Tian Lan, C. W. Li, O. Hellman, D. S. Kim, J. A. Munoz, H. Smith, D. L. Abernathy and B. Fultz

Phys. Rev. B 92, 054304 (2015).

Although the rutile structure of TiO_2 is stable at high temperatures, the conventional quasiharmonic approximation predicts that several acoustic phonons decrease anomalously to zero frequency with thermal expansion, incorrectly predicting a structural collapse at temperatures well below 1000 K. Inelastic neutron scattering was used to measure the temperature dependence of the phonon density of states (DOS) of rutile TiO_2 from 300 to 1373 K. Surprisingly, these anomalous acoustic phonons were found to increase in frequency with temperature. First-principles calculations showed that with lattice expansion, the potentials for the anomalous acoustic phonons transform from quadratic to quartic, stabilizing the rutile phase at high temperatures. In these modes, the vibrational displacements of adjacent Ti and O atoms cause variations in hybridization of 3d electrons of Ti and 2p electrons of O atoms. With thermal expansion, the energy variation in this "phonon-tracked hybridization" flattens the bottom of the interatomic potential well between Ti and O atoms, and induces a quarticity in the phonon potential.

F. Koermann, B. Grabowski, B. Dutta, T. Hickel, L. Mauger, B. Fultz and J. Neugebauer

Phys. Rev. Lett. 113, 165503 (2014).

An ab initio based framework for quantitatively assessing the phonon contribution due to magnon-phonon interactions and lattice expansion is developed. The theoretical results for bcc Fe are in very good agreement with high-quality phonon frequency measurements. For some phonon branches, the magnon-phonon interaction is an order of magnitude larger than the phonon shift due to lattice expansion, demonstrating the strong impact of magnetic short-range order even significantly above the Curie temperature. The framework closes the previous simulation gap between the ferro- and paramagnetic limits.

L. Mauger, M.S. Lucas, J. A. Munoz, S. J. Tracy, M. Kresch, Yuming Xiao, Paul Chow and B. Fultz

Phys. Rev. B 90, 064303 (2014).

Phonon densities of states (DOS) of bcc alpha-57Fe were measured from room temperature through the 1044 K Curie transition and the 1185 K fcc gamma-Fe phase transition using nuclear resonant inelastic x-ray scattering. At higher temperatures all phonons shift to lower energies (soften) with thermal expansion, but the low transverse modes soften especially rapidly above 700 K, showing strongly nonharmonic behavior that persists through the magnetic transition. Interatomic force constants for the bcc phase were obtained by iteratively fitting a Born-von Karman model to the experimental phonon spectra using a genetic algorithm optimization. The second-nearest-neighbor fitted axial force constants weakened signicantly at elevated temperatures. An unusually large nonharmonic behavior is reported, which increases the vibrational entropy and accounts for a contribution of 35 meV/atom in the free energy at high temperatures. The nonharmonic contribution to the vibrational entropy follows the thermal trend of the magnetic entropy, and may be coupled to magnetic excitations. A small change in vibrational entropy across the alpha-gamma structural phase transformation is also reported.

S.J. Tracy, L. Mauger, H.J. Tan, J. A. Munoz, Y.M. Xiao and B. Fultz

Phys. Rev. B 90, 094303 (2014).

Valence fluctuations of Fe2+ and Fe3+ were studied in a solid solution of Li_xFePO_4 by nuclear resonant forward scattering of synchrotron x-rays while the sample was heated in a diamond-anvil pressure cell. The spectra acquired at different temperatures and pressures were analyzed for the frequencies of valence changes using the Blume-Tjon model of a system with a fluctuating Hamiltonian. These frequencies were analyzed to obtain activation energies and an activation volume for polaron hopping. There was a large suppression of hopping frequency with pressure, giving an activation volume for polaron hopping of 5.8+-0.7 A^3. This big, positive value is typical of ion diffusion, which indicates correlated motions of polarons and Li+ ions that alter the dynamics of both. Monte Carlo simulations were used to estimate the strength of the polaron-ion interaction energy.

Maxwell Murialdo, Nicholas P. Stadie, Channing C. Ahn and Brent Fultz

Journal of Physical Chemistry, 119, 944 (2015).

