Physics and Chemistry of Minerals

, Volume 40, Issue 3, pp 207–227 | Cite as

An alternative use of Kieffer’s lattice dynamics model using vibrational density of states for constructing thermodynamic databases

  • Michael H. G. Jacobs
  • Rainer Schmid-Fetzer
  • Arie P. van den Berg
Original Paper


We use Kieffer’s model to represent the vibrational density of states (VDoS) and thermodynamic properties of pure substances in pressure–temperature space. We show that this model can be simplified to a vibrational model in which the VDoS is represented by multiple Einstein frequencies without significant loss of accuracy in thermodynamic properties relative to experimental data. The resulting analytical expressions for thermodynamic properties, including the Gibbs energy, are mathematically simple and easily accommodated in existing computational software for making thermodynamic databases. We show for aluminium, platinum, orthoenstatite and forsterite that thermodynamic properties can be represented with comparable accuracy as with Kieffer’s model with the same number of fitting parameters as in the Mie–Grüneisen–Debye model. We demonstrate that the method enables to achieve thermodynamic properties with superior accuracy relative to the Mie–Grüneisen–Debye method. The method is versatile in the sense that it allows incorporating dispersion of Grüneisen parameters. It is possible to extend the formalism to include other physical effects, such as intrinsic anharmonicity in the same way as in vibrational models based on Kieffer’s formalism.


Equation of state Vibrational density of states Aluminium Platinum Forsterite Orthoenstatite 



MHG Jacobs gratefully acknowledges financial support by the German Research Foundation (DFG) under Grant no. JA 1985/1. Collaboration between A. van den Berg and M. Jacobs has been supported through The Netherlands Research Center for Integrated Solid Earth Science (ISES) project ME-2.7. We wish to express our gratitude to M. Ghiorso and an anonymous reviewer for constructive suggestions which improved the quality of the manuscript significantly.


