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 the equation of state for the total energy. The energy ${\displaystyle E}$ required to compress a solid to volume ${\displaystyle V}$ 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 ${\displaystyle V_{0}}$ are related to material properties, where

${\displaystyle \beta }$ is the bulk modulus ${\displaystyle K_{0}}$ at equilibrium volume ${\displaystyle V_{0}}$.

${\displaystyle n}$ correlates with the Grüneisen parameter ${\displaystyle n=-{\frac {1}{6}}-\gamma _{G}}$.[2][3]

However, the fitting parameter ${\displaystyle E_{\infty }}$ 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]