Unified analyses for P-V-T equation of state of MgO: A solution for pressure-scale problems in high P-T experiments

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2008jb005813, 2009 Unified analyses for P-V-T equation of state of MgO: A solution for pressure-scale problems in high P-T experiments Yoshinori Tange, 1 Yu Nishihara, 2,3 and Taku Tsuchiya 1 Received 19 May 2008; revised 24 October 2008; accepted 30 December 2008; published 20 March [1] In order to determine an accurate and reliable high-pressure and high-temperature equation of state (EOS) of MgO, unified analyses were carried out for various pressure-scale-free experimental data sets measured at 1 atm to 196 GPa and K, which are zero-pressure thermal expansion data, zero-pressure and high-temperature adiabatic bulk modulus (K S ) data, room temperature and high-pressure K S data, and shock compression data. After testing several EOS models based on the Mie-Grüneisen-Debye description for the thermal pressures with the Vinet and the third-order Birch-Murnaghan equations for the 300-K isothermal compression, we determined the K 0 T0 and g(v) using a new functional form g = g 0 {1 + a[(v/ ) b 1]} to express the volume dependence of the Grüneisen parameter. Through least squares analyses with prerequisite zero-pressure and room temperature properties of, K S0, a 0, and C P0, we simultaneously optimized a set of parameters of K 0 T0, g 0, a, and b required to represent the P-V-T EOS. Determined new EOS models of MgO successfully reproduced all the analyzed P-V-T-K S data up to 196 GPa and 3700 K within the uncertainties, and the total residuals between calculated and observed pressures were found to be 0.8 GPa in root mean squares. These EOS models, even though very simple, are able to reproduce available data quite accurately in the wide pressure-temperature range and completely independent from other pressure scales. We propose these models for primary pressure calibration standards applicable to quantitative high-pressure and high-temperature experiments. Citation: Tange, Y., Y. Nishihara, and T. Tsuchiya (2009), Unified analyses for P-V-T equation of state of MgO: A solution for pressure-scale problems in high P-T experiments, J. Geophys. Res., 114,, doi: /2008jb Introduction [2] Periclase (MgO) is one of the most common materials in the Earth sciences, physics, and chemistry. It has the simple NaCl structure with no phase transition at least up to 200 GPa [e.g., Svendsen and Ahrens, 1987; Duffy et al., 1995; Alfè et al., 2005] and also has very high melting temperatures above 3000 K [e.g., Zerr and Boehler, 1994; Dubrovinsky and Saxena, 1997]. These wide stability ranges of MgO cover high-pressure and high-temperature conditions corresponding to the Earth s mantle. For these reasons, various models of P-V-T equation of state (EOS) of MgO have been proposed as an internal pressure standard for high-pressure experiments [e.g., Jamieson et al., 1982; Matsui et al., 2000]. Among the previously proposed EOS models for MgO, an EOS model of Speziale et al. [2001] is popularly used as the pressure scale in high P-T experiments 1 Geodynamics Research Center, Ehime University, Matsuyama, Ehime, Japan. 2 Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan. 3 Now at Geodynamics Research Center, Ehime University, Matsuyama, Ehime, Japan. Copyright 2009 by the American Geophysical Union /09/2008JB005813$09.00 [e.g., Fei et al., 2004a; Fei et al., 2004b; Hirose et al., 2006; Hirose et al., 2008; Kuwayama et al., 2008]. [3] Speziale et al. [2001] proposed the EOS model based on the experimentally determined static compression data up to 52 GPa at 300 K performed with a diamond anvil cell (DAC). Importantly, they concluded that the q value to represent the volume dependence of the Grüneisen parameter (gamma) should have a strong volume dependence in the Mie-Grüneisen-Debye model with the third-order Birch- Murnaghan equation for the 300-K isothermal compressions. They introduced an additional parameter q 1 to further express the volume dependence of the q value as q = q 0 (V/ ) q1, where the q 0 value was determined with the thermodynamic and thermoelastic properties at 1 atm and 300 K. The q 1 value was estimated from the limited data of the 300-K isothermal compression and shock compression; therefore, Speziale et al. s P-V-T EOS is not able to reproduce the thermal expansion correctly at 1 atm and temperatures above 1200 K. In addition, there is another problem in the EOS model to be used as the primary pressure scale. [4] The 300-K isothermal compression was determined using the other pressure scale proposed for ruby [e.g., Mao et al., 1986] in the EOS model of Speziale et al. [2001]. The ruby pressure scale is frequently used in high-pressure experiments using DAC and the ruby scale is a secondary scale based on a reduced shock EOS with an assumption of 1of16

