Anton–Schmidt Equation Of State

Short description: Scientific equation

The Anton–Schmidt equation is an empirical equation of state for crystalline solids, e.g. for pure metals or intermetallic compounds.^[1] Quantum mechanical investigations of intermetallic compounds show that the dependency of the pressure under isotropic deformation can be described empirically by

p (V) = - β {(\frac{V}{V_{0}})}^{n} \ln (\frac{V}{V_{0}})

.

Integration of $p (V)$ leads to the equation of state for the total energy. The energy $E$ required to compress a solid to volume $V$ is

E (V) = - \int_{V}^{\infty} p (V^{'}) d V^{'}

which gives

E (V) = \frac{β V_{0}}{n + 1} {(\frac{V}{V_{0}})}^{n + 1} [\ln (\frac{V}{V_{0}}) - \frac{1}{n + 1}] - E_{\infty}

.

The fitting parameters $β, n$ and $V_{0}$ are related to material properties, where

β

is the bulk modulus

K_{0}

at equilibrium volume

V_{0}

.

n

correlates with the Grüneisen parameter

n = - \frac{1}{6} - γ_{G}

.^[2]^[3]

However, the fitting parameter $E_{\infty}$ does not reproduce the total energy of the free atoms.^[4]

The total energy equation is used to determine elastic and thermal material constants in quantum chemical simulation packages.^[4]^[5]

The equation of state has been used in cosmological contexts to describe the dark energy dynamics.^[6] However its use has been recently criticized since it appears disfavored than simpler equations of state adopted for the same purpose.^[7]

References

↑ Mayer, B.; Anton, H.; Bott, E.; Methfessel, M.; Sticht, J.; Harris, J.; Schmidt, P.C. (2003). "Ab-initio calculation of the elastic constants and thermal expansion coefficients of Laves phases". Intermetallics 11 (1): 23–32. doi:10.1016/S0966-9795(02)00127-9. ISSN 0966-9795.
↑ Otero-de-la-Roza, et al., Gibbs2: A new version of the quasi-harmonic model code. Computer Physics Communications 182.8 (2011): 1708-1720. doi:10.1016/j.cpc.2011.04.016
↑ Jund, Philippe, et al., Physical properties of thermoelectric zinc antimonide using first-principles calculations., Physical Review B 85.22 (2012) arXiv:1207.1670.
↑ ^4.0 ^4.1 Atomic Simulation Environment documentation of the Technical University of Denmark, Department of Physics [1]
↑ Gilgamesh chemical software documentation of the Department of Chemical Engineering of Carnegie Mellon University "7.2. Equations of State — Gilgamesh documentation v0.01 documentation". Archived from the original on 2014-04-14. https://web.archive.org/web/20140414073503/http://gilgamesh.cheme.cmu.edu/doc/software/jacapo/appendices/appendix-eos.html. Retrieved 2014-05-30.
↑ Salvatore Capozziello, Rocco D'Agostino, Orlando Luongo, Cosmic acceleration from a single fluid description, Physics of the Dark Universe 20 (2018) 1-12, arXiv:1712.04317.
↑ Kuantay Boshkayev, Talgar Konysbayev, Orlando Luongo, Marco Muccino, Francesco Pace, Testing generalized logotropic models with cosmic growth, Physical Review D 104 (2021) 2, 023520, arXiv:2103.07252.

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[Mayer2003-1] Mayer, B.; Anton, H.; Bott, E.; Methfessel, M.; Sticht, J.; Harris, J.; Schmidt, P.C. (2003). "Ab-initio calculation of the elastic constants and thermal expansion coefficients of Laves phases". Intermetallics 11 (1): 23–32. doi:10.1016/S0966-9795(02)00127-9. ISSN 0966-9795.

[2] Otero-de-la-Roza, et al., Gibbs2: A new version of the quasi-harmonic model code. Computer Physics Communications 182.8 (2011): 1708-1720. doi:10.1016/j.cpc.2011.04.016

[3] Jund, Philippe, et al., Physical properties of thermoelectric zinc antimonide using first-principles calculations., Physical Review B 85.22 (2012) arXiv:1207.1670.

[dof-4] 4.0 ^4.1 Atomic Simulation Environment documentation of the Technical University of Denmark, Department of Physics [1]

[5] Gilgamesh chemical software documentation of the Department of Chemical Engineering of Carnegie Mellon University "7.2. Equations of State — Gilgamesh documentation v0.01 documentation". Archived from the original on 2014-04-14. https://web.archive.org/web/20140414073503/http://gilgamesh.cheme.cmu.edu/doc/software/jacapo/appendices/appendix-eos.html. Retrieved 2014-05-30.

[6] Salvatore Capozziello, Rocco D'Agostino, Orlando Luongo, Cosmic acceleration from a single fluid description, Physics of the Dark Universe 20 (2018) 1-12, arXiv:1712.04317.

[7] Kuantay Boshkayev, Talgar Konysbayev, Orlando Luongo, Marco Muccino, Francesco Pace, Testing generalized logotropic models with cosmic growth, Physical Review D 104 (2021) 2, 023520, arXiv:2103.07252.

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Anton–Schmidt Equation Of State

See also

References