It follows from the identity 2 V/ T P = 2 V/ P T that [Birch, 1952]
This fundamental relation is written in the dimensionless form
where the isothermal Anderson-Grüneisen parameter dT is introduced [O. Anderson, 1966a, 1967; Barron, 1979].
The variation of a with temperature at P = const is characterized by the parameter a Fürth, 1944; O. Anderson, 1966b; Birch, 1986; O. Anderson et al., 1993]
can be related to the derivatives of CV and KT, by making use of the identity 2S/ V T = 2 S/ T V, which leads to
Similarly, the identity 2S / P T = 2S / T P yields
where the convenient dimensionless product is used
Birch  pointed out that parameter dT for various materials usually lies between 4 and 8 (at normal conditions), which was borne out by subsequent studies [see, e.g., Sumino and O. Anderson, 1984; O. Anderson et al., 1992a] some exclusions are also encountered: e.g., dT 1 for KMnF 3 and dT 77 for Re 2 O 3.
The parameter a values at normal conditions are commonly greater than the dT values [Birch, 1952; O. Anderson et al., 1992a]. The data and estimates listed in the tables of Pankov et al.  for 25 minerals fall into the range 10 a 270. However, at high temperatures ( T > Q ), the dT and a values become closer to each other, and their concidence would mean that a were dependent only on volume (i.e., the intrinsic anharmonicity were suppressed).
The following three assumptions and their consequences are of interest:
(1) The specific heat CV is independent of pressure, i.e., CV = CV(T), as in the Van der Vaals or Hildebrand EOS's [O. Anderson, 1979a], or alternatively, CV = const, as in the classical limit at T > Q. Then, from (32) and (33),
At P = 0, dT K is the common case.
(2) KT depends only on volume; the KT(V) approximation is often warranted at T > Q [O. Anderson, 1982; D. Anderson, 1988, 1989]. Then, ( KT/ T)V = 0, and from (29) and (31), we have K = dT; i.e., dT either depends only on volume or is a constant (leading to the Murnaghan EOS (44)).
(3) If both conditions (1) and (2) take place, then a = a(V) and K = dT = a (that is either volume-dependent or a constant).
In most of the interiror of the Earth, T > Q and a minimally depends on temperature. Consider a few approximate relations for evaluating the isothermal or adiabatic variation of a.
By using the general EOS form of (28) and the formula (12), we find the expansion for a
If only two first terms are retained in (36), then the formula of Birch [1952, 1968] derives. It should be noted that a in this formula changes its sign at KT/P = dT0 (the condition that may be achieved in the lower mantle at KT/P = 4.7 ). Moreover, in this case, dT given by most of the type (28) EOS's increases with pressure instead of its usual decrease (see section 9). It was shown, however, that such a change in the a sign is forbidden thermodynamically [Pankov, 1992].
As an example of using (36), we calculated a(P/K0) with the help of the EOS form proposed by Ullmann and Pankov [1976, 1980], for which
First, we set dK0/ T = 0 and neglect terms containing K0, K0,. The a/a0 versus P/K0 curves, obtained for resonable values K0 = 3 and 4 and dT0 = 2, 4, and 6, are shown in Figure 1. For change from P to volume, Figure 2 gives the variation of P/K0 with x. We see, in particular, that the dT0 values significantly affect the estimated a under lower-mantle conditions (at the base of the mantle compressed along its "hot" adiabat, x 0.7 and P/K0 0.70 for K0 1.9-2.0 Mbar and K0 = 3.8-4.1 [D. Anderson, 1989]). Furthermore, the approximation used for a may lead to the nonrealistic result a< 0 within the lower mantle.
To illustrate the influence of the non-zero dK0/ T values, now we allow for the third term in (36), with the setting dK0/ T = 2 10-3 K -1 (the values that we estimated for NaCl from data discussed by Birch [Birch, 1978]). The value 2 10-3 K -1 is not realistic since it results in the increase of a with pressure (Figures 1 and 3). On the other hand, the negative value -2 10-3 K -1 is too small, since it considerably lessens the pressure at which the condition a 0 mentioned above is reached.