2 g/v = const. Additionally, Dewaele et al. [2004] examined the validity of the ruby scale comparing EOS of six metals reduced from the shock compression data with ab initio calculations or the assumption of g/v = const. The new ruby scale proposed by Dewaele et al. [2004] yields approximately 10 GPa higher pressures at 150 GPa than Mao et al. s [1986] ruby scale even at room temperature. These results suggest the uncertainty between the ruby scales and then Speziale et al. s MgO scale based on the Mao et al. s ruby scale should also have some uncertainties. [5] Previously proposed several pressure scales were made by using experimental data measured depending on other pressure scales [e.g., Shimetal., 2002; Fei et al., 2004a; Hirose et al., 2008]. By means of these approaches, in principle, it is impossible to evaluate absolute reliabilities of the EOS models and also absolute values of pressure. In order to analyze a primary pressure standard, it is most important that we must use the original data sets which have been measured without any other pressure standards. Here we report unified analyses for various pressure-scale-free data sets to model a primary P-V-T EOS of MgO applicable to a pressure standard in high P-T experiments. 2. Details of Analytical Procedures 2.1. Pressure-Scale-Free Data Sets [6] Previously reported four kinds of pressure-scale-free data set were selected and simultaneously analyzed to determine the primary P-V-T EOS of MgO. The first was a thermal expansion data set measured at 1 atm (P 0 -V-T), which is, in general, determined more accurately than other physical properties obtained at high pressures. MgO has a high melting temperature greater than 3000 K at 1 atm, and this means that even at zero pressure, thermal volume expansions can cover wide V-T conditions. We employed the thermal expansion data reported by Dubrovinsky and Saxena [1997] and Fiquet et al. [1999]. They determined the volumes with X-ray diffraction (XRD) measurements and the temperatures with emission thermometers or thermocouples, and their results are consistent with each other up to the melting temperature. [7] Second, we employed a high-temperature adiabatic bulk modulus (K S ) data measured at 1 atm (P 0 -V-T-K S ). K S strongly relates to the Grüneisen parameter (g) because K S is expressed as K S =(1+gaT)K T, in contrast to K T derived from the P-V-T states as ak T =(@P/@T) V, where a is the thermal expansivity and K T is the isothermal bulk modulus. Thus, the temperature and volume dependence of K S is able to tightly constrain g(v) [e.g., Jackson, 1998]. Details of the analyzed EOS models such as g(v) are described in section 2.2. The 1-atm and high-t K S data reported by Isaak et al. [1989] and Sinogeikin et al. [2000] were used for the analyses. Isaak et al. [1989] measured the temperature dependence of the resonant frequencies up to 1800 K and Sinogeikin et al. [2000] measured the temperature dependence of acoustic velocities using the Brillouin scattering method up to 1510 K, and their results are consistent in the temperature dependence of K S. Although volume data are required to determine elastic moduli from the acoustic data, volumes were not measured simultaneously with the acoustic data in their studies. Isaak et al. [1989] employed the thermal expansivity of Suzuki [1975], and Sinogeikin et al. [2000] used that of Dubrovinsky and Saxena [1997] to determine the elastic moduli, respectively. Suzuki [1975] measured the thermal expansion up to 1273 K, and the results were consistent with those of Dubrovinsky and Saxena [1997] and Fiquet et al. [1999]. Hence, the 1-atm and high-t K S data and the 1-atm thermal expansion data employed in the present analyses should be internally consistent. [8] The 300-K and high-p K S data were used as the third data sets (V-T 0 -K S ). The 300-K and high-p K S data reported by Zha et al. [2000] and Li et al. [2006] were employed. They measured the acoustic velocities using the Brillouin scattering method and the ultrasonic method by combining high-pressure generating techniques using DAC and multianvil apparatus, respectively. Zha et al. [2000] directly determined P ruby -K S data and converted it into the V-K S relationships using P ruby -V relationships determined in other experiments [Speziale et al., 2001]. In contrast, Li et al. [2006] simultaneously measured the acoustic velocities and volumes using in situ XRD measurements with a synchrotron radiation. The V-T 0 -K S data were determined by Zha et al. [2000] up to 55 GPa and up to 11 GPa by Li et al. [2006], who both reported absolute pressures integrating the V-K S relationships. Actually, their pressure values were determined depending on some assumptions in the volume dependence of g. Zha et al. [2000] integrated the pressures from the V-K S relationships with the assumption of g/v = const, and also Li et al. [2006] fixed the g 0 and q values in the formulation of g = g 0 (V/ ) q to integrate pressures from the V-K S data. Therefore, we directly analyzed the primary V-T 0 -K S data reported by Zha et al. [2000] and Li et al. [2006] instead of the integrated pressure values. [9] Finally, we employed shock compression data (see Marsh [1980] for Los Alamos Scientific Laboratory (LASL) shock Hugoniot data and Duffy and Ahrens [1993]). In the shock compression experiments, P-V relations are determined independently from other pressure scales and are simply based on the fundamental conservation laws of momentum and mass. LASL shock Hugoniot data reported the P-V relationships up to 122 GPa, and Duffy and Ahrens [1993] reported data to further higher pressures between 169 and 196 GPa. Although Duffy and Ahrens [1993] measured the shock temperatures in addition to the P-V data, we analyzed the primary P-V-DE H relationships for both the data sets. These four kinds of pressure-scale-free data set are summarized in Table 1, and the P-V-T data points are shown in Figure 1. [10] In addition to these P-V-T-K S data sets, primary thermoelastic and thermodynamic parameters were selected to determine a consistent EOS parameter set thorough the analyses. The lattice volume ( ), adiabatic bulk modulus (K S0 ), thermal expansivity (a 0 ), and the isobaric specific heat (C P0 ) at 1 atm and 300 K were fixed as the primary parameters. The fixed parameters are summarized in Table Models of the P-V-T Equation of State [11] The selected P-V-T-K S data were analyzed in Mie- Grüneisen-Debye (MGD) models [e.g., Jackson and Rigden, 1996]. Here we summarize the MGD models in order to make the backgrounds of the analyses distinct. When the phonon contribution is dominant in the thermal effect, P-V-T 2of16

