The empirical linear relation between volume and logarithm of bulk modulus, discovered by ... K \[V. AdP. dV. B. T. (2) and di†ers from the de–nition of bulk modulus used in clas- .... (isobaric cubic expansion coefficient a0) and adiabatic bulk.
A new equation of state based on Grover, Getting and KennedyÏs empirical relation between volume and bulk modulus. The high-pressure thermodynamics of MgO Michel H. G. Jacobs*a and Harry A. J. Oonkb a Geodynamics Research Institute, Utrecht University, Budapestlaan 4, NL -3584 CD, Utrecht, T he Netherlands. E-mail : jacobs=geo.uu.nl b Debye Institute and Geodynamics Research Institute, Utrecht University, Padualaan 8, NL -3584 CH Utrecht, T he Netherlands Received 17th December 1999, Accepted 31st March 2000 Published on the Web 12th May 2000
The empirical linear relation between volume and logarithm of bulk modulus, discovered by Grover, Getting and Kennedy, is taken as the basis for a new pressureÈvolume equation of state. This linear relation and the equation of state have the same two substance-dependent parameters, the values of which can be derived from low-pressure data. For MgO, in the temperature range 100È3100 K and the pressure range 0È225 GPa, it is shown that the new methodology allows the careful calculation of high-pressure thermodynamic properties : all available experimental data are reproduced with great precision.
1 Introduction In high-pressure solid-state physics equations of state are important tools. They are needed to transform thermodynamic properties measured at (relatively) low pressure to conditions of high-pressureÈto conditions inaccessible to accurate experimental studies. This is particularly true in our work, which aims at evaluating phase behaviour and thermodynamic properties of materials of geophysical relevance, and for conditions of temperature and pressure that prevail in the interior of the earth. We were faced with the problem that the canonical equations of state in geophysics start to produce imperfections at pressures of 25 GPa, i.e. at about 10 percent of the pressure in the centre of the earth. These imperfections are Ðrstly an unsatisfactory agreement between calculated properties and (the scarce) experimental data, and secondly anomalous behaviour of speciÐc thermodynamic quantities. As an example of such anomalous behaviour, Fig. 1 shows the calculated entropy of MgO as a function of pressure. The fact that the entropy is going through a minimum would mean, by virtue of the Maxwell equation (dS/dP) \ [(dV /dT ) , that T P the volume is going through a maximum, the thermal expansivity becoming negative. Although there is no law forbidding that the thermal expansivity should become negative, it is not expected for substances having a close-packed structure like MgO (see refs. 2È4). From Fig. 1, and other indications of anomalous behaviour, it became clear that some kind of magic relationship would be needed to stabilise the thermodynamic rigor of the computational system. Thus we decided to study the stabilising possibilities of the compressibility relation discovered by Grover, Getting and Kennedy5 in the early seventies. From shockwave and static compression measurements on a variety of metals they found ““ a nearly precise linear relation ÏÏ between the logarithm of the bulk modulus and the volume, up to volume changes of 40%. We write this relationship as
A
K0(T ) V 0 (T ) \ V 0 (T ) ] b ln m m 0 K0(T ) 0 DOI : 10.1039/a910247g
B
(1)
where V 0 denotes molar volume (which we will express in m3 m mol~1), K the isothermal bulk modulus, T a reference tem0 perature and the superscript ““ 0 ÏÏ refers to standard pressure (1 bar). The isothermal bulk modulus, which is the inverse of the isothermal compressibility, is deÐned as K \ [V
A B dP dV
(2) T and di†ers from the deÐnition of bulk modulus used in classical mechanics K \ [V (dP/dV ) in which V is a reference 0 T 0 volume. Throughout this paper we adhere to the deÐnition given by eqn. (2). In eqn. (1), b is a material-dependent constant. As a matter of fact, eqn. (1) is a di†erential equation in P and V with a known solution. That solution, representing a new equation of state, has the form of a power series and can be made operational for a modern personal computer. Here,
Fig. 1 Calculated entropy curve for MgO at 298.15 K generated by a third-order BirchÈMurnaghan equation of state using the description of Saxena et al.1
Phys. Chem. Chem. Phys., 2000, 2, 2641È2646 This journal is ( The Owner Societies 2000
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we present the new equation of state and its empirical background, and demonstrate its strength by presenting results for the substance MgO. The choice of MgO is more or less obvious : it is one of the key materials for geodynamic research ; it is monomorphous ; in particular there are experimental data, to judge the signiÐcance of our computations and validate them, for high pressures up to 225 GPa. For the sake of overall transparency of the paper, we Ðrst give a short overview of the thermodynamic background.
