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 equation of the state for the total energy. The energy $E$ required to compress a solid to volume $V$ 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 $V_0$ are related to material properties, where

$\beta$ is the bulk modulus ${\displaystyle K_{0}}$ 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]

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) .
4. Atomic Simulation Environment documentation of the Technical University of Denmark, Department of Physics
5. Gilgamesh chemical software documentation of the Department of Chemical Engineering of Carnegie Mellon University "Archived copy". Archived from the original on 2014-04-14. Retrieved 2014-05-30.