nearly constant with temperature up to 1800 K, with only a slight decrease above 1000 K. Utilizing our new data we present calculations showing the temperature dependence of thermodynamic parameters important in studies of earths interior.

The dependence of the Anderson-Grüneisen parameter δT upon compression at extreme conditions

Abstract The parameter δt is a dimensionless thermoelastic parameter important in thermodynamic studies involving high temperature at high pressure. We give a brief history of attempts to define δT in terms of fundamental interatomic potentials. We show that ab initio calculations provide a way to find δT(η, T) (where η = V V 0 ), since it is possible to find the minimum in the isochoric BT vs T data arising from the ab initio analysis (BT is the isothermal bulk modulus). The method is demonstrated for MgO, where δT is calculated over a wide η, T field. We find δT decreases at high compression, but is independent of T. Knowledge of δT(η, T) is important to find the high P-high T behavior of a number of important properties in thermal physics, including entropy, thermal expansivity, thermal pressure and the Gruneisen parameter.

A model for the computation of thermal expansivity at high compression and high temperatures: MGO as an example

We compute the value of the thermal expansivity α over a wide range of compression (η ≡ V/V0, 0.6 ≤ η ≤ 1.0) and temperature (300 K ≤ T ≤ 2500 K). Three methods are combined to find α. We utilize: 1) the high P,T data base from the PIB ab initio model for MgO, which specifies the Helmholtz energy and the volume and temperature derivatives; 2) the measured thermal expansivity and its various pressure and temperature derivatives; and 3) several thermodynamic identities relating T and V dependence of various physical properties. We present a simple equation relating α to η along isochores and suggest values of α. for MgO over V, P, T conditions, including those of the earths lower mantle. The parameters in the equation are evaluated by using the ab initio data base. We find that α varies from about 1.40αa to 0.40αa along a geotherm through the upper and lower mantle, where αa is α at ambient conditions.

High-temperature thermal expansion and elasticity of calcium-rich garnets

We present new high temperature elasticity data on two grossular garnet specimens. One specimen is single-crystal, of nearly endmember grossular, the other is polycrystalline with about 22% molar andradite. Our data extend the high temperature regime for which any garnet elasticity data are available from 1000 to 1350 K and the compositional range of temperature data to near endmember grossular. We also present new data on the thermal expansivity of calcium-rich garnet. We find virtually no discernable differences in the temperatureT derivatives at ambient conditions of the isotropic bulkKS and shearμ moduli when comparing our results between these two specimens. These calcium-rich garnets have the lowest values of ¦(∂KS/∂T)P¦ = (1.47,1.49) x 10-2GPa/K, and among the highest values of ¦(∂μ/∂T)P¦ = 1.25 x 10-2GPa/K, when compared with other garnets. Small, but measurable, nonlinear temperature dependences of most of the elastic moduli are observed. Several dimensionless parameters are computed with the new data and used to illustrate the effects of different assumptions on elastic equations of state extra-polated to high temperatures. We discuss how dimensionless parameters and other systematic considerations can be useful in estimating the temperature dependence of some properties of garnet phases for which temperature data are not yet available. While we believe it is premature to quantitatively predict the temperature variation ofKS andμ for majorite garnets, our results have bearing on the amount of diopside required to explain the shear velocity gradients in Earths transition zone.

Measurement of elastic properties of single-crystal CaO up to 1200 K

The elastic moduli of a single-crystal calcium oxide, CaO, are measured in the temperature range from 300 to 1200 K (1.8 times of the Debye temperature) by the resonant sphere technique (RST). The lowest 18 modes are identified in the frequency range from 0.6 to 1.4 MHz for the vibrating spherical specimen, which is 5.6564 mm in diameter and 3.3493 g/cm3 in density at room temperature, and the resonant frequencies are traced as a function of temperature. The adiabatic elastic moduli are determined in the present temperature range from the observed frequencies by inversion calculations. Most of the elastic moduli, except forC12 modulus, decrease as temperature increases. The temperature curves ofCs andC44 moduli cross at 372 K. This means that the CaO specimen has an isotropic elasticity at the temperature. The temperature derivatives (∂C11/∂T)P and (∂Cs/∂T)P become slightly less negative with temperature increase and (∂Cs/∂T)P and (∂C44/∂T)P are almost constant. Combining the present elastic data with thermal expansion and specimen heat capacity data of CaO, we present the temperature dependence of thermodynamic parameters important in the studies of earths interior.

The relationship between shear and compressional velocities at high pressures: Reconciliation of seismic tomography and mineral physics

The value of the parameter ν from low pressure laboratory measurements disagrees with the value from seismic observations. The parameter ν relates the isobaric change in shear velocity Vs to the change in compressional velocity Vp. Seismic evidence indicates ν exceeds 2.0 in the lower mantle, whereas data on a variety of minerals at high temperature T and ambient pressure P result in lower values. We reconcile these differences. Our ab initio model calculations on MgO show that ν increases with P and is 2.0–2.5 at lower mantle pressures. There is no need to assume partial melting to explain the seismic data. These calculations also provide insight into the P and T dependence of the dimensionless parameter Γ ≡ −(1/αG)(∂G/∂T)p, where G is the isotropic shear modulus and α is the volume thermal expansivity. Using measured values of thermoelastic parameters coupled with thermodynamic identities, we seek constraints on δs ≡ −(1/αKs)(∂Ks/∂T)p, where Ks is the adiabatic bulk modulus, and confirm that P causes δs to decrease. We find Poissons ratio increases with P and T. Altogether these results show that for MgO, ν increases from 1.3 at ambient conditions to over 2 at lower mantle conditions. We expect other mantle minerals to behave similarly. Therefore we find that reconciliation of the mineral physics approach with that of seismic tomography concerning ν does not require special assumptions about the state of the lower mantle.

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

The Grüneisen ratio, γ, is defined as γy=αKTV/Cv. The volume dependence of γ(V) is solved for a wide range in temperature. The volume dependence of αKT is solved from the identity (∂ ln(αKT)/∂ ln V)T ≡ δT-K′. α is the thermal expansivity; KT is the bulk modulus; CV is specific heat; and δTand K′ are dimensionless thermoelastic constants. The approach is to find values of δT and K′, each as functions of T and V. We also solve for q=(∂ ln γ/∂ ln V) where q=δT-K′+ 1-(∂ ln CV/∂ ln V)T. Calculations are taken down to a compression of 0.6, thus covering all possible values pertaining to the earths mantle, q=∂ ln γ/∂ ln V; δT=∂ ln α/∂ ln V; and K′= (∂KT/∂P)T. New experimental information related to the volume dependence of δT, q, K′ and CV was used. For MgO, as the compression, η=V/V0, drops from 1.0 to 0.7 at 2000 K, the results show that q drops from 1.2 to about 0.8; δT drops from 5.0 to 3.2; δT becomes slightly less than K′; ∂ ln CV/∂ 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.