2 Thermodynamic background In understanding the phase behaviour of matter as a function of thermodynamic temperature (T ), pressure (P) and chemical composition, a key role is played by the Gibbs energy function (G). It is composed of the fundamental quantities energy (U), entropy (S) and volume (V ) as G \ H [ T S \ U ] PV [ T S
(3)
where H represents enthalpy. When the change of G (dG) in terms of eqn. (3) is substituted in the fundamental equation for the change of the energy of a closed system in a reversibly carried out experiment, which is dU \ T dS [ P dV
(4)
the following expression is obtained : dG \ [S dT ] V dP
(5)
One could say that, going from eqn. (4) to eqn. (5), the experimentally inconvenient variables V and especially S have been replaced by the experimentally preferred variables P and T . It means, among other things, that the criterion for equilibrium, in terms of U and its variables, dU O 0
for constant S and V
(6)
dG O 0
for constant T and P.
(7)
changes into
When combined, eqns. (4) and (6), and (5) and (7), give rise to dU [ T dS ] P dV O 0,
(8)
dG ] S dT [ V dP O 0
(9)
and where the sign \ is for a reversible change and the sign \ for a spontaneous change. It means that, for given T and P, the stable state of a closed system will be the one for which the Gibbs energy is at its minimum. In (experimental) thermodynamic practice, the determination of G as a function of T and P, via the integration of dG according to eqn. (5), is generally carried out as follows. First, at ambient pressure, G is determined as a function of temperature, by calorimetric methods. Next, for the second part of the integration, where the change of pressure comes into action, the volume of the system has to be known, and must be determined as a function of pressure at each temperature of interest. Here we arrive at the key difficulty of (our) research into the phase behaviour of matter under conditions that exist in the interior of the earth, where pressures run up to several hundreds of GPa. That difficulty is the lack of accurate determinations of volume as a function of pressure, over a large range of temperature, for MgO. One of the possibilities to cope with the volume problem is to make use of an equation of state (EOS), a relationship between a systemÏs P, V and T revealing its volume as a function of pressure and temperature. An equation of state can have an empirical as well as a theoretical background. In any event, a powerful equation of state not only allows a reliable calculation of Gibbs energies, it also has enough strength to allow the calculation of thermodynamic properties from the 2642
Phys. Chem. Chem. Phys., 2000, 2, 2641È2646
(characteristic, see below) Gibbs energy. An example of an EOS, widely used in solid state physics, is the (family of ) equation(s) named after Birch and Murnaghan. The so-called third order BirchÈMurnaghan equation can be formulated as P \ P0 ] 3 K0 2
G
CA B
A B D CA B DH
V ~7@3 V ~5@3 [ V0 V0
] 1 [ 3 (4 [ (K0)@) 4
V ~2@3 [1 V0
(10)
where P0 is the reference pressure (usually P0 \ 1 bar), V 0 the volume at P \ P0, K0 the isothermal bulk modulus at P \ P0 and (K0)@ its pressure derivative. V 0 and K0 are functions of temperature. The former is related to the thermal expansion coefficient a0 as
AP
V 0(T ) \ V 0(T \ T )exp 0
T
a0 dT
B
(11) T0 Normally, power series in T are used for a0, K0 and also (K0)@. Another consequence of the change from U to G is that U, as a characteristic function of its “ natural variables Ï S and V , is replaced by G as a characteristic function of T and P : all thermodynamic properties can be found from G as a function of T and P. The import of this quality in the context of the underlying paper is, that, after having evaluated G as a function of T and P, its accuracy can be checked by matching the properties it generates with available experimental data.
