Thermal equation of state of (Mg 0.9 Fe 0.1 ) 2 SiO 4 olivine

Physics of the Earth and Planetary Interiors 157 (2006) 188 195 Thermal equation of state of (Mg 0.9 Fe 0.1 ) 2 SiO 4 olivine Wei Liu, Baosheng Li Mineral Physics Institute, Stony Brook University, Stony Brook, NY 11794-2100, USA Received 26 December 2005; received in revised form 15 March 2006; accepted 6 April 2006 Abstract In situ synchrotron X-ray diffraction measurements have been carried out on San Carlos olivine (Mg 0.9 Fe 0.1 ) 2 SiO 4 up to 8 GPa and 1073 K. Data analysis using the high-temperature Birch Murnaghan (HTBM) equation of state (EoS) yields the temperature derivative of the bulk modulus ( K T / T) P = 0.019 ± 0.002 GPa K 1. The thermal pressure (TH) approach gives αk T = 4.08 ± 0.10 10 3 GPa K 1, from which ( K T / T) P = 0.019 ± 0.001 GPa K 1 is derived. Fitting the present data to the Mie Grüneisen Debye (MGD) formalism, the Grüneisen parameter at ambient conditions γ 0 is constrained to be 1.14 ± 0.02 with fixed volume dependence q = 1. Combining the present data with previous results on iron-bearing olivine and fitting to MGD EoS, we obtain γ 0 = 1.11 ± 0.01 and q = 0.54 ± 0.36. In this study the thermoelastic parameters obtained from various approaches are in good agreement with one another and previous results. 2006 Elsevier B.V. All rights reserved. Keywords: Equation of state; Olivine; X-ray diffraction; High pressure and high temperature 1. Introduction Olivine is believed to be the major component in the upper mantle with an average composition of 90% mol Mg 2 SiO 4 and 10% mol Fe 2 SiO 4 (Fo 90 Fa 10 ) (e.g., Agee, 1998 and references therein). Its thermoelastic properties are important for a better understanding of the composition and dynamics of the upper mantle. So far, many experimental studies of the elasticity of iron-bearing olivine have been conducted using different methods, including ultrasonic interferometry at elevated pressure (P) and temperature (T) (e.g., Kumazawa and Anderson, 1969; Webb, 1989; Knoche et al., 1995; Darling et al., 2004; Li et al., 2004; Liu et al., 2005), Brillouin spectroscopy measurements to 32 GPa at ambient Corresponding author. Tel.: +1 631 632 8338; fax: +1 631 632 8140. E-mail address: weiliu3@notes.cc.sunysb.edu (W. Liu). temperature (Zha et al., 1998), impulsively stimulated laser scattering (ISLS) to 17 GPa (Zaug et al., 1993; Abramson et al., 1997), and resonant ultrasound spectroscopy (RUS) measurement to 1500 K (Isaak, 1992). However, thermal EoS studies based on in situ high P T measurements are still limited, despite the results on forsterite (Meng et al., 1993; Guyot et al., 1996). Meng et al. (1993) determined the molar volume of forsterite at about 7 GPa and 1100 1400 K. Guyot et al. (1996) measured the cell volumes of forsterite and San Carlos olivine up to 7 GPa and 1300 K, and concluded that the product of thermal expansion and bulk modulus (αk T ) is volume-independent within the investigated P T range for both forsterite and San Carlos olivine (see also Anderson, 1999). The absence of a thermal EoS derived using high P T experimental data highlights the need for further investigation of the P V T data for this important mantle mineral. In this paper we report in situ X-ray diffraction measurements on San Carlos olivine, under pressure 0031-9201/$ see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2006.04.003

W. Liu, B. Li / Physics of the Earth and Planetary Interiors 157 (2006) 188 195 189 and temperature conditions up to 8 GPa and 1073 K. Thermoelastic parameters are derived from the current P V T data using various thermal equations of state, including the high-temperature Birch Murnaghan (HTBM) EoS, thermal pressure (TH) approach, and Mie Grüneisen Debye (MGD) formalism. 