Silicate perovskite analogue ScAlO3: temperature dependence of elastic moduli

Physics of the Earth and Planetary Interiors 120 Ž2000. 299–314 www.elsevier.comrlocaterpepi

Silicate perovskite analogue ScAlO 3: temperature dependence of elastic moduli Jennifer Kung a,b,) , Sally M. Rigden a,c , Ian Jackson a b

a Research School of Earth Sciences, Australian National UniÕersity, Canberra, ACT 0200 Australia Department of Geosciences, State UniÕersity of New York at Stony Brook, Stony Brook, NY, 11794-2100 USA c Molecular Mining Corporation, Kingston, Ontario, Canada

Received 8 February 2000; accepted 17 April 2000

Abstract The elastic moduli of ScAlO 3 perovskite, a very close structural analogue for MgSiO 3 perovskite, have been measured between 300 and 600 K using high precision ultrasonic interferometry in an internally heated gas-charged pressure vessel. This new capability for high temperature measurement of elastic wave speeds has been demonstrated on polycrystalline alumina. The temperature derivatives of elastic moduli of Al 2 O 3 measured in this study agree within 15% with expectations based on published single-crystal data. For ScAlO 3 perovskite, the value of ŽEK SrET . P is y0.033 GPa Ky1 and ŽEGrET . P is y0.015 GPa Ky1. The relative magnitudes of these derivatives agree with the observation in Duffy and Anderson wDuffy, T.S., Anderson, D.L., 1989. Seismic velocities in mantle minerals and the mineralogy of the upper mantle. J. Geophys. Res. 94, 1895–1912.x that <ŽEK SrET . P < is typically about twice <ŽEGrET . P <. The value of ŽEK SrET . P for ScAlO 3 is intermediate between those inferred less directly from V Ž P,T . studies of Fe-free and Fe- and Al-bearing MgSiO 3 perovskites wWang, Y., Weidner, D.J., Liebermann, R.C., Zhao, Y., 1994. P–V–T equation of state of ŽMg,Fe.SiO 3 perovskite: constraints on composition of the lower mantle. Phys. Earth Planet. Inter. 83, 13–40; Mao, H.K., Hemley, R.J., Shu, J., Chen, L., Jephcoat, A.P., Wu, Y., Bassett, W.A., 1991. Effect of pressure, temperature and composition on the lattice parameters and density of ŽMg,Fe. SiO 3 perovskite to 30 GPa. J. Geophys. Res. 91, 8069–8079; Zhang, Weidner, D., 1999. Thermal equation of state of aluminum-enriched silicate perovskite. Science 284, 782–784x. The value of <ŽEGrET .< P for ScAlO 3 is similar to those of most other mantle silicate phases but lower than the recent determination for MgSiO 3 perovskite wSinelnikov, Y., Chen, G., Neuville, D.R., Vaughan, M.T., Liebermann, R.C., 1998. Ultrasonic shear wave velocities of MgSiO 3 perovskite at 8 GPa and 800K and lower mantle composition. Science 281, 677–679x. Combining the results from the previous studies and current measurements on ScAlO 3 perovskite, we extracted the parameters Ž q and g 0 . needed to fully specify its Mie–Gruneisen–Debye equation-of-state. In this study, we have ¨ demonstrated that acoustic measurements of K SŽT ., unlike V Ž P,T . data, tightly constrain the value of q. It is concluded that ScAlO 3 has ‘normal’ g 0 Ž; 1.3. and high q Ž; 3.6.. The high value of q indicates that ScAlO 3 has very strong intrinsic

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Corresponding author. Department of Geosciences, State University of New York at Stony Brook, Stony Brook, NY 11794-2100 USA. Fax: q1-631-632-8240. E-mail address: [email protected] ŽJ. Kung.. 0031-9201r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 1 - 9 2 0 1 Ž 0 0 . 0 0 1 5 9 - X

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J. Kung et al.r Physics of the Earth and Planetary Interiors 120 (2000) 299–314

temperature dependence of the bulk modulus; similar behaviour has been observed in measurements on Fe- and Al-bearing silicate perovskites ŽMao et al., 1991; Zhang and Weidner, 1999.. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Perovskite; Thermoelastic properties; Temperature dependence; Mantle

1. Introduction Pressure and temperature derivatives of elastic moduli are the key parameters for extrapolating elastic properties measured in the laboratory to the conditions of the deep mantle. The application of opto-acoustic techniques at high P–T conditions in diamond-anvil apparatus and parallel developments in ultrasonic interferometry in the multi-anvil environment are increasingly providing such information, even for the silicate perovskites ŽSinelnikov et al., 1998.. However, the continuing inaccessibility of the stability fields of high pressure minerals for acoustic measurements and potential complications associated with ultrasonic measurements in solid pressure-transmitting media provide the motivation for complementary studies of mantle minerals and their close structural analogues over the relatively wide ranges of temperature achievable under hydrostatic pressure conditions in gas-medium apparatus. Accordingly, new procedures have been developed to allow ultrasonic measurements to be performed by high precision ultrasonic interferometry in an internally heated gas-charged pressure vessel. At this stage, such measurements of the elastic properties are restricted to moderate temperatures Ž- 800 K. at 300 MPa confining pressure, although further refinements of the technique are expected to provide access to temperatures as high as 1600 K. In this study, we describe the experimental set-up and report new measurements of the temperature dependence of elastic wave speeds in polycrystalline samples of Al 2 O 3 and ScAlO 3 , the former serving as a standard and the latter as a close structural analogue for MgSiO 3 perovskite. Ever since its discovery by Reid and Ringwood Ž1975., ScAlO 3 perovskite has been regarded as a close structural analogue for MgSiO 3 perovskite. The two structures share the same orthorhombic space group ŽPbnm. and similar molar volumes and exhibit closely comparable degrees of distortion away from the cubic parent structure. Equally important, the two structures respond similarly to changing

