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High Pressure ResearchAn International JournalPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713679167

Equations of state of MgO, Au, Pt, NaCl-B1, andNaCl-B2: Internally consistent high-temperaturepressure scalesP. I. Dorogokupets a; A. Dewaele ba Institute of the Earth's Crust, Siberian Division, Russian Academy of Sciences,Irkutsk, Russiab DIF/Département de Physique Théorique et Appliquée, CEA, France

Online Publication Date: 01 December 2007To cite this Article: Dorogokupets, P. I. and Dewaele, A. (2007) 'Equations of state ofMgO, Au, Pt, NaCl-B1, and NaCl-B2: Internally consistent high-temperature

pressure scales', High Pressure Research, 27:4, 431 - 446To link to this article: DOI: 10.1080/08957950701659700URL: http://dx.doi.org/10.1080/08957950701659700

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High Pressure ResearchVol. 27, No. 4, December 2007, 431–446

Equations of state of MgO, Au, Pt, NaCl-B1, and NaCl-B2:Internally consistent high-temperature pressure scales

P. I. DOROGOKUPETS*† and A. DEWAELE‡

†Institute of the Earth’s Crust, Siberian Division, Russian Academy of Sciences,Irkutsk, 664033, Russia

‡DIF/Département de Physique Théorique et Appliquée, CEA, BP 12, 91680Bruyères-le-Châtel, France

(Received 29 May 2007; revised 22 August 2007; 23 August 2007)

Semiempirical equations of state (EoS) of Au, Pt, MgO, NaCl-B1, and NaCl-B2 based on expandedMie–Grüneisen–Debye approach, which are consistent both with the Mie–Grüneisen–Bose–Einsteinapproach and the thermochemical, X-ray, ultrasonic and shock-wave data in a wide pressure-temperature range, have been constructed. It is shown that to determine the volume dependence of theGrüneisen parameter, not only shock-wave and static compression data, but also experimental infor-mation on heat capacity, bulk moduli, volume, and thermal expansion coefficient at zero pressure needto be taken into account. Intrinsic anharmonicity is of great importance at construction of EoS at hightemperatures and x = V /V0 > 1. Cross-comparison of the current equations of state with independentmeasurements shows that these EoS may be used as the internally consistent and independent pressurescales in a wide range of temperatures and pressures.

Keywords: Expanded Mie-Grüneisen-Debye EoS; Pressure scales; Grüneisen parameter; Intrinsicanharmonicity; Gold; Platinum; Magnesium oxide; Sodium chloride

1. Introduction

Correct measurements of pressure in high-pressure experiments at high temperature are one ofthe most important problems of the recent physics of minerals and solids. Jamieson et al. [1]proposed to useAu, Pt, and MgO as high-temperature pressure calibrants, using the PVT tablescalculated from shock-wave data in the Mie–Grüneisen approach and under the assumptionthat γV = const. Since then, a number of equations of state (EoS) of gold [2–10], plat-inum [8, 10, 11], and MgO [8, 12, 13], as pressure standards in high-pressure experimentswere published. However, as repeatedly indicated [5, 6, 8], differences in EoS of gold reach2 GPa at temperatures ∼2000 K and pressures of 20–30 GPa. Differences increase with pres-sure (see figure 4 in ref. [6] and figure 11 in ref. [14]) and exceed 10 GPa at compressionx = V/V0 = 0.7 (pressure about 170 GPa). The EoS of Au and MgO have been observed

*Corresponding author. Email: dor@crust.irk.ru

High Pressure ResearchISSN 0895-7959 print/ISSN 1477-2299 online © 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/08957950701659700

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432 P. I. Dorogokupets and A. Dewaele

to be consistent in the pressure range of 20 GPa and temperatures up to ∼1800 K [8, 14].However Hirose et al. [15] measurements at pressure higher than 100 GPa and at temperatureup to 2330 K show that Au pressure scale [6, 8] underestimate pressure with more than 10 GPain comparison with MgO pressure scale (see figure 10 in [14]).

Recently, the EoS of Al, Au, Ag, Cu, Pt, Ta, W, MgO, and diamond were proposed inrefs. [14, 16], which are consistent with one another at room-temperature and which determi-nate the pressure at higher temperatures. It was shown there that the Bose–Einstein approachprovides a very good reproduction of thermodynamic functions (heat capacity, entropy,enthalpy) at temperatures from 10 to 15 K up to melting temperature at zero pressure. At hightemperature, the Bose–Einstein approach is equivalent to the Debye approach, which providesthe opportunity to construct more simple EoS with minimum number of fitting parameters.