2. Sample and experimental The polycrystalline olivine sample used in this study was back-transformed in situ from a dense polycrystalline wadsleyite (99.8% theoretical density) hot-pressed using natural San Carlos olivine with Fe/(Mg + Fe) of 0.1, determined using electron microprobe. X-ray diffraction confirmed that the olivine sample was a single phase (Fig. 1). The high pressure, high temperature experiments were performed using a DIA-type cubic-anvil apparatus (SAM85) installed at the superconducting beam line X17B2 of the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory (BNL). Details of this experimental set-up have been described elsewhere (Weidner et al., 1992; Kung et al., 2002; Li et al., 2004). The cube-shaped pressure medium was made of pre-compressed boron epoxy (4:1 wt% ratio), with an edge length of 6.15 mm. Well-sintered polycrystalline specimen of about 1.1 mm thick and 2 mm in diameter was embedded in the NaCl BN powder mixture (10:1 wt% ratio), which provided a pseudo-hydrostatic stress environment for the specimen. The cell pressure was determined using the equation of state for NaCl (Decker, 1971), and the uncertainty resulting from the measured cell parameters is about less than ±0.2 GPa. Temperature was measured using W/Re3% W/Re25% thermocouple wires placed immediately adjacent to the specimen. The pressure effect on the emf reading of the thermocouple was not considered. In this experiment, after the olivine sample was back-transformed from wadsleyite, we performed five heating/cooling cycles at different pressures (up to 8.2 GPa) on decompression. To minimize the effect of non-hydrostatic stress built up during compression/decompression at room temperature, in each heating/cooling cycle X-ray diffraction data were measured only during cooling, at 200 K steps under constant ram load. X-ray diffraction data from the specimen and the NaCl were recorded in energy dispersive mode using a solid state Ge detector. The incident X-ray beam was collimated to 0.2 mm by 0.1 mm and the diffraction angle was set at 2θ = 6.5. Fig. 1 shows the X-ray diffraction patterns of the starting wadsleyite and the backtransformed olivine phase at peak P T conditions. After the phase transition, the full width at half maximum (FWHM) of the diffractions lines of olivine changed less than ±5% through the whole experiment, indicating that the olivine sample was under quasi-hydrostatic condition. We used 12 olivine diffraction lines to determine the cell volume (V), with a standard deviation less than 0.1%. The unit-cell volume at ambient conditions (V 0 ) was determined to be 292.13 ± 0.10 Å 3, in good agreement with the value of 292.0 ± 0.1 Å 3 reported by Zha et al. (1998) and 291.9225 ± 0.1188 Å 3 from Guyot et al. (1996) within mutual uncertainties. 3. Results Unit-cell volumes of olivine obtained along various isotherms from 298 to 1073 K at pressures up to 8.2 GPa are listed in Table 1. The isothermal compression at room temperature showed excellent agreement with the results of Abramson et al. (1997) and Zha et al. (1998) beyond the current experimental pressure range up to 15 and 20 GPa, respectively (Fig. 2). We applied three commonly employed methods, high-temperature Birch Murnaghan EoS (e.g., Duffy and Wang, 1998), thermal pressure approach (e.g., Anderson, 1995, 1999; Jackson and Rigden, 1996), and Mie Grüneisen Debye formalism (e.g., Jackson and Rigden, 1996), to derive thermoelastic parameters from our current P V T data for olivine. 3.1. High-temperature Birch Murnaghan EoS Fig. 1. X-ray diffraction patterns of olivine at high P T and starting material wadsleyite at ambient conditions. The high-temperature Birch Murnaghan EoS incorporates temperature effect into the thirdorder Birch Murnaghan EoS, and is expressed as

190 W. Liu, B. Li / Physics of the Earth and Planetary Interiors 157 (2006) 188 195 Table 1 Pressure, temperature, lattice parameter, and unit-cell volume measurement of San Carlos olivine P (GPa) T (K) a (Å) b (Å) c (Å) V (P, T) (Å 3 ) P th (GPa) 8.2 (2) 1073 4.7253 (15) 10.0537 (27) 5.9275 (16) 281.60 (20) 3.08 6.7 (2) 1073 4.7364 (28) 10.0997 (51) 5.9504 (30) 284.65 (17) 3.12 6.4 (2) 1073 4.7385 (19) 10.1091 (35) 5.9573 (21) 285.37 (12) 3.24 5.4 (2) 1073 4.7428 (29) 10.1487 (64) 5.9775 (36) 287.72 (20) 3.34 7.7 (2) 873 4.7223 (19) 10.0393 (35) 5.9238 (21) 280.84 (12) 2.15 5.9 (2) 873 4.7322 (35) 10.0994 (65) 5.9511 (38) 284.42 (21) 2.23 4.8 (2) 873 4.7415 (36) 10.1405 (66) 5.9697 (39) 287.03 (22) 2.39 4.0 (2) 873 4.7504 (15) 10.1649 (28) 5.9809 (16) 288.81 (9) 2.49 7.3 (2) 673 4.7192 (20) 10.0351 (37) 5.9175 (21) 280.24 (12) 1.39 5.4 (2) 673 4.7314 (18) 10.0924 (32) 5.9454 (19) 283.90 (10) 1.49 4.3 (2) 673 4.7360 (49) 10.1294 (90) 5.9638 (52) 286.10 (25) 1.46 3.5 (2) 673 4.7310 (28) 10.1658 (50) 5.9783 (26) 287.52 (15) 1.40 2.7 (2) 673 4.7530 (17) 10.1808 (38) 5.9891 (21) 289.81 (12) 1.63 6.8 (2) 473 4.7164 (18) 10.0282 (34) 5.9124 (20) 279.64 (12) 0.58 5.1 (2) 473 4.7277 (22) 10.0854 (41) 5.9398 (24) 283.21 (13) 0.78 3.9 (2) 473 4.7256 (40) 10.1309 (47) 5.9584 (23) 285.25 (15) 0.69 3.2 (2) 473 4.7445 (14) 10.1480 (26) 5.9690 (15) 287.39 (08) 0.97 2.2 (2) 473 4.7499 (21) 10.1783 (45) 5.9857 (25) 289.38 (14) 0.95 6.5 (2) 298 4.7142 (20) 10.0233 (37) 5.9104 (22) 279.28 (12) 10 4 4.7 (2) 298 4.7251 (26) 10.0769 (47) 5.9367 (28) 282.67 (15) 10 4 3.6 (2) 298 4.7332 (21) 10.1177 (39) 5.9528 (23) 285.08 (13) 10 4 2.8 (2) 298 4.7427 (11) 10.1426 (20) 5.9644 (12) 286.90 (7) 10 4 0.0 298 4.7656 (17) 10.2142 (31) 6.0014 (18) 292.13 (10) 10 4 Numbers in brackets are 1σ error in last digit (s). the following, [ P(V, T ) = 3 (V0T ) 7/3 ( ) ] 5/3 2 K V0T 0T V V { [ 1 + 3 (V0T ) 2/3 4 (K 0T 4) 1]} V (1) Fig. 2. Volume comparison for olivine at ambient temperature. Line is calculated using the third-order Birch Murnaghan equation of state with fixed K T0 = 129.0 GPa and K T 0 = 4.61 reported by Liu et al. (2005). where K 0T, K 0T, and V 0T are isothermal bulk modulus, its pressure derivative, and unit-cell volume at temperature T and ambient pressure, respectively. Assuming that the second- and higher-order pressure derivatives of the bulk modulus are negligible, then K 0T and K 0T are given by the properties at ambient conditions, ( ) KT K 0T = K 0 + T (T 298) (2) P K 0T = K 0 (3) where K 0 and K 0 are the isothermal values at ambient conditions. The temperature derivative of the bulk modulus ( K T / T) P is assumed to be constant throughout the whole temperature range. Zero-pressure unit-cell volume V 0T is expressed as V 0T = V 0 exp T 298 α T dt (4) α T = α 0 + α 1 T + α 2 T 2 (5) where V 0 is the unit-cell volume at ambient conditions, and α T is the thermal expansivity at ambient pressure, empirically expressed by constant parameters, α 0, α 1, and α 2. We calculated the thermoelastic parameters by fixing K T0 = 129.0 GPa (K T0 has been converted from the adia-

Table 2 Thermoelastic parameters derived from high-temperature Birch Murnaghan (HTBM) EoS, thermal pressure (TP) approach and Mie Grüneisen Debye (MGD) formalism for San Carlos olivine compared with the previous studies a Reference This study 1 2 b 3 4 5 Fo 90 Fa 10 c Fo 93 Fa 7 c Fo c Fo 93 Fa 7 c Fo 90 Fa 10 c Fo c Fo c HTBM d TP d MGD d Ultrasonic d e RUS d RUS d TP d INS-model d K T0 129.0 f 129.0 f 129.0 f 129.0 f 129.0 f 128.3 127.7 (20) 129.8 128.1 127.4 g 127.4 g K T 0 4.61 f 4.61 f 4.61 f 4.61 f 4.61 f 5.16 4.8 h 4.8 h ( K T / T) P 0.038 (6) 0.019 (2) 0.019 (1) 0.021 0.020 (2) 0.0215 i 0.0224 i 0.0214 (30) 0.02 α 0 ( 10 5 K 1 ) 2.73 (34) 3.034 j 2.77 (9) 3.06 j 2.7 α 1 ( 10 8 K 2 ) 2.22 (81) 0.742 j 0.97 (9) 0.796 j 0.9 α 2 (K) 0.538 j 0.32 (54) 0.5782 j 0.31 ( K T / T) V 0 0.005 (4) q 5.64 (142) 1 θ D0 729 f 729 f 760 738 731 γ 0 1.27 (5) 1.14 (2) 1.15 1.28 1.26 RMS misfit 0.11 0.15 0.15 0.12 0.15 1 = Kumazawa and Anderson (1969), single crystal, ultrasonic, up to 0.2 GPa and 306 K. 2 = Gillet et al. (1991), Calorimetric measurements, up to1850 K at 1 bar; Raman spectroscopic measurement, up to 1150 K and 10 GPa. 3 = Isaak (1992), single crystal, resonant ultrasound spectroscopy (RUS), up to 1500 K at 1 bar. 4 = Meng et al. (1993), X-ray diffraction measurement, 6.6 7.6 GPa and 1019 1569 K. 5 = Guyot et al. (1996), X-ray diffraction measurement, up to 7 GPa and 473 1272 K. a Numbers in brackets are 1σ error in the last digits. Italic numbers indicate values being fixed. K T0 and RMS misfit are in GPa; ( K T / T) P and ( K T / T) V are in GPa K 1. Thermal expansion α = α 0 + α 1 T + α 2 T 2. b Initial estimates of α, K T0, and ( K T / T) P taken from Fei and Saxena (1987) and Isaak et al. (1989), respectively. c These are samples. d Methods. e Calorimetric and Raman spectroscopic measurements. f Liu et al. (2005). g Isaak et al. (1989). h Graham and Barsch (1969). i Result from a linear fit of K T reported by Isaak (1992). j Suzuki et al. (1975). W. Liu, B. Li / Physics of the Earth and Planetary Interiors 157 (2006) 188 195 191