pressure and temperature. High temperature X-ray diffraction ŽHill and Jackson, 1990. indicates that ScAlO 3 perovskite survives metastably on heating to 1370 K at atmospheric pressure, and that its thermal expansion is attributable entirely to the expansion of the AlO6 octahedral, i.e. the inter-octahedral Al–O– Al angles do not vary significantly with temperature. Similarly, the volumetric compression Žto 5 GPa. is achieved by the contraction of the constituent AlO6 octahedral without any significant change of the degree of distortion of the inter-octahedral linkages ŽRoss, 1998.. Analogous behaviour has been observed in MgSiO 3 perovskite. Ross and Hazen Ž1990. showed that below 5 GPa, the primary response of the structure is through compression of the Si`O and Mg`O bonds resulting in comparable fractional decreases of the volumes of the Si octahedron and Mg dodecahedron. Ross and Hazen Ž1989. observed that the response of MgSiO 3 perovskite to changing temperature mirrors its behaviour under compression. The close consistency between pressure derivatives of elastic moduli recently measured for polycrystalline ScAlO 3 ŽKung et al., 2000. and those previously reported for MgSiO 3 perovskite ŽKnittle and Jeanloz, 1987; Sinelnikov et al., 1998. further reinforces the analogy. In order to extrapolate the results obtained in the present study beyond the experimental conditions, a Mie–Gruneisen–Debye equation-of-state Žhereafter ¨ MGD EOS. has been used to extract the thermal parameters Ž q and g 0 .. The MGD EOS combines the Birch–Murnaghan finite-strain theory and the Debye description of thermal energy within the Mie– Gruneisen framework and has been widely used in ¨ previous studies ŽStixrude and Bukowinski, 1990; Stixrude et al., 1992; Hemley et al., 1992; Jackson and Rigden, 1996; Jackson, 1998.. Explicit incorporation of a model for the thermal energy provides a physically robust basis for extrapolation beyond the experimental conditions by allowing appropriate temperature dependence for the thermal expansivity and other thermoelastic properties and ready conversion between isothermal and adiabatic conditions.

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The well-established high temperature behaviour of Al 2 O 3 serves to demonstrate the process by which the parameters of the EOS are constrained, before applying the same approach to ScAlO 3 .

2. Experimental details 2.1. Arrangements for ultrasonic measurement Two-way travel times through polycrystalline specimens were measured using the phase comparison method of ultrasonic interferometry with a buffer rod assembly. The use of a buffer rod allows measurements on samples placed in an environment that is hostile to the transducer, such as the high temperature conditions of the current study. The principles of ultrasonic measurement and reduction of travel time have been described in Rigden et al. Ž1988. and Niesler and Jackson Ž1989.. High temperature measurements are carried out in a gas-medium high-pressure apparatus equipped with an internal furnace ŽFig. 1.. The pressure chamber is sealed at each end by a threaded piston supporting a seal, which is a combination of polyurethane O-rings, a hardened steel ‘Paterson’ ring and hardened Be–Cu or phosphor–bronze mitre rings. The argon gas confining pressure Žto 300 MPa. is measured by a Manganin resistance gauge. The furnace is 150 mm long and consists of molybdenum wire wound around an alumina ceramic tube ŽDegussa, AL23., which is embedded in a Zircar alumina–silica insulating sleeve. The temperature profile is such that there is 15 K offset in temperature between sample and thermocouple Žtype K. at the highest experimental temperature Ž820 K., but no more than 2 K variation across the sample. The temperature fluctuation during travel time measurement was ; 0.2 K. The acoustic buffer rod is made of hardened steel ŽAISI M2., with a total length of 90 mm. Circumferential grooves Žnot shown in Fig. 1. machined along the length of the buffer rod serve to eliminate the acoustic noise otherwise caused by elastic waves interacting with the cylindrical wall of the buffer rod. Thin gold layers Ž1 mm thick. are evaporatively deposited on the surfaces across which the buffer rod and specimen make contact. These gold layers, when subjected to the normal stress of 300 MPa, provide

Fig. 1. Gas-pressure vessel configuration for measurement of elastic wave on specimen at high temperature. The left-hand panel is an overview of the vessel. The right-hand panel is the enlarged illustration of buffer rod, sample and thermocouple insert, which is not to scale.

adequate mechanical coupling between buffer rod and sample. The elastic waves are generated and detected by the same LiNbO 3 overtone-polished parallel-plated transducer of 3.2 mm diameter and 40 MHz resonant frequency. The transducer orientations used in this work are 368-rotated Y-cut for compressional waves and 418-rotated X-cut for shear waves. The transducer is attached with Aremco Crystalbond to the cool pressure-free end of the buffer rod. In this study, temperatures are limited to the range below 800 K by softening of the Aremco Crystalbond at about 400 K. Electrical connection to the transducer is effected by a coaxial cable inserted through the lower hollow piston and terminated at the transducer with a 50-V impedance-matching resistor ŽFig. 1.. The sample was placed inside a cup machined from soft metal — either copper or lead. The buffer rod, sample and sample cup were inserted into a

