Relation between Elastic Constants and Second- and Third-Order Force Constants for Face-Centered and Body-Centered Cubic Lattices

Coldwell-Horsfall, Rosemary A.

doi:10.1103/physrev.129.22

Cited by 29 publications

(6 citation statements)

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“…Care must be taken in evaluating these terms since multiple integrations are required over q-vector shifted Brillouin zones. The calculation time can also be reduced by a careful analysis of the symmetries present in a particular crystal which can greatly reduce the number of independent anharmonic terms [57].…”

Section: Ab Initio Computation Of Interatomic Force Constants and Thementioning

confidence: 99%

“…Given that the anharmonic IFCs are symmetrical with respect to the indexes (Ä;˛; q) [57], an anharmonic IFC can be expanded into six terms. From a physical perspective, these different mathematical terms represent the possible three phonon interactions where either two phonons combine to create a third phonon or else a given phonon generates two phonons.…”

Section: Ab Initio Computation Of Interatomic Force Constants and Thementioning

confidence: 99%

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Ab Initio Thermal Transport

Mingo

Stewart

Broido

et al. 2013

Topics in Applied Physics

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Section: Ab Initio Computation Of Interatomic Force Constants and Thementioning

confidence: 99%

Section: Ab Initio Computation Of Interatomic Force Constants and Thementioning

confidence: 99%

Ab Initio Thermal Transport

Mingo

Stewart

Broido

et al. 2013

Topics in Applied Physics

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“…With this notion in mind we therefore choose to express the two-body potentials empirically in power series of the interatomic separation and match the potential with the elastic properties of the material. Explicit relationships exist between a two-body interatomic potential and the second and third order elastic constants for bodycentered cubic and face-centered metals [23]. The interatomic potential so constructed for body-centered cubic iron is shown in Fig.…”

Section: Description Of Methodsmentioning

confidence: 99%

An atomistic study of fracture

Chang

1970

Int J Fract

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The atomistic relaxation method developed by the author to study the core structures of edge and screw dislocations in crystalline solids was used to investigate the configurations of atoms and crack-tip atom cohesion and stress fields in BCC (body-centered cubic) and FCC (face-centered cubic) iron. Single crystallites containing initially an atomically sharp and through crack (a vacancy sheet) of various orientations were subjected to tensile strain applied normal to the crack. It is shown that the crack tip stress fields depend importantly on the orientation of the crack with respect to the parent lattice, a result unexpected of existing linear elasticity and continuum theories.

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“…Analogously, the frequencies of all normal modes of a crystal are determined by the force constants, thus the macroscopic elastic constants can be related to the microscopic force constants, and the elastic constants can therefore be calculated from atomistic calculations of the interatomic force constants. 106,79,80 In the continuous limit, the atomic displacements vary slowly from cell to cell, and can be described as a continuous displacement field u(x), that takes values u i , at site i centered at r i . The equations of motion for these long wavelength elastic waves are,…”

Section: Elasticity and Lattice Dynamicsmentioning

confidence: 99%

“…105 Real materials have complicated microstructures that are impossible to describe using first principles atomistic calculations, so from a theoretical perspective we concentrate on single crystal properties, for which the mechanical properties are described by the elastic constants. The elastic constants can be defined either in the long wave limit of phonons 106,79,80 or as coefficients in an expansion of the potential energy.…”

Section: Vibrational Entropy In Alloysmentioning

confidence: 99%

Vibrations in solids : From first principles lattice dynamics to high temperature phase stability

Shulumba¹

2015

Linköping Studies in Science and Technology. Dissertations

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In this thesis I introduce a new method for calculating the temperature dependent vibrational contribution to the free energy of a substitutionally disordered alloy that accounts for anharmonicity at high temperatures. This method exploits the underlying crystal symmetries in an alloy to make the calculations tractable. The validity of this approach is demonstrated by constructing the phase diagram via direct minimization of the Gibbs free energy of a notoriously awkward and technologically important system, Ti 1-x Al x N. The vibrational entropy including anharmonic effects is shown to be large and comparable to the configurational entropy at high temperatures, and with its inclusion, the theoretical miscibility gap of Ti 1-x Al x N is reduced from 6560 K to 2860 K, in line with atom probe experiments. A similar treatment of Zr 1-x Al x N and Hf 1-x Al x N alloys suggests that mass disorder has a minimal effect on phase stability compared with chemical ordering. My method is also capable of demonstrating that Hf 1-x Al x N, which is dynamically unstable at room temperature, is stabilised at high temperatures. Moreover I develop a new method of computing temperature dependent elastic constants for alloys from their phonon spectra, and show that for Ti 1-x Al x N, the elastic anisotropy is found to increase with temperature, helping to explain the spinodal decomposition.The effects of lattice dynamics on phase stability, mechanical, magnetic and transport properties on other materials are also examined. Four specific systems are discussed in detail. Firstly, in the case of CrN, lattice vibrations are shown to decrease the antiferromagnetic to paramagnetic phase transition temperature from 500 K to 380 K, in line with experimental evidence. Secondly, a temperature/pressure induced phase transition in AlN becomes much more facile than in the quasiharmonic approximation, and the thermal conductivity of the rocksalt phase is shown to be much lower than that of the wurtzite phase, as a result of the increased anharmonicity in the rocksalt structure. Thirdly, the temperature dependence of elastic constants of TiN becomes more isotropic as the temperature increases. Finally, iron carbides are evaluated as potentially important phases at the Earth's core; specifically, calculating the Gibbs free energy of a recently discovered orthorhombic phase of Fe 7 C 3 demonstrates that it is not stable relative to the known hexagonal phase at extreme pressure and temperatures.V Popular science summary Solid materials can be described using a simple mathematical model from the theory of lattice dynamics. In a solid, atoms are arranged in a regular and repeating structure (a lattice), and the forces between them can essentially be modelled as springs. Despite the simplicity of this model, it is possible to derive a surprising amount of information on materials from it. One can determine which arrangements of atoms, or phases, form the stable microscopic structures that we encounter in the real world, their mechanical properti...

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Relation between Elastic Constants and Second- and Third-Order Force Constants for Face-Centered and Body-Centered Cubic Lattices

Cited by 29 publications

References 1 publication

Ab Initio Thermal Transport

Ab Initio Thermal Transport

An atomistic study of fracture

Vibrations in solids : From first principles lattice dynamics to high temperature phase stability

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