JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B10, 2462, doi:10.1029/2003JB002446, 2003

First-principles prediction of the P-V-T equation of state of gold and the 660-km discontinuity in Earth’s mantle Taku Tsuchiya Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan Received 13 February 2003; revised 4 July 2003; accepted 14 July 2003; published 9 October 2003.

[1] The P-V-T equation of state (EOS) of gold is the most frequently used pressure calibration standard in high-P-T in situ experiments. Empirically proposed EOS models, however, severely scatter under high-P-T conditions, which is a serious problem for studies of the deep Earth. In this study, the EOS of gold is predicted using a first-principles electronic structure calculation method without any empirical parameters. The calculated thermoelastic properties of gold compare favorably to experimental data at ambient 0 are 166.7 GPa and 6.12, respectively. Up to V/Va = 0.7, conditions so that BT0 and BT0 the calculated Gru¨neisen parameter of gold depends on volume according to the function g/ga = (V/Va)z with ga of 3.16 and z of 2.15. On the basis of these data, the validity of previous EOS models is discussed. It is found that the present ab initio EOS provides a 1.3 GPa higher pressure than Anderson’s scale at 23 GPa and 1800 K and largely reduces the discrepancy observed between conditions at the transition of Mg2SiO4 and the 660-km seismic discontinuity. However, a discrepancy of about 0.7 GPa still remains INDEX TERMS: 1025 between the 660-km discontinuity and the postspinel transition. Geochemistry: Composition of the mantle; 3630 Mineralogy and Petrology: Experimental mineralogy and petrology; 3919 Mineral Physics: Equations of state; 3939 Mineral Physics: Physical thermodynamics; 8124 Tectonophysics: Earth’s interior—composition and state; KEYWORDS: first-principles density functional calculation, pressure calibration standard, PVT thermal equation state, postspinel transition, 660-km seismic discontinuity Citation: Tsuchiya, T., First-principles prediction of the P-V-T equation of state of gold and the 660-km discontinuity in Earth’s mantle, J. Geophys. Res., 108(B10), 2462, doi:10.1029/2003JB002446, 2003.

1. Introduction [2] Gold is important metal in Earth science, because its pressure-volume-temperature (P-V-T ) equation of state (EOS) is the most frequently used pressure calibration standard for in situ high-pressure and high-temperature experiments [Mao et al., 1991; Fei et al., 1992; Meng et al., 1994; Funamori et al., 1996; Irifune et al., 1998; Kuroda et al., 2000; Hirose et al., 2001a, 2001b; Ono et al., 2001]. The characteristic properties of gold, its low rigidity and chemical stability, make it particularly suitable for this role [Tsuchiya and Kawamura, 2002a]. However, some recent in situ experiments have noted that the pressure values estimated by the thermal EOS of gold show significant gap depending on the model employed [Hirose et al., 2001a, 2001b; Shim et al., 2002]. Using the EOS proposed by Anderson et al. [1989], Irifune et al. [1998] first reported in their in situ study that the postspinel phase boundary of Mg2SiO4 shifted to about 2 GPa lower than the pressure corresponding to the depth of the 660-km seismic discontinuity (23 – 24 GPa and 1700 – 2000 K), implying that the decomposition of spinel occurs at a depth of 60 km

shallower than the seismic discontinuity. Similar results were observed for the MgSiO3 system [Hirose et al., 2001a, 2001b]. However, Hirose et al. [2001a] noted that the pressures at which the phase changes occur in these mantle minerals are more consistent with seismic observations when the EOS proposed by Jamieson et al. [1982] is used. [3] Such uncertainty in pressure measurements is a serious problem for high-P-T in situ experiments of mantle constituents. Because it is difficult to reliably determine the EOS model of gold using only empirical data under limited P-T conditions, it is meaningful to carry out a theoretical investigation. In the present study, the finite temperature thermodynamic properties of gold and its P-V-T thermal EOS are predicted from first-principles with no empirical parameters. On the basis of this ab initio EOS model, the validity of previous empirical models of gold are investigated in detail and, implications for the phase boundaries near the 660-km seismic discontinuity in Earth’s mantle are discussed.

