Hitoshi Oda

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.

Thermoelastic parameters for six minerals at high temperature

The complete high-temperature data for six minerals measured in the University of California, Los Angeles, Mineral Physics Laboratory are analyzed and presented as a group. These minerals include CaO and a calcium rich garnet, grossular. We find no significant differences in the thermoelastic parameters for the calcium rich minerals when compared with the calcium deficient minerals. Evidence of anharmonicity is shown in the specific heat, CV, of corundum, forsterite, pyrope, and grossular garnet, but no evidence is found for the case of MgO and CaO. The patterns for the thermoelastic parameter γ show that in general γ decreases with temperature T and that γ at constant volume always decreases rather markedly with T. The thermoelastic parameters, δT, δS, and αKT tend to remain independent of T at high T (above Θ), whereas they are not independent of T near room temperature. The thermal pressure at P = 0 is quite linear with T at temperatures above Θ for all six minerals. We find no evidence of anharmonic behavoir for the thermoelastic parameters of forsterite, correcting an earlier report by Anderson and Suzuki (1983) that anharmonicity appeared in γ, δT;, δS, and αKT at high T for this mineral.

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. n nWe 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.

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.

Elasticity of fused silica spheres under pressure using resonant ultrasound spectroscopy

Resonant ultrasound spectroscopy (RUS) is a powerful tool for determining the elastic properties of solids at high temperature. Before RUS can be used to measure the pressure derivatives of elasticity, however, effects of boundary conditions between the pressurizing gas and specimen must be understood. Data are presented that show effects of different pressurizing gases on the apparent or measured values of (∂G/∂P)T of fused silica spheres, where G is the shear modulus, P is pressure, and T is temperature. The value of (∂G/∂P)T is found to depend on the molecular mass, M, of the pressurizing gas via −3.425(0.032)−5.9(1.6)×10−3M, when M is in grams. Extrapolating to zero gas mass gives (∂G/∂P)T=−3.42(0.03), a value bracketed by results from previous plane-wave, ultrasonic measurements. An alternative analysis in which effects of the pressurizing gas are removed by theoretical concerns is also presented and suggests that (∂G/∂P)T is −3.32(0.03). Our results quantify the effect of pressurizing gas on measure...

Elastic constants and anelastic properties of an anisotropic periclase sphere as determined by the resonant sphere technique

The resonant sphere technique (RST) is applied to measurements of elastic and anelastic properties of single-crystal periclase (MgO) in order to demonstrate the potential value of this method as an experimental technique. The lowest 22 resonant frequencies and their attenuation properties were measured in the range 1.0 to 2.6 MHz. The oscillation modes were identified by comparing the observed resonant frequencies with computed ones. Using inversion theory, we find perturbations from an assumed set of elastic constants, so that the computed frequencies agree, in a least squares sense, with the measured ones. The final values of the elastic constants were obtained by adding the perturbations to the initial values of elastic constants. Furthermore, the internal friction coefficients were determined from the observed vibrational attenuation data by a least squares method. The values of elastic constants and internal friction coefficients are in good agreement with those determined by other methods. We find better accuracy, by up to 1 order of magnitude, than that found by using the rectangular parallelepiped resonance method. The RST can be an accurate method for measuring elastic and anelastic properties of small anisotropic materials. An applicability of the RST to the Earths free oscillation is considered. Free oscillation of an elastically anisotropic sphere shows different frequency spectrum, depending on the degree of the elastic anisotropy. The frequency spectrum of an anisotropic sphere can be interpreted as the result of the split of degenerate modes of an isotropic sphere. Therefore the anomalous splitting of the Earths free oscillation, which cannot be explained by the rotation and ellipticity of the Earth, may be attributed to the elastic anisotropy of the Earth.

Pressure dependence of the elasticity of a steel sphere measured by the cavity resonance method

The pressure derivatives of elastic moduli of a steel sphere were measured by the cavity resonance method, a modified resonant sphere technique under gas pressure using a spherical three-layered structure (3LS) consisting of a sample-thin gas layer–cavity container system. The pressure-induced shifts of resonance peaks of both toroidal and spheroidal modes were observed up to 100 MPa (1 kbar) under gas pressure with helium gas. The resultant pressure derivatives of frequencies of toroidal modes yielded a pressure derivative of shear modulus of ∂G/∂P=2.01±0.08. The pressure derivative of the bulk modulus was determined from the data of spheroidal modes, ∂K/∂P=5.0±0.4, by analyzing these data as free oscillations of the 3LS superimposed on the static compression. The results demonstrate the efficiency of the cavity resonance method for measuring pressure derivatives of elastic moduli of solids.

Measurements for pressure and temperature dependencies of elastic moduli by the resonant sphere technique, RST.

The resonant sphere technique, RST, has been developed for measurements of elasticity and anelasticity of small crystal specimens. This method has advantages over other methods, especially in high‐temperature and high‐pressure measurements. New methods of data acquisition have made it convenient to measure resonance frequency at high temperatures (using a buffer rod), making a 1 K temperature interval measurement possible; this opens up many possibilities in the determination of physical properties of solids. Preliminary measurements of resonance frequency were performed by RST up to 100 MPa, which showed interference of vibration modes between the specimen and the pressure medium, even for helium gas as the pressure medium. This interference may become more serious at higher pressures or under liquid pressure. In order to evaluate such effects, the cavity resonance method was developed by Ohno in 1993, with a spherical shell structure with a spherical specimen at the center. This gives clear boundary con...