7. Specific Heat

The lattice specific heat of minerals at T 1000 K is close to the classic limit 3Rn = 3RM/m (the molar value, where R is the gas constant and n is the number of atoms in chemical formula). From calorimetry, we have information on the isobaric heat capacity C_P, which exceeds C_V by 1-3% at 300 K and 10-15% at T > Q (Table 2 and Pankov et al., 1997]. Since m 20-22 g/mole for mantle minerals, the classic value of C_V for them is 1.13-1.25 J/g K. Depending on mineral, the high-temperature anharmonic corrections to C_V become singnificant either near the melting point or even at room temperature (sometimes, at T equal 1/6 of the melting point) [Mulargia and Boschi, 1980; Quareni and Mulargia, 1988; Reynard et al., 1992; Fiquet et al., 1992].

7.1. P- V- T derivatives of specific heat

From the identities ²E/ V T = ²E/ T V and ²S/ P T = ²S/ T P, using the Maxwell relations, we find

(57)

(58)

Note that (57) and (58) are the alternate forms of (33) and (34), repectively. The logarithmic volume derivatives at T = constant can be expressed as follows:

(59)

(60)

These identities were used to compute the derivative values given in Tables 3 and in Pankov et al. [1997]. O. Anderson et al. [1993] noted that, for the Debye model, ( C_V/ P)_T 0 and therefore a 2d_T - K.

The difference C_P - C_V satisfies the identity

(61)

i.e., it decreases with pressure for usual values of K and d_T (see Table 3 and Pankov et al. [1997]).

Birch [1952] estimated the decrease in C_P with pressure in the lower mantle, setting a 4 and agT 0.1. By the power law for C_P, this gives a 13-16% decrease in C_P along an isotherm, for x descending from 1.0 to 0.7. According to Birch, the maximum decrease in C_P in the mantle does not appear to exceed 20%.

The adiabatic volume derivative of C_P is

(62)

Substituting ( ln C_P/ ln T)_P 0.15 (the typical value for minerals for T 1000 K), g 1-1.5, agT 0.1, and ( ln C_P/ ln V)_T 0.5 (for a 4 ), we find ( ln C_P/ ln V)_S 0.3-0.4. Consequently, the power law for C_P yields a 8-13% decrease of this value along the mantle adiabat (to x 0.7 ).

By using (29), the temperature derivative of C_P can be represented in the form

(63)

The second term arising from the extrinsic anharmonicity can be estimated by making use of (58), so that (63) in conjuction with data for ( C_P/ T)_P allows the first term coming from the intrinsic anharmonicity to be evaluated. At room temperature, ( ln C_P/ ln V)_T is on the order of ( ln C_P/ ln T)_P, but nevertheless, the contribution of the second term to the sum (63) is small because of the small factor agT. At high temperatures, this contribution generally increases to 15-30% (perhaps, 60-70% for ilmenite and perovskite, according to our estimates) [Pankov et al., 1997]. In the classical limit, C_V = constant, assuming that aa(V) and gg(V), we find ( C_P/ T)_V agC_V.

Further, from (17) and (18), it is easy to obtain the identity

(64)

which we used to estimate the values of this derivative presented in Table 3 and Pankov et al. [1997]. Then, with the help of the identity

(65)

it is possible to compute (( ln C_V)/( ln T))_V, provided that the second term in (65) is given by (59).

An explicit dependence C_V(V, T) can be derived from models and measurements of the vibrational spectra of solids (e.g., Pitzer and Brewer, 1961]):

(66)

where k is the Boltzman constant, y = hn/kT, g(n) is the spectrum density, and n are the lattice frequencies including optic and acoustic modes [Kieffer, 1979a, 1979b, 1979c, 1980; Hofmeister, 1991a, 1991b; Richet et al., 1992]. This method provides information on the inadequacy of the Debye theory and the related approximation g = -d lnQ/d ln V. The characteristic temperature Q in the Debye theory is usually estimated from acoustic data, but Q found from data for C_V at T 300 K (labelled Q_th ), on average, exceeds the acoustic Q (labelled Q_a ) by about 20% (larger deviations are common to quartz and coesite, see Table 2 and Pankov et al. [1997] and Watanabe [1982]). Chopelas [1990b] found, however, good agreement of the Debye model with the spectrum data for MgO at pressures to 200 kbar, provided that Q V^g, i.e., q = 0.