2. Previous Models of the EOS for Gold [4] Several EOS models for gold have been proposed [Jamieson et al., 1982; Heinz and Jeanloz, 1984; Anderson

Copyright 2003 by the American Geophysical Union. 0148-0227/03/2003JB002446$09.00

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et al., 1989; Holzapfel et al., 2001; Shim et al., 2002] (hereafter these scales are referred by the abbreviation of their initials such as JFM, HJ, AIY, HHS and SDT, respectively). However, these models are not in agreement with each other at high pressures and temperatures [Shim et al., 2002], due to uncertainties in the experimental data or limitations inherent in the extrapolation of experimental data obtained under limited P-T conditions to much higher P-V-T conditions. In addition, experimental determination of the pressure dependence of thermodynamic properties such as thermal expansion, thermal pressure, and the Gru¨neisen parameter are essentially difficult. Hence simple assumptions are usually applied to estimate the pressure effect on such thermal properties, although their validity has not been established. [5] In the case of the JFM model, the EOS was determined using only shock compression (Hugoniot) data and heat capacity at ambient pressure. These limited data are clearly insufficient to obtain a thermal EOS with complete thermodynamic consistency. In the JFM model, the Hugoniot data were reduced to an isotherm using the simple but nontrivial assumption for the thermodynamic Gru¨neisen parameter of g/ga = V/Va, where g is the Gru¨neisen parameter and the subscript a indicates the value at ambient conditions. This relationship is modified to the form g/V = const. and has been often employed to analyze the Hugoniots of metal. However, its validity under a wide range of P-T conditions has never been established. [6] In contrast, the HJ model was constructed by blending a static and a shock wave data in addition to incorporating data of the thermal expansion, elastic constants and thermodynamic parameters at ambient pressure. The AIY model improved on the HJ EOS by taking into account the hightemperature anharmonicity to ensure better thermodynamic consistency. However, in terms of consistency with the shock data, the AIY model is worse than those of JFM and HJ. Moreover, it is likely that the room temperature static compression data used by HJ and AIY are not accurate at high pressure, since they were obtained using a diamondanvil cell with the pressure transmitting medium of alcohol. It is well known that the alcohol exhibits severe nonhydrostaticity at pressures over 20 GPa [Takemura, 2001]. In these models, the Gru¨neisen parameter was assumed to depend on volume as g/ga = (V/Va)z, which is more versatile than the JFM model constraint. The exponent z, however, differs considerably between the HJ EOS (1.7) and the AIY EOS (2.5). [7] The EOS of HHS was derived only from shock wave and ultrasonic data. On the basis of their original equations, the volume dependence of g was extrapolated up to the strong compression limit of V/Va = 0. In this model, g showed a complicated behavior as a function of volume and hence, these authors claimed that the simple approximation of g/V = const. was unfavorable for gold even under small compression. However, the validity of their formulations for g is not well established. Most recently, the SDT model was obtained by using different static data from that used in the HJ model, which was measured by taking care of hydrostaticity. However, in order to compare it to the shock data, the SDT EOS was extrapolated to more than 550 GPa, based on the third-order Birch-Murnaghan equation [Birch, 1978]. It is not likely that a simple equation can adequately

fit the entire pressure region from 0 to 550 GPa. Even if there is a good trend up to 6 megabar, this does not ensure that the EOS model is accurate to within a few GPa in the range of the Earth’s mantle P-T conditions. Hence the temperature dependence of the adiabatic bulk modulus is too large even at ambient pressure. Moreover, the simplification that g/V = const. was employed again to construct this EOS. Thus the validity of the SDT model for the EOS of gold is also unclear. [8] Disagreement in g causes the largest uncertainty in the thermal properties of gold at high pressure. Differences between the proposed thermal pressures of gold at V/Va = 1 are actually quite small (see Figure 4) and, except for the JFM EOS, completely agree with each other. The slight difference in the JFM model probably originates in the fact that no thermal property data at ambient pressure were used to determine this EOS. However, the deviation in thermal pressure does increase significantly with compression. At V/Va = 0.9, the JFM and AIY models show the highest and lowest values of thermal pressure, respectively, and their difference reaches 3 GPa at 2000 K. Moreover, this difference increases to more than 5 GPa at V/Va = 0.8 and 2000 K.

