Thermal equation of state of stishovite

Physics of the Earth and Planetary Interiors 112 Ž1999. 257–266

Thermal equation of state of stishovite Jun Liu a

a,)

, Jianzhong Zhang b, Lucy Flesch a , Baosheng Li b, Donald J. Weidner a , Robert C. Liebermann a

Center for High Pressure Research 1 and Department of Geosciences, State UniÕersity of New York at Stony Brook, Stony Brook, NY 11794-2100, USA b Center for High Pressure Research and Mineral Physics Institute, State UniÕersity of New York at Stony Brook, Stony Brook, NY 11794-2100, USA Received 1 June 1998; received in revised form 9 January 1999; accepted 9 January 1999

Abstract The pressure–volume–temperature Ž P–V–T . behavior in SiO 2-stishovite has been studied using a DIA-type, cubic-anvil apparatus ŽSAM85. interfaced with in situ synchrotron X-ray diffraction at the superconducting wiggler beamline ŽX-17B. of the National Synchrotron Light Source, using a polycrystalline specimen previously hot-pressed in a 2000-ton uniaxial split-sphere apparatus ŽUSSA-2000.. The P–V–T data up to 10 GPa and 1273 K were analyzed using several approaches based on the Birch–Murnaghan equation-of-state ŽEOS. and related thermodynamic relations. The results obtained from different approaches are internally consistent. With the pressure derivative K TX 0 fixed at the ultrasonically determined value of K 0X s 5.3, we obtained the isothermal bulk modulus K TX 0 s 294Ž2. GPa and its temperature derivative ŽEK TrET .P s y0.041Ž11. GPa Ky1, and the thermal expansion coefficients a0 s 1.40Ž8. = 10y5 Ky1 and a1 s 10.9Ž20. = 10y9 Ky2 . The room temperature bulk modulus Ž K T0 . of the present study is in agreement with the adiabatic values obtained previously from Brillouin scattering and diamond-anvil cell X-ray diffraction studies on single crystal stishovite and ultrasonic studies on polycrystalline specimens. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Thermal equation; Stishovite; Pressure–volume–temperature

1. Introduction Stishovite, the rutile-structure polymorph of SiO 2 , is stable at pressures above 9 GPa and is a potential component of petrological phase assemblages in the transition zone of the Earth’s mantle. As such, its elastic and thermodynamic properties are important for interpretations of seismic observations in this )

Corresponding author CHiPR: a NSF Science and Technology Center for High Pressure Research. 1

region of the Earth’s interior. With its remarkable physical properties Že.g., low compressibility, large elastic anisotropy and high yield strength., stishovite presents an extreme challenge to experimentalists trying to determine its elastic behavior at high pressures and high temperatures. Previous acoustic and compression studies have constrained the bulk modulus K 0 of stishovite and its pressure derivative K 0X Že.g., Bassett and Barnett, 1970; Ito et al., 1974; Sato, 1977; Weidner et al., 1982; Sugiyama et al., 1987; Ross et al., 1990; Li et al., 1996.. But the temperature dependence of K 0 as well as the behav-

0031-9201r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 1 - 9 2 0 1 Ž 9 9 . 0 0 0 3 7 - 0

258

J. Liu et al.r Physics of the Earth and Planetary Interiors 112 (1999) 257–266

ior of the cell parameters at eleÕated temperatures have not been well studied. Recently, Suito et al. Ž1996. reported thermal expansion measurements of stishovite at high pressure using synchrotron radiation; however, serious experimental difficulties result in unproven assumptions being made in their analysis. Thus, the results of Suito et al. Ž1996. may be compromised. We have performed a study of the P–V–T behavior of stishovite over the temperature range from 300 K to 1273 K and pressures from 1 bar to ; 10 GPa under nearly hydrostatic stress conditions. In this paper, we report our new P–V–T data for stishovite, compare our data with previous elasticity studies, and determine for the first time the temperature derivative of the bulk modulus ŽEK TrET .P . The cause for the discrepancy between the results of Suito et al. Ž1996. and most other previous studies is also briefly discussed.