This paper deals with the EoS of Au, Pt, MgO, NaCl-B1, and NaCl-B2 on the basis ofexpanded Mie–Grüneisen–Debye (EMGD) model, which is valid above room temperature.The formalism used in our previous work [14] has been simplified, in order to provide a con-venient EoS formulation, which is also accurate in the following pressure–temperature range:0–200 GPa and 300–3000 K. Au, Pt, MgO, and NaCl are often used as pressure calibrantsor pressure transmitting media in that pressure and temperature range. It will be shown thatthis simplified formalism can approximate the published [14] EoS of Au, Pt, and MgO inthat range. Then we consider EoS of these substances in respect to their compatibility withmeasured thermodynamic functions at high temperatures and zero pressures. It will be shownthat the form of volume dependence of the Grüneisen parameter and contribution of intrinsicanharmonicity are critical in construction of thermodynamically equilibrated EoS of solids.In conclusion, we will conduct a cross-comparison of various EoS of Au, Pt, and MgO withindependent measurements.

2. Thermodynamics

The Helmholtz free energy F(V, T ) can be represented as the sum:

F = U0 + E(V ) + Fqh(V, T ) + Fanh(V , T ) + Fel(V , T ), (1)

where U0 is the reference energy, E(V ) is the potential (cold) part of the free energy on thereference isotherm, which depends only on volume; Fqh(V, T ), Fanh(V , T ), and Fel(V , T )

present the quasiharmonic part of the Helmholtz free energy, and terms describing intrinsicanharmonicity and electronic contribution. We neglect the contribution of thermal defects.Contribution of intrinsic anharmonicity is taken into account in the form proposed in ref. [17],which is correct above ambient temperature [18].

The following thermodynamic functions can be obtained by differentiating equation (1):entropy, S = −(∂F/∂T )V , internal energy E = F + TS, heat capacity at constant vol-ume, CV = (∂E/∂T )V , pressure, P = −(∂F/∂V )T , isothermal bulk modulus, KT =−V (∂P/∂V )T , slope of pressure at constant volume (∂P/∂T )V = αKT , whereα = 1/V (∂V/∂T )P . The enthalpy and Gibbs energy can be calculated using H = E + PV ,G = F + PV .

Cold energy, pressure, and bulk modulus may be written [19] as:

E(V ) = 9K0V0η−2{1 − [1 − η(1 − y)] exp[(1 − y)η]}, (2a)

P(V ) = 3K0y−2(1 − y) exp[(1 − y)η], (2b)

KT (V ) = K0y−2[1 + (ηy + 1)(1 − y)] exp[(1 − y)η], (2c)

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Equations of state of MgO, Au, Pt, NaCl-B1, and NaCl-B2 433

where y = x1/3 = (V/V0)1/3 and η = 1.5(K ′ − 1), K ′ = dK/dP, V0 and K0 are molar

volume and bulk modulus at reference conditions (T0 = 298.15 K, P0 = 1 bar), or as [20]

E(V ) = 9

2K0V0f

2[1 + (K ′ − 4)f

](3a)

P(V ) = 3f K0 (1 + 2f )52

(1 + 3

2(K ′ − 4)f

), (3b)

KT (V ) = K0 (1 + 2f )52

[1 + (

7 + 3(K ′ − 4))f + 27

2(K ′ − 4)f 2

], (3c)

where f = 12

(x− 2

3 − 1).

The Debye model [17] is used to describe the quasiharmonic phonon part of the Helmholtzfree energy:

Fqh = 3nR

[3�

8+ ln

(1 − 1

e�T

)− 1

3D

(�

T

)](4)

where R is the gas constant, n is the number of atoms per a unit cell, � is the Debye temperaturedepending on volume,

D

(�

T

)= 3

(�

T

)3

×∫ �

T

0

z3dz

ez − 1(5)

is the Debye function.From equation (4) follows (see [17]):

Sqh = 3nR

[4

3D

(�

T

)− �/T

e�T − 1

], (6)

Eqh = 3nR

[3�

8+ T D

(�

T

)], (7)

CV qh = 3nR

[4

3D

(�

T

)− �/T

e�T − 1

], (8)

Pqh = Eqh × γ

V(9)

KT qh = Pqh(1 + γ − q) − T CV qh × γ

V, (10)

(∂P

∂T

)V qh

= CV qh × γ

V, (11)

α = CV qh

KT qh

× γ

V, (12)

CPqh = CV qh × (1 + αγT ), KSqh = KT qh × (1 + αγT ). (13)

The volume dependence of the Grüneisen parameter was taken from Al’tshuler et al. [21]:

γ = γ∞ + (γ0 − γ∞)xβ, (14)

where γ0 is the Grüneisen parameter at ambient conditions, γ∞ is the Grüneisen parameter atinfinite compression (x = 0) and β is a fitted parameter. From equation (14) it is possible to

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434 P. I. Dorogokupets and A. Dewaele

calculate the volume dependence of the Debye temperature and q parameter:

� = �0x−γ∞ exp

[γ0 − γ∞

β

(1 − xβ

)], (15)

q = d ln γ

d ln V= βxβ γ0 − γ∞

γ. (16)

The assumption that γ∞ = 0 leads to the usual form

γ = γ0xβ. (17)

Equations (2)–(6) are classical Mie–Grüneisen–Debye EoS (MGD EoS) [17, 22]. We exceedthe bounds of this formalism because we take into account the additional contributions ofintrinsic anharmonicity and free electrons in the Helmholtz free energy and obtain the EMGDEoS. That is why, in addition to volume dependence of the Grüneisen [equation (14)], thethermodynamic Grüneisen parameter:

� = αV KT

CV

= αV KS

CP

, (18)

which coincides with (14) in the absence of contributions of intrinsic anharmonicity and freeelectrons, should be considered.