3. Calculation of Thermodynamics [9] For nonmagnetic metal, pressure can be represented as the sum of three terms: PðV ; T Þ ¼ P0 ðV Þ þ Pph ðV ; T Þ þ Pel ðV ; T Þ:

ð1Þ

Here, the first, second, and third terms are static pressure, lattice thermal pressure, and electronic thermal pressure, respectively. These are represented by the thermodynamic definition of pressure
 @fi ðV ; T Þ Pi ðV ; T Þ ¼  ; @V T

ð2Þ

where f is the Helmholtz free energy density with respect to each degree of freedom. For static pressure, f is equivalent to the total energy usually called in first-principles study. fel is the electronic free energy density, which has been evaluated for gold from its electronic structure [Tsuchiya and Kawamura, 2002b]. fph is the phonon free energy density, which is calculated in this study as follows. [10] The linear response method based on the densityfunctional theory (DFT) [Hohenberg and Kohn, 1964] and the density-functional perturbation theory (DFPT) [Baroni et al., 1987; Savrasov, 1996; Savrasov and Savrasov, 1996] has been successfully applied to the calculation of the lattice contribution to the free energy [Pavone et al., 1998] and other thermodynamic properties of solids [Karki et al., 2000]. The basic idea of DFPT for the phonon calculation is to accurately evaluate the second-order energy variation d2E caused by the nuclear displacement. The central purpose is to find the linear response of the charge density induced by the phonon. The dynamical matrix at any q vector is determined by these linear response calculations with explicit account of only the primitive lattice. [11] The finite temperature thermodynamic properties of a solid can be calculated by combination of the DFPT and the quasiharmonic approximation. Once the phonon dispersion

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Figure 1. Calculated phonon dispersion relations at a = 7.671 a.u. (solid curves), 7.3 a.u. (dashed curves), and 7.1 a.u. (dash-dotted curves). Filled circles are experimental results of the neutron inelastic scattering at ambient pressure [Lynn et al., 1973].

relation is obtained from the lattice dynamical calculation, the phonon energy density (uph) and the phonon free energy density ( fph) can be calculated as follows: uphðV ; T Þ ¼

X q;i

fph ðV ; T Þ ¼ kB T

  1 hwi ðq; V ; T Þ þ fBE ðwi ; T Þ ; 2

X q;i


 hwi ðq; V ; T Þ ; ln 2 sinh 2kB T

ð3Þ

ð4Þ

where q is the phonon wave vector, i the band index, fBE(w, T ) = 1/(ehw=kB T  1) the Bose-Einstein distribution function and kB the Boltzmann constant. The factor 1/2 in equation (3) is the contribution from zero-point vibration. Within the normal harmonic approximation (HA) for insulators, the frequency w is treated solely as a function of volume and is independent of temperature. However, w of metal does depend on temperature because of electronphonon coupling that increases with increasing temperature. Irrespective of this fact, we can expect that normal HA treatment does not bring a serious loss of accuracy for gold, since the thermal excitation of electrons is actually small in this metal [Tsuchiya and Kawamura, 2002b]. In this work, it was assumed that w = w(q, V). Using equation (2), we can then calculate the phonon thermal pressure using the fph obtained here. Furthermore, the phonon entropy density (sph) may be calculated from the thermodynamic relationship: sph ðV ; T Þ ¼

uph ðV ; T Þ  fph ðV ; T Þ : T

ð5Þ

4. Computational Details [12] In this study, the electronic structure of an fccformed Au crystal was calculated from the first-principles within the DFT and the local density approximation [Kohn