3 The equation of state In Table 1 numerical values are given for the molar volume and the isothermal bulk modulus of MgO, as a function of temperature and at atmospheric pressure. The background of these data can be detailed as follows. The available data for MgO at atmospheric pressure consist of heat capacity at constant pressure (C0), thermal expansivity P (isobaric cubic expansion coefficient a0) and adiabatic bulk modulus (K0), all three as a function of temperature. At (T , P) S the isothermal bulk modulus (K) follows from the adiabatic K , as S C K\K V (12) SC P where C is the heat capacity at constant volume. V Table 1 Thermodynamic properties of the substance MgO. C0 , a0 P, m and V 0 are the result of an optimisation, K0 are original measureS calculated. C0 and ments mof Isaak et al.,6 and K0 and C0 are P, m C0 in J K~1 mol~1 and a0 in K~1 V, m V, m T /K
C0 P, m
105 a0
V 0 /cm3 m
K0/GPa S
K0/GPa
C0 V, m
300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800
37.105 42.475 45.571 47.521 48.848 49.818 50.576 51.207 51.765 52.280 52.776 53.266 53.759 54.262 54.780 54.316
3.1741 3.5497 3.7904 3.9663 4.1071 4.2280 4.3372 4.4398 4.5389 4.6366 4.7344 4.8334 4.9343 5.0377 5.1440 5.2537
11.2507 11.2887 11.3303 11.3744 11.4204 11.4681 11.5174 11.5680 11.6201 11.6735 11.7283 11.7846 11.8423 11.9015 11.9622 12.0245
163.93 162.33 160.73 158.87 157.11 155.10 153.06 151.07 148.91 146.69 144.42 142.03 139.68 137.31 134.94 132.74
161.50 158.88 156.25 153.37 150.58 147.57 144.53 141.55 138.42 135.26 132.07 128.79 125.57 122.35 119.15 116.10
36.5554 41.5715 44.2990 45.8749 46.8178 47.3977 47.7574 47.9794 48.1195 48.2071 48.2624 48.3016 48.3281 48.3492 48.3684 48.3801
As a matter of fact, the numerical values appearing in Table 1 (for a0 and C0 ) are the outcome of an optimisation proP, m cedure, in which data from various sources were weighed and combined for maximal mutual consistency and thermodynamic stability. The allowable temperature range from T \ 300 K to T \ 1800 K, is determined by the K0(T ) data, taken S from Isaak et al.6 The a0(T ) data are from several sources (Suzuki,7 Dubrovinsky and Saxena,8 Austin,9 Durand,10 Ganesan,11 White and Anderson12 and Skinner13) ; a polynomial Ðt of the original experimental data gives rise to the following formula, which is valid from T \ 100 K to T \ 3100 K (see ref. 14) : a0/K~1 \ 4.5248 ] 10~5 ] 8.4711 ] 10~10(T /K) [ 4.1959 ] 10~3(K/T ) ] 2.4984 ] 10~12(T /K)2 (16)
Fig. 2 Plot of the linear relationship between molar volume and logarithm of the bulk modulus for periclase (MgO) using the experimental data of Isaak et al.,6 Suzuki,7 and Dubrovinsky and Saxena.8 There is also a linear relationship for Forsterite (Mg Si O ) using 2@3 1@3 4@3 the data of Suzuki,7 Kajiyoshi,20 and Isaak et al.21 The data for both substances are plotted in the temperature range 300È1800 K.
