Adiabatic-Isothermal moduli

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From version 7.3, it is possible to use the EosFit7c program to refine the EoS parameters directly to moduli data, or to moduli and volume/linear data. For a description of the methods used in EosFit7 to fit moduli and volumes simultaneously please read Milani et al. (2017).


All EoS inside the program are calculated with isothermal moduli. The listing of moduli is always the same as the input data (you can even mix them). But to use adiabatic moduli, you must give the program the parameters to convert the adiabatic moduli to isothermal values. This page describes the basic theory and methods. For the data file format, look at the example datasets.


In this section we use the notation KT to mean isothermal bulk moduli, and  KS to mean adiabatic moduli.  MiT and MiS  are the isothermal and adiabatic linear moduli, for direction i.


Bulk Moduli


If adiabatic moduli are provided as input data, they are converted to isothermal values by

where the value of thermal expansion (αV) is taken from the current EoS at the pressure and temperature of interest. The Grüneisen parameter γ  can be expressed in terms of measurable quantities, for example:

 


However, since heat capacity data is not available at elevated pressures, in EosFit7c we use the simple approximation (Anderson 1996) that 


 with the value of q close to 1 for ‘normal’ solids under modest P,T conditions. The values of γ0  and q can be set with the thermal part of the input utility in EosFit7c.


Linear Moduli

For linear moduli that describe the variation of the unit-cell parameters with pressure as , the procedure is analogous to that used for linear dimensions in EosFit7. Internally the linear moduli are converted to volume-like bulk moduli as and treated with volume EoS. For the conversion of adiabatic to linear moduli, we use the relationship between isothermal and adiabatic compressibilities , which can be derived from the fundamental relationship between isothermal and adiabatic elastic compliances:, with the Cp in units of Jm-3K-1. The substitution gives where the β are the compressibilities which can be rewritten in terms of moduli as. It is important to notice that for cubic materials and so that only for cubic materials this can be written as , which is the same relation as for the bulk moduli. 


We can use this result for non-cubic materials by writing . The factor of ‘3’ makes the thermal expansion term ‘volume like’ and the new ‘linear Gruneisen parameter’ is , which should be close in value to the volume Gruneisen parameter γ. If the material is isotropic, it will be identical. It is not unreasonable to approximate the linear parameter as .


In EosFit7c the user can assign values of  γ0  and q for a linear EoS with the input utility. The values should be close to those used for the volume EoS of the same material.