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blind, close-fitting copper jacket. The other end of the buffer rod, with the attached transducer, protruded slightly beyond the open end of the copper jacket, where a seal against the piston is effected by a polyurethane O-ring and mitre ring within a steel retaining nut. The compound ceramic assembly containing the thermocouple is similarly sealed within a blind copper jacket extending into the pressure vessel from the top. The gas pressure Ž300 MPa. causes the Cu jacket to collapse tightly around the buffer rod and sample cup and the cup to collapse around the sample, transmitting essentially hydrostatic stress and forcing the gold-coated ends of buffer rod and sample into intimate contact. The pressure also helps suppress any minor crack porosity in the sample. Fig. 2. Travel times vs. frequency for S wave in lead pressure medium at different temperature. The travel times are frequency independent at high frequencies Ž ) 40 MHz..

2.2. Data analysis The travel times used to calculate the elastic wave speeds were corrected for the influence of the gold bond between buffer rod and specimen through the procedure described by Niesler and Jackson Ž1989.. The sample length L at high temperature T and pressure P is related to its value L0 at zero pressure and T0 s 300 K by the expression L0 L

s 1 y a Ž T y T0 . q

P K

1 3

Ž 1.

where a and K, respectively, are the thermal expansivity and bulk modulus of the tested material. However, the influence of pressure is negligible for P s 300 MPa. A constant Žaverage. value of a was used for the entire measured temperature range Ž300–800 K.. The compressional ŽP. and shear ŽS. wave travel time measurements have been carried out at ambient conditions ŽKung et al., 2000. and at temperatures up to 800 K at a confining pressure of 300 MPa. The travel times were determined from the frequencies of individual interference extrema located within a broad range of carrier frequency Ž25–55 MHz.. The preferred value of the travel time at each temperature is the average over a common range of frequencies Žnormally above 40 MHz, Fig. 2. where the travel time is essentially frequency independent. The apparent dispersion Žvariation of travel time with frequency. is attributed to interaction of the elastic

waves with the cylindrical surface of the small specimen Ž3 mm diameter.. Since the dispersion is largely independent of temperature, it contributes negligibly to the inferred temperature dependence. The travel time thus determined at each temperature, along with the temperature-corrected specimen length and density, is used to calculate the elastic wave speed and corresponding modulus. The high temperature elastic wave speeds and elastic moduli are described as functions of temperature by second-order polynomial Žquadratic. fits. Differentiation of the quadratic fits yields the temperature derivatives of elastic wave speeds and elastic moduli. 2.3. Specimens Lucalox alumina Ža commercial grade of translucent polycrystalline alumina from the General Electric. is of very high density and acoustic quality and has been used as a standard specimen to demonstrate the experimental precision in the previous high pressure study ŽKung et al., 2000.. The existence of elasticity data for single-crystal a-Al 2 O 3 to 1825 K ŽGoto et al., 1989. means that Lucalox alumina is also an ideal standard for use in the high-temperature experiment. Indeed, the same cylindrical specimen was employed as in the high-pressure study. The

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Table 1 The elasticity of alumina at ambient conditions and high pressures Reference .a

Goto et al. Ž1989 Gieske and Barsch Ž1968. a Kung et al. Ž2000.

Specimen

Density Žg cmy3 .

K S ŽGPa.

G ŽGPa.

EK S rEP

EGrEP

single crystal single crystal Lucalox

3.982 3.986 3.972

253.6 254.4 253.1

164.4 164.6 162.8

4.32 4.00

1.7 1.74

a The elastic properties deriving from the single-crystal studies of Goto et al. Ž1989. and Gieske and Barsch Ž1968. are averages of the Hashin–Shtrikman bounds.

results of the previous high-pressure study ŽKung et al., 2000. on Lucalox alumina are compared with single-crystal data in Table 1. Consistency between results obtained in this high temperature study and the expectations based on single-crystal data would demonstrate that there is no significant influence from non-hydrostatic stresses associated with the finite strength of the metal cup andror the contrast in thermal expansivity between the steel of the buffer rod and the alumina standard specimen. The experiments have been carried out with multiple heatingrcooling cycles on one assembly and with alternative copper and lead cup materials. ScAlO 3 polycrystals were synthesized from mixed oxide and gel origin precursors at 10 GPa, 1470 K in the multi-anvil press. The best two of these specimens Žk160 and k182 with 1.7% and 0.5% porosity, respectively. were measured in the high pressure study ŽKung et al., 2000; Table 2. where it was shown that the small amount of porosity may slightly reduce the absolute value of the elastic wave speeds but has negligible effect on their pressure derivatives. Owing to the presence of microcracks in the specimen k182, it is considered unsuitable for mea-

surements at the lower pressure of this study. Therefore, specimen k160, of slightly greater porosity ŽTable 2., was used here. Examination of the specimens following excursions to high temperature revealed no evidence of deformation or cracking.