The approximations K = dT (see above) and dT constant at T > Q imply a very weak temperature dependence of K. The lattice dynamics models show that K for MgSiO 3 perovskite varies less than 10% in the temperature range of 300-2000 K ( dK0/ T 2 10-4 K -1 ). The theoretical PIB model for MgO [Isaak et al., 1990; O. Anderson et al., 1993] shows that dK0/ T somewhat increases with temperature in the same interval of 300-2000 K, with the values ranging, on average, from 2.8 10-4 to 4.2 10-4 K -1. Values of a similar order follow from the approximation d ln K0/d lnr = -1 indicated by D. Anderson  for PREM.
The derivative dK0/d T can also be estimated by the approximation KT = KT(V) (i.e., K = dT and (70) are allowed for) noted above
where the primes indicate pressure derivatives. With a 3 10-5 K -1 and -KTK 5-10 [e.g., Pankov and Ullmann, 1979a; Hofmeister, 1991b], we find dK0/ T (1-3) 10-4 K-1, which is close to the estimates found above.
Finally, we can use identity (91) from section 9, which of course leads to (39) for K = dT (see (71)). Although the terms in (91) are close to each other, the reasons given below justify the inequality (dT/ P)T 0, and consequently,
Substituting the parameter values from Tables 2 and 3 (see also tables in Pankov et al. ) into the right side of the above, we find
It should be emphasized that the correction to a in (36) related to this derivative enables us to avoid negative or increased values of a at high pressures. The effect of the value dK0/dT = 2 10-4 K -1 on the a curve for K0 = dT0 = 4 is shown in Figure 1.
O. Anderson  derived the power law
by integrating (30) and (8), provided that
where parameter q is defined as
Moreover, he assumed that dT K const. The constancy of K (or alternatively, K = K(T) ) leads to the Murnaghan EOS (of a type of (28))
It is clear that (41) simply follows from the definition of dT by (31) on the condition that dT is either only temperature-dependent, dT = dT(T), or is a constant; moreover, (44) can be replaced with any suitable approximation to the isotherm P(V, T0).
The two functions a(V, T) and P(V, T), however, cannot be picked independently. For example, the EOS can apparently be defined by specifying a(V, T) and P(V, T0). At this point, it is appropriate to discuss the following generalization of the results mentioned in various papers [Birch, 1968; Clark, 1969; O. Anderson, 1986; D. Anderson, 1989]. Consider four statements: ( 1) the Murnaghan EOS (44) is valid, where it is assumed that K0 is a function of temperature, K0(T), and K0 is either a constant or depends only on temperature; ( 2) (dT/ P)T = 0 (i.e., dT = const or dT = dT(T) ); ( 3) ( K/ T)P = 0 ; ( 4) dT = K that, according to (71), is equivalent to KT = KT(V) (i.e., t(T) = aKT or t = const).
Then, by making use of identities (44), (71), (91), and (92), it can be proved that, if any two (except the pair ( 1) and ( 3)) of the four statements above hold true, then the other two statements are also valid. Moreover, then dT = K = const and (44) always takes place. If, in addition to these two statements, it is assumed that CV = const or CV = CV(T), then we have a = a(V), K = dT = a = const and t = const (see section 6.1 and (69) and (71)).
It is clear that such statements place constraints on the EOS formulation. For example, when an equation of type (28) is accepted instead of the Murnaghan EOS, only one of the above statements can strictly be true. Specifically, the concurrent use of the Birch-Murnaghan EOS and the assumption dT = dT(T) (or dT = const) is incompatible with the condition dT = K or K = K(P).
Another not obvious inference is that the Murnaghan equation (44) uniquely follows from the assumptions KT = f(P) + aT ( a being a constant) and t = t(T) or const.
