The fundamental equation (1) relates five variables two of which
are independent. A simple system can therefore be
completely described, given knowledge of its thermal
*P*(*V*, *T*) and caloric
*E*(*V*, *T*) EOS's. The thermal EOS relates
the experimental
*P*-*T* and theoretical
*V*-*T* variables and is
necessary for transforming these variables in analysis of any
thermodynamic property [*Zharkov and Kalinin,* 1971].
The
parameters determined by this EOS kind are termed thermal,
whereas the quantities derived either from only the caloric EOS
or from both thermal and caloric EOS's are thermed caloric. The
latter, in particular, include
*K*_{S},
*g*,
*C*_{P}, and
*t*_{S}.

The two EOS kinds are related by the equation

(21) |

whose integral form is

where the transformation

is used, the integration constant is

and the integral is taken along an isotherm. In view of (21),
the caloric EOS,
*E*(*V*, *T*), is completely determined by
the given thermal EOS and function
*E*(*T*) or
*H*(*T*) at
*P* = 0.
It is clear that any of the caloric functions
*H*(*T*),
*S*(*T*),
*G*(*T*), and
*C*_{P}(*T*) at
*P* = 0 can used for the same
purpose, since the following identities take place

(22) |

(23) |

The latter formula can be written in another useful form

(24) |

where
*T*^{} is a fixed
temperature.

For a mineral whose composition can be expressed by a sum of oxides (component), the Gibbs energy is formulated in difference terms

(25) |

where
*D**H*_{f} = *H*
- *H*_{ ox},
*D**S*_{f} = *S*
- *S*_{ ox}, and
*D**C*_{P} = *C*_{P}
- *C*_{P ox} are the differences of enthalpy,
entropy and heat capacity
between the mineral and oxide sum, respectively (with allowance
for the stoichiometric coefficients). Expressions of type (25)
are often used in calculating phase equilibria
[e.g., * Navrotsky and Akaogi,* 1984;
* Kuskov and Galimzyanov,* 1986;
* Kuskov et al.,* 1989;
* Fabrichnaya and Kuskov,* 1991;
* Fei and Saxena,* 1986;
* Fei et al.,* 1990;
* Sobolev and Babeiko,* 1989].
Some authors use an approximation
*D**C*_{P} = 0
(or const
0 ;
the functions
*D**C*_{P}(*T*)
are sometimes found from
empirical formulas of type (67)). In any case, the term
*D**H*_{f}(*T*^{}) in (25) implicitly contains an arbitrary
normalizing constant [*Kalinin et al.,* 1991].

Integrating (3) gives

(26) |

where
*G*(*T*) = *G*(*T*, 0) is defined by (24) or (25) and can
be written in the reduced form

(27) |

Methods for determining EOS's in geophysics can be classified as follows.

(1) The macroscopic approach suggested by
* Murnaghan* [1951] and
* Birch* [1952] gives the volume dependence of pressure
at
*T* (or
*S* ) = const in the form [*Ullmann and Pankov,* 1976]

(28) |

Hereafter, the values with the subscript 0, unless otherwise
specified, are taken at
*P* = 0 and an arbitrary temperature, the
moduli
*K*_{0} *K*_{T0},
*K*^{}_{0}
= ( *K*_{T}/
*P*)_{T0},...
are material parameters, and
*x* = *V*/*V*_{0} = *r*_{0}/*r* is the compression ratio parameter. Most data for the
material parameter values were obtained at room temperature
[e.g., * Sumino and O. Anderson,* 1984]. Among the
last
experimental achievements are ultrasonic measurements at high
pressures [e.g., * Fujisawa,* 1987;
* Webb,* 1989;
* Yoneda,* 1990;
* Liebermann et al.,* 1993],
*X* -ray data of high pressures and high
temperatures
[e.g., * Yagi et al.,* 1987;
* Mao et al.,* 1991;
* Fei et al.,* 1992a, 1992b;
* Boehler et al.,* 1989],
spectoscopic observations of minerals
[e.g., * Chopelas,* 1990a, 1990b,
1991a, 1991b, 1993;
* Hofmeister,* 1987, 1991a], and
high-temperature
*P*=0 measurements of elastic constants by the
rectangular parallelepiped technique [*O. Anderson et al.,* 1992a;
* O. Anderson,* 1995].