The C0 data from several sources (Barin,15 JANAF thermoP chemical tables,16 Barron et al.,17 Krupka et al.18 and Richet and Fiquet19) give rise to the following formula, valid from T \ 100 K to T \ 1800 K (see ref. 14) : C0 /JK~1 mol~1 \ 180.7689 ] 8.7874 ] 10~3(T /K) P, m [ 1.5081 ] 104(K/T ) ] 6.0367 ] 105(K/T )2 ] 6.1868 ] 10~7(T /K)2 [ 18.0218 ln(T /K)
The two heat capacities are related by the general equation C [ C \ a2KV T P V From eqns. (12) and (13), K and C are solved as V C K P S K\ C ] a2K V T P S and,
(13)
(14)
C2 P (15) C \ V C ] a2K V T P S Starting from the molar volume at T \ 298.15 K, the molar volume is calculated by means of eqn. (11). Thereafter K0(T ) and C0(T ) can be calculated by means of eqns. (14) and (15). V
(17)
Fig. 2 shows a plot of the molar volume data in Table 1 vs. the logarithm of the corresponding isothermal bulk modulus. There is, indeed, a nearly precise linear relation between the two properties, and for MgO it implies the following values of K0(T \ 298.15 K) and b in eqn. (1) : 0 K0(T ) \ (161.5 ^ 0.6) GPa, (18) 0 b \ ([2.359 ^ 0.003) ] 10~6 m3 mol~1 (19) As a next step, note, the most important one in this work, we adopt the hypothesis that eqn. (1) with the values of the constants given by eqns. (18) and (19) will be valid (not only at atmospheric pressure, but) at any pressure. Dropping the subscript ““ m ÏÏ in volume we write : (K/Pa) \ [V
A BN dP dV
A
Pa \ K0(T )exp 0
B
V [ V 0(T ) 0 b
(20) T The solution of this di†erential equation is a power series in V:
G A CA B A
P/Pa \ P0/Pa [ K0(T ) exp [ 0
V 0(T ) 0 b
BH
BD
= b~j(V j [ (V 0)j) V ] ; (21) V0 jj ! j/1 where V 0(T ) \ V (P0, T ) and V 0 \ V (P0, T ). 0 0 In other words, eqn. (21), the solution of eqn. (20), is the equation of state generated by the empirical relation. This equation of state is the focus of this work. The molar Gibbs energy of a substance at arbitrary pressure and temperature conditions is given by : ] ln
P G P
Fig. 3 Deviation of the Gibbs energy from the Ðnal value as a function of the number of terms (““ n ÏÏ) in the summation of eqn. (23). The Ðnal value is determined by selecting a high number for ““ n ÏÏ such that the computer (an Intel Pentium CPU was used) cannot detect a di†erence between the Gibbs energies calculated with n and n ] 1 terms. A pressure of 50 GPa and a temperature of 2000 K were selected.
T
C0 dT [ T P, m T0 T C0 P P, m dT ] V dP (22) ] S0 ] m m T T0 P0 where *H0 is the heat of formation and S0 the absolute f, m m entropy both at T and reference pressure P0. With the equa0 tion of state, eqn. (21), and dropping the subscript ““mÏÏ, the G (T , P) \ *H0 ] m f, m
H P
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last term of eqn. (22) becomes :
P
P
P0
V dP \
P
P, V
d(PV ) [
P
P0, V0
V
P dV V0
G A
\ V (P [ P0) ] K0(T ) exp [ 0
G A B C A
V 0(T ) 0 b
BH
V = [ (V [ V 0) ] ; V0 j/1 b~j V j`1 [ (V 0)j`1 [ (V 0)j(V [ V 0) jj ! j]1
] V ln
BDH (23)
where V 0(T ) \ V (P0, T ) and V 0 \ V (P0, T ). 0 0 Owing to the fact that the (substance dependent) constant b is negative, the terms of the power series are alternatingly negative and positive. Fig. 3 shows how the Gibbs energy calculated for a given T , P pair is behaving as a function of the number (n) of terms of the series, in the range from 20 to 30 terms. This Ðgure gives the di†erence in Gibbs energy G (T \ 2000 K ; P \ 50 GPa ; n \ n) [ G (T \ 2000 K ; m m P \ 50 GPa ; n \ 50). The Gibbs energy value for n \ 50 terms is calculated as G \ [225 455.774 J mol~1. With m n \ 34 the di†erence is reduced to 1 pJ, and this precision is sufficient to allow the numerical calculation of Ðrst and higher order partial derivatives of the Gibbs energy. Although we write the solution of eqn. (20) as a power series, it is perhaps related to an incomplete C (gamma) function.