At high temperatures, the contribution of intrinsic anharmonicity to the Helmholtz freeenergy was expressed using Zharkov and Kalinin [17] form:

Fanh = −3

2nRaxmT 2. (19)

The electronic component of the Helmholtz free energy was taken as [17]:

Fel = −3

2nRexgT 2. (20)

Pressure on the shock-wave adiabats was calculated as follows [17]:

PH = P(V ) − γ /V [E(V ) − E0]1 − (γ (1 − x)/2x)

. (21)

We could carry out a simultaneous processing of all the available measurements of the heatcapacity, thermal expansion coefficient, volume and adiabatic and isothermal bulk moduliat zero pressure, static measurements of volume at room and high temperature, shock-wavedata, and calculate any thermodynamic functions vs. T and V or vs. T and P . In practicalrealization, we write the Helmholtz free energy relative to the reference conditions T0 =298.15 K and P0 = 1 bar; consequently, the fitted parameters that we find correspond toambient conditions.

3. Calculated thermodynamic functions of MgO, Au, and Pt at high temperatureand pressure

Table 1 presents the parameters obtained by fitting published high-pressure–high-temperatureEoS for MgO, Au, and Pt with the current EMGD EoS. The following EoS have beenused: Jamieson et al. [1], Heinz and Jeanloz [2], Anderson et al. [3], Holmes et al. [11],

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Equations of state of MgO, Au, Pt, NaCl-B1, and NaCl-B2 435

Table 1. Parameters for the EoS of Au, Pt, MgO, and NaCl.

a(10−6K−1) m STDK0 (GPa) K ′ �0 (K) γ0 γ∞ β or q e(10−6K−1) g (GPa)

Au [1] 170.9† 5.31† 170 3.215 0 1 – – 0.33Au [2] 166.65‡ 5.48‡ 170 2.95 0 1.7 – – –Au [3] 166.65‡ 5.48‡ 170 2.95§ −1.65§ 1.66§ −10.3§ 1.40§ 0.008Au [5] 166.7† 6.12† 180 3.16 0 2.15 −23.9§ 0.67§ 0.27Au [6] 167.0‡ 5.0‡ 170 2.97 0 1 – – –Au [8] 167.0‡ 5.0‡ 170 2.97 0 0.7 – – –Au [10] 167.0† 6.00† 170 2.97 0 0.6 – – –Au, this 167.0† 5.90† 170 2.89 1.54 4.36 – – –study 173.0¶ 5.65¶

Pt [1] 266.6† 5.54† 200 2.40 0 1 – – 0.23Pt [11] 266.0 5.81† 230 2.55 0 1 – – 0.09Pt [8] 273.0† 4.8‡ 230 2.69 0 0.5 – – –Pt [10] 277.0† 5.08† 230 2.72 0 0.5 – – –Pt, this 277.3† 5.12† 220 2.82 1.83 8.11 −166.9 4.32 –study 283.3¶ 4.93¶ 260.0 2.4MgO [1] 154.7† 4.69† 760 1.32 0 1 – – 0.21MgO [13] 159.2‡ 4.11‡ 760 1.53 0.16 1.22 −11.7 3.50 0.20MgO [8, 12] 160.2‡ 3.99‡ 773 1.52 1.32 7.20 –MgO, this 160.3† 4.18† 760 1.50 0.75 2.96 −14.9 5.12 –study 162.5¶ 4.10¶

NaCl-B2 [10] 26.86† 5.25† 290 1.70 0 0.5 – – –24.90

NaCl-B1|| 23.83† 5.09† 270 1.64 1.12 4.36 −24.0 7.02 –this study 25.18¶ 4.80¶

27.015

NaCl-B2||, 29.72† 5.14† 270 1.64 1.23 6.83 −24.0 7.02 –this study 31.21¶ 4.84¶

24.53

†Fitted parameters for Vinet et al. [19] EoS.‡Third order Birch-Murnaghan [20] EoS.§Our fitting of PVT data from ref. [3, 5].¶Calculated KS and (∂KS/∂P )S .Molar volume in cm3.||EoS of NaCl-B1 and NaCl-B2 will be considered elsewhere.

Speziale et al. [12], Matsui et al. [13], Shim et al. [6], Tsuchiya [5], Fei et al. [8, 10]. �0, γ0, q

have been fixed and the calculated pressure residuals (STD) due to the approximation by ourEMGD EoS are presented in table 1. Note that q parameter coincides with β parameter ifγ∞ = 0. The MgO, Au, and Pt isotherms reduced from shock-wave data in ref. [1] are betterapproximated using Vinet et al. [19] equation. In refs. [1, 6, 8], the intrinsic anharmonicity isimplicitly taken into account by changing the Dulong–Petit limit of heat capacity at constantvolume. This may be used as a first approximation, but it is not physically correct.

