A Few Fultz Group Recent Abstracts

Quantum Anharmonicity and Intermodulation Phonon Sidebands in NaBr

Y. Shen, C.N. Saunders, C.M. Bernal, D.L. Abernathy, M.E. Manley, and B. Fultz


A quantum Langevin model, similar to models from optomechanics, was used to predict intermodulation phonon sidebands (IPS). Ab initio calculations of anharmonic phonons in rocksalt NaBr showed these spectral features as "many-body effects." Modern inelastic neutron scattering measurements on a crystal of NaBr at 300 K revealed diffuse intensity at high phonon energy from a predicted upper IPS. The transverse optical (TO) part of the new features originates from phonon intermodulation between the transverse acoustic (TA) and TO phonons. The longitudinal optical (LO) spectral features originate from three-phonon coupling between the TA modes and the TO lattice modes. The partner lower IPS proves to be an "intrinsic localized mode." Interactions with the thermal bath broaden and redistribute the spectral weight of the IPS pair. These sidebands are a probe of the anharmonicity and quantum noise of phonons in NaBr.

The Anharmonic Origin of the Large Thermal Expansion of NaBr

Y. Shen, C.N. Saunders, C.M. Bernal, D.L. Abernathy, M.E. Manley, B. Fultz

Phys. Rev. Lett. 125, 085504 (2020). DOI: 10.1103/PhysRevLett.125.085504.

All phonons in a single crystal of NaBr were measured by inelastic neutron scattering at temperatures of 10, 300 and 700 K. Even at 300 K the phonons, especially the longitudinal-optical (LO) phonons, showed large shifts in frequencies, and showed large broadenings in energy owing to anharmonicity. Ab initio computations were first performed with the quasiharmonic approximation (QHA), in which the phonon frequencies depend only on V, and on T only insofar as it alters V by thermal expansion. This QHA was an unqualified failure for predicting the temperature dependence of phonon frequencies, even 300 K, and the thermal expansion was in error by a factor of four. Ab initio computations that included both anharmonicity and quasiharmonicity successfully predicted both the temperature dependence of phonons and the large thermal expansion of NaBr. The frequencies of longitudinal-optical (LO) phonon modes decrease significantly with temperature owing to the real part of the phonon self-energy from explicit anhamonicity, originating from the cubic anharmonicity of nearest-neighbor Na-Br bonds. Anharmonicity is not a correction to the QHA predictions of thermal expansion and thermal phonon shifts, but dominates the behavior.

Temperature-dependent phonon lifetimes and thermal conductivity of silicon by inelastic neutron scattering and ab initio calculations

D.S. Kim, O. Hellman, N. Shulumba, C.N. Saunders, J.Y.Y. Lin, H.L. Smith, J.E. Herriman, J.L. Niedziela, D.L. Abernathy, C.W. Li, and B. Fultz,

Phys. Rev. B 102, 174311 (2020).

Inelastic neutron scattering on a single crystal of silicon was performed at temperatures from 100 K to 1500 K. These experimental data were reduced to obtain phonon spectral intensity at all wavevectors Q and frequencies w in the first Brillouin zone. Thermal broadenings of the phonon peaks were obtained by fitting, and by calculating with an iterative ab initio method that uses thermal atom displacements on an ensemble of superlattices. Agreement between the calculated and experimental broadenings was good, with possible discrepancies at the highest temperatures. Distributions of phonon widths versus phonon energy had similar shapes for computation and experiment. These distributions grew with temperature, but maintained similar shapes. Parameters from the ab initio calculations were used to obtain the thermal conductivity from the Boltzmann transport equation, which was in good agreement with experimental data. Despite the high group velocities of longitudinal acoustic phonons, their shorter lifetimes reduced their contribution to the thermal conductivity, which was dominated by transverse acoustic modes.

Thermodynamic stability and contributions to the Gibbs free energy of nanocrystalline Ni3Fe

S.H. Lohaus, M.B. Johnson, P.F. Ahnn, C.N. Saunders, H.L. Smith, M.A. White and B. Fultz

Phys. Rev. Mater 4, 086002 (2020).

The heat capacities of nanocrystalline Ni3Fe and control materials with larger crystallites were measured from 0.4-300 K. The heat capacities were integrated to obtain the enthalpy, entropy, and Gibbs free energy, and to quantify how these thermodynamic functions are altered by nanocrystallinity. From the phonon density of states (DOS) measured by inelastic neutron scattering, we find that the Gibbs free energy is dominated by phonons, and that the larger heat capacity of the nanomaterial below 100 K is attributable to its enhanced phonon DOS at low energies. Besides electronic and magnetic contributions, the nanocrystalline material has an additional contribution at higher temperatures, consistent with phonon anharmonicity. The nanocrystalline material shows a stronger increase with temperature of both the enthalpy and entropy compared to the bulk sample. Its entropy exceeds that of the bulk material by 0.4 kB/atom at 300 K. This is insufficient to overcome the enthalpy of grain boundaries and defects in the nanocrystalline material, making it thermodynamically unstable with respect to the bulk control material.

Nuclear quantum effect with pure anharmonicity and the anomalous thermal expansion of silicon

D. S. Kim, O. Hellman, J. Herriman, H. L. Smith, J. Y. Y. Lin, N. Shulumba, J. L. Niedziela, C. W. Li, D. L. Abernathy, and B. Fultz

Proc. Nat'l Acad. Sci. 115, 1992 (2018).


Despite the widespread use of silicon in modern technology, its peculiar thermal expansion is not well understood. Adapting harmonic phonons to the specific volume at temperature, the quasiharmonic approximation, has become accepted for simulating the thermal expansion, but has given ambiguous interpretations for microscopic mechanisms. To test atomistic mechanisms, we performed inelastic neutron scattering experiments from 100 K to 1,500 K on a single crystal of silicon to measure the changes in phonon frequencies. Our state-of-the-art ab initio calculations, which fully account for phonon anharmonicity and nuclear quantum effects, reproduced the measured shifts of individual phonons with temperature, whereas quasiharmonic shifts were mostly of the wrong sign. Surprisingly, the accepted quasiharmonic model was found to predict the thermal expansion owing to a large cancellation of contributions from individual phonons.

Vibrational Thermodynamics of Materials

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.

Mossbauer Spectrometry

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 57Fe or 119Sn. 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.

Experimental Inelastic Neutron Scattering

Download draft open source textbook of Dec. 2020 (367 pages, 12.8 MB)

Download draft of May 2021 used in course MS 171 (394 pages, 7 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.

Phase Transitions in Materials

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.

Transmission Electron Microscopy and Diffractometry of Materials, Fourth Edition

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.

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