192 W. Liu, B. Li / Physics of the Earth and Planetary Interiors 157 (2006) 188 195 batic bulk modulus K S0 = 130.3 GPa (Liu et al., 2005) using the conversion factor K S /K T =(1+αγT) 1.01 with thermal expansion α = 2.65 10 5 K 1 (Fei, 1995) and Grüneisen parameter γ = 1.26 (Isaak, 1992)), K T 0 = 4.61 (Liu et al., 2005), and V 0 = 292.13 Å 3.A least-squares fit, using program EoS-fit v5.2 (Angel, 2000), yielded ( K T / T) P = 0.036 ± 0.008 GPa K 1, α 0 = 2.75 ± 0.41 10 5 K 1, and α 1 = 2.05 ± 0.97 10 8 K 2 (Table 2). Comparing with previous data, these results show much larger thermal expansion together with greater temperature dependence (negative) for the bulk modulus. This discrepancy is believed to be caused by the trade-off between thermal expansivity and ( K T / T) P in current fit due to the lack of high-temperature data at zero and/or low pressures. Therefore, we adopted the temperature dependence of the thermal expansion α = 3.034 10 5 + 0.742 10 8 T 0.538T 2 (K 1 )byfei (1995) based on the experimental data of Suzuki (1975), who determined the zero-pressure thermal expansion of Febearing olivine up to 1400 K. Fitting the current P V T data (Table 1) to Eq. (1) (5), we obtained a result of ( K T / T) P = 0.019 ± 0.002 GPa K 1, which is in agreement with previous results (Table 2). The slight difference between the current value for ( K T / T) P and those reported earlier measured at ambient/low pressures (references 1 and 3 in Table 2) might result from the pressure effect on ( K T / T) P. When the mixed pressure and temperature derivative of bulk modulus 2 K T / T P = 3.3 ± 0.9 10 4 K 1 (Isaak, 1993) or 2.4 10 4 (Li and Liu, 2006, in preparation) for olivine is considered, the current value of ( K T / T) P will be decreased by 0.0019 0.0026 GPa K 1 over 8 GPa, resulting in a better agreement with the ambient/low pressure acoustic ( K T / T) P values in Table 2. The P V T fitting results and the measured olivine unit-cell volumes were plotted as a function of pressure in Fig. 3. Experimental data are reproduced very well, with a root mean square (RMS) misfit of 0.15 GPa. 3.2. Thermal pressure approach We also analyzed the P V T data using the thermal pressure approach (e.g., Anderson, 1995, 1999; Jackson and Rigden, 1996). The thermal pressure P th was obtained by subtracting the pressure at volume V and at room temperature (derived from Eq. (1)) from the pressure measured at the same V and at temperature T. P th = P(V, T ) P(V, 298) = ( ) KT + T V ln ( V0 V T 298 αk T dt ) (T 298) (6) The results are listed in Table 1 and Fig. 4. As shown in Fig. 4, our thermal pressure results on olivine agree very well with previous results (Anderson and Isaak, 1995; Guyot et al., 1996), though the thermal pressures from the study of Guyot et al. (1996) showed some scatters which could be due to minor chemical destabilization of Fe 2+ at high temperatures. All of the thermal pressures ( P th ) from different sources in Fig. 4 appear to vary linearly with temperature. Meng et al. (1993) has attempted to determine ( K T / T) V for forsterite using the thermal pressure approach, and determined a value of 0.005 ± 0.004 GPa K 1, which is close to zero if considering the uncertainty of the fit, indicating that the Fig. 3. High-temperature Birch Murnaghan equation of state fit for olivine. Measured high-temperature olivine unit-cell volumes (symbols) and estimated error bars (about ±0.2 GPa) of pressures are showed. Fig. 4. A comparison of P th (T) P th (298) for iron-bearing olivine and fitting results using thermal pressure approach as well as the residuals. The residuals are represented with open symbols of the same shape as the corresponding experimental data.

