The thermodynamic Grüneisen parameter is defined by (8) or (14), which further lead to several useful identities

(105) |

The typical values of
*g* by (8) or (14) range from 1 to 2
(see, e.g., Table 2 and * Pankov et al.* [1997]).
Of 54 minerals
treated by * D. Anderson* [1989], only five have
*g* greater
than 2, and none has
*g* over 3. Low values of
*g* are
seldom encountered: e.g.,
*g* = 0.4 for
*a* -quartz,
0.3 for coesite, and even
*g*< 0 for U
_{2} O, AgJ, and
*b* -quartz.

The logarithmic derivatives of
*g* with respect to
*V* (or
*P* ) are characterized by the parameter
*q*, for which from (8)
and (100), we find

(106) |

(107) |

(108) |

As noted earlier (see (57) or (69) and (8)), in general, the
*C*_{V} = constant case leads to
*g* = *g*(*V*),
and
therefore,
*q* = *q*(*V*) or
*q* = constant. If
*C*_{V} is only
temperature-dependent, there are three possibilities: (1)
*q* = *q*(*V*),
(2)
*q* = constant
1 (i.e.,
*d*_{V}^{T} = constant
0 ),
and (3)
*q* = 1 ( *d*_{V}^{T} = 0,
*K*^{} = *d*_{T}(*V*),
and
*t* = *a**K*_{T}
= constant). Thus, both
*C*_{V} = constant
and
*C*_{V} = *C*_{V}(*T*) conditions result
in the case that the two
inequalities are equivalent:

(109) |

If we simply assume that
*g* is only volume-dependent, then
from (14), (33), and (106),

(110) |

Placing in (110) for
(*K*_{S}/ *T*)_{V}
by identity (99),

(111) |

However, in the general case,
*g* = *g*(*V*,
*T*), and
from the formula for
*g* in (18), we find

(112) |

which, upon excluding
*L* by (85), yields the important identity
[*Bassett et al.,* 1968]

(113) |

For
*g* = *g*(*V*),
this identity is reduced to (111).

In section 6, we have already referred to some data on values of
*q*. In general, values of
*q* can be inferred from the following
sources: 1) thermodynamic estimation by (108) or (111), 2) fit
of the Mie-Grüneisen type EOS to data on
*a*(*T*),
*C*_{P}(*T*), and
*K*_{S}(*T*) at
*P* = 0, 3) shock wave data [e.g.,
* McQueen,* 1991;
* Duffy and Ahrens,* 1992a], 4) adiabatic
temperature gradient measurements [*Boehler,* 1982, 1983],
5) spectroscopy of solids [e.g., * Reynard et al.,* 1992;
* Williams et al.,* 1987], 6) theoretical EOS models
[e.g., * Isaak et al.,* 1990], 7) analysis of geophysical
data
[*O. Anderson,* 1979b;
* D. Anderson,* 1989]. The values of
*q* estimated
by (108) and given
in Table 3 and * Pankov et al.* [1997], fall into
the interval
0.5-2, except for the high values for coesite (about 17),
fayalite (2-3), and Fe-perovskite (4-5). Small negative values
were also found for enstatite and FeO (probably, due to
inaccurate input data). With increasing
*T* at
*P* = constant or
with increasing
*P* at
*T* (or
*S* = constant),
*q* decreases (see
also section 6).

For the temperature derivative of
*g*, we again have the
expansion of type (29)

(114) |

where the intrinsic anharmonicity term can be evaluated using (18) and (85)

(115) |

This term is usually negative and completely prevails in (114)
at
*T* < *Q*, but at high temperatures, its value
is
comparable to
*q*. Thus, the frequently used assumption that
*g* = *g*(*V*)
is unsatisfactory in the general case,
and the temperature effect on the Grüneisen
parameter can serve
as a measure of the validity of the Mie-Grüneisen
EOS
[*Molodets,* 1998].

Another suitable representation of
(*g*/ *T*)_{V} follows from (8) and (69)
[*Stacey,* 1977b]

(116) |

If
*g* = *g*(*V*),
then either
*C*_{V} = *C*_{V}(*S*) or
*C*_{V} = constant.
The case
*C*_{V}(*S*) results in

(117) |

Moreover, (117) leads to
*C*_{V}(*V*, *T*) = *C*_{V}(*Q*/*T*) and
*Q*/*T* = *f*(*S*), so that
*g* is represented as
*g* = -*d* ln*Q*/*d*
ln *V*, where
*Q* is a
characteristic temperature.

The frequently used volume dependences of the latice Grüneisen
parameter were given in section 6. The * Rice* [1965]
formula is
also of interest

(118) |

which is derived from (110) under the condition that

The inequality
( *K*_{S}/ *T*)_{V} > 0 (see section 10 and * D. Anderson*
[1989])
holds true of many materials and therefore gives a lower limit
for their dependence
*g*(*V*), i.e.,
*q* 1 + *g*
and
*g* *g*(*x*)
by (118). This limit was previously
found from the Mie-Grüneisen EOS [*Kalinin
and Panov,* 1972],
but it also follows from the given thermodynamic consideration.

