Solving the Christoffel equation: Phase and group velocities

Jaeken, Jan; Cottenier, Stefaan

doi:10.1016/j.cpc.2016.06.014

Cited by 77 publications

(47 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where Mav is the average atomic mass, δ the cubic root of the volume of the primitive cell, θaco the acoustic Debye temperature, n the number of atoms per unit cell, and A the coefficient defined by the relation: In Eq. (7), represents the average speed of sound along the crystallographic direction a, b, and c. By using the elastic constants and density of the material, can be calculated via the Christoffel eigenvalue equation [67][68][69]:…”

Section: Resultsmentioning

confidence: 99%

First-principles prediction of large thermoelectric efficiency in superionic Li₂SnX₃ (X = S, Se)

Haque

Cazorla

2020

Phys. Chem. Chem. Phys.

View full text Add to dashboard Cite

Thermoelectric materials create an electric potential when subject to a temperature gradient and vice versa hence they can be used to harvest waste heat into electricity and in thermal management applications. However, finding highly efficient thermoelectrics with high figures of merit, zT ≥ 1, is very challenging because the combination of high power factor and low thermal conductivity is rare in materials. Here, we use first-principles methods to analyze the thermoelectric properties of Li2SnX3 (X=S,Se), a recently synthesized class of lithium fast-ion conductors presenting high thermal stability. In p-type Li2SnX3, we estimate highly flat electronic valence bands that render high Seebeck coefficients exceeding 400 μVK -1 at 700K. In n-type Li2SnX3, the electronic conduction bands are slightly dispersive however the accompanying weak electron-acoustic phonon scattering induces high electrical conductivity. The combination of high Seebeck coefficient and electrical conductivity gives rise to high power factors, reaching a maximum of 4 mWm -1 K -2 in p-type Li2SnS3 and 8 mWm -1 K -2 in n-type Li2SnSe3 at 300 K. Likewise, the thermal conductivity in Li2SnX3 is low as compared to conventional thermoelectric materials, 2-5 Wm -1 K -1 at room temperature. As a result, we estimate a maximum zT = 1.05 in p-type Li2SnS3 at 700 K and an extraordinary 3.07 (1.5) in n-type Li2SnSe3 at the same temperature (300 K). Our findings of huge zT in Li2SnX3 suggest that lithium fast-ion conductors, typically employed as electrolytes in solid-state batteries, hold exceptional promise as thermoelectric materials.

show abstract

Section: Resultsmentioning

confidence: 99%

First-principles prediction of large thermoelectric efficiency in superionic Li₂SnX₃ (X = S, Se)

Haque

Cazorla

2020

Phys. Chem. Chem. Phys.

View full text Add to dashboard Cite

show abstract

“…We determined the sound velocities and of topaz as a function of propagating direction ( ) by solving the Christoffel equations 31 . Shear waves propagating in the crystal are affected by the propagation direction as well as the direction of particle displacement (polarization).…”

Section: Resultsmentioning

confidence: 99%

Single crystal elasticity of natural topaz at high-temperatures

Tennakoon

Peng

Mookherjee

et al. 2018

Sci Rep

View full text Add to dashboard Cite

Topaz is an aluminosilicate mineral phase stable in the hydrothermally altered pegmatitic rocks and also in subducted sedimentary lithologies. In nature, topaz often exhibits solid solution between fluorine and hydrous end members. We investigated elasticity of naturally occurring single crystal topaz (Al2SiO4F1.42(OH)0.58) using Resonant Ultrasound Spectroscopy. We also explored the temperature dependence of the full elastic constant tensor. We find that among the various minerals stable in the Al2O3-SiO2-H2O ternary system, topaz exhibits moderate elastic anisotropy. As a function of temperature, the sound velocity of topaz decreases with and being −3.10 and −2.30 × 10−4 km/s/K. The elasticity and sound velocity of topaz also vary as a function of OH and F content. The effect of composition () on the velocity is equally important as that of the effect of temperature. We also note that the Debye temperature () of topaz at room temperature condition is 910 K and decreases at higher temperature. The Debye temperature shows positive correlation with density of the mineral phases in the Al2O3-SiO2-H2O ternary system.

show abstract

“…A linear regression of the curves in Seismological observations indicate that the inner core is anisotropic with respect to the speed of sound (Belonoshko et al, 2008;Lincot et al, 2015;Tkalcic, 2015), with the P-waves travelling faster towards the pole than towards the equator. In order to address the pressure dependence of elastic anisotropy in Fe3S we calculated the directional dependence of the P and S wave velocities by solving the Christoffel equation using the method implemented by Jaeken and Cottenier (2016). Using the calculated data for the full elastic tensor we see that Fe3S exhibits a strong anisotropy in both compressional (P) and shear (S) waves (Figure 12).…”

Section: Elastic Moduli and Sound Wave Velocitiesmentioning

confidence: 99%

Ab initio study of structural, elastic and thermodynamic properties of Fe3S at high pressure: Implications for planetary cores

Valencia

Moya

Morard

et al. 2022

American Mineralogist

View full text Add to dashboard Cite

Using density functional theory electronic structure calculations, the equation of state, thermodynamic and elastic properties and sound wave velocities of Fe3S at pressures up to 250 GPa have been determined. Fe3S is found to be ferromagnetic at ambient conditions but becomes nonmagnetic at pressures above 50 GPa. This magnetic transition changes the c/a ratio leading to a more isotropic compressibility, and discontinuities in elastic constants and isotropic sound velocities. Thermal expansion, heat capacity and Grüneisen parameters are calculated at high pressures and elevated temperatures using the quasiharmonic approximation. We estimate Fe-Fe and Fe-S force constants, which vary with Fe environment, as well as the 56 Fe/ 54 Fe equilibrium reduced partition function in Fe3S and compare these results with recently reported experimental values. Finally, our calculations under the conditions of the Earth's inner core allow us to estimate a S content of 2.7 wt%S, assuming the only components of the inner core are Fe and Fe3S, a linear variation of elastic properties between end-members Fe and Fe3S and that Fe3S is kinetically stable. Possible consequences for the core-mantle boundary of Mars are also discussed. This is the peer-reviewed, final accepted version for American Mineralogist, published by the Mineralogical Society of America. The published version is subject to change. Cite as Authors (Year) Title. American Mineralogist, in press.

show abstract

Solving the Christoffel equation: Phase and group velocities

Cited by 77 publications

References 7 publications

First-principles prediction of large thermoelectric efficiency in superionic Li₂SnX₃ (X = S, Se)

First-principles prediction of large thermoelectric efficiency in superionic Li₂SnX₃ (X = S, Se)

Single crystal elasticity of natural topaz at high-temperatures

Ab initio study of structural, elastic and thermodynamic properties of Fe3S at high pressure: Implications for planetary cores

Contact Info

Product

Resources

About

Solving the Christoffel equation: Phase and group velocities

Cited by 77 publications

References 7 publications

First-principles prediction of large thermoelectric efficiency in superionic Li2SnX3 (X = S, Se)

First-principles prediction of large thermoelectric efficiency in superionic Li2SnX3 (X = S, Se)

Single crystal elasticity of natural topaz at high-temperatures

Ab initio study of structural, elastic and thermodynamic properties of Fe3S at high pressure: Implications for planetary cores

Contact Info

Product

Resources

About

First-principles prediction of large thermoelectric efficiency in superionic Li₂SnX₃ (X = S, Se)

First-principles prediction of large thermoelectric efficiency in superionic Li₂SnX₃ (X = S, Se)