W. Liu, B. Li / Physics of the Earth and Planetary Interiors 157 (2006) 188 195 193 thermal pressure is nearly independent of volume (see also Guyot et al., 1996). Based on previous experimental results (Meng et al., 1993; Guyot et al., 1996; Isaak et al., 1989), Anderson (1999) found that the thermal pressure in forsterite shows linear variation with T up to 1600 K independent of compression (V/V 0 ) over the range of V/V 0 = 1 to 0.947 (corresponding to 0 8 GPa), concluding that ( K T / T) V is very close to zero. The same conclusions have been drawn for some other silicate minerals, such as CaSiO 3 perovskite (e.g., Wang et al., 1996; Shim et al., 2000) and MgO (Speziale et al., 2001), and ringwoodite (Mg 0.91 Fe 0.09 ) 2 SiO 4 (Nishihara et al., 2004). Assuming ( K T / T) V = 0 for olivine, the above Eq. (6) is then expressed as (Anderson, 1995, 1999) P th = αk T (T 298) (7) where αk T is the average of αk T at T > 298 K. From the best fit of the thermal pressure data in Table 1, using Eq. (7) we obtain an average value of αk T = 4.08 ± 0.10 10 3 GPa K 1 with a RMS misfit in thermal pressures of 0.15 GPa. Using the thermodynamic identity ( K T / T) P =( K T / T) V αk T (αk T / P) T, a value of 0.019 ± 0.001 GPa K 1 for ( K T / T) P is obtained, which is indistinguishable from the results from current HTBM EoS (Table 2) and previous studies within experimental errors. We also attempted to combine these three datasets on Fe-bearing olivine in Fig. 4, and obtained αk T = 4.03 ± 0.10 10 3 GPa K 1, which is essentially the same as that derived using the current data alone. A comparison of the measured and fitted results is shown in Fig. 4. 3.3. Mie Grüneisen Debye formalism In Mie Grüneisen Debye (MGD) equation of state, the thermal energy is approximated by the Debye lattice vibrational model, with only the acoustic modes taken into account (e.g., Jackson and Rigden, 1996). The pressure P(V, T) at a given volume and temperature can be expressed as the following forms: P(V, T ) = P(V, T 0 ) + P th (8) with P th = γ(v ) V [E th(v, T ) E th (V, T 0 )] (9) where the subscript 0 refers to the principal isotherm (298 K). P(V, T 0 ) is taken from Eq. (1). The thermal free energy E th in Eq. (9) is calculated from the Debye model (Debye, 1912), using E th = 9nRT Θ/T x 3 dx (Θ/T ) 3 0 e x 1 [ ] γ0 γ Θ = Θ 0 exp q (10) (11) γ = γ 0 ( V V 0 ) q (12) where n is the number of atoms per formula unit, R is the gas constant, Θ is the Debye temperature, γ 0 and Θ 0 are the Grüneisen parameter and the Debye temperature at V 0, respectively, and q = (dln γ/dln V) describes its volume dependence. The Debye temperature Θ is assumed to be a function of volume, and temperature independent. Using MGD formalism, it is a common practice to fix Θ 0 during the fit using results from other techniques, such as acoustic measurement and/or calorimetric approach, because it cannot be well resolved in the fit compared to other parameters (e.g., Jackson and Rigden, 1996). The acoustic modes of lattice vibration are related to the compressional and shear wave velocities v p and v s, respectively. The acoustic Debye temperature Θ ac is given by Θ ac = h ( ) 3N 1/3 ( ) ρ 1/3 v m (13) k 4π M/p 3 v 3 = 2 m v 3 + 1 s v 3 (14) p where M is the molecular mass; p is the number of atoms in the molecular formula; and k, h and N are Boltzmann s constant, Plank s constant, and Avagadro s number, respectively. Using the acoustic velocity data measured on the same sample (v p = 8.36 and v s = 4.82 km s 1 at ambient conditions) (Liu et al., 2005), Θ ac0 is determined to be 729 K, in good agreement with the previous result of Θ ac0 = 731 K obtained on olivine