The data sets used to build the current EMGD-EoS are listed below. They include: (i) bulkmoduli measured by ultrasonic or Brillouin scattering techniques, mainly at ambient pressureand high temperature; (ii) heat capacity and thermal expansion coefficient measured at ambientpressure and high temperature; (iii) Hugoniot data measured during shock-wave compressionat high pressure and temperature; (iv) EoS measured by X-ray diffraction in static devicesunder high pressure and ambient or high temperature.

The data used for MgO and Pt are the same as in our previous work [14]. We stress that thestatic high pressure data (iv) have been modified, if necessary, to take into account the recentupdate of the high pressure metrology (see ref. [14] and references therein). Data sets (i) and(ii) are presented in figures 1 and 3, respectively, for MgO and Pt.

In the case of gold, the ambient temperature compression curve has been measured recentlyby Takemura [23], Dewaele et al. [24], and Chijioke et al. [25]. In those experiments, the gold

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436 P. I. Dorogokupets and A. Dewaele

Figure 1. Comparison of experimental data with thermodynamic functions for MgO calculated using our EoS (boldsolid lines) and Speziale et al.’s [12] EoS for MgO. (a) CP and CV , experimental data from refs. [34,35]. (b) Thermalexpansion coefficient in comparison with refs. [36–38]. (c) Variation with temperature of the Grüneisen parametersin comparison with experimental data [39] and calculated from a Kieffer model [40]. (d) Bulk moduli calculatedfrom our EoS and measured in refs. [12, 39, 41–43]. (e) Difference between the current adiabatic bulk modulusand experimental measurements from refs. [44–46]. (f) Variation with compression of the Grüneisen parameterscalculated from different EoSs.

sample was compressed in a soft pressure transmitting medium with a diamond anvil cell andits volume was measured by X-ray diffraction. The data points obtained by Takemura [23]and Dewaele et al. [24] are in disagreement, with a difference in measured pressure for thesame volume reaching 4.5 GPa at 75 GPa. Careful analysis of experimental results showedthat these differences can be explained in part by differences in the stress distribution in thesamples used in these studies (see refs. [26, 27] in the same issue of this journal). Following themethod proposed in ref. [26], the effect of this stress has been compensated and the correcteddata sets are now compatible within 2.5 GPa. A new compression curve has been measuredrecently [28], which is in agreement with the average of the data corrected from refs. [23, 24].These corrected compression curves have been used to build the current EoS, using the updatedpressure calibration of the ruby gauge [14], together with the data published in ref. [25].

NaCl-B1 EoS is based on the data from ref. [29]. Our version of NaCl-B2 EoS is based onrecalibrated data from refs. [10, 30–32] and shock-wave data from ref. [33].

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Equations of state of MgO, Au, Pt, NaCl-B1, and NaCl-B2 437

Figure 2. Comparison of experimental data with thermodynamic functions for gold calculated using our EoS (boldsolid lines) and Fei et al. [8, 10] EoSs for Au. (a) CP and CV , experimental data from refs. [47]. (b) Thermal expansioncoefficient together with the same parameter compiled in refs. [48, 49]. (c) Calculated bulk moduli together with sameparameters from refs. [50–53]. (d) Variation with compression of the Grüneisen parameters calculated from differentEoSs [4, 5, 8, 10].

Figures 1–3 show that the current EMGD EoS of Au, Pt, and MgO describe very well thethermodynamic functions at zero pressure within a wide temperature range. High-pressurebehaviors on the room-temperature isotherm have been discussed in ref. [14].

Quite different results were obtained in the literature, using the MGD approach [1, 2, 6, 8,10, 11]. They are illustrated in figures 1–3 by the example of EoS of MgO from refs. [8, 12],Au and Pt from refs. [8, 10].

High-temperature behavior of calculated thermodynamic functions using MGD EoS ofMgO [12] strongly differs from analogous functions for Au and Pt [8, 10, 11]. Reasons forsuch difference are that Speziale et al. [12] carefully analyzed the volume dependence ofthe Grüneisen parameter, and established that q0 = 1.65 from thermodynamic relations atambient conditions and q approaches to zero at high compression. Therefore, they assumedthe volume dependence of the Grüneisen parameter as γ = γ0 exp(q − q0/q1), q = q0x

q1 ,where q0 and q1 are fitted parameters. However, Speziale et al. [12] used only static andshock-wave measurement for construction EoS for periclase, therefore the calculated CP andα do not conform with experimental measurements at T > 1000 K. Speziale et al.’s [12] EoSfor periclase may be easily improved if an intrinsic anharmonicity term is included, withparameters a0 ≈ −13 × 10−6 K−1 and m ≈ 5.5.

