Author's personal copy 2.06 Theory and Practice – Thermodynamics, Equations of State, Elasticity, and Phase Transitions of Minerals at High Pressures and Temperatures

2.06.1 Thermodynamics of Crystals 122 2.06.1.1 Thermodynamic Potentials 122 2.06.1.2 Differential Relations 122 2.06.1.3 Partition Function 123 2.06.1.4 Harmonic Approximation 124 2.06.1.4.1 Debye model 125 2.06.1.4.2 General harmonic potential 126 2.06.1.5 Quantum Effects in Thermodynamics 127 2.06.1.6 Thermodynamic Perturbation Theory 128 2.06.1.7 Quasiharmonic Approximation 129 2.06.1.8 Beyond the QHA 129 2.06.2 Equations of State and Elasticity 131 2.06.2.1 Equations of State 131 2.06.2.1.1 Mie–Grüneisen EOS 131 2.06.2.1.2 Analytical static EOS 132 2.06.2.1.3 Anharmonicity in static EOS 134 2.06.2.1.4 EOS, internal strain, and phase transitions 134 2.06.2.2 Elastic Constants 135 2.06.2.2.1 Cauchy relations 138 2.06.2.2.2 Mechanical stability 139 2.06.2.2.3 Birch’s law and effects of temperature on the elastic constants 139 2.06.2.2.4 Elastic anisotropy in the Earth’s interior 140 2.06.3 Phase Transitions of Crystals 140 2.06.3.1 Classifications of Phase Transitions 140 2.06.3.2 First-Order Phase Transitions 141 2.06.3.3 Landau Theory of Firstand Second-Order Transitions 141 2.06.3.4 Shortcomings of Landau Theory 142 2.06.3.5 Ginzburg–Landau Theory 143 2.06.3.6 Ising Spin Model 144 2.06.3.7 Mean-Field Treatment of Order–Disorder Phenomena 145 2.06.3.8 Isosymmetric Transitions 145 2.06.3.9 Transitions with Group–Subgroup Relations 146 2.06.3.10 Pressure-Induced Amorphization 146 2.06.4 A Few Examples of the Discussed Concepts 147 2.06.4.1 Temperature Profile of the Earth’s Lower Mantle and Core 147 2.06.4.2 Polytypism of MgSiO3 Post-Perovskite and Anisotropy of the Earth’s D0 layer 147 2.06.4.3 Spin Transition in (Mg, Fe)O Magnesiowüstite 148 References 149