  1. Al’tshuler LV, Brusnikin SE, Kuz’menkov EA (1987) Isotherms and Grüneisen functions for 25 metals. J Appl Mech Tech Phys 28:129–141CrossRefGoogle Scholar
  2. Anderson OL (1996) Anharmonicity of forsterite and the thermal pressure of insulators. Geophys Res Lett 23:3031–3034CrossRefGoogle Scholar
  3. Ashida T, Kume S, Ito E (1987) Thermodynamic aspects of phase boundary among α-, β-, and γ-Mg2SiO4. In: Manghnani MH, Syono Y (eds) High pressure research in mineral physics. Terra Scientific Publishing Company (TERRAPUB), Tokyo/American Geophysical Union, Washington, pp 269–274Google Scholar
  4. Ashida T, Kume S, Ito E, Navrotsky A (1988) MgSiO3 ilmenite: heat capacity, thermal expansivity and enthalpy of transformation. Phys Chem Miner 16:239–245 Google Scholar
  5. Austin JB (1932) A vacuum apparatus for measuring thermal expansion at elevated temperatures with measurements on platinum, gold, magnesium, and zinc. Physics 3:240–267CrossRefGoogle Scholar
  6. Barin I (1989) Thermochemical data for pure substances, Part II. VCH Verlags-gesellschaft mbH, D6940 Weinheim, GermanyGoogle Scholar
  7. Berg WT (1969) The low temperature heat capacity of platinum. J Phys Chem Solids 30:69–72CrossRefGoogle Scholar
  8. Bouhifd MA, Andrault D, Fiquet G, Richet P (1996) Thermal expansion of forsterite up to the melting point. Geophys Res Lett 23:1143–1146CrossRefGoogle Scholar
  9. Chopelas A (1990) Thermal properties of forsterite at mantle pressures derived from vibrational spectroscopy. Phys Chem Miner 17:149–156Google Scholar
  10. Chopelas A (1999) Estimates of mantle relevant Clapeyron slopes in the MgSiO3 system from high-pressure spectroscopic data. Am Miner 84:233–244Google Scholar
  11. Choudury N, Chaplot SL (2000) Free energy and relative stability of the enstatite Mg2Si2O6 polymorphs. Solid State Commun 114:127–132CrossRefGoogle Scholar
  12. Clusius K, Losa CG, Fransozini P (1957) Ergebinisse der Tieftemperaturforschung. Die Bildfehler de XVIII. Die Atom- und Elektronenwärme des Platins zwischen 0°K und 273°K. Z Naturforsch 12A:34–38Google Scholar
  13. Collard SM, McLellan RB (1992) High-temperature elastic constants of platinum single crystals. Acta Metall Mater 40:699–702CrossRefGoogle Scholar
  14. Cynn H, Carnes JD, Anderson OL (1996) Thermal properties of forsterite, including C V, calculated from αK T through the entropy. J Phys Chem Solids 57:1593–1599CrossRefGoogle Scholar
  15. Dewaele A, Loubeyre P, Mezouar M (2004) Equations of state of six metals above 94 GPa. Phys Rev B 70:094112/1-8CrossRefGoogle Scholar
  16. Dorogokupets PI, Oganov AR (2007) Ruby, metals, and MgO as alternative pressure scales: a semiempirical description of shock-wave, ultrasonic, X-ray, and thermochemical data at high temperatures and pressures. Phys Rev B 75:024115/1-16CrossRefGoogle Scholar
  17. Dutton DH, Brockhouse BN, Miller AP (1972) Crystal dynamics of platinum by inelastic neutron scattering. Can J Phys 50:2915–2927CrossRefGoogle Scholar
  18. Eriksson O, Wills JM, Wallace D (1992) Electronic, quasiharmonic, and anharmonic entropies of transition metals. Phys Rev B 64:5221–5227CrossRefGoogle Scholar
  19. Fabrichnaya O, Saxena SK, Richet P, Westrum EF Jr (2004) Thermodynamic data, models, and phase diagrams in multicomponent oxide systems, Springer, New YorkGoogle Scholar
  20. Fradin FY, Koelling DD, Freeman AJ, Watson-Yang TJ (1975) Calculation of the electronic structure and related physical properties of platinum. Phys Rev B 12:5570–5574CrossRefGoogle Scholar
  21. Gasparik T (2003) Phase diagrams for geoscientist. Springer, New YorkGoogle Scholar
  22. Gillet P, Richet P, Guyot F, Fiquet G (1991) High-temperature thermodynamic properties of forsterite. J Geophys Res 96:11805–11816CrossRefGoogle Scholar
  23. Gilvarry JJ (1955) Grüneisen’s law and the fusion curve at high pressure. Phys Rev 102:317–325CrossRefGoogle Scholar
  24. Grimvall G (1968) Temperature dependent effective masses of conduction electrons. J Phys Chem Solis 29:1221–1225CrossRefGoogle Scholar
  25. Hahn TA, Kirby RK (1972) Thermal expansion of platinum from 293 to 1900 °K. AIP Conf Proc 3:87–95CrossRefGoogle Scholar
  26. Holland TJB, Powell R (1998) An internally consistent thermodynamic data ser for phases of petrological interest. J Metamorph Geol 16:309–343Google Scholar
  27. Holmes NC, Moriarty GR, Gathers GR, Nellis WJ (1989) The equation of state of platinum to 660 GPa (6.6 Mbar). J Appl Phys 66:2962–2967Google Scholar