Figures 2 and 3 show that MGD EoS of Au and Pt from refs. [8, 10] gives smaller slopedKS /dT than the slope measured experimentally. This is due to the fact that Fei et al. [8, 10]used incorrect volume dependence of the Grüneisen parameter, and a too low value of q

(q = 0.7 for gold and q = 0.5 for platinum). Anderson et al.’s [3] and Tsuchiya’s [5] EoSs forgold gives a good description of thermodynamic functions at zero pressure, but, as stressedby Tsuchiya, Anderson et al.’s [3] EoS does not agree with shock-wave data.

From figure 3, we can see that platinum EoS cannot be constructed on the basis of simpleMGD model [1, 10, 11, 8]. Derived from shock-wave data, the PVT relations for platinum

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438 P. I. Dorogokupets and A. Dewaele

Figure 3. Comparison of experimental data with thermodynamic functions of platinum calculated using our EoS(bold solid lines) and Fei et al. [8, 10] EoSs for Pt. (a) CP and CV , experimental data from refs. [54, 55], (b) Thermalexpansion coefficient together with the same parameter compiled in ref. [49]. (c) Calculated bulk moduli in comparisonwith ref. [56]. (d) Variation with compression of the Grüneisen parameters calculated from different EoSs.

were approximated by Holmes et al. [11] using the simple following equation:

P(x, T ) = P300(x) + α0K0(T − 300),

where P300(x) calculated from equation (2b) with parameters K0 = 266 GPa, K ′ = 5.81,and α0 = 26.1 × 10−6 K−1. The thermodynamic functions of Pt calculated using Holmeset al.’s [11] EoS disagree with the available experimental measurements of CP and α. OurEMGD EoS for platinum was constructed by taking into account the intrinsic anharmonicityand electronic contribution; it provides a very good description of CP , α, and KS at hightemperatures.

Tables 2–4 give the isochors for Au, Pt, and MgO calculated from our EMGD EoS withparameters from table 1. These isochors are very close to the ones obtained with a more

Table 2. Isochors for MgO (this study).

x = V/V0 298.15 500 1000 1500 2000 2500 3000

1 0.000 1.093 4.116 7.177 10.184 13.120 15.9770.95 9.147 10.190 13.142 16.162 19.149 22.082 24.9540.9 20.990 21.988 24.879 27.868 30.843 33.780 36.6710.85 36.343 37.300 40.142 43.113 46.087 49.037 51.9530.8 56.308 57.227 60.033 63.001 65.988 68.963 71.9140.75 82.396 83.281 86.067 89.048 92.065 95.080 98.0820.7 116.716 117.570 120.351 123.367 126.434 129.510 132.5810.65 162.258 163.085 165.880 168.953 172.095 175.257 178.4220.6 223.356 224.158 226.986 230.147 233.395 236.674 239.963

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Equations of state of MgO, Au, Pt, NaCl-B1, and NaCl-B2 439

Table 3. Isochors for Au (this study).

x = V/V0 298.15 500 1000 1500 2000 2500

1 0.000 1.408 4.921 8.441 11.963 15.4850.95 9.955 11.295 14.645 18.004 21.365 24.7270.9 23.903 25.191 28.419 31.658 34.900 38.1430.85 43.380 44.630 47.776 50.936 54.100 57.2650.8 70.578 71.807 74.911 78.032 81.158 84.2860.75 108.679 109.900 113.002 116.128 119.258 122.3920.7 162.373 163.600 166.742 169.914 173.094 176.2770.65 238.707 239.954 243.179 246.444 249.719 252.998

Table 4. Isochors for Pt (this study).

x = V/V0 298.15 500 1000 1500 2000 2500 3000

1 0.000 1.513 5.259 8.951 12.580 16.142 19.6390.95 16.205 17.615 21.148 24.678 28.192 31.689 35.1680.9 38.120 39.474 42.909 46.383 49.882 53.404 56.9470.85 67.721 69.057 72.483 75.985 79.545 83.160 86.8290.8 107.766 109.110 112.598 116.195 119.876 123.638 127.4780.75 162.150 163.525 167.134 170.879 174.730 178.680 182.7290.7 236.477 237.897 241.675 245.616 249.679 253.855 258.143

advanced formalism [14]. This suggests that the expanded MGD EoS can be used for simul-taneous analysis of thermochemical, ultrasonic, X-ray, and shock-wave data above 300 K.Tables 5–7 give, for comparison, the isochors for Au, Pt, and MgO calculated from EoS by ususing ref. [8]; and tables 8 and 9 give the isochors for Au and Pt calculated using EoS fromref. [10].

