Anton–Schmidt equation of state

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)=-\beta \left({\frac {V}{V_{0}}}\right)^{n}\ln \left({\frac {V}{V_{0}}}\right)

.

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^{\prime })dV^{\prime }

which gives

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

.

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

\beta

is the bulk modulus

K_{0}

at equilibrium volume

V_{0}

.

n

correlates with the Grüneisen parameter

n=-{\frac {1}{6}}-\gamma _{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.
^ ^a ^b 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. 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.

[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] 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. 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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See also

References