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 equation of the 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]

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