Equations of State for Solids in Geophysics and Ceramic Science

CONTENTS......Page 14
PART I. THERMAL PHYSICS......Page 22
1.2. The Helmholtz free energy......Page 24
1.3. Pressure: the equation of state......Page 26
1.4. The Grüneisen parameters......Page 27
1.5. The bulk modulus K......Page 39
1.6. The Debye temperature θ: a lower bound for high T......Page 45
1.7. The Debye temperature of the earth and the moon......Page 47
1.8. γD: the Debye approximation to γ......Page 49
1.9. Electronic heat capacity contribution to γ for iron......Page 50
1.10. Problems......Page 51
2.1. Introduction......Page 52
2.2. The vibrational energy and the thermal energy......Page 53
2.3. The quasiharmonic approximation......Page 55
2.4. The Mie-Grüneisen equation of state......Page 56
2.5. The high-temperature limit of the quasiharmonic approximation......Page 57
2.6. Anharmonic corrections to the Helmholtz energy......Page 66
2.7. The free energy and its physical properties at very low temperature......Page 70
2.8. The Debye theory interpolation......Page 73
2.9. Thermodynamic functions from the partition function......Page 76
2.10. Problems......Page 77
3.1. Thermodynamic identities......Page 78
3.2. The mean atomic mass, μ = M/p......Page 82
3.3 The cases for (∂K[sub(T)]/∂T)[sub(v)] = 0 and (∂(αK[sub(T)]) /∂T)[sub(p)] = 0......Page 83
3.4. Theoretical insight into the change of K\' with T......Page 84
3.5. The condition δ[sub(T)] = K\' in η,T space......Page 87
3.6. The variation of δ[sub(T)] with compression......Page 90
3.7. The variation of αK[sub(T)] with compression......Page 95
3.8. The variation of γ and q with compression......Page 97
3.9. Experimental insight into the value of ∂[sup(2)]K[sub(T)]/∂P∂T......Page 102
3.11. Problems......Page 103
4.2. Thermal expansivity at high T and constant η......Page 104
4.3. Thermal expansivity versus T at high temperature and constant pressure......Page 105
4.4. Thermal expansivity versus η7 at constant T......Page 106
4.5. Measurements of V versus T for silicate perovskite......Page 111
4.6. Grüneisen\'s theory of thermal expansivity (P = 0)......Page 113
4.7. Suzuki\'s theory of thermal expansivity......Page 115
4.8. High temperature expansivity of NaCl......Page 117
4.9. The uncompressed value of α in the lower mantle......Page 118
4.10. Obtaining α from γ using data from seismic models......Page 120
4.11. Finding α from the assumption αK[sub(T)] = constant......Page 123
4.13. Thermal expansivity of silicate perovskite at high P and T......Page 125
4.14. Problems......Page 133
5.2. Packing fraction and coordination......Page 134
5.3. Polyhedral groups in crystal chemistry and V[sub(r)]......Page 138
5.4. Comparing θ[sub(ac)] with θ from calorimetry θ[sub(cal)]......Page 139
5.5. The moments of the vibrational density of states......Page 140
5.6. The vibrational spectra (density of states) g(ω)......Page 142
5.7. Velocity systematics......Page 148
5.8. The Grüneisen ratio γ and γ[sub(ac)]......Page 160
5.9. dK[sub(T)]/dP for closely packed oxides and silicates......Page 161
5.10. The Grüneisen ratio of the earth\'s lower mantle......Page 162
5.11. The seismic equation of state......Page 165
5.13. The Kieffer model for density of states g(ω)......Page 166
REFERENCES......Page 168
PART II. ISOTHERMAL EQUATIONS OF STATE......Page 178
6.1. Introduction......Page 180
6.2. Basic assumption: a series in strain (ε) for the energy E(ε)......Page 183
6.3. Finite strain equations of state based on ε(η)......Page 188
6.4. Problems with truncation of the series......Page 191
6.5. The fourth order isothermal equation of state......Page 192
6.6. On the instability of the Eulerian equation of state......Page 193
6.7. More on the volume strain relation, ε = f(η)......Page 194
6.8. Problems......Page 195
7.2. The Keane EoS: dK[sub(T)]/dP → K\'∞ at high P......Page 196
7.3. The Brennan–Stacey EoS and the Barton–Stacey EoS......Page 198
7.5. The K[sub(To)]K\'o parameter......Page 200
7.6. Other relationships leading to an EoS......Page 201
7.7. Compression in the earth\'s lower mantle......Page 202
8.1. Introduction......Page 204
8.2. The attractive interatomic potential φ[sub(a)]......Page 205
8.4. The repulsive energy term......Page 206
8.5. The Born–Mie equation of state......Page 207
