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

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

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

${\displaystyle E(V)=-\int _{V}^{\infty }p(V^{\prime })dV^{\prime }}$

which gives

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

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

${\displaystyle \beta }$ is the bulk modulus ${\displaystyle K_{0}}$ at equilibrium volume ${\displaystyle V_{0}}$.

${\displaystyle n}$ correlates with the Grüneisen parameter ${\displaystyle n=-{\frac {1}{6}}-\gamma _{G}}$.^[2][3]

However, the fitting parameter ${\displaystyle E_{\infty }}$ 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]

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. 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 . 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.