Mie-Grüneisen-Debye Thermal-pressure EoS
This EoS is based on the idea that the spectrum of thermally-induced vibrations in the material can be described by the Debye model whose energy is represented by a characteristic temperature θD, the Debye temperature. The Debye model differs from the Einstein model in that it includes phonon dispersion. The heat capacity of the Debye model expressed as the molar heat capacity of a single phonon branch is :
This expression involves the third Debye function D(θD/T) which is a complicated integral:
Thus, the thermal pressure from a Debye model arising from an increase in temperature from Tref along an isochor is given by the integral of the heat capacity, for 3N phonon modes, thus:
The variation of the Debye temperature can be modeled in two ways, both of which are consistent with the QHA, as described in the page Thermal Pressure.
The big advantage of the MGD EoS is that the parameters γ0 and q now control the value of the Debye temperature θD , which means that V-T data provide constraints on the values of these parameters in addition to the constraints from the measurements of the adiabatic bulk modulus KS.
In EosFit the isochoric molar heat capacity is calculated directly from the expression for the Debye heat capacity (given above for one phonon), multiplied by 3N. The isobaric heat capacity CP is calculated as:
which is obtained by using the Grüneisen relationship to substitute for α in CP = (1+αγT)CV