Anton–Schmidt equation of state explained

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( (V)/(V _{0}) \right) ^{n} ln \left( (V)/(V _{0}) \right) .

Integration of

p(V) leads to the equation of state for the total energy. The energy E required to compress a solid to volume V isE(V) = - \int ^{inf ty} _{V} p(V ^{\prime)} dV ^{\prime} which gives E(V) = (\beta V _{0})/(\left( n +1) (V)/(V _{0}) \right) ^{n+1} \left[ ln \left( (V)/(V _{0}) \right) - (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 =- (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]}

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

Notes and References

1. Mayer. B.. H. . Anton . E. . Bott . M. . Methfessel . J. . Sticht . J. . Harris . P.C. . Schmidt. 2003. Ab-initio calculation of the elastic constants and thermal expansion coefficients of Laves phases. Intermetallics. 11. 1. 23–32. 0966-9795. 10.1016/S0966-9795(02)00127-9.
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.
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 https://archive.today/20140629160956/https://wiki.fysik.dtu.dk/ase/epydoc/ase.utils.eosase2-module.html
5. Gilgamesh chemical software documentation of the Department of Chemical Engineering of Carnegie Mellon University Web site: 7.2. Equations of State — Gilgamesh documentation v0.01 documentation . 2014-05-30 . dead . https://web.archive.org/web/20140414073503/http://gilgamesh.cheme.cmu.edu/doc/software/jacapo/appendices/appendix-eos.html . 2014-04-14 .
6. Salvatore Capozziello, Rocco D'Agostino, Orlando Luongo, Cosmic acceleration from a single fluid description, Physics of the Dark Universe 20 (2018) 1-12, .
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, .