From the preceding discussion, it is clear that simple expressions in temperature for the thermal expansion coefficient do not simultaneously meet the thermodynamic requirement at T = 0 and match the experimental observation that α(T) becomes linear with temperature at high temperatures. The solution is to use an equation for thermal expansion that explicitly relates the volume to lattice energy of the material. Kroll et al. (2012) show that the Kumar (2003, and references therein) version of this approach is superior to that of Suzuki (1975) and Wallace (1972). Holland & Powell (2011) developed a similar function that is expressed in terms of parameters at a reference temperature. Although this formulation is not quite as robust in extrapolation as the Kumar expression when the underlying data are sparse, when the data are sufficient it produces fits and parameters that are indistinguishable from those of the Kumar equation (Kroll et al., 2012). Tribaudino et al. (2011) and Kroll et al. (2012) give different but equivalent expressions, of which the latter is perhaps clearer:
The two expressions for A and B are:
In the expression for A, the factor . The Einstein temperature, θE, in the coth functions provides the saturation in α at low temperatures, below
. The value of α0 is the thermal expansion coefficient at Tref. The value of θE can be approximated from the molar standard state entropy (e.g. Holland & Powell, 2011), but tests indicate that its precise value is not critical for the correct description of the volume variation with temperature. Consequently it often cannot be reliably determined by refinement to data and its refinement may be unstable.