From Hamiltonians to Phase Diagrams

1. Introduction.- 1.1 Why This Book Was Written and What It Contains.- 1.2 The Thermodynamic Origin of Phase Diagrams.- 1.2.1 First-Order Transitions Without Compositional Change — The pT Phase Diagram.- 1.2.2 First-Order Transitions with Compositional Change — The Alloy Phase Diagram.- 1.2.3 Second-Order Transitions.- 1.3 Adiabatic Decoupling of the Ionic and Electronic Degrees of Freedom.- 1.4 The One-Electron Approximation.- 1.5 Tightly-Bound and Nearly-Free Electrons; Potentials and Pseudopotentials.- 1.6 Response Theory and Interatomic Interactions.- 1.7 The Statistical Mechanics of a Vibrating Lattice.- 1.8 Periodic, Aperiodic and Quasi-Periodic Structures.- 1.9 Elementary Excitations in Aperiodic Structures.- 1.9.1 Effective Medium Approximations.- 1.9.2 The Recursion Method and Related Techniques.- 1.10 Configurational Thermodynamics of Solids and Liquids.- 1.10.1 Order-Disorder Transitions.- 1.10.2 From the Interatomic Force Law to the Structure of a Liquid.- 1.10.3 Order-Parameter Approach to Freezing and Melting.- 2. Interatomic Forces in Metals and Alloys.- 2.1 Pseudopotentials.- 2.1.1 The Operator Approach.- 2.1.2 The Scattering Approach.- 2.1.3 Model Potentials.- 2.2 Response Theory.- 2.2.1 Nonlocality.- 2.2.2 Screening Beyond the Random Phase Approximation.- 2.3 Effective Pair Potentials in Pure Metals.- 2.4 Effective Pair Potentials in Binary Alloys.- 2.4.1 Chemical Compression.- 2.4.2 Chemical Ordering.- 2.5 Interatomic Forces in Non-Simple Metals and Alloys.- 2.6 Beyond Perturbation Theory.- 3. Phase Stability of Crystalline Metals.- 3.1 Simple-Metal Cohesion.- 3.1.1 An Excursion into Transition-Metal Cohesion.- 3.2 Structural Stability.- 3.3 Trends in Crystal Structures.- 3.3.1 A Brief Remark on Transition Metal Structures.- 3.3.2 The Crystal Structures of the Lanthanides.- 3.3.3 Charge-Density Analysis of Bonding.- 3.4 Pressure-Induced Phase Changes.- 3.5 Thermodynamics of Crystalline Metals.- 3.5.1 Harmonic Lattice Dynamics.- 3.5.2 Anharmonicity.- 3.5.3 Variational Method for Calculating Thermodynamic Properties (Gibbs-Bogoljubov Method).- 3.6 Temperature-Induced Phase Changes.- 4. Structure and Thermodynamics of Liquid Metals.- 4.1 Computer Simulations.- 4.2 Integral Equation Approach.- 4.3 Thermodynamic Perturbation Theories.- 4.3.1 Thermodynamic Variational Method for Liquids (Gibbs-Bogoljubov Method).- 4.3.2 Repulsive Forces: Weeks-Chandler-Andersen Theory.- 4.3.3 Attractive Forces: Random Phase Approximation, Optimized Random Phase Approximation and Mean Spherical Approximation.- 4.4 Trends in Liquid Structures.- 4.5 Expanded Fluid Metals.- 4.6 Structure and Thermodynamics of Liquid Transition and Rare-Earth Metals.- 4.7 Atomic Motion in Liquid Metals.- 5. The pT Phase Diagram of Pure Metals.- 5.1 Solid-Liquid Trnasitions: The Total Energy Approach.- 5.2 Microscopic Theories of Melting and Freezing.- 5.3 The Liquid-Vapour Transition.- 6. Alloy Formation and Stability.- 6.1 Nearly-Free-Electron Approach to Alloy Formation.- 6.1.1 Criteria for Solubility in Homovalent Systems.- 6.1.2 Variations of the Atomic Volume in Heterovalent Alloy Systems.- 6.1.3 The Chemical Potential Model for the Heat of Formation.- 6.1.4 Band Picture for Simple-Metal Alloys.- 6.1.5 Real-Space Picture for Alloy Formation.- 6.1.6 A Brief Summary.- 6.2 Miedema’s Semiempirical Theory of Alloy Formation.- 6.2.1 Microscopic Interpretation of Miedema’s Alloying Rules: Simple Metals.- 6.2.2 Microscopic Interpretation of Miedema’s Alloying Rules: Transition Metals.