Table 5. Isochors for MgO from bibliography [8, 12].

x = V/V0 298.15 500 1000 1500 2000 2500 3000

1 0.000 1.154 4.413 7.802 11.225 14.662 18.1060.95 9.103 10.234 13.492 16.900 20.349 23.814 27.2870.9 20.831 21.949 25.243 28.716 32.236 35.775 39.3240.85 36.022 37.133 40.497 44.073 47.707 51.364 55.0330.8 55.840 56.947 60.407 64.124 67.911 71.727 75.5580.75 81.927 83.029 86.607 90.500 94.480 98.497 102.5310.7 116.650 117.742 121.454 125.557 129.770 134.030 138.3120.65 163.494 164.565 168.420 172.767 177.257 181.805 186.3840.6 227.722 228.759 232.758 237.383 242.195 247.085 252.015

Table 6. Isochors for Au from bibliography [8].

x = V/V0 298.15 500 1000 1500 2000 2500

1 0.000 1.431 5.000 8.575 12.153 15.7310.95 9.739 11.188 14.808 18.438 22.070 25.7030.9 22.913 24.378 28.052 31.740 35.430 39.1230.85 40.809 42.290 46.020 49.769 53.523 57.2780.8 65.291 66.786 70.575 74.389 78.210 82.0330.75 99.104 100.609 104.458 108.342 112.235 116.1310.7 146.378 147.889 151.797 155.755 159.726 163.7020.65 213.490 214.998 218.965 223.002 227.057 231.119

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Table 7. Isochors for Pt from bibliography [8].

x = V/V0 298.15 500 1000 1500 2000 2500 3000

1 0.000 1.463 5.134 8.818 12.505 16.193 19.8820.95 15.840 17.334 21.093 24.871 28.653 32.436 36.2210.9 37.068 38.591 42.445 46.323 50.207 54.093 57.9810.85 65.657 67.208 71.161 75.148 79.142 83.140 87.1390.8 104.441 106.018 110.076 114.179 118.294 122.413 126.5340.75 157.574 159.171 163.338 167.567 171.812 176.064 180.3190.7 231.269 232.879 237.156 241.522 245.910 250.307 254.709

Table 8. Isochors for Au from bibliography [10].

x = V/V0 298.15 500 1000 1500 2000 2500

1 0.000 1.450 5.065 8.688 12.312 15.9370.95 9.980 11.455 15.142 18.838 22.537 26.2370.9 24.027 25.527 29.289 33.065 36.843 40.6240.85 43.724 45.249 49.090 52.950 56.815 60.6820.8 71.341 72.889 76.814 80.765 84.723 88.6840.75 110.180 111.748 115.760 119.809 123.868 127.9310.7 165.126 166.709 170.810 174.965 179.133 183.3070.65 243.545 245.133 249.323 253.592 257.880 262.176

Table 9. Isochors for Pt from bibliography [10].

x = V/V0 298.15 500 1000 1500 2000 2500 3000

1 0.000 1.480 5.191 8.916 12.644 16.374 20.1040.95 16.171 17.681 21.482 25.302 29.126 32.952 36.7780.9 38.000 39.539 43.436 47.357 51.284 55.214 59.1450.85 67.435 69.002 72.999 77.030 81.069 85.111 89.1550.8 107.187 108.780 112.883 117.031 121.191 125.356 129.5240.75 161.088 162.702 166.913 171.189 175.482 179.780 184.0830.7 234.637 236.262 240.585 244.998 249.435 253.881 258.331

Tables 10 and 11 give the isochors for NaCl-B1 and NaCl-B2 calculated from our EoSswith parameters from table 1. For comparison, in table 12, the PVT relations for NaCl-B2calculated from EoS [10] are given.

In the next section, we make a cross-comparison of our EoS ofAu, Mg, MgO, NaCl-B2 withthose of [8, 10] recommended as internally consistent pressure scales at high temperatures andpressures.

Table 10. Isochors for NaCl-B1 (this study).

x = V/V0 298.15 500 1000 1500 2000

1 0.000 0.573 1.962 3.281 4.5240.95 1.392 1.960 .356 4.705 5.9980.9 3.271 3.839 5.250 6.630 7.9720.85 5.807 6.378 7.813 9.231 10.6230.8 9.233 9.812 11.282 12.746 14.1930.75 13.881 14.474 15.991 17.511 19.0220.7 20.226 20.838 22.416 24.007 25.5930.65 28.965 29.600 31.257 32.935 34.613

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Table 11. Isochors for NaCl-B2 (this study).

V ,cm3 298.15 500 1000 1500 2000 2500

17 26.763 27.462 29.270 31.093 32.913 34.72616 35.891 36.620 38.523 40.449 42.376 44.30015 47.862 48.625 50.638 52.683 54.733 56.78214 63.704 64.506 66.645 68.827 71.019 73.21213 84.903 85.747 88.030 90.372 92.727 95.08712 113.644 114.533 116.982 119.509 122.056 124.60911 153.225 154.161 156.801 159.546 162.317 165.099

Table 12. Isochors for NaCl-B2 from bibliography [10].