It is interesting to consider the use of the Murnaghan formula as the potential (lattice) part of the P-V-T EOS in the classical high-temperature approximation. In general, for CV = const, we have a linear dependence of KT(V, T) on T (see (69) and (70)), and only for g/x = const ( q = 1 ) and the Murnaghan potential, we obtain
where a, b, and c are constants ( K = const). Hence, it is seen that all isotherms (T 0) are also represented by formula (44), but for a nonlinear g (x) behavior, this is, strictly speaking, not the case.
The consideration presented above concerns also the Birch's law, which for minerals with the mean atomic mass of m = 20-22 g/mole can be written in the power form KT = aVb (where a and b are constants and the distinction between KS and KT is neglected). Since here K = const and KT = KT(V) and (44) is used, this law gives rise to dT = K = const, formula (41), and for CV = const, a = a (V).
Figure 1 shows the behavior a determined by (41), where the EOS is found by (37) with K0 = 3 and 4 (the respective a curves pass through the ends of the bars in Figure 1).
The original assumption of O. Anderson  dT = const was justified by the ultrasonic and shock-wave data of that time and was seemingly corroborated by later ultrasonic and resonance measurements [O. Anderson et al., 1990; Chopelas and Boehler, 1992]. In particular, based on data for seven minerals, the value of dT = 4-6 was recommended to be representative of the lower mantle. However, analyzing seismological and geoid data, D. Anderson [1987, 1989] found dT = 2-3 for in situ lower-mantle conditions. The assumption can therefore be made that dT must decrease with pressure. Evidence for this can also be found in shock-wave and static compression data [Birch, 1986; O. Anderson et al., 1993].
Ab initio calculations for MgO by Reynard and Price  give a constant value in the range 0.7 x 1.0. Another ab initio results [Isaak et al., 1990] reveal, however, that dT actually decreases by decreasing x.
To determine the a (x) more accurately than given by the power law (41), Chopelas and Boehler  used data on the adiabatic pressure gradient tS and specific heat CP. From the Maxwell relation (14), they derived
where n ( ln tS/ ln V)T. Then, they set n = mx, where m = 6 1 from measurements for weakly compressible materials, and ( ln CP/ ln V)T = 1 or 0 for T < Q or T > Q, respectively (compare with the data listed in Table 3 and the paper by Pankov et al. ). Thus, the Chopelas and Boehler' formula for a can be written in the form
O. Anderson et al. [1992a, 1992b, 1993] favored the power law dT = dT0xk (for T Q ) that yields
with only small deviation from values by (47). The value of k = 1.1-1.4 in (48) was inferred from the theoretical PIB model of Isaak et al. .
Applying (47) or (48) to the lower mantle, we find that a decreases 4-5 times along the "hot" lower-mantle adiabat, from the state P = 0 and T 1700-2000 K to the base of the mantle. The power law (41) with dT = 5-6 gives a greater decrease in a (6-8 times), and the same law with dT = 2-3 results in a smaller decrease of a (2-3 times). Although approximations (47) and (48) are more preferable than (41), they require additional confirmation and information on parameters m, k, and ( ln CP/ ln V)T. a in the lower mantle by (36) and (37), as described in section 6.2.1, with dK0/d T = 2.3 10-4 K-1, gives the results close to those derived from (47) or (48) (Figure 3).
Similar results for a with dT decreasing under compression were obtained by Zharkov  from his analysis of EOS's at extremely high pressures. Still earlier, Zharkov  showed that the lower mantle thermodynamics quantified on the basis of the Debye model and seismic data gives the 4-5-fold decrease in a at the mantle base compared to the value at P = 0.