Ethane adsorption was measured on zeolite-templated carbon (ZTC) and compared to superactivated carbon MSC-30. Isotherms measured at temperatures between 252 and 423 K were fitted using a superposition of two Langmuir isotherms and thermodynamic properties were assessed. Unlike typical carbon adsorbents, the isosteric heat of adsorption on ZTC increases by up to 4.6 kJ/mol with surface coverage. This increase is attributed to strong adsorbate-adsorbate intermolecular interactions, a hypothesis that is shown to be consistent with fundamental estimates of intermolecular interactions. Furthermore, the molar entropy of the adsorbed phase was measured and compared to an estimate derived from statistical mechanics. While the measured and estimated entropies of the adsorbed phase of ethane on MSC-30 are in agreement, they differ significantly on ZTC at high coverage, indicative of the atypical properties of ethane adsorption on ZTC.

Download corrected draft of July 6, 2009 (4.5 MB)

Progress in Materials Science, 55, 247-352 (2010).

Brent Fultz

The literature on vibrational thermodynamics of materials is reviewed. The emphasis is on metals and alloys, especially on the progress over the last decade in understanding differences in the vibrational entropy of different alloy phases and phase transformations. Some results on carbides, nitrides, oxides, hydrides and lithium-storage materials are also covered. Principles of harmonic phonons in alloys are organized into thermodynamic models for unmixing and ordering transformations on an Ising lattice, and extended for non-harmonic potentials. Owing to the high accuracy required for the phonon frequencies, quantitative predictions of vibrational entropy with analytical models prove elusive. Accurate tools for such calculations or measurements were challenging for many years, but are more accessible today. Ab-initio methods for calculating phonons in solids are summarized. The experimental techniques of calorimetry, inelastic neutron scattering, and inelastic x-ray scattering are explained with enough detail to show the issues of using these methods for investigations of vibrational thermodynamics. The explanations extend to methods of data analysis that affect the accuracy of thermodynamic information.

It is sometimes possible to identify the structural and chemical origins of the differences in vibrational entropy of materials, and the number of these assessments is growing. There has been considerable progress in our understanding of the vibrational entropy of mixing in solid solutions, compound formation from pure elements, chemical unmixing of alloys, order-disorder transformations, and martensitic transformations. Systematic trends are available for some of these phase transformations, although more examples are needed, and many results are less reliable at high temperatures. Nanostructures in materials can alter sufficiently the vibrational dynamics to affect thermodynamic stability. Internal stresses in polycrystals of anisotropic materials also contribute to 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 to high temperatures by accounting for the effects of thermal expansion against a bulk modulus. Non-harmonic effects beyond the quasiharmonic approximation originate from the interactions of thermally-excited phonons with other phonons, or with the interactions of phonons with electronic excitations. In the classical high temperature limit, the adiabatic electron-phonon coupling can have a surprisingly large effect in metals when temperature causes significant changes in the electron density near the Fermi level. There are useful similarities in how temperature, pressure, and composition alter the conduction electron screening and the interatomic force constants. Phonon-phonon "anharmonic" interactions arise from those non-harmonic parts of the interatomic potential that cannot be accounted for by the quasiharmonic model. Anharmonic shifts in phonon frequency with temperature can be substantial, but trends are not well understood. Anharmonic phonon damping 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. Since vibrational entropy originates at the level of electrons in solids, such rules of thumb prove no better than similar rules devised for trends in bonding and structure, and tend to be worse. Fortunately, the required tools for accurate experimental investigations of vibrational entropy have improved dramatically over the past few years, and the required ab-initio methods have become more accessible. Steady progress is expected for understanding the phenomena reviewed here, as investigations are performed with the new tools of experiment and theory, sometimes in integrated ways.

Download draft of Feb. 6, 2011 (0.9 MB)

Book chapter in __Characterization of Materials__. Elton Kaufmann, Editor (John Wiley, New York, 2011).

Brent Fultz

Mossbauer spectrometry gives electronic, magnetic, and structural information from within materials. A Mossbauer spectrum is an intensity of gamma-ray absorption versus energy for a specific resonant nucleus such as ^{57}Fe or ^{119}Sn. For one nucleus to emit a gamma-ray and a second nucleus to absorb it with efficiency, both nuclei must be embedded in solids, a phenomenon known as the "Mossbauer effect." Mossbauer spectrometry looks at materials from the "inside out," where "inside" refers to the resonant nucleus.