V ,cm3 298.15 500 1000 1500 2000 2500

17 26.588 27.348 29.353 31.395 33.447 35.50216 35.542 36.313 38.370 40.471 42.584 44.70215 47.302 48.084 50.195 52.361 54.541 56.72614 62.893 63.683 65.853 68.089 70.342 72.60313 83.794 84.591 86.822 89.135 91.470 93.81412 112.190 112.990 115.286 117.684 120.110 122.54611 151.383 152.184 154.547 157.040 159.567 162.108

4. Direct comparison of Fei et al. [8, 10] and our EoSs of Au, Pt, and MgO

Until now, several works have been published [8, 15, 30, 57–61] in which measurements ofpressure were carried out using two or more pressure gauges. This allows us to comparethe pressures given by the different pressure gauges, with different calibrations, which areexpected to be equal. We used only the works where tables with measured atomic volumeswere given. Using these data, the pressures from our EoS of Au, Pt, MgO, and NaCl-B2(table 1) and EoS from [8, 10] have been calculated and are plotted in figures 4–7. Figure 4compares ours and Fei et al.’s [8, 10] EoSs for MgO and Au. Figure 5 shows the results ofcomparison of MgO and Pt EoSs. Figure 6 compares the Pt and Au EoSs and figure 7 showsthe results of comparison of Pt and Au on room isotherm at quasihydrostatic [24] and non-hydrostatic measurements [61]. Figure 8 shows the calculated isothermal compression curvesof the NaCl-B2 phase at 300, 1000, and 3000 K. Results of this comparison are obvious anddo not demand special comments.

5. Discussion

We compared the thermodynamic functions of MgO, Au, and Pt calculated by various formsof EoS. The most controversial ambient temperature EoS is the EoS of Au for which K ′varies between 5 [6, 8] and 6.12 [5]. The value of K ′ = 5.9 (our fit) averages the recentdeterminations of compression curves obtained with diamond anvil cell, in helium [23, 24, 28]or hydrogen [25] pressure transmitting media (see figure 8 in [14]). It should be noted that in aprevious work, Heinz and Jeanloz [2] used methanol–ethanol pressure medium. Therefore, thequasihydrostatic ruby pressure calibration from ref. [62] would have been more suitable forthese measurements than the non-hydrostatic scale [63], which would increase K ′ up to ∼5.8.This value is close to the one derived here. The very low value of parameter q recommendedfor EoS of Au in refs. [6, 8, 10] is surprising. Anderson et al. [3] unambiguously showed thatthe volume dependence of the Grüneisen parameter can be evaluated from thermodynamic

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Figure 4. Comparison between MgO and Au pressure scales. (a) Ref. [57]. (b) Ref. [8]. (c) Ref. [58]. (d) Ref. [59].(e) Ref. [15]. (f) Ref. [60].

Figure 5. Comparison between MgO and Pt pressure scales. (a) Ref. [8]. (b) Ref. [30].

data and calculated the temperature dependence of q at ambient pressure. It follows from theiranalysis that q parameter cannot be less than 1.5 under ambient conditions. Anderson et al. [3]also took into consideration the intrinsic anharmonicity, therefore the EoS of Au from ref. [3]is in best agreement with our analysis, even with the relatively low value of K ′ used (5.5).

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Figure 6. Comparison between Pt and Au pressure scales. (a) Ref. [8]. (b) Ref. [58].

Figure 7. Comparison between Pt and Au pressure scales on room isotherm. (a) Ref. [24]. (b) Ref. [61].

Figure 8. Calculated isotherms for the NaCl-B2 phase at 300, 1000, and 3000 K. Experimental data for roomisotherm are from refs. [10, 30, 32], based on the Au, Pt, and MgO EoSs of this study.

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The EoS of Au from ref. [4], in which a complex dependence of γ as a function of volume isused and intrinsic anharmonicity is taken into account, agrees well with our EoS for Au.

Our MgO EMGD-EoS conforms well with the MgO EoS from refs. [8, 12] at ambientcondition, but not at high temperature, which is due to the neglect of intrinsic anharmonicityin refs. [8, 12]. Platinum exhibits the largest deviation from a Debye-like behaviour, which isquite evident in figure 3.

There are three main reasons for the differences between EoS of MgO, Au, and Pt from thisstudy and from refs. [8, 12].

First, as it is shown in ref. [14], the room-temperature isotherms of Ag, Al, Au, Cu, Pt,Ta, W, diamond, and periclase on the basis of measurements [23, 24, 45, 64] are consistentwith a new ruby pressure scale [14]. The values of K ′ obtained with this scale agree withavailable ultrasonic measurements. The room-temperature isotherm of ε-Fe [65], which wasdetermined using this ruby pressure scale and a tungsten pressure calibrant from [14, 16], maybe added to this list. Therefore, consensus on the new ruby pressure scale is emerging at leastup to 200 GPa [24, 25, 64, 66–68]. However, there is an opposite point of view [6, 8, 69, 70],maintaining the Mao et al. [62] calibration of ruby scale. The basic argument that confirms theMao et al. [62] ruby scale is the self-consistent measurement of the EoS of MgO by combinationof Brillouin and X-ray diffraction measurements up to 55 GPa [12, 45]. This EoS has beenused to calculate a ruby scale, which was found to be close to the calibration of Mao et al. [62]up to 55 GPa. However, our EoS of MgO also agrees with the Brillouin measurement [45] (seefigure 1(e)), a small systematic difference in pressure on a room-temperature isotherm (seefigure 8 in [14]) lying within the scatter of measurements scattering [24] (see figures 3 and 4in ref. [14]).

