Vibrational Thermodynamics of MaterialsBrent FultzCalifornia Institute of Technology, W. M. Keck Laboratory, Pasadena CA 91125 USAJuly 6, 2009Abstract. The literature on vibrational thermodynamics of materials is reviewed. The emphasisis on metals and alloys, especially on the progress over the last decade in understanding differencesin the vibrational entropy of different alloy phases and phase transformations. Some results oncarbides, nitrides, oxides, hydrides and lithium-storage materials are also covered.Principles of harmonic phonons in alloys are organized into thermod ynamic models for unmixingand ordering transformations on an Ising lattice, and extended for non-harmonic potentials. Owingto the high accuracy required for the phonon frequencies, quantitative predictions of vibrationalentropy with analytical models prove elusive. Accurate tools for such calculations or measurementswere challenging for many years, but are more accessible today. Ab-initio meth ods for calculatingphonons in solids are summarized. The experimental techniques of calorimetry, inelastic neutronscattering, and inelastic x-ray scattering are explained with enough detail to show the issues ofusing these methods for investigations of vibrational thermodynamics. The explanations extend tomethods of data analysis that affect the accuracy of thermodynamic information.It is sometimes possible to identify the structural and chemical origins of the differences invibrational entropy of materials, and the number of these assessments is growing. There has beenconsiderable progress in our understanding of the vibrational entropy of mixing in solid solutions,compoun d formation from pure elements, chemical unmixing of alloys, order-disorder transforma-tions, and martensitic transformations. Systematic trends are available for some of these phasetransformations, although more examples are needed, and many results are less reliable at hightemperatures. Nanostructures in materials can alter sufficiently the vibrational dynamics to affectthermodynamic stability. I nternal stresses in polycrystals of anisotropic materials also contributeto the heat capacity. Lanthanides and actinides show a complex interplay of vibrational, electronic,and magnetic entropy, even at low temperatures.A “quasiharmonic model” is often used to extend the systematics of harmonic phonons tohigh temperatures by accounting for t he effects of thermal expansion against a bulk modulus.Non-harmonic effects beyond the quasiharmonic approximation originate from the interactions ofthermally-excited phonons with oth er phonons, or with the interactions of phonons with electronicexcitations. In the classical high temperature limit, the adiabatic electron-phonon coupling canhave a surprisingly large effect in metals when temperature causes significant changes in the elec-tron density near the Fermi level. There are useful similarities in how temperature, pressure, andcomposition alter the conduction electron screening and the int eratomic force constants. Phonon-phonon “anharmonic” interactions arise from those non-harmonic parts of the interatomic potentialthat cannot be accounted for by the quasiharmonic model. Anharmonic shifts in phonon frequencywith temperature can be substantial, but trends are not well understood. Anharmonic phonondamping does show systematic trends, however, at least for fcc metals.Trends of vibrational entropy are often justified with atomic properties such as atomic size,electronegativity, electron-to-atom ratio, and mass. S ince vibrational entropy originates at thelevel of electrons in solids, such rules of thumb prove no better than similar rules devised fortrends in bonding and structure, and tend to be worse. Fortunately, the required tools for accurateexperimental investigations of vibrational entropy have improved dramatically over the past fewyears, and the required ab-initio methods have become more accessible. Steady progress is expectedfor understanding the ph enomena reviewed here, as investigations are performed with the new toolsof experiment and theory, sometimes in integrated ways.II B. FultzTable of ContentsI. Principles and Methods1 Overview 12 Harmonic Lattice Dynamics 22.1 Partition Function 32.2 Hamiltonian for Lattice Dynamics 52.3 Equations of Motion 62.4 The Eigenvalue Problem for the Polarization Vector 72.5 Calculation of the Phonon Density of States 83 Predictions with the Harmonic Model 93.1 Mass, Local Modes, and Resonance Mod es 93.2 Long Wavelength Limit and the Debye Model 113.3 Disordered Systems 134 Bond Proportion Model 134.1 Bond Proportion Model and Unmixing on the Ising Lattice 144.2 Bond Proportion Model and Ordering on the Ising Lattice 184.3 Monte Carlo Results 205 Bond-Stiffness-versus-Bond-Length Model 215.1 Phonon Frequencies and Bond Lengths 215.2 Extending the Bond Proportion Model 245.3 Chemical Effects on Bond Stiffness 256 Heat Capacity 256.1 Harmonic Heat Capacity 256.2 Quasiharmonic Thermodynamics 276.3 Anharmonic Heat Capacity 296.4 Thermodynamic Entropy 307 Calorimeters 318 Neutron Scattering 328.1 Elastic and Inelastic Scattering 328.2 One-Phonon and Multiphonon Scattering 358.3 Neutron Weighting 378.4 Direct-Geometry Fermi Chopper Spectrometer 388.5 Triple-Axis Spectrometer 389 Ab-Initio Methods 399.1 Density Functional Theory 409.2 First-Principles Phonon Calculations 409.3 Molecular Dynamics 41II. Vibrational Entropy of Materials at Lower Temperatures10 Earlier Investigations 4310.1 Polymorphism 4310.2 Solubilities and Defect Concentrations 4410.3 Carbides, Nitrides, and Oxides 4711 First-Principles Studies of Solutions an d S tu dies of Ordered Compounds 48Vibrational Thermodynamics of Materials III11.1 Solubility in Al Alloys 4811.2 Aluminum Compounds 4911.3 Low-Symmetry Phases at High Temperatures 5012 Chemical Order-Disorder Transformations 5212.1 Transition Metal Aluminides 5212.2 B2 Ordering in CuZn and FeCo 5612.3 L12Ordering 5713 Unmixing 6213.1 Computational Studies of Unmixing 6213.2 Cluster Expansion Method for Solid Solutions 6413.3 Fe-Cr Solid Solutions 6614 Interstitial Alloys 6814.1 Interstitial Oxygen and Nitrogen 6814.2 Hydrogen and Hydrides 6914.3 Lithium Intercalation and Insertion 7015 Martensite and Twin ning 7316 Microstructural Contributions 7616.1 Elastic Energy 7716.2 Nanostructures 7917 f-Electron Metals 8417.1 Cerium 8417.2 Uranium and Plutonium 87III. Vibrational Entropy of Materials a t Higher Temperatures18 Interactions between