EQUATIONS OF STATE OF SOLIDS FOR GEOPHYSICS AND CERAMIC SCIENCE ORSON L. ANDERSON Institute of Geophysics and Planetary Physics University of California at Los Angeles New York Oxford OXFORD UNIVERSITY PRESS 1995 CONTENTS PART I. THERMAL PHYSICS, 1 1. THE FREE ENERGY AND THE GRUNEISEN PARAMETER, 3 1.1. Introduction, 3 1.2. The Helmholtz free energy, 3 1.3. Pressure: the equation of state, 5 1.4. The Griineisen parameters, 6 1.5. The bulk modulus K, 18 1.6. The Debye temperature 9: a lower bound for high T, 24 1.7. The Debye temperature of the earth and the moon, 26 1.8. jjj: the Debye approximation to 7, 28 1.9. Electronic heat capacity contribution to 7 for iron, 29 1.10. Problems, 30 2. STATISTICAL MECHANICS AND THE QUASI- HARMONIC THEORY, 31 2.1. Introduction, 31 2.2. The vibrational energy and the thermal energy, 32 2.3. The quasiharmonic approximation, 34 2.4. The Mie-Griineisen equation of state, 35 2.5. The high-temperature limit of the quasiharmonic approxi- mation, 36 2.6. Anharmonic corrections to the Helmholtz energy, 45 2.7. The free energy and its physical properties at very low temperature, 49 2.8. The Debye theory interpolation , 52 2.9. Thermodynamic functions from the partition function, 55 2.10. Problems, 56 XIV 3. THERMOELASTIC PARAMETERS AT HIGH COMPRESSION, 57 3.1. Thermodynamic identities, 57 3.2. The mean atomic mass, fi = M/p, 61 3.3. The cases for (dKT/dT)v = 0 and (d (aKT) /dT)p = 0, 62 3.4. Theoretical insight into the change of K' with T, 63 3.5. The condition ST = K' in r},T space, 66 3.6. The variation of ST with compression, 69 3.7. The variation of aKj with compression, 74 3.8. The variation of 7 and q with compression, 76 2 3.9. Experimental insight into the value of d KT/dPdT, 81 3.10. Comments, 82 3.11. Problems, 82 4. THERMAL EXPANSIVITY AT HIGH P and T, 83 4.1. Introduction, 83 4.2. Thermal expansivity at high T and constant t], 83 4.3. Thermal expansivity versus T at high temperature and constant pressure, 84 4.4. Thermal expansivity versus 77 at constant T, 85 4.5. Measurements of V versus T for silicate perovskite, 90 4.6. Griineisen's theory of thermal expansivity (P = 0), 92 4.7. Suzuki's theory of thermal expansivity, 94 4.8. High temperature expansivity of NaCl, 96 4.9. The uncompressed value of a in the lower mantle, 97 4.10. Obtaining a from 7 using data from seismic models, 99 4.11. Finding a from the assumption aKr = constant, 102 4.12. or versus p in the lower mantle from three thermal models, 104 4.13. Thermal expansivity of silicate perovskite at high P and T, 104 4.14. Problems, 112 5. OXIDES THAT ARE DEBYE-LIKE SOLIDS, 113 5.1. Introduction, 113 XV 5.2. Packing fraction and coordination, 113 5.3. Polyhedral groups in crystal chemistry and Vr, 117 5.4. Comparing 0ae with 6 from calorimetry 0eai, 118 5.5. The moments of the vibrational density of states, 119 5.6. The vibrational spectra (density of states) g(u), 121 5.7. Velocity systematics, 127 5.8. The Griineisen ratio 7 and fac, 139 5.9. dlKf/dP for closely packed oxides and silicates, 140 5.10. The Griineisen ratio of the earth's lower mantle, 141 5.11. The seismic equation of state, 144 5.12. Entropy of closely packed minerals, 145 5.13. The Kieffer model for density of states g{u), 145 REFERENCES, 147 PART II. ISOTHERMAL EQUATIONS OF STATE, 157 6. FINITE STRAIN, 159 6.1. Introduction, 159 6.2. Basic assumption: a series in strain (e) for the energy E(e), 162 6.3. Finite strain equations of state based on e(rj), 167 6.4. Problems with truncation of the series, 170 6.5. The fourth order isothermal equation of state, 171 6.6. On the instability of the Eulerian equation of state, 172 6.7. More on the volume strain relation, e = f(rj), 173 6.8. Problems, 174 7. CONSTRAINING PARAMETERS TO GET THE EQUATION OF STATE, 175 7.1. Introduction, 175 7.2. The Keane EoS: dKT/dP -» K'^ at high P, 175 7.3. The Brennan-Stacey EoS and the Barton-Stacey EoS, 177 7.4. The Murnaghan EoS, 179 7.5. The KTOK'J parameter, 179 XVI 7.6. Other relationships leading to an EoS, 180 7.7. Compression in the earth's lower mantle, 181 8. EQUATIONS OF STATE FROM THE INTER- ATOMIC POTENTIAL, 183 8.1. Introduction, 183 8.2. The attractive interatomic potential <j>a, 184 8.3. A simplification in the Madelung constant, 185 8.4. The repulsive