3 Table 1. Pressure-Scale-Free Data Sets Used in the Analyses and the Measured Conditions Reference Pressure (GPa) V/ Temperature (K) Number of Data The 1-atm Thermal Expansion Data Dubrovinsky and Saxena [1997] Fiquet et al. [1999] The 1-atm and High-T K S Data Isaak et al. [1989] Sinogeikin et al. [2000] The 300-K and High-P K S Data Zha et al. [2000] (0 55) a Li et al. [2006] (2 11) a Shock Compression Data LASL shock Hugoniot data [Marsh, 1980] Duffy and Ahrens [1993] ( ) a 4 a Values in parentheses were not used in the analyses. EOS of solids are generally represented by the Mie- Grüneisen formulation, PV; ð TÞ ¼ P T0 ðvþþdp th ðv ; TÞ: ð1þ Here, P T0 represents pressures at a reference temperature T 0. The Vinet equation [Vinet et al. 1987], P T0 " V 2=3 ðvþ ¼ 3K T0 1 V # 1=3 ( " exp 3 #) 2 K0 T0 1 V 1=3 1 ; ð2þ or the third-order Birch-Murnaghan equation, P T0 " ðvþ ¼ 3 2 K V 7=3 T0 V # 5=3 ( 1 þ 3 4 K0 T0 4 " # ) V 2=3 1 ; ð3þ is widely used to describe the isothermal P-V relation at T 0. Thermal pressures DP th are expressed by the variation of internal thermal energies DE th between T 0 and T as DP th ðv ; T Þ ¼ g V DE thðv; T Þ ¼ g ½ V E thðv ; TÞ E th ðv; T 0 ÞŠ; ð4þ where g is the Grüneisen parameter. On the basis of the Debye model, the internal thermal energy E th at a given temperature can be calculated as E th ðv ; TÞ ¼ 9nRT Q 3 Z QD=T D x 3 T 0 e x 1 dx; where R is the gas constant, n is the number of atoms in the formula unit of the concerned material (n = 2 in MgO), and Q D is the Debye temperature. As is well known within the ð5þ Debye model, the specific heat C V asymptotically approaches the Dulong-Petit limit of 3nR at temperatures above the Q D. [12] In the MGD model, g is defined as g ln Q ln V ; representing a volume dependence of Q D.Ifg is constant (g = g 0 ) with respect to volume change, Q D is directly derived from equation (6) as ð6þ V g0 Q D ðv Þ ¼ Q 0 : ð7þ However, generally, g varies with volumes and is assumed, similar to equation (6), as q ln ln V ; where q is a parameter indicating the volume dependence of g. When q has no volume dependence, g and Q D can be obtained by integrating equations (8) and (6) as and ð8þ V q gðvþ ¼ g 0 ; ð9þ Q D ðvþ ¼ Q 0 exp gðvþ g 0 ; ð10þ q respectively. Combining equations (4), (5), (9), and (10), the thermal pressure DP th can be calculated with three parameters (Q 0, g 0, and q) as functions of volume and temperature. This simple MGD model with constant q is widely used to analyze P-V-T EOS [e.g., Jackson and Rigden, 1996] and also applied in some pressure-scale models [e.g., Shimetal., 2002; Fei et al., 2004a]. However, 3of16

4 Figure 1. P-V-T data points of the analyzed data with an EOS model determined in this study (Fit3- Vinet) shown as isotherms at K. Pressures in the 300-K and high-p K S data and temperatures in the shock compression data are calculated using the determined EOS model (Fit3-Vinet). Gray symbols and lines represent the projections onto the two-dimensional planes. the validity of constant q is still not well understood, and the volume dependence of q might also be necessary in some materials [e.g., Speziale et al., 2001]. [13] Speziale et al. [2001] employed the q(v) model proposed by Jeanloz [1989] in the P-V-T EOS of MgO, where an additional parameter q 1 was introduced to represent the volume dependence of q similarly to equation (8) as q 1 ln ln V : Integrating equation (11), the q is obtained as ð11þ V q1 qv ð Þ ¼ q 0 : ð12þ Applying equation (12) to equation (8), we can obtain gðvþ ¼ g 0 exp qv ð Þ q 0 q 1 ¼ g 0 exp q 0 q 1 V q1 1 : ð13þ In order to acquire the g in the volume-dependent q model, one must not apply the q values calculated from equation (12) directly to equation (9). One must also take care when calculating the Q D in the volume-dependent q model. In general cases with volume-dependent q, Q D must be determined by numerically integrating equation (6) as " Z # ln ð V=V0 Þ Q D ðvþ ¼ Q 0 exp gðþdx x ; ð14þ 0 Table 2. Primary Parameters Fixed in the Analyses a Parameter Values Reference (Å 3 ) (16) LASL shock Hugoniot data [Marsh, 1980], Isaak et al. [1989], Dubrovinsky and Saxena [1997], Fiquet et al. [1999], Sinogeikin et al. [2000], Zha et al. [2000], and Li et al. [2006] K S0 (GPa) (18) Isaak et al. [1989], Yoneda [1990], Sinogeikin et al. [2000], Zha et al. [2000], and Li et al. [2006] a 0 (10 5 K 1 ) 3.17(3) Suzuki [1975], Dubrovinsky and Saxena [1997], and Fiquet et al. [1999] C P0 (J mol 1 K 1 ) 37.4(4) Watanabe [1982] a Subscript zero means the value at 1 atm and 300 K. 4of16