and Sham, 1965]. For this purpose, I adopted the allelectron full-potential linear muffin-tin-orbital (FPLMTO) method that can simulate core state relaxation [Weyrich, 1988] and, hence, is especially suitable for the calculation of electronic and mechanical properties of solids under high pressure [Tsuchiya and Kawamura, 2001, 2002a, 2002b]. The detailed calculation conditions for the static lattice energy are fundamentally the same as those used previously [Tsuchiya and Kawamura, 2002a]. The Vosko-Wilk-Nasairtype formulation [Vosko et al., 1980] was applied to represent the exchange and correlation energy functional. I used 3k-spd LMTO basis set (27 orbitals) with tail energies (k2) of 0.1, 1.0, and 2.5 Ry. Moreover, a semicore panel for 5p at k2 = 3.5 Ry was set to take into account the interatomic interactions of this state. Fully and scalar relativistic corrections for the core and valence states were recalculated after each self-consistent iteration, respectively. Static pressure P0(V ) was evaluated according to equation (2) by linearly interpolating the total energy variations with a cell parameter of ±0.001 at each volume. [13] On the other hand, for the phonon calculation, charge densities and potentials inside the muffin-tin spheres (MTS) were expanded using spherical harmonics up to l = 6. The d2E obtained has the same precision as setting lmax to 8. The dynamical matrix was calculated as a function of the wave vector for a total of 29 q points for the irreducible Brillouin zone of the fcc cell. That corresponds to the (8, 8, 8) reciprocal lattice grid defined as qijk = (i/I )G1 + ( j/J )G2 + (k/K )G3, where Ga is the primitive translation in reciprocal space. Throughout the calculations, nonoverlapping MTS with radii of 2.3 a.u. were applied. [14] Phonon dispersions were calculated for a total of 13 cell parameters from 6.8 a.u. to 7.9 a.u. at intervals of ˚ ) in addition to the zero0.1 a.u. (1 a.u. = 0.529177 A pressure cell parameter. EOS parameters of zero-pressure volume V0(T ), isothermal bulk modulus BT0(T, P) and its 0 (T, P ) were determined by least pressure derivative BT0

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squares fit of the P-V relationship to Vinet’s EOS function [Vinet et al., 1989].

5. Results and Discussion 5.1. Phonon Dispersion and the Thermodynamics of Gold [15] The calculated phonon dispersion curves at cell parameters of 7.671 a.u., 7.3 a.u. and 7.1 a.u. are plotted along the high symmetry direction in Figure 1. Here, a of 7.671 a.u. is the predicted zero-temperature equilibrium lattice constant and 7.3 a.u. and 7.1 a.u. correspond to static pressures of 42 GPa and 83 GPa, respectively. Experimental results for neutron inelastic scattering at ambient pressure [Lynn et al., 1973] are also shown for comparison. We can see the excellent agreement between theory and experiment across the zone at zero pressure. This agreement is typical of FPLMTO+LDA+DFPTbased calculations for simple metals and semiconductors [Savrasov, 1996; Savrasov and Savrasov, 1996] and it is likely that the results for a wide range of volume have a similar accuracy. [16] I confirmed the temperature dependence of the X point phonon frequencies by Fermi-Dirac fermi surface smearing. At any volume, differences of only about 1% were found between frequencies at 0 K and 3000 K, even taking into account the electronic excitation. This results in a negligible contribution to the free energy and, consequently, to the thermal pressure. The assumption that w = w(q, V ) can, therefore, be adequately applied to gold, as expected. In Figure 2, the phonon energy density, the phonon free energy density and the phonon entropy density calculated according to equations (3 – 5) at several volumes, are shown as a function of temperature. At low temperature, the phonon total energy increases with increasing compression due to an increase of the zero point vibration energy (ZPVE), whereas it converges at higher temperatures (Figure 2a and its inset). The calculated entropy (Figure 2c) at 0 GPa and 300 K is 47 J/(mol K). More than 99% of this comes from the phonon contribution and is in remarkable agreement with the measured standard entropy of 47.4 J/(mol K). This illustrates the quantitative reliability of the FPLMTO+DFPT+QHA method. These are the first data on the thermodynamic properties of gold calculated from first-principles theory.

Figure 2. Temperature dependencies of (a) the phonon energy density uph, (b) the phonon free energy density fph, and (c) the phonon entropy density sph at 9 volumes from 7.985 cm3/mol (a = 7.1 au) to 11.000 cm3/mol (a = 7.9 au) with
a = 0.1 au. The upper curve is at smaller volume in uph and sph, and the lower curve is at smaller volume in fph, as shown by arrows. In the inset of Figure 2a, uph from 0 to 250 K is enlarged.