- 7. Solid Substitutional Alloys.- 7.1 Primary Solid Solutions.- 7.1.1 The Homovalent Case.- 7.1.2 The Heterovalent Case.- 7.2 Hume-Rothery Phases.- 7.3 Static Lattice Distortions.- 7.4 Ordering in Substitutional Alloys.- 7.4.1 Long-Range Order.- 7.4.2 Short-Range Order.- 7.4.3 Ordering in Substitutional Transition-Metal Alloys..- 7.5 Thermodynamics of Alloys.- 7.5.1 Lattice Dynamics of Substitutional Alloys.- 7.5.2 Thermodynamic Perturbation Theory.- 7.5.3 Vibrational Dynamics and the Ordering Transition..- 8. Intermetallic Compounds.- 8.1 Structure Maps.- 8.2 Empirical Pair-Potential Analysis of Intermetallic Phases.- 8.3 Classification of Intermetallic Phases According to Building Principles and Properties.- 8.4 Topologically Close-Packed Intermetallic Compounds (Frank-Kasper Phases).- 8.4.1 Heat and Volume of Formation of Laves Phases.- 8.4.2 Structural Stability of Laves Phases.- 8.4.3 Other Topologically Close-Packed Compounds.- 8.4.4 Charge-Density Analysis of Bonding.- 8.4.5 Lattice Dynamics of Topologically Close-Packed Compounds.- 8.5 Intermetallic Phases with Large Band-Structure Stabilization.- 8.5.1 Charge-Density Analysis of Bonding in Zintl Phases.- 8.5.2 Lattice Dynamics of Zintl Phases.- 9. Liquid Alloys.- 9.1 Computer Simulations of Binary Liquid Alloys.- 9.2 Thermodynamic Variational Calculations.- 9.2.1 Systems with a Nearly Ideal Mixing Behaviour.- 9.2.2 Liquid Alloys with Strong Chemical Short-Range Order.- 9.2.3 Liquid Alloys with a Miscibility Gap.- 9.3 Thermodynamic Perturbation Theory.- 9.3.1 Repulsive Forces: Weeks-Chandler-Andersen Theory.- 9.3.2 Long-Range Forces: Optimized-Random-Phase Approximation.- 9.4 Structure and Thermodynamics of Liquid Transition-Metal Alloys.- 9.5 Collective Excitations in Liquid Alloys.- 10. Alloy Phase Diagrams.- 10.1 First Principles Calculations of Alloy Phase Diagrams.- 10.2 Chemical Short-Range Order and Alloy Phase Diagrams.- 10.2.1 Melting Extrema.- 10.2.2 Eutectic Diagrams.- 10.2.3 Compound Formation.- 10.2.4 Phase Separation in the Liquid State.- 10.3 Molecular Theory of the Freezing of Liquid Alloys.- 11. Beyond the Phase Diagram: The Formation and Properties of Metastable Phases.- 11.1 Amorphous Alloys — Metallic Glasses.- 11.1.1 Glass-Forming Ability.- 11.1.2 Atomic Structure of Metallic Glasses.- 11.1.3 Elementary Excitations in Metallic Glasses.- 11.2 Quasi-Crystals.- 12. Conclusions and Outlook.- Appendices.- A. Density-Functional Pseudopotentials.- A.1 Optimized Pseudopotentials — the Operator Approach.- A.2 Norm-Conserving Pseudopotentials — the Scattering Approach.- B. Linear Response Theroy.- C. Electrostatic Energies of Crystals and Liquids.- C.1 The Madelung Constants of the Elemental Structures.- C.2 The Madelung Constants of Binary Alloys and Intermetallic Compounds.- C.3 Electrostatic Energies of Model Liquids and Liquid Mixtures.- D. Liquid State Theory: Integral Equations, Variational Principles and Exactly Soluble Models.- D.1 Correlation Functions and Equations of State.- D.2 Integral Equations and Variational Principles for the Total and Direct Correlation Functions.- D.3 Analytical Solutions for Model Liquids and Mixtures.- D.3.1 Solution of the PY Equation for the Hard-Sphere Fluid.- D.3.2 Solution of the PY Equation for Hard-Sphere Mixtures.- D.3.3 The Solution of the MSA for Charged Hard Spheres with Yukawa Interactions.- D.3.4 The Solution of the MSA for a Symmetric Mixture of Charged Hard Spheres with Yukawa Interactions.- References.