8.6. The Born–Meyer equation of state: the method of potentials......Page 209
8.8. Van der Waals bonds in the potential φ[sub(a)]......Page 210
8.9. The Decker equation of state for NaCl......Page 211
8.10. Equations of state for metals......Page 212
8.11. EoS parameters for iron at core pressures......Page 213
8.13. How to choose the best EoS: a general discussion......Page 217
8.14. The virial theorem equation of state......Page 219
8.15. Choosing an EoS for the earth\'s lower mantle......Page 220
9.2. Elastic constant relationships in cubic solids (centrosymmetry)......Page 222
9.3. Pressure derivatives for the repulsion model, v = b/r[sup(n)]......Page 231
9.4. Averaging to get isotropic moduli and velocity......Page 234
9.5. dv,/dP can be negative......Page 237
9.7. Shear velocity versus pressure......Page 238
9.8. Poisson\'s ratio in closely-packed cubic metals at high pressure......Page 241
9.9. Experiments for C[sub(44)] versus P and v[sub(3)] versus P for NaCl: a test for the central force ionic equation......Page 244
9.11. Calculating the velocity of sound near melting......Page 251
9.12. The intrinsic (∂G/∂T)[sub(v)] for oxides and silicates......Page 252
REFERENCES......Page 254
PART III. THERMAL PROPERTIES AT HIGH PRESSURE......Page 262
10.1. Introduction......Page 264
10.2. Is there anharmonicity in thermal pressure?......Page 267
10.3. Anharmonicity effect for thermal pressure at V < V[sub(0)]......Page 268
10.4. Experimental results of the dependence of P[sub(TH)] on V......Page 270
10.5. The volume dependence of αK[sub(T)]......Page 275
10.6. (∂K[sub(T)]/∂T)[sub(v)] for noble gas solids......Page 277
10.7. General comments on the behavior of P[sub(TH)]......Page 279
10.8. The thermal pressure of the lower mantle......Page 280
10.9. The thermal pressure of the inner core......Page 289
10.10. Thermal pressure and the EoS of silicate perovskite......Page 292
10.12. Summary......Page 295
11.1. Introduction......Page 296
11.2. The Clausius–Clapeyron equation......Page 297
11.3. Development of the Lindemann law for melting......Page 299
11.4. The Simon law: a special case of the Lindemann law......Page 302
11.5. The Kraut–Kennedy law based on the Lindemann law......Page 303
11.7. The Lindemann law at P = 0......Page 307
11.8. Improvements on the Lindemann formulation......Page 311
11.9. Verification of the Lindemann law for a dense oxide......Page 312
11.10. The Lindemann law for oxides and silicates......Page 313
11.11. The elastic constant instability criterion for melting......Page 317
11.12. Compressibility divergence......Page 319
11.13. Compressibility divergence in the Lindemann law......Page 323
11.14. Melting of iron......Page 324
11.15. The fundamental two–phase theory of phase transition......Page 326
12.1. Introduction......Page 328
12.2. The hydrostatic Hugoniot......Page 331
12.3. The Hugoniot variables......Page 333
12.4. The isentropic bulk modulus......Page 336
12.5. Differentials along the Hugoniot......Page 338
12.6. Changing from EoS parameters to shock parameters......Page 339
12.7. Computing the temperature along the Hugoniot......Page 341
12.8. The temperature of shocked iron along the Hugoniot......Page 342
12.9. T[sub(m)] of iron at 330 GPa: the inner–outer core interface......Page 345
12.10. What is the dominant iron phase in the inner core?......Page 346
13.1. Introduction......Page 348
13.2. Basic equations for entropy......Page 349
13.3. The internal energy as a function of V and T for MgO......Page 355
13.4. The entropy versus P and T......Page 359
13.5. The enthalpy versus P and T......Page 360
13.6. The Helmholtz free energy......Page 363
13.7. The Gibbs free energy......Page 364
13.8. Isentropes for MgO and the lower mantle......Page 365
13.11. Important sources of uncertainty......Page 366
REFERENCES......Page 368
APPENDIXES......Page 376
Table A–7. Physical properties and thermoelastic parameters of oxides and silicates at high T......Page 20
GLOSSARY OF SYMBOLS......Page 392
B......Page 400
K......Page 401
S......Page 402
Z......Page 403
A......Page 404
B......Page 406
D......Page 407
E......Page 409
G......Page 411
H......Page 413
I......Page 414
L......Page 415
M......Page 416
N......Page 417
P......Page 418
Q......Page 420
S......Page 421
T......Page 423
W......Page 425
Z......Page 426