3. Results and discussion 3.1. Lucalox alumina In order to test reproducibility obtained with the current procedure, the S wave run to 650 K in the copper pressure medium consisted of three successive heating cycles with the same sample assembly and gold bond. Highly consistent results were obtained ŽFig. 3. and there was no sign of deterioration of the gold bonding layer on close examination following the run. The high temperature ultrasonic measurements on the Lucalox specimen comprised five runs in all: one

Table 2 The elasticity of ScAlO 3 at ambient conditions and high pressure Density Porosity K S ŽGPa. G ŽGPa. EK S rEP EGrEP Žg cmy3 . k182 4.267 k160 4.215

0.5 1.7

218.9 216.2

129.4 125.8

3.80 3.75

1.89 1.87

The starting material for k182 is 1:1 Žmolar. mixed oxides and k160 amorphous ScAlO 3 . The elastic wave speed and elastic moduli for k182 and k160 are the values extrapolated from high pressure. The X-ray density for ScAlO 3 in this study is 4.285 g cmy3 , calculated using unit cell parameters refined by Sinclair et al. ˚ 3. Ž1979., 185.91 A

Fig. 3. Travel times as a smooth function of temperature for Lucalox with three heating cycles.

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closely consistent with those calculated from singlecrystal data ŽFig. 4.. S wave speeds measured at high temperature in different pressure media are consistent within the uncertainties with expectations based on single-crystal data. The similarity of the measurements in the lead and copper media have demonstrated that copper and lead provide comparable pseudo-hydrostatic environments for the specimen during high temperature ultrasonic measurement. The compressional and shear wave speeds measured in Cu cups have been combined with the temperature-corrected density to infer the temperature dependence of bulk and shear moduli ŽFig. 5., which may be compared with the averaged Hashin–

Fig. 4. Ža. Measured P wave speeds as functions of temperature for Lucalox. The circle symbols represent the measurements carried out in the copper-pressure medium and square symbols in the lead-pressure medium. Although, the velocities of three P wave runs are systematically 0.5% lower than those of Hashin–Shtrikman bounds, the relationship of velocity vs. temperature for three P wave runs and Hashin–Shtrikman bounds are similar. Žb. Measured S wave speeds as smooth functions of temperature for Lucalox. The symbols represented here are the same as those in Ža.. The S wave data fall between Hashin–Shtrikman and Reuss bounds.

P wave run in copper and two in lead cups, and one S wave run in each type of metal cup. The measured elastic wave speeds vary smoothly with temperature and are compared in Fig. 4 with the Reuss, Voigt and Hashin–Shtrikman bounds calculated from single-crystal elastic constants of Goto et al. Ž1989.. The wave speeds of the two P wave runs in lead cups display a high degree of internal consistency but are systematically 0.3% lower than the average of the essentially coincident Hashin–Shtrikman bounds, perhaps as the result of imperfect coupling between the buffer rod and sample. Nevertheless, the slopes of the wave speed–temperature trajectories which yield the temperature derivatives of wave speeds have apparently not been affected and are

Fig. 5. Ža. Bulk modulus of Lucalox as a function of temperature. The elastic moduli here are those measured in the copper-pressure medium due to the measurements extended to higher temperature range. The solid line is Hashin–Shtrikman bounds. Žb. Shear modulus of Lucalox as a function of temperature. The shear modulus is 0.6% lower than Hashin–Shtrikman bounds at near room temperature Ž300 K., but agrees well with Hashin–Shtrikman bounds well at high temperature Žabove 500 K.. The shear moduli are assumed to be the same as those calculated from single-crystal data Žshaded circles; Goto et al., 1989. as the S wave measurements are carried out up to 650 K. The bulk moduli from 650 to 800 K are calculated using measured P wave data and S wave data calculated from single-crystal data Žshaded circles..

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Table 3 Comparison of temperature derivatives of elastic properties for Lucalox alumina with single-crystal data of Goto et al. Ž1989. Specimen

Lucalox Single-crystal Single-crystal ŽHT. a

Temperature ŽK.

300–800 300–800 300–1800

EVrET Ž10y4 . Žkm sy1 Ky1 .

EMrET Ž10y2 . ŽGPa Ky1 .

P

KS

G

y2.3 y1.9 y2.1

y1.8 a y2.1 y2.4

y4.9 y4.4

S a

y3.1 y3.5

The high temperature measurements of S wave travel time of ScAlO was only carried out up to temperature 640 K.

Shtrikman bounds Žabove 650 K, the unmeasured shear wave speed for the Lucalox alumina is approximated by this average.. The variation of shear modulus for Lucalox with temperature shows more curvature at low temperature Žbelow 400 K. than the Hashin–Shtrikman bounds calculated from singlecrystal data. The shear modulus for Lucalox alumina closely approaches the Hashin–Shtrikman average at higher temperature ŽFig. 5b.. The bulk modulus– temperature relationship for Lucalox alumina ŽFig. 5a. shows a trend approximately parallel with the ‘single-crystal’ trend, but displaced downwards by 1.6% as a result of the systematically lower P wave speeds ŽFig. 4.. Average values of EVrET and EMrET for Lucalox alumina, calculated from the quadratic fits are compared in Table 3 with those from the single-crystal data. The mean temperature derivatives Ž300–800 K. of elastic wave speeds and elastic moduli for Lucalox agree with those calculated for the same temperature interval from single-crystal data within 15%. The average temperature derivatives of the Hashin– Shtrikman elastic moduli are also calculated for the wider temperature range of 300–1800 K. It is noted that the average temperature derivatives of elastic moduli at moderate temperatures Ž300–800 K. are closely representative of the wider range 300–1800 K, as a consequence of the approximately linear M ŽT . variation.