Spectroscopic measurements at high pressures allow us to estimate the derivative ( C_V/ P)_T. For example, the data of Chopelas [1990a, 1990b] to 200 kbar show that C_V linearly decreases with P, so that the gradient -( C_V/ P)_T is 17.6 10^-3 ( T = 300 K) and 0.91 10^-3 ( T = 1800 K) for MgO and 49.8 10^-3 ( T = 300 K) and 3.4 10^-3 ( T = 1800 K) J/(mole K kbar) for forsterite. With these values, C_V being extrapolated (by the power volume dependence) to the maximum pressure P = 1357 kbar in the mantle will decrease 40-60, 8-12, and 2-3% on the 300, 1000, and 1700 K isotherms, respectively. Comparing these results with the decrease in C_P estimated above, we verify that the difference C_P-C_V in the lower mantle must be exceedingly small. From the same Chopelas' data, using also the K_T and C_V values from Table 2, we find ( ln C_V/ ln V)_T = 0.77 ( T = 300 K) and 0.022 ( T = 1800 K) for MgO and 0.54 ( T = 300 K) and 0.022 ( T = 1700 K) for forsterite. These results are comparable to our estimates of this derivative from thermodynamic data (Table 3).

7.2. An explicit temperature dependence of C _P at P = 0

Calorimetric data for C_P versus temperature at P = 0 are commonly fitted to various empirical expressions [e.g., Fei and Saxena, 1987; Berman, 1988; Saxena, 1989; Richet and Fiquet, 1991]:

(67)

Richet and Fiquet [1991] showed that the last of the above formalas are favored but no one of them provides an accurate description of C_P over a wide temperature range.

In addition, Figure 6 compares C_P(T) found by the simpler formula used by Watanabe [1982] to fit the measurement in the temperature interval of 350-700 K. An example of MgO shows that the extrapolation by this formula can lead to series errors.

As well as in the analysis of thermal expansivity in section 6.3, we calculated C_P(T) (Figure 6) from the same Mie-Grüneisen EOS as was used to compute a. One can see that the theoretical curves can be reconciled with the data shown by varying parameter q in the limits 1-2. In so doing, we find q 0 for MgO, q = 1-2 for Al ₂ O ₃, and q 0.5 for forsterite. However, these values of q are not always consistent to data on a (see section 6.3), and this fact also suggests a certain inaccuracy of the Mie-Grüneisen EOS model.

8. Thermal Pressure

8.1. P- V- T derivatives of the thermal pressure coefficient

The thermal pressure coefficient defined by (9) or (12) is the basic characteristic of the thermal pressure and can also be defined as t = ( P_th/ T)_V. Note that t has also the meaning of the latent heat of expansion per 1 K. O. Anderson and his co-workers [O. Anderson, 1982, 1984, 1988; O. Anderson and Sumino, 1980; O. Anderson and Goto, 1989; O. Anderson et al., 1982, 1991, 1992a] paid special attention to this parameter, in particular, in relation to their development of the rectangular parallelepiped resonance technique for measuring elastic properties of minerals at high temperatures.

The basic identities for the derivatives of t can easily be derived from those given in sections 6 and 7. The following identities are especially suitable [Brennan and Stacey, 1979; Birch, 1978; O. Anderson and Yamamoto, 1987]:

(68)

(69)

(70)

(71)

Formula (68) is obtained by expanding the derivative at the left side and then by using (30); (69) is a consequence of (57), and (70) is easily derived by equating the second derivatives of P with respect to V and T taken in one order or another; finally, (71) follows from (31) and (70).

Combining (68) and (69), we have

(72)

According to (71), for the common inequality d_T > K, parameter t decreases with pressure along an isotherm. However, at high pressure, the decrease can change to an increase, as, e.g., in the PIB model for MgO [O. Anderson et al., 1993]. It is clear from (68) that t increases with temperature at constant pressure, at least for T < Q.

As follows from (29), the logarithmic temperature derivative of t at P = constant is represented as

(73)

where the first, intrinsic anharmonic term is positive due to (69) and can be written as the sum of two terms

(74)

and the second term in (73), in view of (70) and (71), is

(75)

Here, the symbol d_V^T is introduced for convenience (see a further analysis in section 9). Although due to large values of a, the intrinsic anharmonic term in (73) is dominant in value at T < Q, the sign of (73) at high temperatures can be either positive or negative.

The temperature behavior of t resembles that of C_V, so that at T > Q, t tends to be independent of temperature [O. Anderson, 1984]. Accordingly, the thermal pressure P_th tends to a linear dependence on T. Our estimates of t (Table 2 and Pankov et al. [1997]) show that the nonlinear terms in P_th versus T makes a contribution not greater than 1-3% at the highest temperatures indicated in Table 3 and Pankov et al. [1997]. The linear temperature behavior of P_th is considered to be the universal property of solids at high temperatures [O. Anderson et al., 1992a]. Unlike the temperature dependence, the extent to which t depends on volume at T = constant varies from one type of solid to another. According to O. Anderson et al. [1992a], the earth minerals fall into an intermediate group between materials with significant (e.g., gold) and relatively weak (e.g., sodium chloride, alkali metals, noble elements) volume dependences of t. It is important that these inferences are based on both P-V-T data (analysis of P_th(V, T) ) and high-temperature data for K_T and aK_T at P = 0 (analysis of (70)).