  28. Hugh-Jones D (1997) Thermal expansion of MgSiO3 and FeSiO3 ortho- and clino pyroxenes. Am Miner 82:689–696Google Scholar
  29. Isaak DG, Anderson OL, Goto T (1989) Elasticity of single-crystal forsterite measured to 1700 K. J Geophys Res 94:5895–5906Google Scholar
  30. Jackson JM, Palko JW, Andrault D, Sinogeikin SV, Lakshtanov DL, Wang J, Bass JD, Zha C-S (2003) Thermal expansion of natural ortho enstatite to 1473 K. Eur J Miner 15:469–473CrossRefGoogle Scholar
  31. Jacobs MHG, de Jong BHWS (2005) Quantum-thermodynamic treatment of intrinsic anharmonicity; Wallace’s theorem revisited. Phys Chem Miner 32:614–626CrossRefGoogle Scholar
  32. Jacobs MHG, de Jong BHWS (2007) Placing constraints on phase equilibria and thermophysical properties in the system MgO-SiO2 by a thermodynamically consistent vibrational method. Geochim Cosmochim Acta 71:3630–3655CrossRefGoogle Scholar
  33. Jacobs MHG, de Jong BHWS (2009) Thermodynamic mixing properties of olivine derived from lattice vibrations. Phys Chem Miner 36:365–389Google Scholar
  34. Jacobs MHG, Oonk HAJ (2012) Equilibrium between phases of matter. Supplemental text for material science and high-pressure geophysics. Springer, DordrechtCrossRefGoogle Scholar
  35. Jacobs MHG, Schmid-Fetzer R (2010) Thermodynamic properties and equation of state of fcc aluminum and bcc iron, derived from a lattice vibrational method. Phys Chem Miner 37:721–739CrossRefGoogle Scholar
  36. Jacobs MHG, van den Berg AP (2011) Complex phase distribution and seismic velocity structure on the transition zone: convection model predictions for a magnesium-end member olivine-pyroxene mantle. Phys Earth Planet Int 186:36–48CrossRefGoogle Scholar
  37. Jacobs MHG, van de Berg AP, de Jong BHWS (2006) The derivation of thermo-physical properties and phase equilibria of silicate materials from lattice vibrations: application to convection in the Earth’s mantle. Calphad 30:131–146CrossRefGoogle Scholar
  38. Jaeger FM, Rosenbohm E (1939) The exact formulae for the true and mean specific heats of platinum between 0° and 1600°C. Physica 6:1123–1125CrossRefGoogle Scholar
  39. Kajiyoshi K (1986) High-temperature equation of state for mantle minerals and their anharmonic properties, MS Thesis. Okayama University, OkayamaGoogle Scholar
  40. Kanzaki M (1991) Ortho/Clino enstatite transition. Phys Chem Miner 17:726–730Google Scholar
  41. Kelley KK (1943) Specific heats at low temperatures of magnesium orthosilicate and magnesium metasilicate. J Am Chem Soc 65:339–341CrossRefGoogle Scholar
  42. Kendall WB, Orr RL, Hultgren R (1962) Heat content of platinum. J Chem Eng Data 7:516–518CrossRefGoogle Scholar
  43. Kieffer SW (1979) Thermodynamics and lattice vibrations of minerals: 3 lattice dynamics and an approximation for minerals with application to simple substances and framework silicates. Rev Geophys Space Phys 17:35–59CrossRefGoogle Scholar
  44. Knapp GS, Jones RW (1972) Determination of the electron-phonon enhancement factor from specific-heat data. Phys Rev B 6:1761–1767CrossRefGoogle Scholar
  45. Kresch M, Lucas M, Delaire O, Lin JYY, Fultz B (2008) Phonons in aluminum at high temperature studied by inelastic neutron scattering. Phys Rev B 77:024301/1-9CrossRefGoogle Scholar
  46. Krupka KM, Richard A, Hemingway BS, Ito J (1985a) Low-temperature heat capacities and derived thermodynamic properties of anthophyllite, diopside, enstatite, bronzite, and wollastonite. Am Miner 70:249–260Google Scholar
  47. Krupka KM, Hemingway BS, Robie RA, Kerrick DM (1985b) High-temperature heat capacities and derived thermodynamic properties of anthophyllite, diopside, dolomite, enstatite, bronzite, talc, tremolite, and wolastonite. Am Miner 70:261–271Google Scholar
  48. Li L, Wentzcovitch RM, Weidner DJ, Da Silva CRS (2009) Vibrational and thermodynamic properties of forsterite at mantle conditions. J Geophys Res 112:B05206/1-8Google Scholar
  49. Lukas HL, Fries SG, Sundman B (2007) Computational thermodynamics, the Calphad method. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  50. Manghnani MH, Matsui T (1981) Temperature dependence of pressure derivatives of single-crystal elastic constants of pure forsterite. ISPEI Symposium on properties of materials at high pressures and temperatures. Toronto, CanadaGoogle Scholar
  51. Marsh SP (1980) LASL shock hugoniot data. University of California Press, BerkeleyGoogle Scholar
  52. Matsui T, Manghnani MH (1985) Thermal expansion of single-crystal forsterite to 1023 K by Fizeau interferometry. Phys Chem Miner 12:201–210CrossRefGoogle Scholar