specimens with iron content similar to the current study (Isaak, 1992). The resulting best-fit Grüneisen parameter γ 0 and its volume dependence q are 1.27 ± 0.05 and 5.64 ± 1.42, respectively, at fixed Θ 0 = 729 K. The RMS misfit is 0.12 GPa. The value of γ 0 is in accord to the value of 1.26 determined by Isaak (1992), while the value q is much larger than previous results fits on other mantle minerals, e.g., CaSiO 3 perovskite (Shim et al., 2000), ironbearing wadsleyite, and MgSiO 3 perovskite (Jackson and Rigden, 1996). For a wide range of materials, the volume dependence of γ has been shown to hold with q 1(Stixrude and Bukowinski, 1990), and most of the

194 W. Liu, B. Li / Physics of the Earth and Planetary Interiors 157 (2006) 188 195 Fig. 5. The best-fitting Mie Grüneisen Debye models for the combined P V T datasets of iron-bearing olivine and residuals. The residuals are represented with open symbols of the same shape as the corresponding experimental data. previous fit have been carried out with q fixed at values between 0.5 and 1.5 for metals and minerals (e.g., Duffy and Ahrens, 1995; Jackson and Rigden, 1996). Therefore, we fixed the value of Θ 0 = 729 K and q =1 and we obtained γ 0 = 1.14 ± 0.02 with a RMS misfit of 0.15 GPa, in agreement with γ 0 = 1.16 from the study of Anderson (1988). We also constrained q by combining our data with high-temperature results for iron-bearing olivine from previous studies (Anderson and Isaak, 1995; Guyot et al., 1996). The high P T volumes from these two studies were normalized to their respective starting volumes. In this case, fitting the combined datasets to Eq. (8) (12) yield 1.11 ± 0.01 and 0.54 ± 0.36 for γ 0 and q, respectively, with a RMS misfit of 0.17 GPa. The parameter q appears to be improved when experimental data covering a wider temperature range are used. Fig. 5 showed the three combined datasets and fitting results as well as their respective residuals. 4. Conclusion P V T measurements on San Carlos olivine (Mg 0.9 Fe 0.1 ) 2 SiO 4 at pressures up to 8 GPa and temperatures of 298 1073 K were carried out using a DIA-type apparatus. Thermoelastic properties have been derived using the P V T data set by different equation of state approaches, namely HTBM equation of state, thermal pressure approach, and MGD formalism with constraints on K T0, K T 0, and Θ 0 from acoustic measurement on the same sample and α from Suzuki (1975). The results from different methods showed excellent agreement with each other as well as with previous results. The temperature derivatives of the bulk modulus of San Carlos olivine, derived from HTBM EoS and thermal pressure approach are 0.019 ± 0.002 and 0.019 ± 0.001 GPa K 1, respectively. The thermal pressure approach gave αk T = 4.08 ± 0.10 10 3 GPa K 1. Fitting the present data with the MGD approach yields γ 0 = 1.14 ± 0.02 with fixed q = 1. Combining the current data (P = 2.2 8.2 GPa, T = 473 1073 K) with prior experimental data on olivine with similar iron content, including Anderson and Isaak (1995) (P = 1 bar, T = 400 1500 K) and Guyot et al. (1996) (P = 3 7 GPa, T = 473 1272 K) and fitting them to MGD EoS, we obtained γ 0 = 1.11 ± 0.01 and q = 0.54 ± 0.36. Acknowledgements We thank two anonymous reviewers for their helpful and thorough comments on the manuscript. We are grateful for discussion with Yangbin Wang. We also thank Michael Vaughan, Liping Wang, and Zhong Zhong for their technical support at the X17B2 beamline. This work was supported by NSF grant (EAR00135550). The experiments were carried out at the National Synchrotron Light Source (NSLS), which is supported by the US Department of Energy, Division