Equation (118) can be considered a partial case of the more
general representation
*g* = *g*(*V*,
*S*). We introduce
a parameter
*l* defined as

(119) |

where
*F* = *K*_{S}/*r*.
Assuming that
*l* = *l*(*S*)
or
*l* = constant and using (119) and (105),
we find by integration that

(120) |

where
*g*_{0} = *g*_{0}(*S*)
and
*V*_{0} = *V*_{0}(*S*). These
dependences of
*g*(*x*) for various
*l* are
illustrated in Figure 10. One can see that they are quite
sensitive to variations of
*l* in the interval from 0 to 1.

In geophysics, the conditions close to adiabatic are realized in
the convecting mantle and core, as well as in seismic wave
propagation. Furthermore, the state at the initial part of
Hugoniot are close to adiabatic. Adiabats of a given material
form a one-parametric family of curves. In this case, the
temperature and pressure are related by the adiabatic gradient
*t*_{S}, which, considering its definition
by (10) and
relations in section 2, can be written in the form

(121) |

Typical values of
*t*_{S} found by (121) are given
in Table 2
and * Pankov et al.* [1997].

Direct measurements of
*t*_{S} at high pressures and
temperatures were made in a series of works
[*Dzhavadov,* 1986;
* Boehler and Ranakrishnan,* 1980;
* Boehler,* 1982, 1983].
* Chopelas and Boehler* [1992]
reported corrections to the * Boehler* [1982]
initial results on
*t*_{S.}

We consider the basic identities and approximaions for the
derivatives. Denoting by
*n* the logarithmic volume derivative of
*t*_{S} and using
*q* by (106), we have

(122) |

(compare with (45)).

Formula (122) can be represented in various forms, using
*q* by
(113), (115), and (58). It is clear that
*n* decreases by
isothermal or adiabatic compression. The simplest estimate of
*n* is given by assuming that

hence,

(123) |

The typical values of
*q* = 1-2 and
*K*^{} = 4-5 yield
*n* 5-7.
If we neglect the last term in (122) at
*T* > *Q*, then
*n* 1 + *d*_{T}
[*Chopelas and Boehler,* 1992].

Changing from variables
(*V*, *S*) to
(*V*, *T*), the adiabatic
derivative with respect to volume takes the form

(124) |

Approximation (89) and
*n* 1 + *d*_{T}
give
*n*_{S} *n* - *g*.
Writing the derivative of
*t*_{S} with respect to
*T* in the form of (29),

(125) |

or after substituting
( ln*t*_{S}/ *T*)_{V} by (124),

(126) |

The values of
(( ln*t*_{S})( ln *V*))_{P} and
*n* (an extrinsic anharmonic contribution)
calculated by (126) and (122) are given in Table 2 and
* Pankov et al.* [1997].
They show that the intrinsic anharmonic term
dominates in (125).

Note that the
*t*_{S} parameter occurs in any expression
when
changing variables
*P*,
*S* to
*P*,
*T* : for example,

(127) |

which was used in deriving (98).

For a moderate compression, the volume dependence of
*t*_{S} can be described by the power
law

(128) |

where
*n* = *n* (*T*) or constant,
*t*_{S0} = *t*_{S0}
(*T*) and
*V*_{0} = *V*_{0} (*T*). This formula was used to
fit the measured
*t*_{S} values to
*P* = 50 kbar and
*T* = 1000 K
[*Boehler and Ramacrishnan,* 1980;
* Boehler,* 1982].

However, * Chopelas and Boehler* [1992], accounting for
the
variation of
*d*_{T} with
*V* (see (46) and (47)), found that
the linear dependence of
ln*t*_{S} on
*V* (*n* = *mx*) better
describes teir data on
*t*_{S} than the power law, and
consequently,

(129) |

where constant
*m* is determined by the approximation

(130) |

(see (122), where
*C*_{P} is approximately
*C*_{V} ). Thus, on the
condition that
( ln *C*_{V}/ ln *V*)_{T} is independent of
*V*, the derivative
(*d*_{T}/ *x*)_{T} can be
found given knowledge of the
*m* value. * Isaak* [1993]
applied this
method to evaluate the derivative
^{2}*K*_{S}/ *P*
*T* with the help of (91).

Measured values of
*t*_{S}(*P*, *T*) allow
us first to find the
isobaric specific heat [*Dzhavadov,* 1986]

(131) |

which is deduced from the identity
^{2} *T*/ *P* *S* =
^{2} *T*/ *S* *P*.
The integral in (131)
is taken over an adiabat, and the specific heat versus
temperature,
*C*_{P}(*T*), for
*P* = 0, is assumed to be known. Then,
given a reference isotherm
*V*(*P*, *T*_{0} ), from (14), we can find
the thermal EOS in the form

(132) |

Conversely, given the thermal EOS and
*t*_{S}(*T*) at
*P* = 0,
(14) gives
*C*_{P}(*T*) at
*P* = 0, and thus, the caloric EOS can be
determined.