To this point, considering a at high compression, we have not applied to the Grüneisen parameter g. However, the problem of thermal expansivity at high pressures and temperatures is intimately related to the problem of a similar variation of the Grüneisen parameter. D. Anderson [1987, 1989] characterized the lower-mantle thermodynamics by using the acoustic or Brilloin g. For the adiabatic lower mantle, he found from PREM that g0 = 1.4 and a0 = 3.8 10-5 K -1 at P = 0 and T = 1700 K; the value of g was determined by the thermodynamic relation (8) for CV = const ( T > Q ). Given function g (V), the variation of a with volume in the classical temperature range can be evaluated by the formula derived from (8)
Note that the thermodynamic parameter g, generally speaking, is different from the so-called lattice Grüneisen parameter [e.g., Mulargia, 1977; D. Anderson, 1989; O. Anderson, 1968, 1979b, 1980]. However, assuming that the latter depends only on volume, both parameters were found to coincide (the same inference follows from the quasiharmonic atomistic model of EOS at high temperatures, when, on the other hand, we come up with the purely thermodynamic consequence g = g (V) for CV = const (see section 11).
The three most familiar formulas for the lattice g can be written in the general form [Zharkov and Kalinin, 1971]
where m = 0, 1, or 2 gives the formulas of Slater, Dugdale-Macdonald, and Zubarev-Vashcheno (or Irvine and Stacey ), respectively. The latter of these formulas appears to be the most favored, at least at T > Q, for high symmetry crystals. Following Leibfried and Ludwig, 1961], g can approximately be expressed in terms of the root-mean-square frequency of atomic oscillations. For cubic crystals with the central interaction, when only the nearest neighbors are allowed for, this approximation also leads to the Zubarev-Vashchenko formula [Pankov, 1983; Hofmeister, 1991a].
Calculation by (50) requires knowledge of the P(V) dependence at T = 0 K, but the replacement of the T = 0 K isotherm by any isotherm at T > 0 K is not significant for this case. Using the EOS from (37) at K0=4 and determining g by (50) at m = 2 and then a by (49), we obtain a0/a = 1.7 for x = 0.7 (that is approximately at the mantle base). Such a small decrease in a compared to the 4-5-fold decrease found above is due to the fact that the EOS by (37), like many other P(V) relationships [Pankov and Ullmann, 1979b], results in a low value of the slope
calculated from (50). This either tells us that a more flexible EOS involving the independent parameter K0K0 K2 (such as in model 2 by Ullmann and Pankov  or the Birch-Murnaghan fourth-order EOS) must be introduced, or some amendments to (50) are required. The Zubarev-Vashchenko formula was somewhat improved by Stacey [1981, 1992], but nevertheless, the slope q for most two-parametric ( K0 and K0 ) EOS's appears to remain low).
Another useful approximation for g is the empirical power law [e.g., O. Anderson, 1968, 1974; McQueen et al., 1970]
where q is often assumed to be one, according to shock-wave data [McQueen, 1991] or studies of the mantle [O. Anderson, 1979b; D. Anderson, 1989]. From (49) with q = 1, we find Birch's formula aKT = a0K0 = const, which gives a0/a = 3.5-4.0 at the mantle base (for x = 0.7, K0 = 4, KT/K0 = 3.51, and T = 2000-3000 K). The value of q = 1.5-2.0 may be more favored for the mantle perovskite [Pankov et al., 1998], yielding, however, a0/a = 4.2-5.0 that is close to the result obtained from (47) and (48).
Note that, according to (49), the assumption of the power laws for a (41) and g (51) again gives the Murnaghan EOS (44). Since the latter fits data well over a range of P/K0 0.3 ( x 0.82 ), we expect (41) to be a sufficient approximation for a in the same compression range.
Duffy and Ahrens  estimated a from shock wave data for MgO, CaO, CaMgSi 2 O 6 and e -Fe at pressures to P > 140 GPa. By using (49) and (51) with q = const and KT/K0 from the PIB model for MgO [Isaak et al., 1990], they found q = 0.5 0.5 that is smaller than q = 0.83-1.26 in the compression range x = 0.67-1.0 along the PIB isotherm. Periclase is more compressible than the lower mantle matter and has x = 0.67 at P = 134 GPa near the mantle base (according to the PIB 2000 K isotherm of MgO, P/K0 = 1.047, KT/K0 = 4.699, and K = 4.74 ). With these values, the shock wave results of Duffy and Ahrens for MgO give a0/a = 3.1-4.7 at the mantle base ( x = 0.67 ), i.e., the value 1-1.6 times less than a0/a by (47) and (48) at the typical value of dT0 = 5 1 (if the value dT0 = 4 is used in (47) and (48), the resulting a0/a value will be closer to the shock-wave estimate above). These results can be viewed as an argument for the decrease of both dT and q under compression.