4 Results and discussion In the subject of this paper we can distinguish three levels of a di†erent character. These are (i) the empirical relation between volume and bulk modulus ; (ii) the equation of state built on the empirical relation ; and (iii) the Gibbs energy built on the equation of state. In this section we will make an assessment of our new approach, paying attention to each of these three levels. Most importantly, we have to examine to what extent the Gibbs function will come up to our requirements : being able to provide all thermodynamic functions of the system with accuracy and precision, and to generate no anomalies. The empirical relation
makes the linear relation the basis of an equation of state, having the advantage of containing just two system-dependent parameters, which can be derived from data obtained at relatively low pressure. The equation of state can be used to calculate the molar volume of the substance, at a speciÐc temperature and pressure, from the molar volume at atmospheric pressure for the given temperature. For MgO, experimental data on molar volume are available for T \ 298.15 K and pressures up to 225 GPa. The data, which are shown in Fig. 4, comprise shock compression data by Vassilou and Ahrens,22 Du†y and Ahrens,23 and Carter et al.,24 and static compression data by Mao and Bell,25 Du†y et al.,26 and Perez-Albuerne and Drickamer.27 The curve, which is drawn in Fig. 4, represents the relation between volume and pressure calculated with the new equation of state from no more than the volume at atmospheric pressure and the values of the substance-dependent parameters K0(T ) and b. The 0 agreement between the experimental data and the calculated function is virtually perfect. Next we consider the pressure derivative, K@, of the (isothermal) bulk modulus, an important property in geophysics. It follows from our eqn. (20) as K@ \
A B dK dP
V \[ m b
(24) T Owing to the fact that the constant b is always negative, K@ is always positive : a material becomes less compressible as pressure increases. The value of K@ is 4.769 ^ 0.005 for MgO at ambient conditions, for which V 0 (T \ 298.15 K) \ 11.25 ] 10~6 m3 mol~1 m 0 (Robie et al.28), and follows from eqns. (24) and (19) for constant b. This value is close to the experimentally determined value by Richet et al.,29 and also close to the value found by Inbar and Cohen30 by means of molecular dynamics ; see Table 2. It follows from eqn. (24) that K@ becomes zero at inÐnite pressure (V ] 0). According to eqn. (20) this corresponds to a Ðnite constant value of the isothermal bulk modulus, with the (somewhat inconvenient) side e†ect that the pressure may become larger than the bulk modulus. In our computations the numerical value of the bulk modulus becomes smaller than the pressure above 12 000 GPa, which (at any rate) is considerably larger than the pressure in the centre of the earth (about 300 GPa). To proceed further, we consider the second derivative, KA, of the bulk modulus with respect to pressure. Using eqn. (24)
The empirical linear relation between volume and logarithm of bulk modulus was discovered by Grover, Getting, and Kennedy5 by studying the compression characteristics of a series of (more than 25) metals. They spoke of a ““ nearly precise linear relation ÏÏ, and as can be observed in Fig. 2, that same designation is applicable to the substance MgO. From the numerical values in Table 1 it follows that, in terms of volume, the maximum deviation of a data point from the linear Ðt of the data is ca. 0.005 cm3 mol~1 (ca. 0.04%). This is commensurate with the uncertainty in numerical values as a result of the procedure followed and the uncertainties in the original experimental data. On closer inspection, the deviation of the data points from linearity reveals a sinusoidal structure. It is a remaining artefact, due to the use of the power series, eqns. (16) and (17). Anyhow, the remaining deviations are such that the existence of a really precise linear relation can neither be proved nor disproved. We conclude that the evidence is strong enough to adopt linearity between V and log K as a working hypothesis. The equation of state We made an additional step and adopted as hypothesis that the linear relationship will be valid at any pressure. Doing so 2644
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Fig. 4 Calculated molar volume of MgO at room temperature together with experimental data of : (]) Vassilou and Ahrens,22 (]) Du†y and Ahrens,23 (K) Carter et al.,24 (L) Mao and Bell,25 (|) Du†y et al.,26 ()) Perez-Albuerne and Drickamer.27
Table 2 Isothermal Ðrst pressure derivative of bulk modulus with respect to pressure at ambient conditions (K0)@
Ref.