of Materials Sciences and Division of Chemical Sciences under Contract No. DE-AC02-76CH00016. The operation of X-17B2 is supported by COMPRES, the Consortium for Materials Properties Research in Earth Sciences. Mineral Physics Institute Publication No. 363. References Abramson, E.H., Brown, J.M., Slutsky, L.J., Zaug, J., 1997. The elastic constants of San Carlos olivine to 17 GPa. J. Geophys. Res. 102, 12253 12264. Agee, C.B., 1998. Phase transformations and seismic structure in the upper mantle and transition zone. In: Hemley, R.J. (Ed.), Ultrahigh- Pressure Mineralogy: Physics and Chemistry of Earth s Deep Interior. Reviews in Mineralogy, vol. 37. Mineralogical Society of America, Washington, DC, pp. 165 200. Anderson, D.L., 1988. Temperature and pressure derivatives of elastic constants with application to the mantle. J. Geophys. Res. 93, 4688 4700. Anderson, O.L., 1995. Equation of State of Solids for Geophysics and Ceramic Science. Oxford University Press, New York, pp. 243 274. Anderson, O.L., 1999. The volume dependence of thermal pressure in perovskite and other minerals. Phys. Earth Planet. Int. 112, 267 283. Anderson, O.L., Isaak, D.G., 1995. Elastic constants of mantle minerals at high temperatures. In: Ahrens, T.J. (Ed.), Mineral Physics and Crystallography: A Handbook of Physical Constants (Reference Shelf 2). Am. Geophys. Union, Washington, DC, pp. 64 97.

W. Liu, B. Li / Physics of the Earth and Planetary Interiors 157 (2006) 188 195 195 Angel, R.J., 2000. Equations of state, in high-pressure, hightemperature crystal chemistry. In: Hazen, R.M., Downs, R.T. (Eds.), Rev. in Mineral. Geochem., vol. 41. Mineral. Soc. of Am., Washington, DC, pp. 35 60. Darling, K.L., Gwanmesia, G.D., Kung, J., Li, B., Liebermann, R.C., 2004. Ultrasonic measurements of the sound velocities in polycrystalline San Carlos olivine in multi-anvil, high-pressure apparatus. Phys. Earth Planet. Int. 143 144, 19 31. Debye, P., 1912. Zur Theorie der spezifischenwärmen. Annalen der Physik 39, 789 840. Decker, D.L., 1971. High-pressure equations of state for NaCl, KCl, CsCl. J. Appl. Phys. 42, 3239 3244. Duffy, T.S., Ahrens, T.J., 1995. Compressional sound velocity, equation of state, and constitutive response of shock-compressed magnesium oxide. J. Geophys. Res. 100, 529 542. Duffy, T.S., Wang, Y., 1998. Pressure volume temperature equations of state. In: Hemley, R.J. (Ed.), Ultrahigh-Pressure Mineralogy: Physics and Chemistry of Earth s Deep Interior. Reviews in Mineralogy, vol. 37. Mineralogical Society of America, Washington, DC, pp. 425 457. Fei, Y., 1995. Thermal expansion. In: Ahrens, T.J. (Ed.), Mineral Physics and Crystallography: A Handbook of Physical Constants (Reference Shelf 2). Am. Geophys. Union, Washington, DC, pp. 29 44. Fei, Y., Saxena, S.K., 1987. A thermochemical data base for phase equilibria in the system Fe-Mg-Si-O at high pressure and temperature. Phys. Chem. Miner. 13, 311 324. Gillet, P., Richet, P., Guyot, F., Fiquet, G., 1991. High-temperature thermodynamic properties of forsterite. J. Geophys. Res. 96, 11805 11816. Graham, E.K., Barsch, G.R., 1969. Elastic constants of single-crystal forsterite as a function of temperature and pressure. J. Geophys. Res. 74, 5949 5960. Guyot, F., Wang, Y., Gillet, P., Ricard, Y., 1996. Quasiharmonic computations of thermodynamic parameters of olivines at high pressure and high temperature: a comparison with experiment data. Phys. Earth Planet. Int. 98, 17 29. Isaak, D.G., 1992. High-temperature elasticity of iron-bearing olivine. J. Geophys. Res. 97, 1871 1885. Isaak, D.G., 1993. The mixed P, T derivatives of elastic moduli and implications on extrapolating throughout