Mie–Grüneisen equation of state

The Mie–Grüneisen equation of state is an equation of state that relates the pressure and volume of a solid at a given temperature. It is used to determine the pressure in a shock-compressed solid. The Mie–Grüneisen relation is a special form of the Grüneisen model which describes the effect that changing the volume of a crystal lattice has on its vibrational properties. Several variations of the Mie–Grüneisen equation of state are in use.

The Grüneisen model can be expressed in the form

${\displaystyle \Gamma =V\left({\frac {dp}{de}}\right)_{V}}$

where V is the volume, p is the pressure, e is the internal energy, and Γ is the Grüneisen parameter which represents the thermal pressure from a set of vibrating atoms. If we assume that Γ is independent of p and e, we can integrate Grüneisen's model to get

${\displaystyle p-p_{0}={\frac {\Gamma }{V}}(e-e_{0})}$

where ${\displaystyle p_{0}}$ and ${\displaystyle e_{0}}$ are the pressure and internal energy at a reference state usually assumed to be the state at which the temperature is 0K. In that case p₀ and e₀ are independent of temperature and the values of these quantities can be estimated from the Hugoniot equations. The Mie–Grüneisen equation of state is a special form of the above equation.