Einstein Thermal-pressure EoS
This EoS uses a very simple model for the phonon density of states. It assumes that all of the phonons have the same frequency or energy, that is represented by the Einstein temperature θE, without any dispersion.
The molar isochoric heat capacity CVm,i associated with each individual phonon mode i within a solid is given by:
in which yi is a related to the phonon frequency ω and temperature by:
If ω is in units of wavenumbers (cm-1), it is numerically equal to 0.695θE, when θE is in K.
The heat capacity of a solid of N atoms in the molar volume Vm is thus 3N times the heat capacity given above, and the thermal pressure is just:
The variation of the Einstein temperature can be modeled in two ways, both of which are consistent with the QHA, as described in the page Thermal Pressure.
In EosFit the isochoric molar heat capacity is calculated directly from the expression for the heat capacity CVm,i 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
The Einstein EoS yields heat capacities that do not match the measured variation of Cp with temperature for most solids, because the model does not include the variation in phonon mode frequencies found in solids. It is included in EosFit to serve as a comparison to the Holland-Powell thermal-pressure EoS and to provide the basis for future more realistic models for phonon density of states of solids.