energy term, 185 8.5. The Born-Mie equation of state, 186 8.6. The Born-Meyer equation of state: the method of potentials, 188 8.7. The Demarest equation of state for NaCl: nnn repulsion, 189 8.8. Van der Waals bonds in the potential <f>a, 189 8.9. The Decker equation of state for NaCl, 190 8.10. Equations of state for metals, 191 8.11. EoS parameters for iron at core pressures, 192 8.12. Equations of state of iron at inner core pressures, 196 8.13. How to choose the best EoS: a general discussion, 196 8.14. The virial theorem equation of state, 198 8.15. Choosing an EoS for the earth's lower mantle, 199 9. SHEAR VELOCITY AT HIGH PRESSURE, 201 9.1. Introduction, 201 9.2. Elastic constant relationships in cubic solids (centrosym- metry), 201 9.3. Pressure derivatives for the repulsion model, v = b/rn, 210 9.4. Averaging to get isotropic moduli and velocity, 213 9.5. dv,/dP can be negative, 216 9.6. Finite strain, 217 9.7. Shear velocity versus pressure, 217 9.8. Poisson's ratio in closely-packed cubic metals at high pressure, 220 9.9. Experiments for C44 versus P and v, versus P for NaCl: xvu a test for the central force ionic equation, 223 9.10. Negative values of dva/dP for silicates and oxides, 230 9.11. Calculating the velocity of sound near melting, 230 9.12. The intrinsic (dG/dT)v for oxides and silicates, 231 REFERENCES, 233 PART III. THERMAL PROPERTIES AT HIGH PRESSURE, 241 10. THE THERMAL PRESSURE, 243 10.1. Introduction, 243 10.2. Is there anharmonicity in thermal pressure?, 246 10.3. Anharmonicity effect for thermal pressure at V < VQ, 247 10.4. Experimental results of the dependence of PTH on V, 249 10.5. The volume dependence of OCKT, 254 10.6. (dKT/dT)v for noble gas solids, 256 10.7. General comments on the behavior of PTH, 258 10.8. The thermal pressure of the lower mantle, 259 10.9. The thermal pressure of the inner core, 268 10.10. Thermal pressure and the EoS of silicate perovskite, 271 10.11. Swenson's law, 274 10.12. Summary, 274 11. MELTING, 275 11.1. Introduction, 275 11.2. The Clausius-Clapeyron equation, 276 11.3. Development of the Lindemann law for melting, 278 11.4. The Simon law: a special case of the Lindemann law, 281 11.5. The Kraut-Kennedy law based on the Lindemann law, 282 11.6. The Lindemann law at high compression, 286 11.7. The Lindemann law at P = 0, 286 11.8. Improvements on the Lindemann formulation, 290 11.9. Verification of the Lindemann law for a dense oxide, 291 xvm 11.10. The Lindemann law for oxides and silicates, 292 11.11. The elastic constant instability criterion for melting, 296 11.12. Compressibility divergence, 298 11.13. Compressibility divergence in the Lindemann law, 302 11.14. Melting of iron, 303 11.15. The fundamental two-phase theory of phase transition, 305 12. SHOCKED SOLIDS, 307 12.1. Introduction, 307 12.2. The hydrostatic Hugoniot, 310 12.3. The Hugoniot variables, 312 12.4. The isentropic bulk modulus, 315 12.5. Differentials along the Hugoniot, 317 12.6. Changing from EoS parameters to shock parameters, 318 12.7. Computing the temperature along the Hugoniot, 320 12.8. The temperature of shocked iron along the Hugoniot, 321 12.9. Tm of iron at 330 GPa: the inner-outer core interface, 324 12.10. What is the dominant iron phase in the inner core?, 325 13. THERMODYNAMIC FUNCTIONS, 327 13.1. Introduction, 327 13.2. Basic equations for entropy, 328 13.3. The internal energy as a function of V and T for MgO, 334 13.4. The entropy versus P and T, 338 13.5. The enthalpy versus P and T, 339 13.6. The Helmholtz free energy, 342 13.7. The Gibbs free energy, 343 13.8. Isentropes for MgO and the lower mantle, 344 13.9. Finding 7 from the entropy, 345 13.10. Extrapolations to absolute zero limits for KT, 345 13.11. Important sources of uncertainty, 345 REFERENCES, 347 XIX APPENDIXES, 355 Table A-l. Physical constants and conversions, 355 Table A—2. Prefix used for fractions and multiples, 355 Table A—3. International system of units (SI units), 356 Table A—4. Atomic mass m of selected elements, 357 Table A-5. Values of the earth model PREM, 358 Table A-6. Debye functions for high T at P = 0, 360 Table A-7. Physical properties and thermoelastic parameters of oxides and silicates at high T, 362 GLOSSARY OF SYMBOLS, 371 AUTHOR INDEX, 379 SUBJECT INDEX, 383.