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

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. ↑ 4.0 4.1 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
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