5 where x =ln(v/ ). We have to use the correct combinations to obtain proper values of g and Q D according to the used models. [14] In this study, we employed another formulation expressing the volume dependence of g with only volume-independent parameters in order to make the EOS analyses simple by avoiding the numerical integration to obtain Q D. We defined the volume dependence of g as ( " # ) V b gðvþ ¼ g 0 1 þ a 1 ; ð15þ where a and b are volume-independent adjustable parameters to be optimized in the analyses and g was assumed to be a function of volume only. The new expression of g(v) is essentially equivalent to the g 1 model proposed by Al tshuler et al. [1987] as g = g 0 +(g 0 g1)(v/ ) b, which is employed in recent EOS models [e.g., Dorogokupets and Oganov, 2007]. In equation (15), g approaches (1 a)g 0 for infinite compression (V! 0); hence, the newly introduced parameter a means 1 g 1 /g 0, and a should be given in the range 0 a 1. Applying equation (15) to equation (6), Q D can be simply integrated as Q D ðvþ ¼ Q 0 V ð1 a Þg0 exp g V ð Þ g 0 b : ð16þ This g(v) form is able to express the various volume dependences and universally represents the models described above with the quite simple parameterization: when a =0orb =0,theg becomes constant (= g 0 ) and then Q D = Q 0 (V/ ) g0 (equation (7)); when a =1,g = g 0 (V/ ) b and Q D = Q 0 exp{ [g g 0 ]/b} (equations (9) and (10)); when a = b =1,g = g 0 (V/ ) and Q D = Q 0 exp{ [g g 0 ]}, which is the g/v = const model frequently used in the shock compression analyses and also used by Zha et al. [2000]. Using this g(v) form in the MGD formulation (equations (4), (5), (15) and (16)), we calculated the thermal pressure DP th with four parameters (Q 0, g 0, a, and b), and the 300-K isothermal compression was calculated using the Vinet EOS (equation (2)) or the third-order Birch-Murnaghan EOS (equation (3)) with two constants (K T0, K 0 T0), in addition to the primary parameter of Analyses of the Shock Compression Data [15] The shock compression data were also analyzed using the MGD model. Internal energy variations are primary quantities in the shock compression data instead of temperatures, and pressures are generally determined on the basis of the Rankine-Hugoniot equation given as DE H ¼ E E 0 ¼ 1 ð 2 P þ P 0Þð V Þ: ð17þ The internal energy variations along the shock compressions (Hugoniot), DE H, are calculated from the energy conservation law DE H ¼ 1 2 u2 p ; ð18þ where u p is a particle velocity. The DE H can be represented as a sum of energy changes associated with isothermal compression and isochoric heating DE H ðv H ; T H Þ ¼ DE T0 ðv H ; T 0 ÞþDE th ðv H ; T H Þ; ð19þ where the second term representing the thermal energy DE th can be applied to equation (4) to calculate the thermal pressure DP th. On the other hand, DE T0 is evaluated on the basis of the thermodynamic relation Z V DE T0 ðv H ; T 0 Þ ¼ ½T 0 ds P T0 ðv ÞdVŠ: ð20þ Here, entropy S is calculated from the relation of Z T C V SV; ð TÞ ¼ 0 T dt; ð21þ where C V is the specific heat at constant volume and given as C V ðv ; TÞ ¼ 9nR Q 3 Z QD=T D x 4 e x dx ð22þ T 0 ðe x 2 1Þ in the Debye model, and C V /T is numerically integrated. The thermal energy variation in the shock compression process is then represented as DE th ðv H ; T H Þ ¼ DE H ðv H ; T H Þ T 0 ½SV ð H Þ SV ð 0 ÞŠ: Z VH P T0 ðvþdv ; ð23þ where the third term on the right-hand side is also numerically integrated using the P T0 -V relation according to the used EOS model (equation (2) or (3)). Through these operations, the shock compression P-V data were reduced to P T0 and DP th in the analyzed EOS models Least Squares Analyses [16] In the MGD models defined above, we have six adjustable parameters (K T0, K 0 T0, Q 0, g 0, a, and b) with to represent the P-V-T EOS. In those parameters, K T0 and Q 0 were simultaneously calculated from g 0 with the fixed parameters shown in Table 2 on the basis of the thermodynamic relations. The K T0 were calculated from the relation of K S K T ¼ 1 þ gat ; ð24þ where a is the thermal expansivity and the Q 0 calculated using equation (22) and C P was C V ¼ 1 þ gat : ð25þ On the other parameters, K 0 T0 and g 0, a, and b [g(v)] express the pressure or volume dependencies of principal 5of16

6 Table 3. Fixed Parameters in the Analyzed EOS Models Parameter Fit1 Fit2 Fit3 K T0 (GPa) (fixed) a variable b variable b K 0 T0 free free free Q 0 (K) 763 (fixed) a variable b variable b g (fixed) a free free a free 1 (fixed) free b free free free a Calculated from the primary parameters in Table 2. b Calculated with g 0 and the primary parameters in the analyses. physical properties at 1 atm and 300 K. K 0 T0 is the pressure derivative of K T0 at 1 atm and constrains K T at high pressures according to the used EOS models. The g(v) expresses the volume dependence of Q D and relates to the DP th under compressed conditions (equation (4)). [17] In principle, K 0 T0 and g(v) should be determined independently from each other to make a physically reasonable EOS, for example, determining K 0 T0 first from isothermal compression measurements at T 0 and then determining g(v) with high-temperature experiments. However, since it is generally difficult to determine K T0 without any pressure scale, K 0 T0 of MgO is significantly unclear to represent the P-V relations at extreme high-pressure conditions. Instead, we tried to determine K 0 T0 and g(v) simultaneously thorough the unified analyses using the P-V-T-K S data sets in the wide V-T conditions. [18] On the basis of the EOS models defined previously, we calculated pressures with the reported volumes and temperatures (more correctly, DE th in the shock compressions). Through the least squares analyses with respect to pressures, we optimized the combinations of P T0 and DP th to reproduce the measured data. The measured P-V-T data were weighted according to the errors, and all the analyses were carried out in terms of DP residual /dp obs, where DP residual = P calc P obs. For the thermal expansion data, errors in pressure are evaluated from the uncertainties in the volume and temperature measurements as 0.3 GPa, and for the LASL shock Hugoniot data we employed the same errors as the typical errors reported by Duffy and Ahrens [1993] (3 GPa). For the K S data, DP residual and dp obs were evaluated from DK S (residual) and dk S (obs) with the relation of dp/dk S = 1/[(1 + gat) K 0 T], where the g, a, and K 0 T were sequentially calculated for each data points using the analyzed EOS models. 