hence the precision of the travel time determinations. The high temperature data for ScAlO 3 are therefore limited to a temperature range of 300–600 K in this study. The cause of the deterioration of the signal ratio with increasing temperature remains unknown. The high temperature ultrasonic measurements on ScAlO 3 involved a number of successive heating cycles for both P and S waves ŽFig. 6.. For P waves, the difference of travel time at each temperature

3.2. ScAlO3 The high temperature ultrasonic measurements were carried out using specimen k160, embedded in a copper pressure medium. The ratio of buffer to sample echo amplitudes was 4 1 at 300 K and increased further with increasing temperature. It is this echo amplitude ratio that determines the sharpness of the minima of the interference pattern, and

Fig. 6. Travel time as function of temperature for P and S wave measurements subjected to a number of heating cycles. The results show that there is 0.1% difference between travel times in the first two heating cycles; however, the trends remain the same. In the S wave Ža., the 0.8% difference of travel time between first and third heating cycle maybe caused by the decay of the gold bond between buffer rod and sample.

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between the two heating cycles is less than 0.1%, and the t ŽT . curves are parallel ŽFig. 6b.. S wave travel times ŽFig. 6a. show a 0.8% discrepancy between the third Žand last. cycle and the previous two — attributed to deterioration of the gold bond. The travel times from the third S wave heating cycle were therefore excluded from subsequent analysis. Bench-top measurements of the elastic wave speeds for specimen k160 before and after the high temperature experiments are consistent. However, the elastic wave speeds ŽFig. 7a,b. measured in the high temperature environment and extrapolated to room temperature are 0.6% lower for VP and VS than those measured on the bench top, suggesting that the confining pressure Ž300 MPa. is insufficient to provide perfect welded contact boundary conditions between buffer rod and sample. This does not significantly affect the temperature derivatives of elastic wave

Fig. 8. Bulk and shear moduli of ScAlO 3 perovskite as function of temperature. A deficit of ;1% of moduli of bench-top and those extrapolated from high temperature is caused by the imperfect welding condition between the buffer rod and sample, as the confining pressure Ž300 MPa. is insufficient. Detail is explained in the text. The quadratic fitting of elastic moduli have been described in Table 4b.

Fig. 7. P and S wave speeds of ScAlO 3 perovskite as function of temperature. A comparison with bench-top measurement shows the elastic wave speeds extrapolated from high temperature are 0.6% lower, suggesting that the confining pressure Ž300 MPa. is insufficient to provide perfect coupling between the buffer rod and sample. The quadratic fits of elastic wave speeds have been described in Table 4a.

speed and elastic moduli as demonstrated in the measurements on Lucalox alumina ŽFigs. 4 and 5.. The coefficients of the second-order polynomials fitted to the temperature dependence of the elastic wave speeds and elastic moduli ŽFigs. 7 and 8. are presented in Table 4. Although quadratic equations have been used to fit the data of Figs. 7 and 8, the fact that the K S ŽT . curve is Žunphysically. concave upwards, giving a positive value of ŽE 2 K SrET 2 ., suggests that the curvature must be treated with caution. The measurements on Lucalox have demonstrated that the aÕerage temperature derivatives of elastic wave speed and elastic moduli for the temperature range Ž300–800 K. agree with those calculated from single-crystal data within ; 15%. As the measured temperature range is limited Ž300–600 K. and given the scatter in the data, the temperature deriva-

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307

Table 4 Ža. Polynomial fits Ž V s a q bT q cT 2 . describing the temperature dependence of wave speeds for ScAlO 3 and its mean temperature derivatives ŽEVrET . up to 600 K a a Žkm sy1 . VS V Pb

5.397 9.616

b Ž10y4 . Žkm sy1 Ky1 .

c Ž10y7 . Žkm sy1 Ky2 .

EVrET Ž10y4 . Žkm sy1 Ky1 .

2.83 y3.45

y5.97 y2.34

y2.5 Ž3. y5.6 Ž1.1.

Žb. Polynomial fits Ž M s a q bT q cT 2 . describing the temperature dependence of plastic moduli Ž M s K S and G . for ScAlO 3 and its mean temperature derivatives ŽEMrET . up to 600 K c Modulus

a ŽGPa.

b Ž10y2 . ŽGPa Ky1 .

c Ž10y5 . ŽGPa Ky2 .

EMrET Ž10y2 . ŽGPa Ky1 .