5.2. Thermal Pressure [17] The phonon free energy converted as a function of volume is shown for several temperatures from 0 K+ZPVE to 2500 K in Figure 3. By using these free energy data, phonon thermal pressures can be predicted using equation (2). The total thermal pressure can be represented by the sum of the phonon contribution plus the electronic contribution (Pth = Pth,ph + Pth,el). Using data from previous work [Tsuchiya and Kawamura, 2002b] for the latter term, the thermal pressures of gold at volumes of V/Va = 1.0, 0.9 and 0.8 are shown in Figure 4, together with values from previous empirical models. [18] Figure 4 shows that the calculated thermal pressure agrees well with empirical values at V/Va = 1.0. Particularly, it appears to agree well with the JFM EOS. However, at ambient volume, other scales are likely to be more reliable

than the JFM, since this EOS was determined without taking into account the physical properties of gold at ambient pressure. It should be noted that the present calculation somewhat overestimates thermal pressure at ambient volume, with a discrepancy of about 0.6 GPa at 1500 K. This overestimation may be attributed to an anharmonic effect that cannot be completely included in the QHA level approximation. [19] At V/Va = 0.9 and 0.8, the empirical data scatter widely. The JFM and SDT equations of state clearly show larger thermal pressures than do the present ab initio values, while that of AIY is considerably lower. Although the HHS EOS is close to the ab initio value at V/Va = 0.9, the deviation increases at V/Va = 0.8. Consequently, among the several empirical models, the HJ model is closest to the

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Figure 3. Volume dependencies of the phonon free energy density fph. The number shown indicates temperature, and dashed lines in the left panel are the results at intermediate temperature between the upper and lower solid lines. ab initio thermal pressure and its volume dependence. Note that in the present calculations, thermodynamic properties are obtained by fully nonempirical procedure in contrast to previous models. Moreover, the quantitative agreement of the phonon dispersion at ambient pressure ensures high reliability of the predicted thermal pressures. The fact that these thermal pressures have values intermediate to the widely scattering experimental values is remarkable. 5.3. Thermal Equation of State [20] On the basis of these thermodynamic data, we can obtain full information about the P-V-T EOS of gold (Table 1) without any empirical assumptions or adjustable parameters. The predicted isotherms at several temperatures

are shown in Figure 5 together with the empirical 300 K isotherms of HJ (=AIY), HHS and SDT. The theoretical 300 K isotherm is in good agreement with HJ up to 20 GPa, whereas HHS and SDT give somewhat larger and smaller volumes, respectively, than the present EOS. The volume of the HJ model, however, gradually becomes smaller than the present isotherm and this deviation grows with pressure. These discrepancies mainly relate to the difference in B00 of each model (Table 1). Reported values of B00 range from 5.0 (SDT) to 6.2 (HHS). HJ and SDT used a B00 value determined from static data, while HHS used a B00 value from ultrasonic data. The present ab initio value of B00 of 6.12 is close to the value of the ultrasonic determination. The JFM model is not shown in Figure 5, since it is close to that

Figure 4. Total thermal pressure at the values V/Va of 1, 0.9, and 0.8. Solid lines are the present results. Previous empirical estimations are shown by circles [Jamieson et al., 1982] (JFM), diamonds [Heinz and Jeanloz, 1984] (HJ), triangles [Anderson et al., 1989] (AIY), inverted triangles [Holzapfel et al., 2001] (HHS), and squares [Shim et al., 2002] (SDT).