An explicit form of function
*f* in (28) (the volume dependence
of pressure) was considered by * Murnaghan* [1951],
* Birch* [1952, 1968, 1978,
1986] and others
[*Thomsen,* 1970, 1971;
* Ahrens and Thomsen,* 1972;
* Davies,* 1973;
* Ullmann and Pankov,* 1976, 1980;
* Pankov and Ullmann,* 1979a;
* Stacey,* 1981;
* Aidun et al.,* 1984;
* Jeanloz,* 1989;
* Bina and Helffrich,* 1992;
* Isaak et al.,* 1992;
* Wall et al.,* 1993]. The most widely used equation of
this type
is the Birch-Murnaghan EOS.

Elastic moduli and sound velocities in minerals depends first of
all on the composition, crystalline structure, pressure, and
temperature. Data on these dependences are generalyzed and
interpreted in terms of empirical laws such as the Birch's law,
the seismic EOS, the law of corresponding states, and a
universal EOS
[*Birch,* 1961;
* O. Anderson and Nafe,* 1965;
* D. Anderson,* 1967, 1987;
* Chung,* 1973;
* Davies,* 1976;
* O. Anderson,* 1973;
* D. Anderson and O. Anderson,* 1970;
* Mao,* 1974;
* Kalinin,* 1972;
* Schankland and Chung,* 1974;
* Campbell and Heinz,* 1992].
These laws enable us to estimate the parameters
*K*_{0} and, to a
lesser accuracy,
*K*_{0}^{}
for unmeasured minerals
[*D. Anderson,* 1988;
* Duffy and D. Anderson,* 1989].

(2) Statistical physics describing the vibrations of atoms in
crystals provides the background for microscopic EOS theory
including the Mie-Grüneisen EOS
[*Grüneisen,* 1926;
* Born and Huang,* 1954;
* Leibfried and Ludwig,* 1961;
* Knopoff,* 1963;
* Knopoff and Shapiro,* 1969;
* Zharkov and Kalinin,* 1971;
* Wallace,* 1972;
* Mulargia,* 1977;
* Mulargia and Boschi,* 1980;
* Hardy,* 1980;
* O. Andrrson,* 1980;
* Gillet et al.,* 1989, 1990, 1991;
* Richet et al.,* 1992;
* Reynard et al.,* 1992]. This approach also uses the
lattice or vibrational Grüneisen parameters,
as well as either
semiempirical potentials of atomic interactions or the
reference (isothermal or adiabatic)
*P*-*V* relations derived
from continuum mechanics
[e.g., * Al'tshuler,* 1965;
* Zharkov and Kalinin,* 1971;
* Ahrens and Thomsen,* 1972;
* McQueen,* 1991]. The
material parameters in these cases are determined using static
and dynamic compression data, elastic constant measurements,
caloric functions, and vibrational spectrums.

(3) Integrating (12) yields the pressure as a sum of two terms:
a reference isotherm and the thermal pressure increment
*D**P*_{th}.
This thermodynamic approach based on experimental
data has been developed by * O. Anderson* [1979a,
1979b,
1979c, 1980, 1982,
1984,
1988, 1995] and was used to describe
the
*X* -ray and resonance data for a set of minerals
[*O. Anderson et al.,* 1982, 1992a;
* O. Anderson and Yamamoto,* 1987;
* O. Anderson and Zou,* 1989;
* Mao et al.,* 1991;
* Fei et al.,* 1992a, 1992b].

(4) More intricate theoretical EOS models are derived from ab
initio calculation using the Hartree-Fock and
Thomas-Fermi-Dirak methods, as well as pseudopotential
theory, many-term contributions in semiempirical potentials, and
molecular dynamics
[*Hemley et al.,* 1985, 1987;
* Isaak et al.,* 1990;
* Wolf and Bukowincki,* 1987, 1988;
* Wall and Price,* 1988;
* Wall et al.,* 1986;
* D'Arco et al.,* 1991;
* Price et al.,* 1989;
* Matsui et al.,* 1987;
* Matsui,* 1988, 1989;
* Reynard and Price,* 1990;
* Agnon and Bukowinski,* 1990a;
* Walzer,* 1992;
* Cohen,* 1987a;
* Dovesi et al.,* 1987;
* Catlow and Price,* 1990;
* Boisen and Gibbs,* 1993;
* Silvi et al.,* 1993;
* Catti et al.,* 1993;
* Barton and Stacey,* 1985].