3. Results and Discussion 3.1. Reproducibility for the Analyzed Data Sets The g(v) Models [19] In order to test the adaptability to MgO, several EOS models were analyzed. The models we considered were based on the Vinet (equation (2)) or the third-order Birch- Murnaghan EOS (3BM; equation (3)) for the 300-K isothermal compression with the MGD models (equations (4), (5), (15) and (16)) for the thermal contributions. In the MGD models, three kinds of g(v) models were tested changing the fixed parameters in equation (15). Here the three EOS models are named Fit1, Fit2, and Fit3, and the fixed parameters are shown in Table 3. In the Fit1 model, we fixed g 0 as the thermodyanamic Grüneisen parameter (g th0 ) obtained from the thermodynamic relation of g ¼ avk S C P ð26þ with the primary fixed parameters shown in Table 2. In the Fit2 and Fit3 models, g 0 was not fixed and considered as an effective Grüneisen parameter. K T0 and Q 0 were treated as g 0 -dependent variables. The volume dependence of g was compared between the Fit2 and Fit3 models. In the Fit2 model, a was fixed as 1, and the conventional g = g 0 (V/ ) b model (equations (9) and (10)) was considered. In the Fit3 model, all of the EOS parameters were not fixed. [20] Figure 2 shows residuals between the calculated and observed pressures for all the analyzed data as a function of V/. In the Fit1 model (Figures 2a and 2b), the DP residual has volume dependence, and DP residual increases with increasing volume in the volume range of V/ > 1 and also increases with decreasing volume in the range V/ <1. The Fit2 model shows slightly positive volume dependence in the DP residual, which decreases with decreasing volume for the extreme compressions of Duffy and Ahrens [1993] (Figures 2c and 2d). The Fit3 model has no volume dependence in the DP residual (Figures 2e and 2f). The total DP residual in root mean squares decreases with increasing the degrees of freedom in the EOS models and is 1.4 GPa, 1.1 GPa, and 0.8 GPa for the Fit1, Fit2, and Fit3 models, respectively. [21] The residuals in the K S data are shown in Figure 3. The similar tendency to the DP residual is observed in the volume dependence of DK S (residual) and the total DK S (residual).itis concluded that fixing g 0 as g th0 (Fit1), it is impossible to reproduce all the P-V-T-K S data simultaneously. In addition, g = g 0 {1 + a[(v/ ) b 1]} model (Fit3) more accurately reproduces the experimental data compared to the conventional g = g 0 (V/ ) b model (Fit2). In the current comparison, there is no significant difference between the models using the Vinet EOS and the third-order Birch-Murnaghan EOS (3BM) for the 300-K isothermal compression in Fit3 (Figures 2e, 2f, 3e, and 3f) P T 0 Models [22] Figure 4 shows the detailed comparison between the EOS models using the Vinet EOS (Fit3-Vinet) and the thirdorder Birch-Murnaghan EOS (Fit3-3BM) for the 300-K isothermal compression in Fit3, along with the analyzed data. Calculated thermal expansions are compared to the 1-atm thermal expansion data as a function of temperature in Figure 4a. The Fit3-Vinet and Fit3-3BM models yield the same thermal expansions and reproduce the experimental results to 3000 K. In contrast, the EOS of Speziale et al. [2001] does not reproduce the observation beyond 1200 K as mentioned before. Both Fit3-Vinet and Fit3-3BM models reproduce the analyzed data within the errors, and the total DP residual is 0.2 GPa for the data of Dubrovinsky and Saxena [1997] and 0.3 GPa for the data of Fiquet et al. [1999]. [23] Figure 4b shows the comparison to the 1-atm and high-t K S data as a function of temperature. Both models are consistent with the analyzed data, and there is a difference at higher temperatures beyond the observation. In the work by Isaak et al. [1989] the total DK S (residual) is 6of16

7 Figure 2. Differences between the calculated and observed pressures as a function of V/ for all the analyzed EOS models. Determined parameters in the EOS models are shown with the total residuals. The fixed parameters in the analyzed EOS models are summarized in Table GPa in Fit3-Vinet and 1.2 GPa in Fit3-3BM model, and in the work by Sinogeikin et al. [2000] the total DK S (residual) is 0.8 GPa in the both models. The total DK S (residual) in both models is comparable to the experimental uncertainties. The Fit3-Vinet model yields the relatively larger (@K S /@T) P than the Fit3-3BM model at high temperatures, and then the total residual in Fit3-Vinet is smaller than the Fit3-3BM model in the work by Isaak et al. [1989]. [24] The calculated 300-K and high-p K S were compared to the experimental results in Figure 4c as a function of V/. Both the Fit3-Vinet and Fit3-3BM models reproduce the experimental results within the errors. The Fit3-Vinet model calculates a slightly smaller K S than the Fit3-3BM model in the volume range of V/ <0.80(P T0 > 55 GPa). The total DK S (residual) in the work by Zha et al. [2000] is 8.5 GPa in Fit3-Vinet and 8.7 GPa in Fit3-3BM, and the residuals are smaller than the uncertainty in the observation (12 GPa). For the K S data of Li et al. [2006], the total DK S (residual) is 1.5 GPa and 1.6 GPa in Fit3-Vinet and Fit3-3BM, respectively, and the residuals are comparable to the reported uncertainties (<1%). In the comparison at 1 atm and high temperature and 300 K and high pressure up to 55 GPa, both the Fit3-Vinet and Fit3-3BM models completely reproduce the analyzed P-V-T-K S data. [25] Next, the two EOS models are compared to the shock compression data (Figure 5). Figure 5a shows differences between shock pressures directly calculated by the Rankine- Hugoniot equations (P H ) and shock pressures calculated 7of16