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Figure 5. Calculated isotherms at five temperatures of 0 K + ZPVE, 300 K, 1000 K, 1500 K, and 2000 K sequentially from the bottom. Experimental 300 K isotherms are also plotted as dashed curves (HJ), long dashed curves (HHS), and dotted curves (SDT) for comparison. of HJ, although it shows a slightly larger volume than all other models up to pressures of 30 GPa. [21] Calculated physical properties of gold, thermal expansivity a = 1/V(@V/@T )P , isothermal bulk modulus BT = [1/V(@V/@P)T]1 and adiabatic bulk modulus BS = [1/V(@V/@P)S]1, are shown in Figure 6, in addition to isobaric specific heat CP = (@u/@T )P , where u is the total energy density from the phonon and electron degree of freedom. These quantities are summarized in Table 2, which shows that all the calculated values at ambient conditions compare well to experimental data. However, Figure 6 shows that errors appear at temperatures higher than the Debye temperature, which is significant for thermal expansion. This is clearly due to neglect of the anharmonic effect in the present calculations. Within the QHA, intrinsic anharmonicity arising from phonon-phonon interactions is not taken into account. However, the error in gold is smaller than that of the previous QHA calculation for MgO [Karki et al., 2000]. It may be expected that the anharmonic effect is smaller in simple metals than in oxides because oxides usually have high frequency optic phonon modes that yield Table 1. Isochors for Gold From This Studya 1  V/Va

300 K

500 K

1000 K

1500 K

2000 K

2500 K

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34

0.00 3.55 7.68 12.42 17.86 24.12 31.30 39.52 48.94 59.76 72.11 86.36 102.65 121.38 142.98 167.77 196.48 229.56

1.52 5.04 9.13 13.83 19.23 25.46 32.60 40.78 50.17 60.95 73.26 87.48 103.73 122.42 143.99 168.74 197.41 230.45

5.35 8.78 12.79 17.40 22.71 28.85 35.90 43.99 53.29 63.98 76.21 90.34 106.50 125.10 146.58 171.24 199.83 232.78

9.19 12.54 16.45 20.98 26.20 32.25 39.22 47.22 56.43 67.03 79.18 93.22 109.29 127.80 149.19 173.77 202.26 235.13

13.04 16.29 20.12 24.56 29.70 35.66 42.54 50.45 59.58 70.09 82.14 96.10 112.08 130.51 151.81 176.30 204.70 237.49

16.88 20.05 23.79 28.14 33.19 39.07 45.86 53.68 62.72 73.15 85.11 98.98 114.88 133.21 154.43 178.83 207.15 239.84

a

Unit of pressure is given in GPa.

Figure 6. Calculated temperature dependence of (a) thermal expansion a, (b) isothermal and adiabatic bulk modulus BT (solid lines) and BS (dashed lines), and (c) isobaric heat capacity CP at pressures of 0 GPa, 24 GPa, and 72 GPa. In Figure 6a, circles are zero-pressure experimental values of Touloukian et al. [1977]. In Figure 6b, filled and open circles are zero-pressure experimental values of BT of Anderson et al. [1989] and BS of Neighbours and Alers [1958] and Chang and Himmel [1966], respectively. In Figure 6c, circles are zero-pressure experimental values of CP of Touloukian et al. [1977]. large vibrational energies at high temperature. Moreover, in general, the anharmonic effect becomes less important with increasing pressure because of ascent of the Debye temperature and the melting temperature. These effects are discussed later in more detail. 5.4. Gru¨neisen Parameter [22] Next, we investigate the validity of the empirical relationship for thermal behavior assumed in the previous studies is investigated. The thermodynamic Gru¨neisen parameter g is defined as g¼

aBS V aBT V ¼ : CP CV

ð6Þ

TSUCHIYA: FIRST-PRINCIPLES PREDICTION OF EQUATION OF STATE OF GOLD Table 2. Several Physical Quantities of Gold at Zero Pressure and 298 Ka Parameter

Theory

Experiment

Va, cm3/mol a, 105/K BT, GPa BT0 CP, J/mol K ga z D, K

10.207 4.52 166.7 6.12 25.5 3.16 2.15 180

10.215 4.26 167 – 171 5.0 – 6.2 25.4 2.95 – 3.215 1 – 2.5 165 – 170

a Experimental data are from Jamieson et al. [1982], Heinz and Jeanloz [1984], Anderson et al. [1989], Holzapfel et al. [2001], and Shim et al. [2002].

