Physics: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
- [math]\displaystyle{ p(V) = - \beta \left(\frac{V}{V_0}\right)^n \ln\left(\frac{V}{V_0}\right) }[/math].
Integration of [math]\displaystyle{ p(V) }[/math] leads to equation of the state for the total energy. The energy [math]\displaystyle{ E }[/math] required to compress a solid to volume [math]\displaystyle{ V }[/math] is
- [math]\displaystyle{ E(V) = - \int_V^\infty p(V^\prime) dV^\prime }[/math]
which gives
- [math]\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 }[/math].
The fitting parameters [math]\displaystyle{ \beta, n }[/math] and [math]\displaystyle{ V_0 }[/math] are related to material properties, where
- [math]\displaystyle{ \beta }[/math] is the bulk modulus [math]\displaystyle{ K_0 }[/math] at equilibrium volume [math]\displaystyle{ V_0 }[/math].
- [math]\displaystyle{ n }[/math] correlates with the Grüneisen parameter [math]\displaystyle{ n = -\frac{1}{6} - \gamma_G }[/math].[2][3]
However, the fitting parameter [math]\displaystyle{ E_\infty }[/math] 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
- ↑ 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.
- ↑ 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
- ↑ Jund, Philippe, et al., Physical properties of thermoelectric zinc antimonide using first-principles calculations., Physical Review B 85.22 (2012) arXiv:1207.1670.
- ↑ 4.0 4.1 Atomic Simulation Environment documentation of the Technical University of Denmark, Department of Physics [1]
- ↑ Gilgamesh chemical software documentation of the Department of Chemical Engineering of Carnegie Mellon University "Archived copy". 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.
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