pure and applied geophysics

, Volume 141, Issue 2–4, pp 379–392 | Cite as

Performance of the complete travel-time equation of state at simultaneous high pressure and temperature

  • Hartmut A. Spetzler
  • Akira Yoneda
Acoustic Studies of the Elasticity and Equation of State of Minerals
Received:
Revised:
Accepted:

Abstract

The complete travel-time equation of state (CT-EOS) is presented by utilizing thermodynamics relations, such as;
$$K_T = K_S (1 + \alpha \gamma T)^{ - 1} , \gamma = \frac{{\alpha K_S }}{{\rho C_P }}, \left. {\frac{{\partial C_P }}{{\partial P}}} \right)_T = - \frac{T}{\rho }\left[ {\alpha ^2 + \left. {\frac{{\partial \alpha }}{{\partial T}}} \right)_P } \right], etc.$$

The CT-EOS enables us to analyze ultrasonic experimental data under simultaneous high pressure and high temperature without introducing any assumption, as long as the density, or thermal expansivity, and heat capacity are also available as functions of temperature at zero pressure. The performance of the CT-EOS was examined by using synthesized travel-time data with random noise of 10−5 and 10−4 amplitude up to 4 GPa and 1500 K. Those test conditions are to be met with the newly developed GHz interferometry in a gas medium piston cylinder apparatus. The results suggest that the combination of the CT-EOS and accurate experimental data (10−4 in travel time) can determine thermodynamic and elastic parameters, as well as their derivatives with unprecedented accuracy, yielding second-order pressure derivatives (∂2M/∂P2) of the elastic moduli as well as the temperature derivatives of their first-order pressure derivatives ∂2M/∂P∂T). The completeness of the CT-EOS provides an unambiguous criterion to evaluate the compatibility of empirical EOS with experimental data. Furthermore because of this completeness, it offers the possibility of a new and absolute pressure calibration when X-ray (i.e., volume) measurements are made simultaneously with the travel-time measurements.

Key Words

Elastic constant thermodynamics equation of state acoustic velocity thermal expansivity heat capacity Grüneisen constant high pressure high temperature 
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References