Earth s mantle. Phys. Earth Planet. Int. 80, 37 48. Isaak, D.G., Anderson, O.L., Goto, T., 1989. Elasticity of single crystal forsterite measured to 1700 K. J. Geophys. Res. 94, 5895 5906. Jackson, I., Rigden, S.M., 1996. Analysis of P-V-T data: constraints on the thermoelastic properties of high-pressure minerals. Phys. Earth Planet. Int. 96, 85 112. Knoche, R., Webb, S.L., Rubie, D.C., 1995. Ultrasonic velocity to 10 GPa and 1500 C in the multi-anvil press: measurements in polycrystalline olivine. Eos 76, 563. Kumazawa, M., Anderson, O.L., 1969. Elastic moduli, pressure derivatives, and temperature derivatives of single-crystal olivine and single crystal forsterite. J. Geophys. Res. 74, 5961 5972. Kung, J., Li, B., Weidner, D.J., Zhang, J., Liebermann, R.C., 2002. Elasticity of (Mg 0.83,Fe 0.17 )O ferropericlase at high pressure: ultrasonic measurements in conjunction with X-radiation techniques. Earth Planet. Sci. Lett. 203, 557 566. Li, B., Kung, J., Liebermann, R.C., 2004. Modern techniques in measuring elasticity of Earth materials at high pressure and high temperature using ultrasonic interferometry in conjunction with synchrotron X-radiation in multi-anvil apparatus. Phys. Earth Planet. Int. 143 144, 559 574. Li, B., Liu, W., 2006. Determination of P and T cross-derivatives for mantle minerals, in preparation. Liu, W., Kung, J., Li, B., 2005. Elasticity of San Carlos olivine to 8 GPa and 1073 K. Geophys. Res. Lett. 32, L16301. Meng, Y., Weidner, D.J., Gwanmesia, G.D., Liebermann, R.C., Vaughan, M.T., Wang, Y., Leinenweber, K., Pacalo, R.E., Yeganeh- Haeri, A., Zhao, Y., 1993. In situ P-T X-ray diffraction studies on three polymorphs (α, β, γ) ofmg 2 SiO 4. J. Geophys. Res. 98, 22199 22207. Nishihara, Y., Takahashi, E., Matsukage, K.N., Iguchi, T., Nakayama, K., Funakoshi, K., 2004. Thermal equation of state of (Mg 0.91 Fe 0.09 ) 2 SiO 4 ringwoodite. Phys. Earth Planet. Int. 143 144, 33 46. Shim, S., Duffy, T.S., Shen, G., 2000. The stability and P-V-T equation of state of CaSiO 3 perovskite in the Earth s lower mantle. J. Geophys. Res. 105, 25955 25968. Speziale, S., Zha, C.-S., Duffy, T.S., Hemley, R.J., Mao, H., 2001. Quasi-hydrostatic compression of magnesium oxide to 52 GPa: implications for the pressure-volume-temperature equation of state. J. Geophys. Res. 106, 515 528. Stixrude, L., Bukowinski, M.S.T., 1990. Fundamental thermodynamic relations and silicate melting with implications for the constitution of D. J. Geophys. Res. 95, 19311 19325. Suzuki, I., 1975. Thermal expansion of periclase and olivine and their anharmonic properties. J. Phys. Earth 23, 145 159. Wang, Y., Weinder, D.J., Guyot, F., 1996. Thermal equation of state of CaSiO 3 perovskite. J. Geophys. Res. 101, 661 672. Webb, S.L., 1989. The elasticity of the upper mantle orthosilicates olivine and garnet to 3 GPa. Phys. Chem. Minerals 16, 684 692. Weidner, D.J., Vaughan, M.T., Ko, J., Wang, Y., Liu, X., Yeganeh- Haeri, A., Pacalo, R.E., Zhao, Y., 1992. Characterization of stress, pressure and temperature in SAM85, a DIA type high pressure apparatus. In: Syono, Y., Manghnani, M.H. (Eds.), High-Pressure Research: Application to Earth and Planetary Sciences, Geophysical Monograph, vol. 67. Terra Scientific Publishing Company and Am. Geophys. Union, Tokyo and Washington, DC, pp. 13 17. Zha, C.-S., Duffy, T.D., Downs, R.T., Mao, H.K., Hemley, R.J., 1998. Brillouin scattering and X-ray diffraction of San Carlos olivine: direct pressure determination to 32 GPa. Earth Planet. Sci. Lett. 159, 25 33. Zaug, J., Abramson, E.H., Brown, J.M., Slutsky, L.J., 1993. Sound velocities in olivine at Earth mantle pressures. Science 260, 1487 1489.