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(VV0)nln⁡(VV0).

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)=−∫V∞p(V′)dV′

which gives

E(V)=βV0n+1(VV0)n+1[ln⁡(VV0)−1n+1]−E∞.

The fitting parameters β,n and V0 are related to material properties, where

β is the bulk modulus K0 at equilibrium volume V0.
n correlates with the Grüneisen parameter n=−16−γG.[2][3]

However, the fitting parameter E∞ 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]

## See also

* Murnaghan equation of state
* Rose–Vinet equation of state
* Birch–Murnaghan equation of state

## References

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.
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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