  1. Anderson, D. L.,Theory of the Earth (Blackwell Scientific Publications, Boston 1989) pp. 91–92.Google Scholar
  2. Anderson, O. L., Isaak, D. L., andOda, H. (1991),Thermoelastic Parameters for Six Minerals at High Temperature, J. Geophys. Res.96, 18,037–18,046.Google Scholar
  3. Bassett, W. A., Shimizu, H., andBrody, E. M.,Pressure dependence of elastic moduli of forsterite by Brillouin scattering in diamond cell. InHigh-pressure Research in Geophysics (eds. Akimoto, S., and Manghanani, M. H.) (Center for Academic Publications Japan, Tokyo 1982) pp. 115–124.Google Scholar
  4. Brown, J. M., Slutsky, L. J., Abramson, E. (1992),Applications of Impulsive Stimulated Scattering in Mineral Physics: Elasticity, Equation of State, Thermal Diffusivity, and Structural Relaxation, Eos Trans. Amer. Geophys. Union, Fall Meeting Supplement, 555.Google Scholar
  5. Brown, J. M., Slutsky, L. J., Nelson, K. A., andCheng, L. T. (1989),Single-crystal Elastic Constants for San Corlos Peridot: An Application of Impulsive Stimulated Scattering, J. Geophys. Res.94, 9485–9492.Google Scholar
  6. Dandekar, D. P. (1970),Iterative Procedure to Estimate the Values of Elastic Constants of a Cubic Solid at High Pressure from the Sound Wave Velocity Measurements, J. Appl. Phys.41, 667–672.Google Scholar
  7. Davies, G. F. (1974),Effective Elastic Moduli under Hydrostatic Stress—I. Quasi-harmonic Theory, J. Phys. Chem. Solids35, 1513–1520.Google Scholar
  8. Getting, I. C., andKennedy, G. C. (1970),Effect of Pressure on the emf of Chromel-alumel and Platinum-platinum 10% Rhodium Thermocouples, J. Appl. Phys.41, 4522–4562.Google Scholar
  9. Getting, I. C., andSpetzler, H. (1993),Gas-charged piston-cylinder apparatus for pressures to 4 GPa. InHigh-pressure Science and Technology (eds. Schmidt, S. C., Shaner, J. W., Samara, G. A., and Ross, M.) (American Institute of Physics, Woodbury, New York 1993) pp. 1581.Google Scholar
  10. Goto, T., andAnderson, O. L. (1988),An Apparatus for Measuring Elastic Constants of Single Crystals by a Resonance Technique up to 1825 K, Rev. Sci. Instrum.59, 1405–1408.Google Scholar
  11. Isaak, D. G., Anderson, O. L., Goto, T., andSuzuki, I. (1989),Elasticity of Single-crystal Forsterite Measured to 1700 K, J. Geophys. Res.94, 5895–5906.Google Scholar
  12. Jackson, I., andNiesler, H.,The elasticity of periclase to 3 GPa and some geophysical implications. InHigh-pressure Research in Geophysics (eds. Akimoto, S., and Manghnani, M. H.) (Center for Academic Publications Japan, Tokyo 1982) pp. 93–113.Google Scholar
  13. Jackson, I., Niesler, H., andWeidner, D. J. (1981),Explicit Correction of Ultrasonically Determined Elastic Wave Velocities for Transducer-bond Phase Shift, J. Geophys. Res.86, 3736–3748.Google Scholar
  14. Jeanloz, R.,Differential finite strain equation of state. InHigh-pressure Research: Application to Earth and Planetary Sciences (eds. Syono, Y., and Manghnani, M. H.) (TERRAPUB, Tokyo/American Geophysical Union, Washington, D.C. 1992) pp. 147–156.Google Scholar
  15. Kittel, C.,Introduction to Solid State Physics, 6th edition (John Wiley & Sons, New York 1986) pp. 83–98.Google Scholar
  16. Mao, H. K., Hemley, R. J., Fei, Y., Shu, J. F., Chen, L. C., andBassett, W. A. (1991), Effect of Pressure, Temperature, and Composition on Lattice Parameters and Density of (Fe, Mg)SiO3-Perovskites to 30 GPa, J. Geophys. Res.96, 8069–8079.Google Scholar
  17. Niesler, H., Jackson, I., andEdwin, C. M. (1988),Calibration and Intercomparison of Minalpha and Manganin Resistance Pressure Gauge to 3 GPa, High Temp.-High Pressures,20, 495–508.Google Scholar
  18. Spetzler, H., Chen, G., Whitehead, S., andGetting, I. C. (1993),A New Ultrasonic Interferometer for the Determination of Equation of State Parameters of Sub-millimeter Single Crystals, Pure Appl. Geophys.,141, 341–377.Google Scholar
  19. Spetzler, H., Sammis, C. G., andO'Connell, R. J. (1972),Equation of State of NaCl: Ultrasonic measurements to 8 kbar and 800°C and Static Lattice Theory, J. Phys. Chem. Solids33, 1727–1750.Google Scholar
  20. Suzuki, I. (1975),Thermal Expansion of Periclase and Olivine, and their Anharmonic Properties, J. Phys. Earth23, 145–159.Google Scholar
  21. Wang, Y., Weidner, D. J., Liebermann, R. C., Liu, X., Ko, J., Vaughan, M. T., Zhao, Y., Yeganen-Haeri, A., andPacalo, R. E. G. (1991), Phase Transition and Thermal Expansion of MgSiO3 Perovskite, Science251, 410–413.Google Scholar
  22. Watanabe, H. (1982),Thermomechanical properties of synthetic high-pressure compounds relevant to the earth's mantle. InHigh-pressure Research in Geophysics (eds. Akimoto, S., and Manghnani, M. H.) (Center for Academic Publications Japan, Tokyo 1982) pp. 441–464.Google Scholar
  23. Weidner, D. J. (1975),Elasticity of Microcrystals, Geophys. Res. Lett.2, 189–192.Google Scholar
  24. Yoneda, A. (1990), Pressure Derivatives of Elastic Constants of Single Crystal MgO and MgAl2O4, J. Phys. Earth38, 19–55.Google Scholar
  25. Yoneda, A., Spetzler, H., andGetting, I. (1993),Implication of the complete travel time equation of state for a new pressure scale. InHigh-pressure Science and Technology (eds. Schmidt, S. C., Shaner, J. W., Samara, G. A. and Ross, M. (American Institute of Physics, Woodbury, New York 1993) pp. 1609.Google Scholar

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Hartmut A. Spetzler
    • 1
  • Akira Yoneda
    • 2
  1. 1.Cooperative Institute for Research in Environmental Sciences and Department of Geological SciencesUniversity of ColoradoBoulderUSA
  2. 2.Cooperative Institute for Research in Environmental SciencesUniversity of ColoradoBoulderUSA

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