By substituting the thermodynamic definition for each quantity and by considering that thermal pressure is the pressure change at constant volume, g can be modified as


@Pth g¼V @uth

 :

ð7Þ

V

The calculated relationships between thermal pressure and internal energy density are shown in Figure 7a. Up to uth = 60 kJ/mol which corresponds to 2405 K (see Figure 2a), linear relationships between thermal pressure and internal energy density are found. This means that the Gru¨neisen parameter is constant with respect to temperature under isochoric conditions. The volume dependence of the Gru¨neisen parameter is plotted in Figure 7b, together with previous empirical data. The present value of g can be perfectly fit to the function (g/ga) = (V/Va)z and give ga = 3.16 and z = 2.15. This means that the assumption that (g/ga) = (V/Va)z used in the HJ and AIY models is plausible, at least for the present volume range. [23] As shown in Figure 7b, the value of g in the JFM model, followed by those of SDT and AIY. HJ’s g is close to the present ab initio value. These g values are reflected in the magnitude of the thermal pressure of each model, since at constant volume, a larger g gives a larger thermal pressure (equation (7)). On the other hand, the logarithmic volume derivative z represents the volume dependence of g. The larger z means a more rapid decrease in thermal pressure with volume compression. If z is close to 1,

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thermal pressure over 300 K hardly depends on volume at all (see JFM and SDT in Figure 4). However, the present ab initio results clearly demonstrate a change of thermal pressure that is dependent on volume. This is caused by the decrease in nonlinear behavior between fph and V with increasing compression (Figure 3). Therefore we conclude that the assumption z = 1 employed by JFM and SDT is unfavorable for gold. In the SDT model, a large g is necessary to reproduce the Hugoniot, since this EOS was based on the compressible room temperature isotherm model (Figure 5). Moreover, z was determined from data at a larger compression than V/Va of 0.75. In fact, highpressure data tend to show z close to 1, because the volume dependence of g becomes constant with increasing compression as shown in Figure 7b. HHS’s g does not seem to be suitable, since the complex volume dependence of g assumed in this model does not satisfy the relationship (g/ga) = (V/Va)z. [24] The ab initio z (=2.15) is closest to the z of AIY (=2.4). However, rather than z, it is the magnitude of g itself that is actually meaningful in the physical sense. Therefore it should be noted that the ab initio g is closest to HJ’s g over a wide pressure range. 5.5. Melting Temperature [25] On the basis of the classical mean field potential (MFP) approach, Wang et al. [2001] proposed the following melting formula Tm ¼ AV 2=3 2 ;

ð8Þ

where Tm is the melting temperature and  is the characteristic temperature. If  is regarded as a generalized Debye temperature, this equation is equivalent to the Lindemann law. Since A is an adjustable parameter determined from a fit to the observed zero-pressure melting temperature, the formula is not first-principles. However, this simple formula may be used to estimate the pressure dependence of the melting temperature in order to discuss the anharmonicity under pressure. Using the well-established value of Tm0 of 1063C and the calculated Debye temperature, the predicted melting curve of gold is shown in Figure 8. It is evident from Figure 8 that the melting temperature increases with increasing pressure (2000 K at

Figure 7. (a) Calculated relationship between thermal pressure and internal energy density at V/V0 = 1 (solid line), 0.9 (dashed line), and 0.8 (dash-dotted line) and (b) volume dependence of the Gru¨neisen parameter g. In Figure 7b the solid curve is the present result, and dashed curves are previous experimental results.