G K Sd

124.26 227.21

0.72 y4.80

y2.45 1.66

y1.5 Ž2. y3.3 Ž5.

a

The numbers in the parentheses are uncertainties of the temperature derivatives of elastic wave speeds for ScAlO 3 , for details, see the

text. b

The quadratic equation of V P is the first cycle of P wave run. The numbers in the parentheses are uncertainties of the temperature derivatives of elastic moduli for ScAlO 3 ,details see the text. d The quadratic equation of K S is the first heating cycle run.

c

tives of elastic properties reported for ScAlO 3 ŽTable 4. are also averages for the interval 300–600 K. The numbers in the parentheses associated with the derivatives are uncertainties estimated from the scatter of the ScAlO 3 data about the fits and the differences between Lucalox alumina results and the expectations based on single-crystal elasticity. Interestingly, the relative magnitudes of temperature derivatives measured in this study accord well with the observation of Duffy and Anderson Ž1989. that <ŽEK SrET . P < is typically about twice <ŽEGrET . P <. The measured ŽEK SrET . P is also similar to that predicted for MgSiO 3 perovskite Žy0.031 GPa Ky1 . by Duffy and Anderson Ž1989., and intermediate between values subsequently inferred from V Ž P,T . studies of MgSiO 3 and ŽMg,Fe.SiO 3 perovskite, respectively Žsee later discussion.. As for <ŽEGrET . P <, ScAlO 3 exhibits an unexceptional value comparable with those for most mantle silicates Že.g. garnet, olivine and orthopyroxene., but much lower than the recent pioneering measurement on MgSiO 3 perovskite Žy0.029 " 0.003 GPa Ky1 , Sinelnikov et al., 1998.. Thus, for the shear mode properties of ScAlO 3 and MgSiO 3 perovskite, direct ultrasonic measurements have indicated similar values of ŽEGrEP . T Ž1.8–1.9., but a factor of two difference in the magnitude of ŽEGrET . P . This difference, if confirmed by more comprehensive studies, would

have important implications for the interpretation of the shear mode properties of the lower mantle. The higher value would admit more silicic compositions for the lower mantle.

4. Thermoelastic properties of Al 2 O 3 and ScAlO 3 In the MGD EOS, the pressure is comprised of two parts: that associated with cool compression, P Ž V, 300 K. and the thermal pressure, Pth Ž V,T . P Ž V ,T . s P Ž V , 300 K . q Pth Ž V ,T . .

Ž 2.

The first term of the right hand side, P Ž V, 300 K., is relatively well constrained and described by a thirdorder Birch–Murnaghan EOS y5

P Ž V , 300 K . s

3 2

K0

V

ž / V0

3

° V ¢ž V /

y2

~

3

y1

0

y2

3

q K 0 Ž K 0X y 4 . 4

V

ž / V0

3

¶ y1 • ß 2

Ž 3.

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where K 0 and K 0X , respectively are the bulk modulus and its pressure derivative and V0 is molar volume, all at 300 K and zero pressure. The MGD equation is employed to describe the thermal pressure: g Pth s E Ž u ,T . , Ž 4. V th where

ž /

Eth s 9nR

gsg0

3

T

u

V

t3

H0 u Ž e y 1. d t ,

Ž 4a .

t

ž / ET

V0

,

Ž 4b .

g0 yg q

.

Ž 4c .

Therefore, the MGD EOS can be written in the form: P Ž V ,T . s P Ž V , 300 K . q Pth s P Ž V , 300 K . q

g ŽV . V

Eth u Ž V . ,T .

Ž 5.

Parameters required to fully specify the MGD EOS are thus V0 , K S 0 , ŽEK SrEP . T , q, g 0 and u 0 . V0 is determined by X-ray diffraction, and K S 0 and ŽEK SrEP . T are preferably determined directly from ultrasonic measurements such as those of this study. The Debye temperature u Žhere the elastic Debye temperature. is calculated1 from the elastic sound speeds VP and VS , q and g 0 are extracted from the pressure and temperature dependencies of bulk modulus from the measured data, obtained by successive differentiation of Eq. Ž5.,

P

yaKT

g ED Eth V

ž

ET

EK T

ž / EP

/ ž yg

E 2D Eth ETEV

V

.

/ Ž 7.

T

The conversion between isothermal and adiabatic conditions required in evaluation of K S and ŽEK SrET . P within the MGD EOS has been described in detail in Jackson and Rigden Ž1996, Appendix B2.. 4.1. Al 2 O3 Al 2 O 3 with its well-established thermoelastic behaviour provides a convenient illustration of the process whereby the parameters of the EOS are constrained. The high temperature thermal expansion coefficient Ž a . and adiabatic bulk modulus Ž K S . are each calculated at a series of temperatures at atmospheric pressure from the MGD EOS incorporating the measured ŽEK SrEP . T , K S , V0 , and u D Ž1045 K, Goto et al., 1989., and trial values of g 0 and q. The goal is to vary g 0 and q to obtain a satisfactory fit ŽEqs. 6 and 7. to the measured data Ž K S ŽT .: Goto et al., 1989; a ŽT .: White and Roberts, 1983.. The optimal set of parameters, g 0 and q, are identified by minimising the root-mean-square misfit Ž M .

Ms

EP K T s yV

s y Ž q y 1.

q

ž /

u s u 0 exp

T

EK T

ž /

s

EV

s K T0 y Ž q y 1.

g D Eth V

yg

EEth

ž / EV

Ž 6. T

1 The Debye temperature is calculated from the Žisotropic. compressional and shear wave speeds as follows: 1r 3

y3 u s Ž h r k .Ž 9r Nr4p M . 1r3 w 2Vy3 S qV P x where h and k, respectively are the Plank’s and Boltzmann’s constants, M is the mean atomic weight and N is Avogadro’s number.

)

1 2

)ž

2

Ý m2j , and m j js1

1 N

N

Ý is1

X j,cal Ž Ti . y X j,exp Ž Ti .

s Ž Xj .