Method
5.40 ^ 0.20 4.85 ^ 0.2 4.769 ^ 0.005 4.68 4.54 4.29 ^ 0.08 4.13 ^ 0.09
Mao and Bell25 Richet et al.29 This work Inbar and Cohen30 Carter et al.24 Chang and Barsch31 Jackson and Niesler32
Static compression (nonhydrostatic, quasi-hydrostatic conditions) Static compression (quasi-hydrostatic conditions)
it is given by KA \
A B d2K dP2
V \ m bK
(25) T and is invariably negative due to the constant b. At ambient conditions we calculate KA \ [0.0295 ^ 0.0001 GPa~1. The fact that KA is negative is a favourable characteristic according to Stacey et al.,33 who pointed out that an equation of state becomes useful, if applied to terrestrial data, when the bulk modulus has a concave curvature. The experimental
Fig. 5 There is no anomaly present in the predicted entropy of MgO. The molar absolute entropy at ambient conditions, 26.94 J K~1 mol~1, has been taken from Barin.15
Fig. 6 Calculated thermal expansivity of MgO together with data from : (|) Isaak et al.6 at 1 atm, (L) Du†y and Ahrens23 with error bars, (K) Isaak et al.,34 ()) Wang and Reeber2 and (]) Chopelas and Boehler.35 The dashed curve is calculated with a third order BirchÈ Murnaghan equation of state according to Saxena et al.1
Molecular dynamics Shock wave Ultrasonic pulse superposition method Ultrasonic pulse interferometry
determination of KA is rather cumbersome. In spite of that, Jackson and Niesler32 managed to indicate the range from [10.1 to ]0.7 for the product KKA at ambient conditions. With their value of K \ 162.5 ^ 0.2 GPa, and taking the centre of their KKA range, this gives rise to KA \ [0.0289 ^ 0.0143 GPa~1. Du†y et al.26 found a value of [0.022 ^ 0.004 GPa~1 using a fourth order BirchÈ Murnaghan Ðt to the data presented in Fig. 4. The Gibbs energy The Gibbs energy of a system or substance is a property which is characteristic for the variables temperature and pressure. It means that, when for a given system the Gibbs energy is known as a function of T and P, all thermodynamic properties of the system are Ðxed, through the Gibbs function and its Ðrst and higher order derivatives with respect to temperature and pressure. As a matter of fact, we have used data for a number of thermodynamic properties to establish the Gibbs energy function. Here we shall examine how well the data, that have not been used, will be reproduced by the Gibbs function. The Ðrst partial derivative with respect to pressure is the volume of the system. That this property is reproduced well has already been demonstrated in Fig. 4. It has also been shown that the isothermal derivatives with respect to pressure, K@ related to (d3G/dP3), and KA related to (d4G/dP4), describe the experiments well and have the correct sign. We will now examine the entropy and mixed partial derivatives of the Gibbs function with respect to temperature and pressure. The Ðrst partial derivative with respect to temperature is the opposite of the entropy. Although high-pressure data on the entropy are not available, Fig. 5 shows a number of isothermal sections of entropy as a function of pressure. With
Fig. 7 Calculated thermal expansivity curves together with experimental data from : (|) Dubrovinsky and Saxena8 and model calculations from (L) Wang and Reeber,2 ()) Reeber et al.3 and (K) Cynn et al.4
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our new equation of state, the calculated curves do not show the anomaly as sketched in Fig. 1. Besides, recalculating the entropy at ambient temperature as a function of pressure, using the isothermal bulk modulus generated by our equation of state and its pressure derivative, along with eqns. (16) and (17), as the input in a BirchÈMurnaghan or a Murnaghan equation of state, we still Ðnd anomalous behaviour of the type of Fig. 1. The remaining experimental data pertain to the thermal expansivity, which is related to the mixed second partial derivative (d2G/dT dP). In Fig. 6, which is valid for T \ 2000 K, the calculated expansivity as a function of pressure is compared with experimental data. Again the agreement is more than satisfactory, and this also applies to Fig. 7, which represents isobaric sections of thermal expansivity as a function of temperature.
6 7 8 9 10 11 12 13 14 15 16 17 18
5 Conclusions The empirical virtually ideal linear relationship between volume and logarithm of bulk modulus, discovered by Grover, Getting and Kennedy5 for a series of metals, is also valid for the substance MgO. The linear relationship between volume and logarithm of bulk modulus can be used as a basis for an equation of state. The equation of state has two system-dependent parameters that can be calculated with the help of data obtained at low pressure. The Gibbs energy function established for the system MgO on the basis of the equation of state is free from anomalies, and high-pressure thermodynamic properties calculated from it are in virtually perfect agreement with experimental data.