In analysis of the volume dependence of g, O. Anderson et al.  proposed the power law
which, similarly to (48), yields
Setting q0 = 1.5-2.0 and n = 1 (as for MgO, according to O. Anderson et al. [1992b]), we find from (49) that a0/a = 4-4.5 at x = 0.7. Note that q0 for MgO descends from 1.72 to 1.26 as temperature increases from 300 K to 2000 K [O. Anderson et al., 1993].
In total, many estimates of a0/a using various methods described above consistently show that the thermal expansion coefficient in the lower mantle decreases 4-5 times along the hot low-mantle adiabat as the pressure increases from zero to the base of the mantle. Nevertheless, the complete consensus on all the parameter values related to these estimates (e.g., for q and dT ) has yet not been achieved.
Most data on thermal expansion refers to the dependence a(T) = a0 at P = 0. The value of a0 is necessary, in particular, to extrapolate the thermal expansivity data to higher pressures in the mantle. The typical behavior of a(T) is illustrated in Figures 4 and 5. Usually, the data at P = 0 are fitted using the empirical formula [e.g., Fei et al., 1990, 1991]
which we used to calculate a presented in Table 2 (and in Pankov et al. ). Note that the applicability of (54) can also be justified by calculations of phase diagrams [Fei et al., 1990, 1991].
A theoretically based approach to calculating a(T) was developed by Suzuki [1975a, 1975b], who used the Mie-Grüneisen EOS yielding
where Et is the Debye thermal energy
k = 1/2(K0-1), Q = K0V0 / g, and D(z) is the Debye function. Here, it is assumed that g = constant, and parameters K0 and K0 are defined at T = 0. The fitted parameters are Q, k, and Q. Formula (55) is derived by expanding the potential pressure in V and truncating at only two first terms. The Q values obtained from this method are given in Table 2 (see also Pankov et al. ).
O. Anderson et al. [1992a] extrapolated a0(T) from a fixed value at T > Q to higher temperature using the relation
where dT = a = constant (see (32)). Formula (56) is easily derived from the condition that a0 at P = 0 varies with density by the power law. Note that (56) has an asymptote close to which a dramatically increase with temperature (reflecting to some extent the fact that the potential energy has an inflection point).
For a more complete consideration of the temperature behavior of a, we calculated a(T) for three minerals from the Mie-Grüneisen EOS (with the Debye model), in which, unlike the Suzuki method, g(x) was found by (51) with q = 0, 1, and 2, and room-temperature isotherms were represented by equations (28) and (37). The material parameters of the EOS's were found from values of r, KS, a, CP, and ( KS/ P)T at normal conditions (Tables 2 and 3).
The results of the computations are shown by the solid lines in Figures 4 and 5. We see that the curves for periclase and particularly forsterite systematically deviate from the experimental points at high temperatures, although there is a considerable uncertainty in data for a at high temperatures. Nevertheless, such deviations can be caused by the fact that the temperature dependence of g (at V = const) is not accounted for in the Mie-Grüneisen EOS [Mulargia, 1977; Mulargia and Boschi, 1980; Mulargia et al., 1984; O. Anderson et al., 1992a; Molodets, 1998]. To gain a better insight into the quality of the Mie-Grüneisen EOS and to construct a self-consistent database on EOS parameters, it is very important to measure the thermal expansivity of minerals at high temperatures, up to their melting points. This conclusion was emphasized by many authors [e.g., Saxena, 1988, 1989; Goto et al., 1989; Isaak et al., 1989b; Gillet et al., 1991; Richet et al., 1992].