2. Experimental techniques 2.1. The specimen A polycrystalline specimen of stishovite was synthesized in a 2000-ton uniaxial split-sphere apparatus ŽUSSA-2000. ŽLiebermann and Wang, 1992. using Puratronic amorphous silica as the starting material. The specimen used in this study was hot-pressed under pressure of 14 GPa and temperature of ; 1300 K for 30 min. The hot-pressed specimen was single phase Žimpurity less than 0.5 wt.%., fully dense Ždensity 4.274 g cmy3 ; porosity less than 0.5%., fine-grained Žgrain size 2–5 mm. and elastically isotropic Žsee Li et al., 1996.. The advantage of using well-sintered polycrystalline instead of powder specimens in the X-ray diffraction experiments is that the former are free of intragranular stresses and thus, are less likely to exhibit diffraction line broadening when compressed at room temperature ŽWeidner et al., 1994; Li et al., 1996.. 2.2. In situ X-ray diffraction experiment Our P–V–T measurements were conducted in a DIA-type, cubic anvil apparatus ŽSAM85. described by Weidner et al. Ž1992. which was installed on the

superconducting wiggler beamline ŽX17B. at National Synchrotron Light Source ŽNSLS. of the Brookhaven National Laboratory. SAM85 consists of six tungsten carbide anvils with 3.5 mm = 3.5 mm tapered square truncations which form a cubic cavity in which the cell assembly was inserted Žas described in Wang et al., 1994.. Amorphous boron epoxy was used as the pressure medium, and graphite, the heating element; temperature was monitored with a WrRe%24–WrRe%6 thermocouple inserted in the center of the cell assembly. A piece Ž0.5 mm. of the polycrystalline stishovite was embedded in NaCl powder, which also acted as both the pressure medium and an internal pressure standard. Cell pressures were determined by fitting the P–V–T data of NaCl using the equation of state ŽEOS. of Decker Ž1971.. The uncertainty of the pressure measurements from cell volume is about 0.1 GPa. The multichannel analyzer used in data collection was calibrated with the characteristic decay energies of radioactive standards 57Co, 129 I and 109Cd. The diffraction angle Ž2 u . was set to about 7.58 and was further calibrated; energy dispersive X-ray diffraction data were collected at this fixed 2 u by focusing the highly collimated X-ray beam Ž100 = 200 mm. into the sample. Data collection usually took 2–3 min at each P–T condition. At the beginning and the end of each experiment, a mixture of Si, Al 2 O 3 and MgO was X-rayed and the diffraction peaks of these standards are used to calibrate the 2 u angle. This standard mixture and the 1-bar stishovite and NaCl data are also used to provide a direct calibration of the channel number and d-spacing. The standard error of the calibration is in the order of 0.0005 to ˚ ŽWang et al., 1994.. 0.0008 A Peak positions were obtained by Gaussian peakfitting of the diffraction profiles. Unit-cell parameters for NaCl and stishovite were calculated by least-squares fitting of the peaks based on the symmetry of the materials Žcubic for NaCl, tetragonal for stishovite.. Five diffraction lines of NaCl, Ž111., Ž200., Ž220., Ž222. and Ž420. were used for pressure determination, and eight peaks of stishovite, Ž110., Ž101., Ž111., Ž210., Ž211., Ž220., Ž301. and Ž112. were usually used for determination of its unit-cell parameters. For stishovite, the relative standard deviations in determinations of unit-cell volume were generally less than 0.05%.

J. Liu et al.r Physics of the Earth and Planetary Interiors 112 (1999) 257–266

In Fig. 1, we show the diffraction patterns in SAM85 at ambient conditions and after compression to 10 GPa at room temperature; note that the peaks at 10 GPa are just as sharp as those prior to compression. In contrast, peaks from a powder specimen under the same conditions Žtop spectrum in Fig. 1 from Li et al., 1996. exhibit pronounced line broadening due to the microscopic deviatoric stress produced when the powder is compressed within the NaCl medium ŽWeidner et al., 1994.. Two separate runs ŽST15 and ST19. have been performed using the same cell design and materials and similar P–T paths. As we will see later, the two runs gave consistent results for the bulk modulus and its temperature derivative. In the first pilot run ŽST15., fewer data were collected and there is some uncertainty in the volume at zero pressure. For these reasons, we treat Run ST15 as complementary to Run ST19 and focus on the latter run in our data analysis. The P–T space which we could explore in these experiments was limited by the capabilities of SAM85 with 3.5 mm tapered anvil truncation Žabout

Fig. 1. X-ray diffraction spectra of stishovite in SAM 85 at ambient conditions and at high pressure. At room temperature, the diffraction peaks from the polycrystalline sample Žbottom two profiles. remain sharp in compression, whereas the powder sample Žtop profile. exhibits pronounced peak broadening Žsee also Li et al., 1996..