8 Figure 3. Differences between the calculated and observed K S as a function of V/ for all the analyzed EOS models along with the total residuals. The fixed parameters in the analyzed EOS models are summarized in Table 3. with the EOS parameters [P H (calc) ]. The P H (calc) are calculated with P T0 (V H ) and DP th derived from the shock energies with the determined EOS. In the calculation of the Hugoniot, relationships between U s and u p for MgO reported by Vassiliou and Ahrens [1981], as U s = 6.65(5) (2)u p, is used. Additionally the results of the Fit2 models are shown for comparison, and the Fit2 models do not reproduce the Hugoniot. Although the Fit3-Vinet model has the larger fluctuation than Fit3-3BM, both of the EOS models reproduce the Hugoniot within the uncertainties up to 200 GPa. The total DP residual for LASL shock Hugoniot data is 1.4 GPa in Fit3-Vinet and is 1.5 GPa in Fit3-3BM. In the work by Duffy and Ahrens [1993], the total DP residual is 1.5 GPa in Fit3-Vinet and 0.15 GPa in Fit3-3BM. The residual in Fit3-3BM is extremely smaller than Fit3-Vinet in the work by Duffy and Ahrens [1993]; however, the Fit3- Vinet model also reproduces the experimental data within the uncertainties (2 3 GPa). [26] Figure 5b shows another comparison between the present models and the shock compression data in terms of temperature. The calculated Hugoniot temperatures based on the U s -u p relationship [Vassiliou and Ahrens, 1981] are compared with the measured temperatures of Duffy and Ahrens [1993]. The calculated temperatures of both of the EOS models are consistent with the measured values within the uncertainties, in spite of the fact that we have not used the measured temperatures in the analyses. The total RMS residuals between the calculated and measured temperatures 8of16

9 are 80 K in Fit3-Vinet and 130 K in Fit3-3BM, which are smaller than the uncertainty in the measurements ( K). These results strongly support that the obtained combinations of P T0 and DP th in Fit3-Vinet and Fit3-3BM are both appropriate to reproduce the shock compression data in terms of pressure and temperature Consistency and Difference Between the Two EOS Models [27] In spite of the detailed comparisons, there was no evidence to choose or reject either Fit3-Vinet or Fit3-3BM model in terms of the reproducibility for the experimental data. Here we directly compare the determined two EOS models. The determined EOS parameters in the Fit3-Vinet and Fit3-3BM models are summarized in Table 4, and the two EOS are compared in Table 5 and Figure 6. Table 5 represents the determined P-V-T states of MgO as the isochors at K in the two EOS models, and Figure 6a shows the P-V-T states as the isotherms at K in the volume range of 0.65 V/ 1.15, where the analyzed data exist (Table 1). The isotherms of Fit3-Vinet and Fit3-3BM are consistent up to 200 GPa and 4000 K within ±2 5 GPa. However, there is a systematic difference, and the difference depends on the P-V-T conditions of the analyzed data. The differences between the two EOS at K are shown as a function of V/ in Figure 6b. [28] Precisely comparing, in the volume range of 1.00 V/ 1.15, the difference is almost less than 0.1 GPa at K (see Table 5). At 300 K, the difference is within ±0.12 GPa in the volume range of 0.78 V/ (up to GPa in Fit3-Vinet), and the two EOS models are extremely consistent each other. However, the difference monotonically increases with decreasing volume below V/ = 0.78 and reaches 4.6 GPa at V/ = 0.65 (168.8 GPa in Fit3-Vinet). In the volume range of V/ > 0.80, we have the 300-K and high-p K S data and the shock compression data; therefore, P T0 is well constrained. However, in the volume range of V/ < 0.78, there is only the shock compression data, and the difference in P T0 becomes prominence. The present EOS analyses were carried out with respect to the totals pressure (P T0 + DP th ), and then it was difficult to constrain P T0 or DP th separately by only the shock data. Carefully comparing Fit3-Vinet and Fit3-3BM in Figure 6a, the two EOS are consistent each other along the P-V conditions of the Hugoniot in the volume range of V/ < 0.8. [29] In the determined EOS parameters in Table 4, g 0 is consistent between the two EOS models because the g 0 value at V/ = 1 has small influence on the volume dependence for the P-V-T states. Reflecting the consistent g 0, K T0 and Q 0 show similar consistencies. In contrast, the parameters a and b expressing the volume dependence of g have large uncertainties, and the absolute values are different beyond the uncertainties between the two EOS models. The inconsistencies in a and b were derived from the inevitable difference in DP th respecting the different P T0 between the two EOS models. Figure 7 shows the deter- 9of16 Figure 4. Comparison between the determined two EOS models (Fit3-Vinet and Fit3-3BM) along with the analyzed data. (a) The 1-atm thermal expansion as a function of temperature, (b) 1-atm and high-t K S as a function of temperature, and (c) 300-K and high-p K S as a function of V/ are compared.