As mentioned above, the complete description of a simple system
requires knowledge of either any of its thermodynamic
potentials or its thermal EOS and one of the caloric functions
(at
*P* = 0 ). Table 1 lists various approaches to the
determination of EOS's, showing which functions must be
found from theory or experiment so as to provide such a
complete description. These approaches can also be formulated in
the form of partial differential equations with appropriately
chosen boundary conditions.

The order of a thermodynamic parameter (characteristic of a
matter) is defined by the maximum order of the thermodynamic
potential derivative involved to define the thermodynamic
parameter. To find all of the third-order parameters
( *P*,
*V*,
*T*, or
*S* derivatives of the second-order parameters), whose
total number for the potentials in (1)-(4) is 16, it is
sufficient to know four independent and appropriately chosen
third-order parameters, in addition to knowledge of the
lower-order parameters. Specifically, experiments often provide
information on the derivatives
( *K*_{S}/ *P*)_{T} (or

The relationships of these derivatives to other parameters are further discussed in later sections.

To extrapolate data on thermodynamic properties to high
pressures and temperatures, the power volume dependence is often
applied stating that the logarithmic volume derivative
of the parameter considered is a constant
[*Zharkov,* 1986;
* D. Anderson,* 1988, 1989]. The
temperature derivative of any
parameter
*A* at
*P* = const is represented in the dimensionless
form

(29) |

where the first term characterizes the so-called intrinsic
anharmonicity and the second is a parameter of the extrinsic
anharmonicity related to thermal expansion
[*Jones,* 1976;
* Smith and Cain,* 1980]. Parameter
*A* can be any physical property,
such as the transport coefficients or mode Grüneisen
parameters
[*Reynard et al.,* 1992;
* Gillet et al.,* 1989].

The database on properties of minerals, required for geophysical
analysis and EOS construction, must include first of all their
density and the second and third order thermodynamic parameters.
An example of such database for three mantle minerals is given
in Tables 2
and 3,
and the database for 25 mantle minerals,
including their high-pressure phases (and some fictive phases),
is presented in Internet [*Pankov et al.,* 1997]. The
parameter
values in these tables refer to the conditions
*P* = 0,
*T* = 300 K or
*P* = 0 and the temperature indicated. Apart from
the second-order thermodynamic parameters, Table 2 includes the
molar mass
*M*, mean atomic weight
*m*, density
*r*, the
melting temperature
*T*_{m}, the Debye temperatures
*Q* ( *Q*_{a} is the acoustic
temperature,
*Q*_{a}
is from fitting the Mie-Grüneisen EOS to
data
on
*a* [*Suzuki,* 1975a,
1975b],
and
*Q*_{th} is our estimate
from data on specific heat), the classical value
*C*_{V} = 3*R*/_{m}
( *R* is the gas constant), the enthalpy
*D**H*_{f} and
entropy
*D**S*_{f} of
mineral formation from oxides, and
the estimated thermal pressure
*P*_{th}
0.5*a**K*_{T}. It is
important to have mutually consistent values of the second (and
higher) order parameters: here, the calculations are based on
the input values of
*K*_{S} (or
*K*_{T} ),
*a*, and
*C*_{P}. At high
temperatures,
*T* > 300 K, the
*a* and
*C*_{P} values were
found by the empirical formulas from * Fei and Saxena* [1987]
and
* Fei et al.* [1990, 1991],
and for
*K*_{S}, we give either
experimental values or our estimates through the
Anderson-Grüneisen parameter
*d*_{S} (at 300 K),
which is
assumed to be a constant (see Table 3 and sections 9 and 10).
The values listed in Table 3 are based on the input values of
the derivatives
( *K*_{S}/ *P*)_{T},
( *K*_{S}/ *T*)_{P} (or
*d*_{S} ),
( *C*_{P}/ *T*)_{P}, and
*a* = *a*^{-2}
(*a*/ *T*)_{P}, as well as on
the second-order parameter values given in Table 2. The
high-temperature values of the third-order parameters were
evaluated using the condition
( *K*_{S}/ *P*)_{T} const.
Finally, in Table 2
are
given the references to sources of thermodynamic data for each
of the minerals.