2

/

Ž 8.

where X j,exp and s Ž X j . are the measured quantities Ži.e. a and K S . and their uncertainties, and X j,cal are the corresponding calculated quantities. The experimental uncertainties are used to estimate the appropriate weight s Ž X j .. The parameters for the best fitting MGD EOS representation of the thermoelastic properties for Al 2 O 3 are displayed in Table 5. The g 0 and q which give the best fits to the measured a ŽWhite and Roberts, 1983. and temperature dependence of

J. Kung et al.r Physics of the Earth and Planetary Interiors 120 (2000) 299–314 Table 5 The EOS and calculated thermoelastic parameters at Ž V0 ,T0 . for corundum Ž a-Al 2 O 3 . from the MGD EOS Parameters

This study

This study

EOS V0 Žcm3 . u D ŽK. K S ŽGPa. ŽEK S rEP . T q g0

25.547 Ž2. 1045a 253.7 b 4.0 1.8 1.35

25.574 Ž2. 1045a 253.7 b 4.3 c 1.6 1.35

Thermoelastic a 0 Ž10y5 . ŽKy1 . K T ŽGPa.

1.52 252.2

1.52 252.2

Anderson and Isaak Ž1995.

253.7 b

1.32

1.62 252

a

Watchman et al. Ž1962. Goto et al. Ž1989. c Gieske and Barsch Ž1968.

309

The fitting of MGD EOS to the Al 2 O 3 data has demonstrated that the EOS parameters Ž q and g 0 . can be constrained tightly by measured thermoelastic quantities, especially a , ŽEK SrEP . T , ŽEK SrET . P . The MGD EOS thus determined describes very accurately the thermoelastic properties of this material at high temperatures T ) 0.8 u ŽFigs. 10 and 11., and provides a secure basis for extrapolation to P–T conditions beyond the range of the measurements. 4.2. ScAlO3 Similarly derived EOS parameters and calculated thermoelastic properties of ScAlO 3 are presented in Table 6. The elastic parameters Ž K S , ŽEK SrEP . T and

b

K S ŽGoto et al., 1989. are 1.35 and 1.8 for ŽEK SrEP . T s 4.0 as measured on polycrystalline Lucalox alumina ŽKung et al., 2000.. For ŽEK SrEP . T s 4.3 inferred from measurements on single-crystal Al 2 O 3 ŽGieske and Barsch, 1968., the calculated g 0 and q are 1.35 and 1.6 ŽTable 5.. It is evident that there is a trade-off between ŽEK SrEP . T and q, but g 0 is insensitive to variation of ŽEK SrEP . T . The calculated results and experimental data are in good agreement although the thermal expansion calculated from the MGD EOS slightly underestimates the measured a for T - u ŽFig 9.. The average ŽEK SrET . P Žy0.021 GPa Ky1 . calculated through the EOS with the parameters of Table 5 is in good agreement with our experimental data Žy0.023 GPa Ky1 . ŽFig. 10., although the absolute values of bulk modulus measured on the polycrystal are systematically lower. The high pressure ŽKung et al., 2000. and current high temperature experiments have demonstrated that although the absolute values of the elastic moduli measured on polycrystalline samples are slightly lower than those calculated from single-crystal data, the pressure and temperature derivatives of the elastic moduli are nevertheless in good agreement. Here, the objective is therefore to adjust the EOS parameters Ž q and g 0 . to fit the experimentally determined temperature derivatives, ŽEK SrET . P , rather than the absolute value of K S ŽT ..

Fig. 9. Ža. A comparison of the measured thermal expansion Žsolid circles, White and Roberts, 1983. and that calculated from the MGD EOS Žsolid and dashed plus cross lines. for Al 2 O 3 at high temperature. The solid line represents the parameters fitted to the EOS with EK S rEP s 4.0, q s1.8 and g 0 s1.35, and the dashed plus cross line with EK S rEP s 4.3, q s1.6 and g 0 s1.35. Žb. Shows the variation of thermal expansion calculated from EOS relative to the measured data, which are in good agreement at high temperature within "1% for curves calculated with both sets of parameters.

310

J. Kung et al.r Physics of the Earth and Planetary Interiors 120 (2000) 299–314

Fig. 10. Ža. A comparison of measured bulk modulus Žsolid circles, Goto et al., 1989. and that calculated from the MGD EOS Žsolid and dashed plus cross lines. for Al 2 O 3 at high temperature. The solid line represents the parameters fitted to the EOS with EK S rEP s 4.0, q s1.8 and g 0 s1.35, and the dashed plus cross line with EK S rEP s 4.3, q s1.6 and g 0 s1.35. The solid squares represent the measurement in the current study and its slope ŽEK S rET . agrees well with the result in Goto et al. Ž1989.. Žb. Shows the variation of bulk modulus calculated from the EOS relative to the measurement, which are in good agreement within "0.5% for curves calculated with two sets of parameters.