259

Fig. 2. Pressure and temperature range of data coverage in experiment ST19. Dashed curve is the kinetic boundary between coesite and stishovite ŽZhang et al., 1996.. Arrows and numbers indicate the chronological P – T path of data collection.

11 GPa. and by the kinetic phase boundary between stishovite and coesite ŽLiu et al., 1996; Zhang et al., 1996.. Our experiments were performed in the stablermetastable field of stishovite as shown in Fig. 2; no back transformation of stishovite to coesite was detected in the course of the experiments. The experiments were carried out along a specific path in the P–T space as discussed below Žalso see numerical sequence in Fig. 2.; during acquisition of the diffraction data for stishovite and NaCl, the pressure and temperature were kept constant. Pressure was first increased at room temperature to about 11 GPa Žsegment 1 to 2 in Fig. 2. while the cell pressure was monitored. Temperature was then increased to 870 K Ž2 to 3. to eliminate the macroscopic deviatoric stress due to the non-liquid NaCl pressure medium and cylindrical sample symmetry ŽWeidner et al., 1992.. Notice that cell pressure has decreased by about 1 GPa in this first heating. Data were collected while decompressing at constant temperature Ž873 K. to about 6.9 GPa Ž3 to 4., after which the temperature was decreased to room temperature under constant ram load Ž4 to 5.. After a dataset at room temperature was collected, the temperature was increased to 673 K Ž5 to 6. and then the cell pressure was increased at constant temperature to about 9 GPa Ž6 to 7.. Finally, the temperature was increased from 673 K to 1273 K at constant ram load

260

J. Liu et al.r Physics of the Earth and Planetary Interiors 112 (1999) 257–266

Ž7 to 9.. The subsequent diffraction data were collected at 200 K intervals, on cooling at constant ram load Ž9 to 19.. After reaching room temperature in each cycle, the oil pressure on the ram was decreased to lower the cell pressure by about 1 GPa, and then the temperature was increased to a maximum Žsee paths 11, 13, 15. and again, P–V–T data were collected on another cooling cycle. Increasing the temperature of the experimental runs at constant ram load changes the cell pressure as the consequence of competing phenomena: Ž1. the thermal expansion of the cell materials increases the cell pressure if the cell volume is constant Žsometimes referred to as the thermal pressure.; and Ž2. relaxation and extrusion of the cell materials Že.g., boron nitride, NaCl, alumina, pyrophyllite, etc.. lead to a reduction in cell pressure. As can be seen in Fig. 2, the result of these combined effects is that the cell pressure decreases at constant ram load Žsee 2 to 3 and 8 to 9. on the first heating to a given peak temperature. On cooling, the cell pressure always decreases due to the loss of the thermal pressure Že.g., 4 to 5 and 9 to 10.; subsequent heating leads to a recovery of this thermal pressure Že.g., 5 to 6 and 7 to 8.. All the data used in the following P–V–T analyses were obtained along isothermal decompression Ž3 to 4. or compression Ž6 to 7. paths at temperature above 573 K, or along cooling paths at constant ram load Ž9 to 10, etc.. To verify that our data were obtained under conditions of minimum macroscopic deviatoric stress, we used the NaCl peaks to calculate the differential stress Ž s 1 y s 3 . following the procedures described by Weidner et al. Ž1992.. The results show that the pressure environment of the sample is nearly hydrostatic with the deviatoric stress less than 0.04 GPa under most of the P–T conditions Žsee Fig. 3.. Data collected while pressurizing before heating show higher deviatoric stress and are not included in P–V–T data analysis.

3. Data analysis of RUN ST19 3.1. Room temperature data Experimental results of the present study for stishovite are summarized in Tables 1 and 2; Fig. 4.

Fig. 3. Deviatoric stress Ž s 1 y s 3 . in the cell determined from peaks of NaCl. Dashed lines indicate the 0.04 GPa range.

The unit cell volume of stishovite obtained at ambi˚ 3, as ent conditions in this experiment is 46.540Ž19. A 3 ˚ compared with 46.615 A in the study by Ross et al. Ž1990. in a static compression study using a single crystal specimen and a diamond anvil cell with liquid pressure medium ŽFig. 5.. The pressure– volume curve calculated using the Birch–Murnaghan EOS: y Ž V0rV .

5r3

= 1 q 3r4 Ž K TX 0 y 4 . Ž V0rV .

3r2

P s 3r2 K T 0 Ž V0rV .

