The variation of the volume of a solid with hydrostatic pressure at fixed temperature is termed its ‘isothermal equation of state’. It is characterised by the bulk modulus of the material, , which is a function of both temperature and pressure. For infinitesimal changes in pressure which give rise to infinitesimal changes in volume, the bulk modulus can also be defined in terms of the elastic tensor of the material by applying linear elasticity theory, or ‘Hooke’s law’. In linear elasticity it is assumed that the strains εi of a solid are linearly related to the magnitude of the applied stress field σj by the matrix equation
. The suffixes run from 1 to 6, with i,j =1,2,3 referring to normal stresses or strains along orthogonal axes, and i,j =4,5,6 referring to shear stresses or strains (e.g. Nye, 1957; Angel, Jackson, Reichmann & Speziale, 2009). The elastic properties of the material are represented by the values of the elements of the compliance matrix sij which is symmetric and can contain up to 21 independent elements for triclinic crystals, less for crystals and materials with higher symmetries. The compliance matrix is a convenient representation of the compliance tensor of the material (Nye, 1957).
Hydrostatic pressure is a special stress state in which the normal stresses are all equal and there are no shear stresses
. Thus, at any pressure P, the strains caused by an infinitesimal increase in pressure δP can be calculated through linear elasticity theory by setting
. (Note that while pressure is considered to be a positive quantity, compressive stresses are by convention considered to be negative; Nye, 1957). Therefore each of the resulting strain elements is given by:
The sum of the three normal strains is, in the infinitesimal limit, equal to the fractional change in the volume , thus:
Re-arrangement of this last equation shows that the bulk modulus for hydrostatic compression at any pressure is defined by six of the elements of the compressibility matrix at that same pressure:
This bulk modulus for hydrostatic compression of a solid, whether a powder or single crystal, is therefore equal to the Reuss bound on the bulk modulus of a polycrystal made of the same material where it represents the volume response when every constituent grain is subject to the same stress.
While linear elasticity thus defines the bulk modulus of a material under hydrostatic compression at any pressure, it cannot define an equation of state which describes the large (finite) changes in volume due to large (finite) changes in pressure. In this sense an equation of state is an extension of linear elasticity; although normally defined in terms of the volume variation with pressure, it can also be seen as a definition of the variation of bulk modulus with pressure. Because there is no absolute thermodynamic basis for specifying how the bulk modulus K varies with pressure, all EoS that have been developed and are in widespread use are based upon a number of assumptions (e.g. Anderson, 1995; Duffy & Wang, 1998; Holzapfel, 2001). The validity of such assumptions can only be judged in terms of whether the derived EoS reproduces experimental data for volume or elasticity. For materials that do not exhibit phase transitions, isothermal equations of state are usually parameterized in terms of the values of the bulk modulus and its pressure derivatives, and
, evaluated at a reference pressure, normally taken as zero pressure. If the material undergoes a structural phase transition, additional parameters are required (e.g. Tröster, Schranz & Miletich, 2002; Schranz, Tröster, Koppensteiner & Miletich, 2007). In order to allow for the description of the variation of the volume with temperature and pressure, we also define a reference temperature for the EoS, Tref. We then denote the values of the parameters describing an equation of state at the reference temperature and pressure with two subscripts “0”, thus:
,
,
Using this notation leads to the value of the room-pressure bulk modulus at some temperature T being denoted , and its isothermal pressure derivatives K'0T and K''0T. The subscript ‘T’ is therefore not to be read as indicating isothermal as opposed to adiabatic moduli, although all of the moduli discussed here are isothermal. Note that the values of adiabatic bulk moduli are typically a few % larger than isothermal moduli, by a factor (1+αVγT) in which αV is the thermal expansion coefficient, γ the Anderson-Grunesien parameter, and T the temperature.
The variation of the linear dimensions of a material with temperature and pressure can be expressed in terms of the linear thermal expansion and compressibility matrices:
and
Both of these can be converted to tensor forms (Nye, 1957; Knight, 2010) with the same factors for the shear terms (i = 4,5,6) as used for the conversion of strain between matrix elements and tensor components. The use of a negative sign in the definition of the compressibility is purely a convention to obtain positive values for the first three (i = 1,2,3) matrix elements. We have already shown that the linear strains due to a small increment in pressure are given by linear elasticity theory as:
Given the definition of the infinitesimal normal strains as , it follows that the linear compressibilities are also given by the sum of elements of the compliance matrix:
. As for the volume variation with pressure and temperature, there is however no absolute thermodynamic basis for specifying the variation in individual distances or cell parameters of a crystal with P and T because this involves finite strains. The only constraint that we can apply is that the sum of the normal strains must be equal to the volume strain in the infinitesimal limit. As a consequence:
and
The only way to ensure such consistency between descriptions of volume variation and variation in individual cell parameters or other distances in the same material is to use the same equations as used for the volume variation (Angel, 2000a). For linear quantities a, the quantity is cubed and then treated as a volume in the same equations described above for the volume variation with P and T. It is clear that for cubic or isotropic materials a description of the volume variation, , will lead to values of
and
. The same is true for all other symmetries, except that the values of cubed lengths are not the true volume. The EoS module of CrysFML handles these inter-conversions of parameter values internally and always returns the thermal expansion in the linear case as
. For linear compression we can define a linear modulus as
. Thus the relationship between the linear moduli and bulk moduli is
. For isotropic and cubic materials this reduces to
and thus
, as required. The linear moduli and their pressure derivatives defined in this way (which correspond to the elastic compliances) have numerical values three times those of the corresponding volume bulk modulus. Again, the EoS module of CrysFML handles the inter-conversions internally and always returns the correct values of the moduli M, and its derivatives for the linear case. Note that this is a change from earlier versions of EosFit (Angel, 2000a) which returned volume-like values for the linear moduli. The consequence of using these conventions is that the corresponding values of finite strain and normalised pressure are those for volume, not for linear quantities. Therefore f-F plots of linear data yield values of intercepts and slopes of lines that correspond to the values of volume bulk moduli and their derivatives, not those of the linear moduli. When an f-F plot is used to display linear data, the moduli derived from its intercept and slope must therefore be multiplied by 3 to obtain the linear moduli and its pressure derivatives.
All of the moduli described above are isothermal; they describe the volume and cell parameter variations with pressure at constant temperature. Some direct measurements of moduli (e.g. Brillouin scattering, ultrasonic wave velocity measurements - see Angel et al. 2009) give adiabatic values. Adiabatic values are typically greater than the isothermal values, and the values must be converted before you fit the data. See the section Adiabatic-Isothermal moduli for a description of how EosFit converts between the two.