The second reason is the underestimation of the role of functional dependence of theGrüneisen parameter on the volume. The Mie–Grüneisen relation is determined by equation (9)which connects pressure and internal energy, the Grüneisen parameter being the coefficientof proportionality. The Grüneisen parameter and its logarithmic derivative q are also includedin equations (10)–(13) which describe the temperature dependence of heat capacity and bulkmoduli; when determining the volume dependence of Grüneisen parameter, not only shock-wave data, but also the complete set of experimental information on heat capacity, bulk moduli,volume, and thermal expansion coefficient at zero pressure should be taken into considera-tion. It is also evident that simple dependences of the Grüneisen parameter on volume, suchas γ = γ0x and γ = γ0x

q , can be used only as the first approach. The analytical form of γ

proposed by Al’tshuler et al. [21] (see equation (16)) is the most appropriate, as it is shown inthis article.

The third component of a thermodynamically correct EoS is the consideration of additionalcontributions such as intrinsic anharmonicity and free electrons to the Helmholtz free energy.As it is shown in Table 1, these contributions are nonzero, having, however, opposite signs forPt. The effect of these contributions on thermodynamic Grüneisen parameter is negligible forAu; but it is very significant for periclase and platinum (see figures 1(c), 1(f), 3(d)). As theanharmonic analogue of the Grüneisen parameter, m, is relatively high for analyzed solids,these contributions provide a significant effect for calculated thermal pressure, heat capacity,thermal expansion coefficient, and bulk moduli at high temperature and at low pressure. Theeffect of these contributions decreases sharply when pressure increases. At a compression of0.6, the anharmonicity contribution to pressure and heat capacity is divided by 10 and more,dependending on m value.

These conclusions have been made before the publication of Fei et al.’s [10] work, whereinternally consistent pressure scales have been proposed. Fei et al. [10] have corrected theprevious versions of EoS of Au and Pt [6, 8, 70]; they have confirmed recent quasihydrostaticruby pressure scales [14, 16, 24, 66–68]. In turn, the consistency in MgO, Au, and Pt pressure

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scales has been cardinally improved (see figures 4–7). However, the questions concerning thehigh temperature formulation of the EoS of Au and Pt from ref. [10] remained.

Fei et al. [10] did not consider additional contributions in the Helmholtz free energy andused the simple form of γ = γ0x

q . Therefore, calculated CP , α, KS , KT at ambient pressureand high temperature do not agree better with experimental measurements than former EoSversions [6, 8] (see figures 2 and 3). This also leads to non-negligible differences in calculatedthermal pressure (see Tables 3 and 8 for Au and Tables 4 and 9 for Pt). As a result, the pressurescalculated using Fei et al.’s [10] EoSs are systematically higher than pressures given by ourEMGD-EoS at high temperature. The differences reached 1–2 GPa at 20–25 GPa and increasesup to 4 GPa at 110–116 GPa (see figure 4(e)).

6. Concluding remarks

Semiempirical EoS of Au, Pt, MgO, NaCl-B1, and NaCl-B2 based on the expanded MGDapproach consistent with both the Mie–Grüneisen–Bose–Einstein formalism [14] and thethermochemical, X-ray, ultrasonic, and shock-wave data in a wide pressure–temperature range,have been constructed. It is shown that in order to determine a volume dependence of theGrüneisen parameter it is necessary to take into account not only shock-wave and staticpressure data, but the complete sets of available experimental information on heat capacity,bulk moduli, and thermal expansion coefficient at zero pressure. The volume variations of theGrüneisen coefficient γ cannot be approximated by the simple form γ = γ0x

q . The intrinsicanharmonicity is of great importance when constructing EoS at high temperatures and x > 1.This contribution dramatically decreases when pressure increases. A cross-comparison ofconstructed EoS with independent data showed that the current EMGD-EoS of Au, Pt, MgO,NaCl-B1, and NaCl-B2 can be used as independent internally consistent pressure scales in awide range of temperatures and pressures.

Acknowledgements

The authors thank Artem Oganov (ETH, Zurich, Switzerland) as the first reader of thismanuscript and Kenichi Takemura (NIMS, Tsukuba, Japan) for sharing raw x-ray diffrac-tion data. P.D. thanks A.D., who took the liberty of presentation of part of this work forCOMPRES workshop in Geophysical Laboratory in January 2007. This work was supportedby the Russian Foundation for Basic Research, grants No. 05-05-64491 and No. 08-08-00147.

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