½

7r3

y1

5;

Ž 1.

is plotted as a curve using the isothermal bulk modulus K T 0 s 302 GPa Žconverted from the adiabatic modulus K s . and its pressure derivative K TX 0 s 5.3 derived from the ultrasonic study of Li et al. Ž1996., which used a similar polycrystalline specimen of stishovite as in the current study. The agreement between these three sets of experimental data is excellent. The single data point of Suito et al. Ž1996. at P s 10.5 GPa is clearly discordant with all of the other experimental data; we note that these authors analyzed their data under the assumption that cell pressure at high temperature is unchanged from the room-temperature value Žunlike the dependence we observe, as seen in Fig. 2.. The ratios of the axial cell parameters, ara0 , crc0 and the cell volume VrV0 , are plotted against pressure at room temperature in Fig. 6. If we assume a linear relationship between these parameters and pressure, we obtain mean axial compressibilities of b a s 1.24Ž4. = 10y3 GPay1 and bc s 0.65Ž4. = 10y3 GPay1 between 1 bar and 7.25 GPa. These values are in excellent agreement with those deter-

J. Liu et al.r Physics of the Earth and Planetary Interiors 112 (1999) 257–266 Table 1 Cell parameters Ž a,c . and volume as a function of pressure Ž P . and temperature ŽT . for Run ST15 and Run ST19 for stishovite P ŽGPa.

T ŽK.

˚ 3. Vcell ŽA

˚. c ŽA

˚. a ŽA

Run ST15 8.40 6.91 8.08 6.42 4.02 9.79 7.77 6.01 3.76 7.44 5.68 3.38 7.21 5.66 5.41 3.80 1.75 y0.03

1071 1080 870 873 876 675 672 671 673 473 472 472 303 303 308 306 303 303

45.820Ž2. 46.095Ž2. 45.733Ž2. 45.989Ž3. 46.277Ž2. 45.332Ž2. 45.616Ž2. 45.863Ž2. 46.149Ž2. 45.516Ž1. 45.779Ž2. 46.065Ž2. 45.438Ž3. 45.706Ž2. 45.700Ž1. 45.972Ž1. 46.487Ž2. 46.503Ž2.

2.659Ž1. 2.665Ž1. 2.658Ž1. 2.664Ž1. 2.665Ž1. 2.650Ž1. 2.656Ž1. 2.661Ž1. 2.660Ž1. 2.660Ž1. 2.657Ž1. 2.659Ž1. 2.653Ž1. 2.659Ž1. 2.658Ž1. 2.659Ž1. 2.669Ž1. 2.667Ž1.

4.151Ž1. 4.159Ž2. 4.148Ž2. 4.155Ž3. 4.167Ž1. 4.136Ž1. 4.144Ž1. 4.152Ž2. 4.165Ž1. 4.141Ž1. 4.151Ž1. 4.165Ž1. 4.139Ž2. 4.146Ž1. 4.147Ž1. 4.158Ž1. 4.174Ž2. 4.175Ž1.

Run ST19 0.00 6.03 7.25 6.23 6.39 5.50 4.52 3.24 2.26 1.14 7.51 5.72 4.75 3.53 2.54 1.40 6.57 7.10 7.64 8.40 9.23 6.00 5.18 3.88 3.10 1.76 9.18 8.23 7.63 6.90 7.35

298 301 304 302 303 302 301 300 300 300 473 473 472 472 473 473 673 673 673 673 673 673 673 673 673 673 872 873 873 873 872

46.540Ž2. 45.614Ž2. 45.482Ž3. 45.598Ž3. 45.610Ž3. 45.733Ž3. 45.868Ž2. 46.026Ž2. 46.199Ž1. 46.379Ž2. 45.559Ž3. 45.799Ž3. 45.953Ž3. 46.128Ž3. 46.277Ž3. 46.477Ž2. 45.847Ž3. 45.738Ž2. 45.684Ž2. 45.571Ž3. 45.438Ž2. 46.913Ž3. 46.055Ž2. 46.234Ž2. 46.365Ž2. 46.559Ž2. 45.664Ž1. 45.792Ž4. 45.879Ž2. 45.989Ž2. 45.916Ž3.

2.667Ž1. 2.658Ž1. 2.656Ž1. 2.657Ž1. 2.657Ž1. 2.659Ž1. 2.660Ž1. 2.664Ž1. 2.666Ž1. 2.666Ž1. 2.658Ž1. 2.659Ž1. 2.662Ž1. 2.665Ž1. 2.667Ž1. 2.668Ž1. 2.662Ž1. 2.655Ž1. 2.657Ž1. 2.658Ž1. 2.651Ž1. 2.659Ž1. 2.663Ž1. 2.665Ž1. 2.668Ž1. 2.669Ž1. 2.659Ž1. 2.661Ž1. 2.662Ž1. 2.663Ž1. 2.663Ž2.