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10 GPa and 6400 K at 130 GPa) and is much larger than the mantle geotherm. This provides supports that the anharmonic effect in gold decreases with pressure and becomes insignificant at mantle P-T conditions. 5.6. 660-km Seismic Discontinuity [26] Recent in situ experiments for determination of the postspinel [Irifune et al., 1998], postilmenite [Hirose et al., 2001b] and postgarnet [Hirose et al., 2001a] phase boundaries in the MgO-SiO2 (with some Al2O3) system were carried out using the gold EOS of AIY. The present results suggest that this model tends to underestimate the pressure value. Phase boundaries modified by the present ab initio scale are shown in Figure 9. In the revised phase diagram, in the PT region of 20– 24 GPa and 1800– 2000 K, the ab initio EOS results in phase boundaries that are shifted about 0.9 GPa lower than JFM, 0.6 GPa higher than HJ, 1.3 GPa higher than AIY, 0.7 GPa higher than HHS and 0.5 GPa higher than SDT. [27] The present EOS for gold greatly reduces the discrepancy between the postspinel transition pressure measured by AIY and the pressure at the 660-km seismic discontinuity (shown by the vertical dashed line in Figure 9). However, the postspinel and the postilmenite transitions occur at still lower pressure. If the postspinel transition in fact occurs at 660 km depth, the transition temperature is too low compared to the typical mantle geotherm at this depth (1800– 2000 K). On the other hand, with respect to the transition pressure, the revised postgarnet transition appears to be a more favorable candidate for the origin of the 660-km seismic discontinuity. This is, however, unlikely, since the postgarnet transition pressure strongly depends on the Al2O3 content with a positive Clapeyron slope [Hirose et al., 2001a] whereas seismological information suggests that the discontinuity is quite sharp and is caused by a phase change with a negative Clapeyron slope. [28] Attribution of the 660-km discontinuity to the postspinel transition still results in a discrepancy that cannot be fully compensated even after applying corrections from the ab initio EOS model of gold. A gap of 0.7– 1 GPa still

Figure 9. Phase boundaries of some important mantle constituents. The dashed lines show boundaries determined using the AIY model, and the solid lines show the results obtained using the present equation of state. Dotted lines are linear extrapolations of dashed lines. The original spinel! perovskite + periclase transition, the garnet + perovskite! perovskite transition, and the ilmenite!perovskite transition are from Irifune et al. [1998], Hirose et al. [2001a], and Hirose et al. [2001b], respectively. The postgarnet transition shown here is for a composition of MgSiO3 + 5 mol% Al2O3. The dashed vertical lines at 23.5 GPa indicate the pressure corresponding to the 660-km seismic discontinuity.

remains between the transition pressure and the pressure at 660 km depth. This is comparable to another computational analysis of MgO using empirical model potentials [Matsui and Nishiyama, 2002], which also suggested that AIY underestimates pressure by about 0.6 GPa. Although this may only be a coincidence, the following considerations should be noted. Matsui and Nishiyama [2002] argued that a source of uncertainty most likely exists in the temperature measurement. In in situ experiments using multianvil apparatus, temperature is measured by a thermocouple placed in the pressure cell. However, no correction of pressure effect on the thermal electromotive force (emf ) of thermocouple metals is applied, since such quantities are not well known. If the discrepancy remaining in the present analysis can be attributed to the temperature measurement, the error in the temperature would need to be 100 –150 K.

6. Conclusion

Figure 8. Melting temperature of gold calculated as a function of pressure based on the mean field potential method.

[29] In this study, I have predicted the thermodynamic properties and the P-V-T equation of state of gold based on fully nonempirical techniques within the framework of the first-principles theory, with following results. (1) a combination of the local density functional theory and the firstprinciples lattice dynamics method allows prediction of the thermodynamics of gold quite accurately with no adjustable parameters; (2) it is confirmed that the relationship g/ga =

TSUCHIYA: FIRST-PRINCIPLES PREDICTION OF EQUATION OF STATE OF GOLD

(V/Va)z, assumed in some previous studies, is adequate for gold, at least up to V/Va = 0.7; (3) the predicted values of the EOS parameters of BTa, B0Ta , ga and z are 166.7 GPa, 6.12, 3.16 and 2.15, respectively, which agree well with experimental values; (4) the ab initio EOS model reduced the discrepancies between the observed phase boundaries of spinel, ilmenite and garnet and the seismic discontinuity. However, a gap of about 0.7 GPa still remains between the postspinel transition pressure and the 660-km discontinuity. Further investigation of possible sources of error, including that associated with the temperature measurement, are important to obtain an exact pressure standard. [ 30 ] Acknowledgments. T. T. thanks K. Hirose, M. Matsui, K. Kondo, E. Ito, E. Ohtani and E. Takahashi for their helpful comments. T. T. also acknowledges D. Alfe` and an anonymous referee for their critical reviews and C. Floss for her cooperation for linguistic editings. This study was supported by Research Fellowships of the Japan Society for the Promotion of Science (JSPS) for Young Scientists.

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T. Tsuchiya, Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan. ([email protected])