Mossbauer spectra give quantitative information on "hyperfine interactions," which are small energies from the interaction between the nucleus and its neighboring electrons. The three hyperfine interactions originate from the electron density at the nucleus (the isomer shift), the gradient of the electric field (the nuclear quadrupole splitting), and the unpaired electron density at the nucleus (the hyperfine magnetic field). Over the years, methods have been refined for using these three hyperfine interactions to determine valence and spin at the resonant atom. Even when the hyperfine interactions are not easily interpreted, they can often be used reliably as "fingerprints" to identify the different local chemical environments of the resonant atom, usually with a good estimate of their fractional abundances. Mossbauer spectrometry is useful for quantitative phase analyses or determinations of the concentrations of resonant element in different phases, even when the phases are nanostructured or amorphous.

Most Mossbauer spectra are acquired with simple laboratory equipment and a radioisotope source, but the recent development of synchrotron instrumentation now allow for measurements on small 10 micron samples, which may be exposed to extreme environments of pressure and temperature. Other capabilities include measurements of the vibrational spectra of the resonant atoms, and coherent scattering and diffraction of nuclear radiation.

This article is not a review of the field, but an instructional reference that explains principles and practices, and gives the working materials scientist a basis for evaluating whether or not Mossbauer spectrometry may be useful for a research problem. A few representative materials studies are presented.

Download draft open source textbook of July 2016 (365 pages, 13.6 MB)

Brent Fultz, Tim Kelley, Jiao Lin, JaeDong Lee, Olivier Delaire, Max Kresch, Mike McKerns, Michael Aivazis

This book is intended for graduate students beginning their doctoral research in inelastic neutron scattering, and also for scientists who need to learn how to use inelastic neutron chopper spectrometers and their data analysis. The book explains the physical principles behind excitations in hard condensed matter, how neutrons are scattered inelastically by these excitations, and to best measure inelastic neutron scattering. Also included are descriptions about how to compute the spectra measured with inelastic neutron spectrometers, and the text offers some development of computational scattering science. The focus of this text, and our heartfelt concern, is for the graduate student who enters the field of inelastic neutron scattering with no experience with instruments, probably only a sketchy understanding of the scientific principles, and perhaps limited knowledge of modern concepts in software engineering. This text was designed to help the reader in all three areas, and do so effficiently. The text continues to evolve, but it presently has enough content to satisfy these expectations. Some improvements in explanations and writing are expected over the next years, but it should be useful today.

*612 pages, 446 figures, 1,502 equations (Cambridge, 2014).*

Brent Fultz

Offering a fresh viewpoint on phase changes and the thermodynamics of materials, this textbook covers the thermodynamics and kinetics of the most important phase transitions in materials science, spanning classical metallurgy through to nanoscience and quantum phase transitions. Clear, concise and complete explanations rigorously address transitions from the atomic scale up, providing the quantitative concepts, analytical tools and methods needed to understand modern research in materials science. Topics are grouped according to complexity, ensuring that students have a solid grounding in core topics before they begin to tackle more advanced material, and are accompanied by numerous end-of-chapter problems. With explanations firmly rooted in the context of modern advances in electronic structure and statistical mechanics, and developed from classroom teaching, this book is the ideal companion for graduate students and researchers in materials science, condensed matter physics, solid state science, and physical chemistry.

Excerpts from the first edition of the book in Adobe Acrobat .pdf format are
here.

*761 pages, 478 figures, 1,300 equations (Springer-Verlag, 2012).*

Brent Fultz and James Howe

This book explains concepts of transmission electron microscopy (TEM) and x-ray diffractometry (XRD) that are important for the characterization of materials. The fourth edition has been updated to cover important technical developments, including electron tomography and new nanobeam methods.
This edition is not substantially longer than the third, but all chapters have been updated and revised for clarity. A new chapter on neutron scattering
follows the chapters on x-ray diffractometry and electron microscopy.
The book explains the fundamentals of how waves and wavefunctions interact with atoms in solids, and the similarities and differences of diffraction measurements with x-rays, electrons, or neutrons. Diffraction effects of crystalline order, defects, and disorder in materials are explained in detail. Both practical and theoretical issues are covered. This textbook can be used in an introductory-level or advanced-level course, since sections are identified by difficulty. Each chapter includes a set of problems to illustrate principles, and the extensive Appendix includes laboratory exercises.

Excerpts from the first edition of the book in Adobe Acrobat .pdf format are
here.