10 Figure 5. Comparison between the determined two EOS models with the shock compression data. (a) Differences between the shock pressures directly calculated from U s -u p relationships (P H ) and the shock pressures calculated from the shock energies with the determined EOS models [P H(calc) ] are shown as a function of P H. (b) Calculated shock temperatures along the Hugoniot with the determined EOS models are compared with the measured shock temperatures of Duffy and Ahrens [1993]. The shock pressures and temperatures are based on the U s -u p relationships determined by Vassiliou and Ahrens [1981]. mined g in the Fit3-Vinet and Fit3-3BM models as a function of volume. In the volume range of V/ >1,g is consistent each other. However, under compressed conditions, g(v) remains uncertain because of the uncertainty in DP th between the two EOS models. This is the limit of not only the present analytical operation but also the currently available experimental data sets. [30] The new EOS models successfully reproduce all the currently available pressure-scale-free experimental data sets within the uncertainties in the wide P-T conditions ranging from 1 atm to 196 GPa and from 300 K to 3700 K. The residuals between the calculations and the observations are totally within errors of the measurement with no volume dependency. Therefore, the both EOS models should be candidates for the primary P-V-T EOS of MgO so far, although there remain some differences between the two EOS. We believe that the present analyses are the best approach to constrain the P-V-T EOS as long as the currently available experimental data are used. The way to improve the present EOS models, in other words to constrain the P T0 -V relation, requires the V-T 0 -K S data such as from Zha et al. [2000] and Li et al. [2006] to much higher pressures above 100 GPa. Additional shock compression data are also required to examine the present EOS models in the pressure range to the Earth s inner core P-T region. Especially, the preheated shock compression experiments should be a promising method to constrain P T0 and DP th, simultaneously overcoming the limit of the present analytical operation [e.g., Molodets, 2006]. We conclude that the present EOS models based on the g(v) model as equation (15) in the MGD formalism with the Vinet and the third-order Birch-Murnaghan EOS, which have the seven parameters (, K T0, K 0 T0, Q 0, g 0, a, and b) in total, are sufficient to reproduce all the observed P-V-T-K S relations of MgO up to 200 GPa and 4000 K Comparison With Other EOS Models [31] The determined EOS models of MgO (Fit3-Vinet, Fit3-3BM) are compared with previously reported EOS, which have been used as pressure scales in high P-T experiments [Jamieson et al., 1982; Matsui et al., 2000; Speziale et al., 2001; Dorogokupets and Oganov, 2007; Wu et al., 2008]. Figure 8 shows differences between the EOS models in terms of the total pressure as a function of V/. The Fit3-Vinet model is employed as a standard in the comparison. The condition of V/ = 0.65 corresponds approximately to 170 GPa at 300 K and 190 GPa at 3000 K in the present models (Table 5). [32] Between the two EOS models determined in this study, Fit3-3BM model gives higher pressures than Fit3- Vinet in the volume range of V/ < 0.8 at 300 K, and the Fit3-3BM model is consistent with the ab initio calculation by Wu et al. [2008] within 1 GPa up to 170 GPa at 300 K (Figure 8a). The differences between the Fit3-Vinet and Fit3-3BM models decrease with increasing temperature (Figures 8b and 8c). Additionally, comparing the EOS Table 4. EOS Parameters Determined in the Fit3 Model Parameter Vinet 3BM (Å 3 ) (fixed) (fixed) K T0 (GPa) (18) (18) K 0 T (13) 4.221(11) Q 0 (K) 761(13) 761(13) g (15) 1.431(14) a 0.138(19) 0.29(4) b 5.4(11) 3.5(5) 10 of 16

11 Table 5. P-V-T States of MgO Determined in This Study a V/ Vinet 3BM Vinet 3BM Vinet 3BM Vinet 3BM Vinet 3BM Vinet 3BM Vinet 3BM Vinet 3BM 300 K 500 K 1000 K 1500 K 2000 K 2500 K 3000 K b 4000 K (6.97) (7.07) (7.61) (7.73) (8.30) (8.45) (9.05) (9.20) (9.85) (10.01) (10.71) (10.87) (11.63) (11.79) (12.62) (12.76) (13.67) (13.79) (14.79) (14.88) (15.98) (16.04) (17.25) (17.27) a Pressures are given in GPa. b P-T conditions in the parentheses are beyond the stability field of the solid-state MgO. 11 of 16

12 Figure 6. The Fit3-Vinet and Fit3-3BM models are compared in terms of the total pressure. (a) The isotherms at K are compared along with the Hugoniot [Vassiliou and Ahrens, 1981]. (b) Differences between the two EOS models are represented at K. model of Wu et al. [2008] with our EOS models at high temperatures, the differences decrease with increasing temperature, especially, the difference between Fit3-3BM and the model of Wu et al. [2008] is approximately less than 1 GPa all over the compared P-T region (up to 190 GPa and 3000 K). In contrast, the early EOS models of Jamieson et al. [1982], Matsui et al. [2000], Speziale et al. [2001], and Dorogokupets and Oganov [2007] give lower pressures than the Fit3-Vinet model, and the differences are monotonously increase with decreasing volumes. The differences remain at high temperatures, and there is no significant temperature effect. Compared to the other EOS models, Speziale et al. s EOS shows deviations in the volume range of V/ >0.9at high temperatures (Figures 8b and 8c). These are caused by the large g in V/ > 1 (Figure 7), and the large g also causes the less reproducibility for the thermal expansion at 1 atm (Figure 4a). [33] The thermal pressures DP th are separately compared between the EOS models as a function of V/ (Figure 9). Within the two EOS models of this study, Fit3-Vinet has larger DP th than Fit3-3BM, and the difference increases with decreasing volume and increasing temperature, in contrast to the smaller P T0 compared to those of Fit3-3BM in the volume range of V/ < 0.8 (Figure 8a). The larger (@DP th /@V) T in Fit3-Vinet is consistent with those of Speziale et al. [2001] in the volume range of V/ < 1.0, and the smaller (@DP th /@V) T in Fit3-3BM is similar to those of Jamieson et al. [1982], Matsui et al. [2000], and Wu et al. [2008]. The EOS model of Dorogokupets and Oganov [2007] has the intermediate DP th and (@DP th /@V) T between those of Fit3-Vinet and Fit3-3BM. Speziale et al. s EOS yields the tremendous DP th in V/ > 1.0, which causes the nonlinear behavior for the total pressures (Figures 8b and 8c). [34] The new EOS models yield the higher total pressures than the early models and are consistent with the ab initio EOS of Wu et al. [2008] at high temperatures (Figures 8b and 8c). The higher total pressures are required to reproduce the shock compression data around 200 GPa. Figure 10 shows the 3000 K and 4000 K isotherms given by this study, Speziale et al. [2001], Dorogokupets and Oganov [2007], and Wu et al. [2008] along with the shock com- Figure 7. Determined g(v) in the Fit3-Vinet and Fit3-3BM models along with the g(v) model of Speziale et al. [2001]. 12 of 16

