The saturation of thermal expansion at low temperature can be addressed using an equation of the form suggested by Salje et al. (1991):
In which θsat is termed the “saturation temperature”. In fact the thermal expansion coefficient only becomes zero below . The equation should only be used to describe volume variation at low temperatures, because it leads to an expression (Kroll et al., 2012) for the thermal expansion coefficient of:
At moderate temperatures, above about 3θsat (θsat is typically 200-500K), the thermal expansion becomes almost independent of temperature, which is not observed for most materials. The value of p1 is approximately 3 times the high-temperature value of α. At T = 0K the coth function has the value of unity, so the volume at T = 0 is . Therefore, in order for the parameter V00 to have the value of the volume at the temperature Tref = 0K, we re-write the equation as:
The Salje equation could also be re-written in terms of a reference temperature, but since it is only applicable to low-temperature thermal expansion, this is not implemented in the EoS module of CrysFML, which therefore sets Tref = 0.