4.177Ž1. 4.146Ž1. 4.139Ž1. 4.143Ž1. 4.143Ž1. 4.147Ž1. 4.153Ž1. 4.157Ž1. 4.163Ž0. 4.171Ž1. 4.141Ž1. 4.150Ž1. 4.155Ž1. 4.160Ž1. 4.166Ž0. 4.173Ž1. 4.150Ž1. 4.150Ž1. 4.146Ž1. 4.141Ž1. 4.140Ž1. 4.154Ž1. 4.159Ž1. 4.165Ž1. 4.169Ž0. 4.177Ž1. 4.145Ž1. 4.148Ž2. 4.152Ž1. 4.155Ž1. 4.153Ž2.

261

Table 1 Žcontinued. P ŽGPa.

T ŽK.

˚ 3. Vcell ŽA

˚. c ŽA

˚. a ŽA

Run ST19 6.43 9.35 8.38 7.71 8.81

875 1071 1072 1074 1272

46.072Ž3. 45.792Ž2. 45.937Ž2. 46.038Ž2. 46.049Ž2.

2.664Ž1. 2.661Ž1. 2.662Ž1. 2.663Ž1. 2.663Ž1.

4.159Ž2. 4.149Ž0. 4.154Ž1. 4.158Ž1. 4.158Ž1.

Numbers in parentheses are uncertainties in the last digit.

mined in the compression study made by Ross et al. Ž1990. Ž b a s 1.28 = 10y3 GPay1 , bc s 0.65 = 10y3 GPay1 , over the pressure range of 0–5 GPa. and calculated from the single crystal elastic compliances Ž b a s 1.2 = 10y3 GPay1 , bc s 0.60 = 10y3 GPay1 . measured at ambient conditions by Weidner et al. Ž1982. using Brillouin spectroscopy on a single crystal of stishovite. The a-axis is, thus, about twice as compressible as the c-axis, confirming the results of many previous studies Že.g., Sato, 1977; Weidner et al., 1982; Sugiyama et al., 1987; Ross et al., 1990.. The room T data are analyzed using the Birch– Murnaghan EOS ŽEq. Ž1.. in several different ways: ˚ 3, obtain Ža. Fix K TX s 5.3 and V0 s 46.540 A 0 K T 0 s 294Ž2. GPa; Žb. Fix K TX s 5.3 and obtain K T s 294Ž2. GPa 0 0 ˚ 3; and V0 s 46.546Ž12. A Žc. Vary all three parameters and obtain K T s 0 ˚ 3. 295Ž12. GPa, K TX 0 s 5Ž4., and V0 s 46.547Ž12. A As can be clearly seen, the pressure derivative of the bulk modulus K TX 0 is not well-resolved from our P–V data due to the limited range of pressure investigated and the low compressibility of stishovite. Thus, our preferred value is K T 0 s 294Ž2. GPa with K TX 0 s 5.3, and V0 fixed at the value calibrated both before and after the experimental run. All subsequent analyses will utilize this ultrasonic value of K TX 0 . We did similar analyses with the room temperature data of Run ST15. By fixing K TX 0 to 5.3, we ˚ 3. The obtain K T 0 s 297Ž5. GPa and V0 s 46.52Ž2. A agreement of the K T 0 and V0 values for the two independent runs demonstrates the reproducibility in our P–V experiments. 3.2. High temperature data We used several approaches to analyze the P–V–T data: Ž1. treating the whole dataset with a high-tem-

J. Liu et al.r Physics of the Earth and Planetary Interiors 112 (1999) 257–266

262

Table 2 Summary of equation of state parameters for stishovite from analysis of room-temperature P–V data and comparison with previous work Cell volume

Bulk modulus

˚3. V0 ŽA

KT0

46.496 46.566 – – 46.591 46.615 – 46.52Ž3. 46.54Ž2.

300 – 298Ž5. 306Ž4. a 313Ž4. 313Ž4. a 302 a 297Ž5. 294Ž5.

ŽGPa.

