Freund & Ingalls (1989) showed that the ‘modified Tait equation’ of Huang & Chow (1974) is a generalised form of the Murnaghan EoS which remains easily invertible:
and
The three parameters a, b, c are defined in terms of the bulk modulus and its derivatives at room pressure:
and the inverse relationships:
If K''0T = 0, then , and the equation is reduced to the Murnaghan form (Freund & Ingalls, 1989). While precise values of K''0T are difficult to measure, it is clear that for most solids they are not zero. Holland & Powell (2011) therefore introduced a ‘truncation’ of the Tait equation by setting
. In the EoS module of CrysFML this estimate of K''0T is named the ‘3rd-order’ form, with the 4th-order form including a refineable K''0T. There is no rationale for a 2nd-order formulation, but for completeness we can define a 2nd-order form as having K'0T = 4, and thus
. Thus all orders of the Tait equation as implemented in the EoS module of CrysFML have
and therefore fit P-V data of solids substantially better than the Murnaghan EoS. We find that the 3rd-order fit of the Tait EoS normally yields parameters that are indistinguishable within the uncertainties to those obtained from the 3rd-order Birch-Murnaghan EoS, typically with marginally worse formal measures of statistical fit. The 4th-order fits of the two equations are normally statistically and numerically indistinguishable.
Because the Tait EoS is invertible, the expressions for the bulk modulus and its pressure derivatives as a function of pressure can be obtained directly (since the parameters a, b, and c are constants) by differentiation with respect to pressure of the expression for the volume:
, or:
Further differentiation with respect to pressure leads to:
Setting a=1 (and thus K''0T = 0) in these equations leads to and K'PT = K'0T as required for the Murnaghan EoS.