Thermal EoS

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The volume thermal expansion of a material is defined as . Integration of this expression yields the volume variation with temperature at constant pressure:



Where Tref is a reference temperature at which the volume is V00

The only thermodynamic constraints on the form of the function for α(T) are that at absolute zero. Consequently many different forms have been proposed in the literature, and several have become widely-used, for example in thermodynamic databases (e.g. Berman, 1988; Gottschalk, 1997; Holland & Powell, 1998; Holland & Powell, 2011). Some simple formulations that describe thermal expansion at high temperatures very well do not include the low-T saturation, but are widely used and are perfectly adequate for thermodynamic databases. On the other hand, some equations explicitly handle the saturation in thermal expansion as the temperature drops towards absolute zero, but these often result in unphysical values of thermal expansion at high temperatures in excess of 1000 K, where experiments indicate that α(T) increases approximately linearly with temperature. Following the reviews of various thermal expansion expressions based on analysis and fitting of data  (Tribaudino, Angel, Camara, Nestola, Pasqual & Margiolaki, 2010; Tribaudino, Bruno, Nestola, Pasqual & Angel, 2011; Kroll, Kirfel, Heinemann & Barbier, 2012), we have implemented the following thermal expansion equations in the EoS module of CrysFML. 



Important: The values of thermal expansion coefficients are typically of the order of 10-5 to 10-6 K-1. In order to display the values to a significant number of digits, all calculated values of thermal expansion displayed by the GUI or EosFit7c are mutliplied by 10-5. That means that if a value is displayed as alpha = 2.134 then the real value is 2.134 x 10-5 K-1.


If you want to input a value of 2.134 x 10-5 K-1, then actually type in 2.134 without the multiplier.


Values of the parameters describing the thermal expansion may also be multiplied by other fixed factors in a similar way. These are always displayed in the output of EosFit7c.