Reference: Authors: type of work ŽEK T rEP .T s K 0X 0 – – 0.7Ž11. – – 1.7Ž6. 5.3Ž1. w5.3x w5.3x

Bassett and Barnett Ž1970.: powder X-ray Ito et al. Ž1974.: powder X-ray Sato Ž1977.: powder X-ray ŽCA. Weidner et al. Ž1982.: Brillouin, single crystal Sugiyama et al. Ž1987.: X-ray, single crystal ŽDA. Ross et al. Ž1990.: X-ray, single crystal ŽDA. Li et al. Ž1996.: ultrasonic, polycrystal This study: X-ray, polycrystal ŽCA., Run ST15 This study: X-ray, polycrystal ŽCA., Run ST19

ŽDA. Diamond anvil. ŽCA. Cubic anvil. Ž – . Not measured. a Adiabatic bulk modulus.

perature version of the Birch–Murnaghan EOS; analyzing the data for constant temperatures; and analyzing the data for constant pressures.

Ž2. Ž3.

VT 0 s V0 exp

3.3. High temperature P–V–T EOS (Birch– Murnaghan) In the first approach, Eq. Ž1. is modified to take into account the effects of temperature dependence of the bulk modulus: K T s K T 0 q Ž EK TrET . P Ž T y 300 . ,

and the cell Õolume T

žH

a Ž T . dT

300

/

,

Ž 3.

where a ŽT . is the volume thermal expansivity and can be expressed as:

a Ž T . s a0 q a1T . By fitting the whole dataset to Eq. Ž1. in its high-temperature modification, we can obtain best

Ž 2.

˚ 3 . vs. pressure. Solid line is the Fig. 4. Unit cell volume Žin A high-temperature Birch–Murnaghan EOS fit to the whole data set.

Fig. 5. Comparison of room temperature V r V0 data with those of Ross et al. Ž1990. obtained on single crystal stishovite in diamond anvil cell and liquid pressure medium. Also shown is the Birch– X Murnaghan trajectory calculated with K T 0 s 302 GPa and K T 0 s 5.3 from Li et al. Ž1996.. The data point of Suito et al. Ž1996. at P s10.5 GPa and 293 K is also plotted Žusing V0 s14.021 cm3 moly1 , Ito et al., 1974..

J. Liu et al.r Physics of the Earth and Planetary Interiors 112 (1999) 257–266

263

conditions, we fix K T 0 s 294Ž2. GPa, K TX 0 s 5.3, and ˚ 3, and obtain ŽEK TrET .P s y0.041Ž11. V0 s 46.54 A y1 GPa K , a 0 s 1.40Ž8. = 10y5 Ky1 and a1 s 10.9Ž20. = 10y9 Ky2 . Our thermal expansion data based on the high-T Birch–Murnaghan EOS analyses are compatible with those of Ito et al. Ž1974. as shown in Fig. 7. For run ST15, if we fix K T 0 s 297 GPa and ˚ 3, ŽEK TrET .P K TX 0 s 5.3, we obtain V0 s 46.51Ž1. A y1 s y0.034Ž26. GPa K , a 0 s 8Ž3. = 10y6 Ky1 and a1 s 6.9Ž33. = 10y9 Ky2 . Thus, the thermoelastic parameters obtained from analysis of the two separate experimental runs are in agreement within their mutual uncertainties. Fig. 6. Comparison of ar a 0 , cr c 0 and V r V0 at room temperature with those calculated from the data of Weidner et al. Ž1982. and Li et al. Ž1996.. It should be pointed out that a direct comparison between the linear compressibilities as functions of pressure of this study and those calculated from b a and bc of Weidner et al. Ž1982. is not possible because b a and bc in Weidner et al. Ž1982. were determined at 1 bar pressure. Nevertheless, we see in this figure a consistency between the two studies.

fits for the variables K T 0 a 0 , a1 and ŽEK TrET .P . We assume that K TX 0 does not change with temperature. If we let all other parameters vary in the fit, we obtain very large uncertainties in both K T 0 and ŽEK TrET .P . Since K T has been obtained from room 0 temperature data, and V0 is measured at ambient

3.4. Isothermal compression If we treat the data along various isotherms in Fig. 4 as separate datasets, we can calculate the bulk modulus Ž K T 0 . and cell volume Ž VT 0 . for each temperature at zero pressure wunder the assumption that the pressure derivative of the bulk modulus K TX 0 is independent of temperaturex. Although there are only a few data at high temperature and lower pressure, and thus there may be large uncertainties in K T 0 and VT 0 , this approach of analyzing data provides a check for self-consistency of the dataset. In Table 3, we show the results of fits of such isothermal analyses, from which we obtain ŽEK TrET .P s y0.042Ž25. GPa Ky1 ; this value is consistent with the results of the high-temperature Birch–Murnaghan approach. The zero pressure cell volume data are also compatible with that of Ito et al. Ž1974. ŽFig. 7.. 3.5. Isobaric thermal expansion A subset of the P–V–T data in Table 1 and Fig. 4 at 7.6 GPa also allows us to estimate the temperature Table 3 Values of bulk modulus Ž K T 0 . obtained by fitting isothermal datasets from Run ST19 at various temperatures ŽT .

Fig. 7. Comparison of our high-temperature Birch–Murnaghan EOS Žhigh-T B–M. and isothermal fit results with the thermal expansion data of Ito et al. Ž1974..

T ŽK.

K T ŽGPa.

˚ 3. Vcell ŽA

298 473 673 873 1073

292Ž5. 284Ž2. 277Ž4. 270Ž8. 260Ž15.

46.544 46.69Ž1. 46.87Ž1. 47.11Ž3. 47.33Ž7.

J. Liu et al.r Physics of the Earth and Planetary Interiors 112 (1999) 257–266

264

3.6. Thermal pressure The thermal pressure Pth of a solid is the excess pressure produced when the solid is heated to high temperature at constant volume. Although the present experiment does not allow us to measure the thermal pressure directly, the thermal pressure can be calculated via our P–V–T data in Table 1 from calculations of the pressure difference between the isothermal curves ŽFig. 4. as a function of volume. As shown in Fig. 8, Pth increases linearly with T at constant V, but is independent of V at constant T. Following the thermodynamic relation of Anderson Ž1987., Pth can be calculated by: Pth s P Ž V ,T . y P Ž V ,300 . Fig. 8. Thermal pressure vs. temperature from 298 K to 1273 K. The thermal pressure increases linearly with T and is thus independent of volume Žand thus pressure. at a given temperature.

s a K T Ž V0 ,T . q Ž d K TrdT . V ln Ž V0rV . = Ž T y 300 . .

From the best fit of thermal pressure data to Eq. Ž5., using the volume data of Fig. 4, we obtain the average value of a K T Ž V300 ,T . and ŽEK TrET . V which are, respectively, 0.0058Ž19. GPa Ky 1 and y0.016Ž13. GPa Ky1 . Within the uncertainty of the fit, ŽEK TrET . V is roughly zero, indicating that the thermal pressure is independent of volume, as is also shown in Fig. 8. From thermodynamic relation:

derivative of the bulk modulus ŽEK TrET .P via the following thermodynamic identity:

Ž EarEP . T s Ž EK TrET . P Ky2 T0 .

Ž 5.

Ž 4.

Using Eq. Ž3. with V0 at 7.6 GPa fixed at 45.44

˚ 3, extrapolated from the room T data of Fig. 4, we A obtain an average thermal expansivity a s 1.77 = 10y5 Ky1 in the range of 300 K–1073 K. From comparison with the average thermal expansion at 1 bar pressure a s 2.13 = 10y5 Ky1 , we find ŽEarEP .T s y4.7 = 10y7 Ky1 GPay1, assuming a 0 linear relationship between a and P. From the thermodynamic identity, ŽEK TrET .P is calculated to be y0.041 GPa Ky1 , from Eq. Ž4., in good agreement with that derived above using the high-temperature Birch–Murnaghan equation.

Ž EK TrET . V s Ž EK TrET . P q Ž EK TrEP . T a K T Ž V0 ,T . ,

Ž 6.

K TX 0 s 5.3, y1

with K T 0 s 294 GPa, we obtain ŽEK Tr . Ž . ET P s y0.047 23 GPa K at zero-pressure, which is in agreement with previous analyses using the high-temperature Birch–Murnaghan EOS, isothermal compression and isobaric thermal expansion approaches.

Table 4 Summary of calculation of the Anderson–Gruneisen parameter Ž d T . Pressure ŽGPa.

Temperature ŽK.

dT

a = 10 5 ŽKy1 .

KT ŽGPa.

ŽEK T rET .Pa 0 ŽGPa Ky1 .

0 0 7.6 7.6

300 1073 300 1073

9.9 7.0 7.8 5.4

1.40 2.24 1.58 2.50

294 262 334 303

y0.041 y0.041 y0.041 y0.041

a

d T Values are calculated under the assumption that ŽEK T 0rET .P s y0.041 GPa Ky1 and is independent of both temperature and pressure.

J. Liu et al.r Physics of the Earth and Planetary Interiors 112 (1999) 257–266

265

Table 5 Summary of analyses of P–V–T data using various approaches Approach

ŽEK T rET .P 0 ŽGPa Ky1 .

EarEP ŽKy1 GPay1 .

a0 ŽKy1 .

a1 ŽKy2 .

High-T Birch–Murnaghan ŽST15. High-T Birch–Murnaghan ŽST19. Isothermal compression Isobaric thermal expansion Thermal pressure

y0.034Ž26. y0.041Ž11. y0.042Ž3. Žy0.041. y0.047Ž23.

Žy4.7 = 10y7 . Žy3.9 = 10y7 . Žy4.9 = 10y7 . y4.7 = 10y7 y5.4 = 10y7

8.0Ž30. = 10y6 1.40Ž8. = 10y5 – – –

6.9Ž30. = 10y9 10.9Ž20. = 10y9 – – –

Values of parameters in parentheses are calculated using Eq. Ž4..

3.7. Anderson–Gruneisen parameter (d T ) The Anderson–Gruneisen parameter can be calculated using our results for a and ŽEK TrET .P with the relation between d T , a and ŽEK TrET .P Že.g., Anderson, 1987.:

d T s y1r Ž a K T . Ž EK TrET . P .

Ž 7.

At 1 bar, our data yield values of d T s 9.9 at 300 K and 7.0 at 1073 K, which are within the range of values for other oxides and silicates summarized by Anderson et al. Ž1990.. At P s 7.6 GPa and T s 300 ˚ 3 ; from Eq. Ž3., a is calculated to K, VP is 45.44 A y5 be 1.58 = 10 Ky1. Assuming ŽEK TrET .P s y1 y0.041 GPa K and independent of pressure, we get d T s 7.8. Table 4 summarizes our calculation of d T and parameters used at different P and T conditions. In contrast, Suito et al. Ž1996. obtained large d T s 55 for P s 10.5 GPa at room temperature and an extremely large temperature dependence of d T ; this is because they obtained an anomalously large cell volume at 10.5 GPa Žsee Fig. 5., which results in a small value for the thermal expansion, a .

4. Discussion and conclusion Generally, the dataset collected in Run ST19 is self-consistent, as is shown by the consistency among the high-temperature Birch–Murnaghan EOS, isothermal and isobaric treatments. Our data are also consistent with previous single crystal hydrostatic compression and Brillouin spectroscopy studies, and with the independent P–V–T run ŽST15.. Different approaches of data analysis ŽTable 5. yield very consistent ŽEK TrET .P values, all close to y0.041

GPa Ky1 , which is close to the one estimated by Watanabe Ž1982. Žy0.047 GPa Ky1 . based on thermodynamic assessments. Suito et al. Ž1996. obtained compression data of stishovite which showed a big discrepancy as compared with previous studies ŽSato, 1977; Ross et al., 1990.. This is because they had overestimated cell pressure by about 2–3 GPa after temperature had been decreased from 1473 K to 300 K; after the temperature is decreased, the cell pressure should have decreased as well because of the relaxation of the thermal pressure, as is indicated by the cooling cycles of our experiments in Fig. 2. This is the first time that the temperature derivative of bulk modulus of stishovite was determined from volume measurements at high pressure and temperature. The Žd KrdT .P value from this study is very similar to that estimated by Watanabe Ž1982. and thus, has little effect on the thermodynamic calculations of the coesite–stishovite phase boundary of Liu et al. Ž1996. based on the calorimetric measurements. Our 1 bar pressure d T values Ž6.4–9.9. fall in the range of those for oxides for temperature below Debye temperature ŽAnderson et al., 1990..

Acknowledgements We are grateful to Prof. Ian Jackson for his advice in our data analyses. We thank Michael Vaughan, and Kenneth Baldwin for their technical assistance in X-ray diffraction experiments and Greg Symmes for his help in scanning electron microscopy ŽSEM. characterization of stishovite samples. This work was carried out at the X17B1 beamline of NSLS at Brookhaven National Laboratory. We also wish to

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thank J.B. Hastings and D.P. Siddons for their technical support at the beamline. We also thank reviewers Francois Guyot and Donald Isaak for their constructive comments. The Stony Brook High Pressure Laboratory is jointly supported by the State University of New York at Stony Brook and the NSF Science and Technology Center for High Pressure Research ŽEAR 89-20239.. This research is also supported by the NSF Division of Earth Sciences under grants to RCL ŽEAR 93-04502 and 96-14612.. MPI Contribution No. 250.

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