Temperature dependence of the normal vibrational modes of hcp Zr

Stassis, C.; Zarestky, J.; Arch, D.K.; McMasters, O.D.; Harmon, B. N.

doi:10.1103/physrevb.18.2632

Cited by 103 publications

(44 citation statements)

References 23 publications

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“…The energies of the zone-center LO and TO phonons are found to be 20.5 and 9.0 meV from the totalenergy calculation, respectively, and 21.8 and 9.7 meV from the force calculation, respectively. These values are in agreement with the experimental results from the neutron inelastic scattering by Stassis et al 9 The phonon energies at 5.5 K were found to be 17.5 and 11.0 meV for the LO and TO modes, respectively. The calculated values ͑closed circles͒ and the phonon dispersion relations of the hcp zirconium in the ͓0001͔ direction measured by Stassis et al are shown in Fig.…”

supporting

confidence: 82%

“…This assumption may be justified because the interlayer forces determined from the experimental phonon dispersion curve in the ͓0001͔ direction become negligible beyond the fourth nearest neighbor. 13,9 To determine the force constants uniquely, we use the following approximation: c 45 Ӎc 56 and c 35 Ӎc 46 . The calculated force constants are shown in Table I.…”

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confidence: 99%

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First-principles calculation of the longitudinal phonon in the surface-normal direction of a zirconium(0001) slab: Localization mode at the subsurface

Yamamoto

Chan

et al. 1996

Phys. Rev. B

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The energies and amplitudes of the longitudinal vibrational mode in the surface-normal ͓0001͔ direction of the fully relaxed zirconium slab with a ͑0001͒ surface are calculated with force constants determined from first-principles total-energy and force calculations. A subsurface vibrational mode with frequency above the bulk continuum is found. We attribute the origin of this mode to the hardness of the force constant between the surface and the second layer.Most transition-metal surfaces exhibit relaxations; i.e., the interlayer spacing between the first and the second layer is reduced and an oscillatory relaxation pattern of the inner layers is usually observed. 1 Since the redistribution of charges on the surface causes the relaxation of the top few layers and the change of the interlayer force constants, the vibrational mode in the surface-normal direction is expected to be changed with the relaxation.Surface lattice dynamics has been studied experimentally by high-resolution electron energy-loss spectroscopy ͑HREELS͒ and helium atom scattering ͑HAS͒ methods. 2,3 In the HAS method, the vibrational motion of the surface layer is detected and the change of the vibrational behavior in the surface-normal direction has been obtained indirectly. The information of the subsurface vibrational motion can be obtained directly by HREELS, but its energy resolution ͑ca. 7 meV͒ is much less than that of the HAS method ͑ca. 0.2 meV͒. In the case of group IV transition metals such as titanium, zirconium, and hafnium, no detailed surface phonon dispersion has been reported. The hydrogen contamination of the surface may change the surface relaxation and also the dynamical properties of the surfaces. 4 The change of the vibrational motion with the surface normal direction has been investigated with Debye-Waller factor analysis for lowenergy electron diffraction ͑LEED͒ from the Ti͑0001͒ surface 5 and electron energy-loss fine-structure ͑EELFS͒ measurements from a polycrystalline titanium surface. 6 The Debye temperature, which is a measure of the rigidity of the lattice, is monotonically decreased with increase in the surface properties of the detected vibrational motion. However, these analyses give the averaged information of the dynamical properties over the several surface layers.First-principles total-energy and force calculations have been shown to give an accurate description of the surface phonon dynamics. 7 However, the phonon dynamics in the surface-normal direction of the slab has not been investigated in detail. In the present study the vibrational properties of the zirconium ͑0001͒ slab are investigated by using the results of the first-principles total-energy and force calculations for this surface.The first-principles calculations have been carried out using a pseudopotential method within the framework of the density-functional theory. The Hedin-Lundqvist form of exchange-correlation functionals has been used. The Bloch wave functions are expanded in a mixed basis of plane waves and numerical localized wave fun...

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supporting

confidence: 82%

mentioning

confidence: 99%

First-principles calculation of the longitudinal phonon in the surface-normal direction of a zirconium(0001) slab: Localization mode at the subsurface

Yamamoto

Chan

et al. 1996

Phys. Rev. B

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“…The origin of the curvature is not well established. Stassis et al 27) have performed an investigation of the temperature dependence of the vibrational modes of -Zr by means of inelastic-neutron-scattering and they found anomalies of the phonon dispersion curves. So it is possible to think in an intrinsic non-Arrhenius behaviour, like in the -phase case.…”

Section: Self-diffusionmentioning

confidence: 99%

Diffusion in α-Ti and Zr

Pérez¹,

Nakajima

Dyment³

2003

Mater. Trans.

141

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Impurity diffusion in -Ti and -Zr is characterized by fast interstitial diffusion of small atoms and slow substitutional diffusion of large atoms. The mechanism of the fast diffusion is discussed on the basis of the several features: the effect of -phase transformation, the diffusion anisotropy, the atomic size effect and the correlation between solubility and diffusivity. The temperature dependence of the diffusion coefficients of all the solutes and solvent Ti studied in -Ti follows the Arrhenius law, whereas in -Zr, all the elements studied with exception of Ti show downward curvatures that follow the self-diffusion behaviour in -Zr. The influence of ultra-fast diffusion impurities on the diffusion process in these matrices, as well as the possibility that different mechanism acts in each matrix is analyzed.

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“…In the few cases tested to date, this approach seems to work well. Eriksson, Wills and Wallace [334] used neutron scattering phonon measurements at elevated temperatures by Stassis, Petry, Heiming, and co-workers [337,338,339,340] on bcc and hcp Ti, and bcc and hcp Zr to calculate the total phonon entropy at temperature (using Eq. 14).…”

Section: Anharmonicity and Heat Capacitymentioning

confidence: 99%

Vibrational thermodynamics of materials

Fultz¹

2010

Progress in Materials Science

570

428

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Abstract. 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. Phononphonon "anharmonic" interactions arise from those non-harmonic parts of the interatomic potential that cannot be accounted for by the quasiharmonic ...

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Temperature dependence of the normal vibrational modes of hcp Zr

Cited by 103 publications

References 23 publications

First-principles calculation of the longitudinal phonon in the surface-normal direction of a zirconium(0001) slab: Localization mode at the subsurface

First-principles calculation of the longitudinal phonon in the surface-normal direction of a zirconium(0001) slab: Localization mode at the subsurface

Diffusion in α-Ti and Zr

Vibrational thermodynamics of materials

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Temperature dependence of the normal vibrational modes of hcp Zr

Cited by 103 publications

References 23 publications

First-principles calculation of the longitudinal phonon in the surface-normal direction of a zirconium(0001) slab: Localization mode at the subsurface

First-principles calculation of the longitudinal phonon in the surface-normal direction of a zirconium(0001) slab: Localization mode at the subsurface

Diffusion in &alpha;-Ti and Zr

Vibrational thermodynamics of materials

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Diffusion in α-Ti and Zr