  53. Mohr PJ, Taylor BN, Newell DB (2012) CODATA recommended values of the fundamental physical constants: 2010. Rev Mod Phys 84:1527–1605CrossRefGoogle Scholar
  54. Oganov AR, Dorogokupets P (2004) Intrinsic anharmonicity in equations of state and thermodynamics of solids. J Phys Condens Matter 16:1351–1360CrossRefGoogle Scholar
  55. Piazzoni AS, Steinle-Neumann G, Bung H-P, Dolejš D (2007) A mineralogical model for density and elasticity of the Earth’s mantle. Geochem Geophys Geosyst 8:1–23CrossRefGoogle Scholar
  56. Price GD, Parker SC, Leslie M (1987) The lattice dynamics and thermodynamics of the Mg2SiO4 polymorphs. Phys Chem Miner 15:181–190CrossRefGoogle Scholar
  57. Robie RA, Hemingway BS (1995) Thermodynamic properties of minerals and related substances a 298.15 K and 1 bar (105 Pascals) pressure and at higher temperatures. United States Geological Survey Bulletin No. 2131Google Scholar
  58. Robie RA, Hemingway BS, Takei H (1982) Heat capacities and entropies of Mg2SiO4, Mn2SiO4 and Co2SiO4 between 5 and 380 K. Am Miner 67:470–482Google Scholar
  59. Stixrude L, Lithgow-Bertelloni C (2005) Thermodynamics of mantle minerals—I. Physical properties. Geophys J Int 162:610–632CrossRefGoogle Scholar
  60. Sumino Y, Nishizawa O, Goto T, Ohno I, Ozima M (1977) Temperature variation of elastic constants of single-crystal forsterite between −190 and 400°C. J Phys Earth 23:377–392Google Scholar
  61. Suzuki I (1975) Thermal expansion of periclase and olivine, and their anharmonic properties. J Phys Earth 23:145–159Google Scholar
  62. Suzuki I, Anderson OL, Sumino Y (1983) Elastic properties of a single-crystal forsterite Mg2SiO4 up to 1200 K. Phys Chem Miner 10:38–46Google Scholar
  63. Suzuki I, Takei H, Anderson OL (1984) Thermal expansion of single-crystal forsterite Mg2SiO4. In: The proceedings of the eight international thermal expansion symposium, sponsored by the national bureau of standards. Plenum Pub Co, New York, pp 79–88Google Scholar
  64. Thiéblot L, Téqui C, Richet P (1999) High-temperature hat capacity of grossular (Ca3Al2Si3O12), enstatite (MgSiO3) and titanite (CaTiSiO5). Am Miner 84:848–855Google Scholar
  65. Tsuchiya T, Kawamura K (2002) First-principles electronic thermal pressure of metal Au and Pt. Phys Rev B 66:094115/1-5CrossRefGoogle Scholar
  66. Vinet P, Ferrante J, Rose JH, Smith JR (1987) Compressibility of solids. J Geophys Res 92:9319–9325CrossRefGoogle Scholar
  67. Vinet P, Ferrante J, Rose JH, Smith JR (1989) Universal features of the equation of state of solids. J Phys Condens Matter 1:1941–1963CrossRefGoogle Scholar
  68. Wallace DC (1972) Thermodynamics of crystals. Wiley, New YorkGoogle Scholar
  69. Watanabe H (1982) Thermochemical properties of synthetic high-pressure compounds relevant to the Earth’s mantle. In: Manghnani MH, Akimoto S (eds) High-pressure research in geophysics. Center for Academic Publications, Japan/Tokyo: Reidel/Dordrecht, pp 441–464Google Scholar
  70. White GK (1972) Thermal expansion of platinum at low temperatures. J Phys 2F(2):L30–L31CrossRefGoogle Scholar
  71. Xu W, Lithgow-Bertelloni C, Stixrude L, Ritsema J (2008) The effect of bulk composition and temperature on mantle seismic structure. Earth Planet Sci Lett 275:70–79CrossRefGoogle Scholar
  72. Yang H, Ghose S (1995) High-temperature single crystal X-ray diffraction studies of the ortho-proto phase transition in enstatite, Mg2Si2O6 at 1360 K. Phys Chem Miner 22:300–310CrossRefGoogle Scholar
  73. Yokokawa H, Takahashi Y (1979a) Analysis of heat capacity of platinum: electron-phonon effects and anharmonic contribution. J Phys Chem Solids 40:445–448CrossRefGoogle Scholar
  74. Yokokawa H, Takahashi Y (1979b) Laser-flash calorimetry II. Heat capacity of platinum from 80 to 1000 K and its revised thermodynamic functions. J Chem Thermodyn 11:411–420CrossRefGoogle Scholar
  75. Yusa H, Akaogi M, Ito E (1993) Calorimetric study of MgSiO3 garnet and pyroxene: heat capacities, transition enthalpies, and equilibrium phase relations in MgSiO3 at high pressures and temperatures. J Geophys Res 98:6453–6460CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael H. G. Jacobs
    • 1
  • Rainer Schmid-Fetzer
    • 1
  • Arie P. van den Berg
    • 2
  1. 1.Institute of MetallurgyClausthal University of TechnologyClausthal-ZellerfeldGermany
  2. 2.Department of Theoretical GeophysicsUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations