Orson L. Anderson

A simplified method for calculating the debye temperature from elastic constants

Abstract The Reuss-Voigt approximations are well known methods whereby the isotropic polycrystalline elastic constants can be calculated from the single crystal elastic constants. It is shown here that the Reuss and the Voigt approximations can be used to estimate, accurately, the mean sound velocity of a crystal. Using this method, the Debye Temperature, which is proportioned to the mean sound velocity, can be determined without recourse to the published tables or high speed computers. This approximation is valid for all crystal classes.

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.

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.

Temperature coefficients of elastic constants of single crystal MgO between 80 and 1,300 K

AbstractElastic constants of single crystal MgO have been measured by the rectangular parallelepiped resonance (RPR) method at temperatures between 80 and 1,300 K. Elastic constants Cij (Mbar=103 kbar) and their temperature coefficients (kbar/K) are:

Equations for the elastic constants and their pressure derivatives for three cubic lattices and some geophysical applications

\begin{gathered} {\text{ }}C_{{\text{11}}} {\text{ }}C_{{\text{12}}} {\text{ }}C_{{\text{44}}} {\text{ }}K_s {\text{ }}C_s \hfill \\ C_{ij} {\text{ 300 K 2}}{\text{.966 0}}{\text{.959 1}}{\text{.562 1}}{\text{.628 1}}{\text{.004}} \hfill \\ \partial C_{ij} {\text{/}}\partial T{\text{100 K }} - {\text{0}}{\text{.259 0}}{\text{.013 }} - {\text{0}}{\text{.072 }} - {\text{0}}{\text{.078 }} - {\text{0}}{\text{.136}} \hfill \\ {\text{ 300K }} - {\text{0}}{\text{.596 0}}{\text{.068 }} - {\text{0}}{\text{.122 }} - {\text{0}}{\text{.153 }} - {\text{0}}{\text{.332}} \hfill \\ {\text{ 800 K }} - {\text{0}}{\text{.619 0}}{\text{.009 }} - {\text{0}}{\text{.152 }} - {\text{0}}{\text{.200 }} - {\text{0}}{\text{.314}} \hfill \\ {\text{ 1,300 K }} - {\text{0}}{\text{.598 0}}{\text{.036 }} - {\text{0}}{\text{.130 }} - {\text{0}}{\text{.223 }} - {\text{0}}{\text{.218}} \hfill \\ \end{gathered}

A universal thermal equation-of-state

By combining the present results with the previous data on the thermal expansivity and specific heat, the thermodynamic properties of magnesium oxide are presented and discussed. The elastic parameters of MgO at very high temperatures in the earths lower mantle are also clarified.

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

Elastic moduli of forsterite were measured between 300 and 1,200 K (≃ 1.6 times the Debye temperature) by the Rectangular Parallelepiped Resonance method. All the moduli decrease regularly with temperature. A summary of the results is as follows:Elastic moduli Cij in GPaT/KC11C22C33C44C55C660337.0205.6241.169.4953.5283.43300328.7199.8235.566.7880.9580.571,200292.9174.7207.155.3769.1467.22Temperature derivatives of elastic moduli, −∂Cij/∂T in MPa/K30038.426.929.412.312.514.11,20040.128.232.112.813.315.0temperature derivatives of elastic moduli, - ∂Cij/∂R in MPa/K whereCsi=(Cjj+Ckk−2·Cjk)/4; (i, j, k=1, 2, 3;i ≠j ≠k), and ρ is density in kg/m3. These data permit for the first time the calculation of elastic and thermal properties well into the classical range far above the Debye temperature. We find, for example, that the elastic constants, including the bulk moduls, closely follow standard equations throughout the measured temperature range. This information aids extrapolations up to the melting point. This data, coupled with thermal expansivity data permit the computations of thermal anharmonic parameters of minerals forT>θ.