At first glance, the observed regularity in t(T) at T > Q is explained by the fact that C_V constant in this temperature range, where, in view of (8) and (69), g is therefore independent of temperature. In other words, the quasiharmonic Mie-Grüneisen EOS is seemingly justified at high temperatures. However, this is not quite true to be the general case when we start with the condition t = constant or t = t(V) which are compatible with the case of C_V(T) (see (69)) and therefore with a dependence of g on both volume and temperature [O. Anderson and Yamamoto, 1987]. Theoretically, the departure of C_V from the Dulong and Petit law [e,g., Mulargia and Broccio, 1983; O. Anderson and Suzuki, 1983; Gillet et al., 1991; Reynard et al., 1992] is in part related to the intrinsic anharmonicity described by the third and higher order terms in the lattice Hamiltonian expansion. In describing experimental data, the thermal EOS generally requires smaller number of terms in this expansion than the caloric EOS [Leibfried and Ludwig, 1961; Wallace, 1972; Davies, 1973]. Thus, when anharmonicity in P_th versus T and the temperature dependence of g are not observed, this may suggest that either the quasiharmonic limit for the vabrational g has not yet been achieved or the higher order terms in the thermal part of the EOS are mutually cancelled [O. Anderson et al., 1982].

In conclusion to the above analysis of t, we formulate the following important assumptions and their consequences that can easily be verified: (1) Let C_V be independent of V, i.e., either C_V = C_V(T) or C_V = constant. (2) Assume that K_T = K_T(V) that is equivalent to d_T = K. From (1), it follows that either t = t(V) or t = constant, and in addition, K_T is either a linear function of T or K_T(V), which leads to q = q(V) (or q = 1 ) and either g = Vt(V)/C_V(T) or g = g(V). The statement (2) is equivalent to either t = t(T) or t = constant. If both (1) and (2) statements are valid (but C_V constant), then g = const V/C_V(T) and q = 1. The conditions C_V = constant and K_T = K_T(V) yield g = const V.

8.2. Thermal pressure model

The thermal EOS resulting directly from integrating (12) is of the form

(76)

where f(V) is the static lattice pressure plus the zero oscillation pressure. The second term in (76) is the total thermal pressure P_th accounting for all anharmonic contributions. This EOS can be rewritten in the form

(77)

where, for example, T₀ = 300 K. In accordance with the behavior of t described above, the thermal pressure can be approximated as [O. Anderson, 1984, 1988]

(78)

where T T₁ Q. As already noted, the variation with volume in (78) is insignificant for some solids.

In the Mie-Grüneisen EOS, the thermal pressure is however defined as

(79)

where the quasiharmonic approximation for C_V is used. Thus, here, at high temperatures T T₂ Q, when C_V constant,

(80)

Even when b(V) = b(V) in some temperature range, the distinction between (78) and (80) is retained since, in the general case, g = g(V, T) in (78) and a a. In practice, for certain minerals and for the present accuracy of measurements, P_th from (78) and the Mie-Grüneisen theory can be indistinguishable [Fei et al., 1992a, 1992b; Mao et al., 1991], especially for minerals with low Debye Temperature.

The term DP_th in (77) can be approximated in various ways. For example, with given volume dependences of K and d_T, by integrating (75), we can find t(V) at T = constant [O. Anderson et al., 1992a, 1993]. The temperature dependence of t is derived from data on a(T) and K_T(T) at P = 0. Another possibility to explicitly approximate the thermal pressure is given by the power law for t(V) on an isotherm.

Fei et al. [1992a, 1992b] and Mao et al. [1991] used a number of models for P_th in order to describe P-V-T X-ray data. Specifically, they assumed ( K_T/ T)_V = constant. Then, (70) yields

(81)

where t(0, T) is determined from data on a provided that (dK_T/dT)_P = constant, the condition assumed over the entire P-V-T range of measurements. From the assumption that both temperature derivatives of K_T are constant, it follows that tK = const. However, in such a case, K will increase with pressure along an isotherm and decrease with temperature along an isobar--the behavior that disagrees with the usual properties of this parameters (see section 6). Note that the estimated d_T values can be very sensitive to the adopted model of P_th [e.g., Mao et al., 1991], although the P_th values themselves from various models can be close.

Finally, in Figure 7, we illustrate the temperature dependences of t at P = 0, calculated from the Mie-Grüneisen EOS for periclase, corundum, and forsterite, described in section 6, with various values of q in the interval 0-2. Although this type EOS gives correct orders of magnitude and the correct regularities in the P-T variations of t, it is difficult to achieve the complete consistency for all of the data given, as well as in the cases of specific heat (section 7) and thermal expansivity (section 6). A better accuracy of the EOS is undoubtedly required than that of the Mie-Grüneisen EOS in order to describe experimental data, to reliably predict unmeasured properties, and in particular, to calculate the phase diagrams at high pressures (when a 10% error in P_th can substantially affect the estimated phase boundary slopes and positions).

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