19 20 21 22 23
24 25 26 27 28
References 1
2 3 4 5
S. K. Saxena, N. Chatterjee, Y. Fei and G. Shen, T hermodynamic Data on Oxides and Silicates, Springer Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, Budapest, 1993, p. 260. K. Wang and R. R. Reeber, Geophys. Res. L ett., 1995, 22, 1297. R. R. Reeber, K. Goessel and K. Wang, Eur. J. Miner, 1995, 7, 1039. H. Cynn, O. L. Anderson, D. G. Isaak and M. Nicol, J. Phys. Chem., 1995, 99, 7813. R. Grover, I. C. Getting and G. C. Kennedy, Phys. Rev. B, 1973, 7, 567.
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29 30 31 32 33 34 35
D. G. Isaak, O. L. Anderson and T. Goto, Phys. Chem. Miner., 1989, 16, 704. I. Suzuki, J. Phys. Earth, 1975, 23, 145. L. S. Dubrovinsky and S. K. Saxena, Phys. Chem. Miner., 1997, 24, 547. J. B. Austin, J. Am. Ceramic Soc., 1931, 14, 795. M. A. Durand, Physics, 1936, 7, 297. S. Ganesan, Philos. Mag., 1962, 7, 197. G. K. White and O. L. Anderson, J. Appl. Phys., 1966, 37, 430. B. J. Skinner, Am. Mineral., 1957, 42, 39. M. H. G. Jacobs and H. A. J. Oonk, Calphad, Computer Coupling of Phase Diagrams and T hermochemistry, in the press. I. Barin, T hermochemical Data of Pure Substances, part II, VCH Verlags-gesellschaft mbH, D6940 Weinheim (FRG), 1989, p. 868. M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald and A. N.Syverud, J. Phys. Chem. Ref. Data, 1985, 14, Suppl. 1, 1469. T. H. K. Barron, W. T. Berg and J. A. Morrison, Proc. R. Soc. L ondon A, 1959, 250, 70. K. M. Krupka, R. A. Robie and B. S. Hemingway, Am. Mineral., 1979, 64, 86. P. Richet and G. Fiquet, J. Geophys. Res., 1991, 96, 445. K. Kajiyoshi, High temperature equation of state for mantle minerals and their anharmonic properties, M.Sc. Thesis, Okayama, Japan, 1986. D. G. Isaak, O. L. Anderson and T. Goto, J. Geophys. Res., 1989, 94, 5895. M. S. Vassilou and T. J. Ahrens, Geophys. Res. L ett., 1981, 8, 729. T. S. Du†y and T. J. Ahrens, in, J. Z. Hu, H. K. Mao, J. Shu and R. J. Hemley, High-Pressure Science and T echnology, ed. S. C. Schmidt, J. W. Shaner, G. A. Samara and M. Ross, AIP, New York, 1994, p. 1107. W. T. Carter, S. P. Marsh, J. N. Fritz and R. G. McQueen, in Accurate Characterization of the High Pressure Environment, ed. E. C. Lloyd, NBS special pub. 326, Washington DC, 1971, p. 147. H. K. Mao and P. M. Bell, J. Geophys. Res., 1979, 84, 4533. T. S. Du†y, R. J. Hemley and H. K. Mao, Phys. Rev. L ett., 1995, 74, 1371. E. A. Perez-Albuerne and H. G. Drickamer, J. Chem. Phys., 1965, 43, 1381. R. A. Robie, B. S. Hemingway and J. R. Fisher, T hermodynamic properties of minerals and related substances at 298.15 K and 1 bar (105 P) pressure and higher temperatures, US Geol. Survey Bull., 1978, 1452, 184. P. Richet, H. K. Mao and P. M. Bell, J. Geophys. Res., 1989, 96, 3037. I. Inbar and R. E. Cohen, Geophys. Res. L ett., 1995, 22, 1533. Z. P. Chang and G. R. Barsch, J. Geophys. Res., 1969, 74, 3291. I. Jackson and H. Niesler, in High Pressure Research in Geophysics, ed. S. Akimoto and M. H. Manghnani, Centre for Academic Publishing, Tokyo, 1982. p. 93. F. D. Stacey, B. J. Brennan and R. D. Irvine, Geophys. Survey, 1981, 4, 189. D. G. Isaak, R. E. Cohen and M. J. Mehl, J. Geophys. Res., 1990, 95, 7055. A. Chopelas and R. Boehler, Geophys. Res. L ett., 1992, 19, 1983.
