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 Grüneisen 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 Grüneisen
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
Grüneisen parameter of g/ga = V/Va, where g is the
Grüneisen 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 Grüneisen 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
TSUCHIYA: FIRST-PRINCIPLES PREDICTION OF EQUATION OF STATE OF GOLD
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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
0.1 a.u. (1 a.u. = 0.529177 Å) in addition to the zeropressure 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
TSUCHIYA: FIRST-PRINCIPLES PREDICTION OF EQUATION OF STATE OF GOLD
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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. Grüneisen Parameter
[22] Next, we investigate the validity of the empirical
relationship for thermal behavior assumed in the previous
studies is investigated. The thermodynamic Grüneisen
parameter g is defined as
g¼
aBS V aBT V
¼
:
CP
CV
ð6Þ
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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 Grüneisen
parameter is constant with respect to temperature under
isochoric conditions. The volume dependence of the
Grüneisen 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 Grüneisen
parameter g. In Figure 7b the solid curve is the present result, and dashed curves are previous
experimental results.
ECV
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TSUCHIYA: FIRST-PRINCIPLES PREDICTION OF EQUATION OF STATE OF GOLD
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. Alfè 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])