13 the P-V data of Duffy and Ahrens [1993] at 196 GPa (Figure 5b). This temperature is more than 700 K higher than the actual temperature reported as 3663 K, which is larger than the uncertainty in the measurement. The model of Dorogokupets and Oganov [2007] would need further higher temperatures due to the smaller DP th compared to Speziale et al. s model (Figure 9b) Applications to the Postspinel and Postperovskite Boundaries [35] In sections , the suitability of the current EOS models to a primary pressure scale has been elucidated in detail. Then, we recalculate some phase boundaries determined by in situ high P-T experiments using Speziale et al. s [2001] MgO scale. [36] First, the postspinel phase boundary in Mg 2 SiO 4 was reassessed. Reassessed phase boundaries with the Fit3-Vinet and Fit3-3BM models for the reported V-T data of Fei et al. [2004b] are represented with the original boundary based on Speziale et al. s [2001] EOS in Figure 11a. The reassessed phase boundaries are consistent each other within the experimental errors and also are coincident with the original boundary based on the Speziale et al. s EOS. The reassessed boundaries are located at T (K) = P (GPa) in Fit3-Vinet and at T (K) = P (GPa) in Fit3-3BM Figure 8. Comparison between the EOS models in terms of the total pressure at (a) 300 K, (b) 1500 K, and (c) 3000 K. The differences from the Fit3-Vinet model are represented. pression data of Duffy and Ahrens [1993]. The isotherms of Speziale et al. [2001] and Dorogokupets and Oganov [2007] are quite consistent at 3000 K, though there are some differences at 300 K. However, these models reproduce the shock compression data less accurately. In contrast, the present models of Fit3-Vinet, Fit3-3BM, and Wu et al. s EOS reproduce the shock compression data between 3000 K and 4000 K, although there are some differences at 300 K. The other EOS models need some excess temperatures to reproduce the shock compression data. For example, the model of Speziale et al. [2001] needs 4400 K to reproduce Figure 9. Comparison between the EOS models in terms of the thermal pressure DP th. The absolute values of the thermal pressure are compared at (a) 1500 K and (b) 3000 K. 13 of 16

14 Figure 10. Comparison between the EOS models in the and 4000-K isotherms with the shock compression data of Duffy and Ahrens [1993]. Gray lines represent the 300-K isotherms. and yield the transition pressures of 22.3 and 22.6 GPa at 2000 K, respectively. These conditions are approximately 1 GPa lower than the condition where the 660-km seismic discontinuity is observed [Flanagan and Shearer, 1998]. The experimentally determined Clapeyron slopes are much steeper than a theoretical investigation [Yu et al., 2007]. The experimental results and the ab initio calculation are consistent for the pressures of the triple junction between wadsleyite, ringwoodite (Rw), and perovskite + periclase (Pv + Per) at 2200 K and the experimentally determined steeper slopes possibly due to the kinetic effect in the experiments. The Clapeyron slope based on the high P-T experiments should be reexamined by considering the transformation kinetics at the lower-temperature conditions. [37] Second, Figure 11b shows postperovskite phase boundaries in MgSiO 3 obtained by reassessing the high P-T experiments of Hirose et al. [2006]. Similar to the case of the postspinel boundary, our two EOS models show the consistent phase boundaries within the errors in the experiments. The differences between the original and reassessed Clapeyron slopes are emphasized rather than the case of the postspinel boundary, and the transition pressures are shifted to higher pressures 3 4 GPa. The phase transition boundaries are now expressed as T (K) = P (GPa) in Fit3-Vinet and as T (K) = P (GPa) in Fit3-3BM, which is consistent with the theoretical prediction from first principle simulations [Tsuchiya et al., 2004] within reasonable errors at 2300 K. 4. Concluding Remarks [38] We have carried out the unified analyses for the pressure-scale-free data sets of the 1-atm thermal expansion data, 1-atm and high-t K S data, 300-K and high-p K S data, and the shock compression data in the P-T range of 1 atm- 196 GPa and K. Fixing the reliable quantities at 1 atm and 300 K (, K S0, a 0, C P0 ; Table 2), we have optimized P T0 and DP th to represent the high P-T properties through the weighted least squares treatments with respect to the total pressure. Applying the formulation of g = g 0 {1 + a[(v/ ) b 1]} in the MGD model with the Vinet EOS and the third-order Birch-Murnaghan EOS for the P T0, we determined K 0 T0 and g(v) simultaneously. The two EOS models determined in this study (Fit3-Vinet and Fit3-3BM; Table 4) can reproduce all the analyzed data with the total RMS residuals of 0.8 GPa in pressure. Among the several proposals for the EOS model of MgO, only our two models and the model of Wu et al. [2008] are able to reproduce the shock compression data correctly. The two EOS models determined in this study are consistent each other within typical uncertainties in high P-T experiments to the condition of the Earth s core-mantle boundary. We propose the new EOS models of MgO as the primary pressure calibration standard because of the simplicity in the formulation and the accurate reproducibility for the reliable experimental data, in addition to the independency from other pressure standards. [39] As a result of the detailed comparison, it was impossible to select the more appropriate one between the two EOS models (Fit3-Vinet and Fit3-3BM). The result reflects the current limits in both the theory and the experiments: the true formulation of P(V, T) has been unknown; precise V-T 0 -K S data have been absent above 55 GPa; shock compression data have been limited to on the Hugoniot. The Fit3-3BM model is coincident with the ab initio EOS of Wu et al. [2008] precisely; however, it is premature to 14 of 16