elastic Debye temperature. used are from the present study and differ from those Ž K S 0 s 249 GPa, ŽEK SrEP . T s 5, ue s 900 K. of Hill and Jackson Ž1990. who used the single-crystal elastic properties of Bass Ž1984. and an estimated ŽEK SrEP . T from Jones Ž1979.. The elastic Debye temperature Ž ue s 860 K. is calculated using the elastic velocities Ž V P s 9.57 km sy1 and VS s 5.50 km sy1 . measured by Kung et al. Ž2000.. The measured temperature dependence of the unit-cell volume Ž283–1373 K; Hill and Jackson, 1990; ‘Run a3’ powder diffraction data analysed with the full-profile Rietveld method. and average temperature derivative of K S for ScAlO 3 Ž300–600 K; present study. are used to determine the appropri-

Fig. 11. Ža. A comparison of molar volume of ScAlO 3 at high temperature fitted from the MGD EOS Žthe solid and dashed lines. and the measurement in Hill and Jackson Ž1990. Žsolid circles.. The dashed line presents the molar volume at high temperature fitted to the EOS with q s1, g 0 s1.37, and the solid line with q s 3.6 and g 0 s1.3. Žb. The residual relative to the measurements ŽHill and Jackson, 1990. shows 0.1% volume difference by assigning high q Ž s 3.6., suggesting that the V – T measurement is not sensitive to q deviating from unity.

ate values of q and g 0 for use in the MGD EOS. V0 is chosen to be 28.026 cm3 moly1 , for reasons of Table 6 The EOS and calculated thermoelastic parameters at Ž V0 ,T0 . for ScAlO 3 from MGD EOS Parameters EOS V0 Žcm3 . u D ŽK. K S ŽGPa. ŽEK S rEP . T q g0

28.026 Ž3. 860 219 3.8 3.6 1.3

Thermoelastic a 0 Ž10y5 . ŽKy1 . K T ŽGPa.

1.82 217.5

J. Kung et al.r Physics of the Earth and Planetary Interiors 120 (2000) 299–314

internal consistency with the high temperature volumes, even though it differs slightly from that obtained in the single-crystal study of Sinclair et al. Ž1979; 27.99 cm3 mole y 1.. A satisfactory fit to the measured temperature dependence of volume is obtained with q s 1.0 and g 0 s 1.37 ŽFig. 11, dashed line.. However, the resulting calculated value for <ŽEK SrET . P < of ScAlO 3 is not nearly large enough to match that measured in this study ŽFig. 12, solid circles.. In order to match the measured ŽEK SrET . P , it is necessary to increase q to 3.6, with a slight reduction in the value of g 0 to 1.3. This combination yields a compromise, albeit with a very high value of q, whereby the experimentally determined value of ŽEK SrET . P is matched without producing more than a slight systematic misfit of the V ŽT . observations ŽFig. 11.. With q s 3.6 and g 0 s 1.3, the mean thermal expansion coefficient Ž2.76 = 10y5 Ky1 . for the temperature interval 300–1300 K calculated from the EOS is comparable to that determined by Hill and Jackson Ž1990.. The calculated a 0 for ScAlO 3 Ž1.82 = 10y5 Ky1 . is low, not markedly dissimilar from the figure of 1.45 = 10y5 Ky1 determined below

311

300 K for MgSiO 3 perovskite ŽRoss and Hazen, 1989.. The average value of ŽEK SrET . P for the interval 300–1300 K computed from the MGD EOS is 0.039 GPa Ky1 intermediate between the experimental values from V Ž P,T . studies on Fe-free Ž; 0.020 GPa Ky1 , e.g. Wang et al., 1994; Utsumi et al., 1995., and Fe- and Al-bearing MgSiO 3 perovskites Ž; 0.060 GPa Ky1 , Mao et al., 1991; Zhang and Weidner, 1999.. The new measurements of the pressure ŽKung et al., 2000. and temperature dependence of the elastic moduli for ScAlO 3 , along with the Hill and Jackson Ž1990. thermal expansion data provide a better-constrained P–V–T EOS than has previously been available for this important silicate perovskite analogue. In particular, it has been shown here that a large value of q Ž3.6. is needed to fit the relatively large value of <ŽEK SrET . P < deriving from the ultrasonic measurements. It had previously been demonstrated by Jackson and Rigden Ž1996. that MGD EOS fits to P–V–T data are insensitive to variations in q away from unity. The V ŽT . behaviour modelled here ŽFig. 11. is similarly insensitive to the value of q.

4.3. Intrinsic temperature deriÕatiÕes of ScAlO3 This high value of q Ž4 1. results in a large intrinsic temperature dependence of bulk modulus for ScAlO 3 as follows. ŽEK TrET . P , given within the framework of the MGD EOS by Eq. Ž7., can also be written as: EK T

EK T

ž / ž / ET

Fig. 12. Bulk modulus as functions of temperature for ScAlO 3 with different fitting parameters of the EOS and measurements in the present study. In order to match the measured ŽEK S rET . P in the present study, Žsolid circle and dashed line. the q has been increased from 1 Žthe heavy dashed line. to 3.6 Žthe solid line.. The average of ŽEK S rET . P from the EOS fitting with q s 3.6 Žy0.029 GPa Ky1 . is comparable to that averaged from the measurement in this study Žy0.033 GPa Ky1 ..

s

P

ET

yaKT V

EK T

ž / EP

Ž 9. T

Že.g. Jackson and Rigden, 1996.. Comparison of Eqs. Ž7. and Ž9. indicates that the intrinsic temperature dependence of K T , given by ŽEK TrET . V , is small if q s 1 ŽEq. Ž7..; however, for q 4 1, <ŽEK TrET . V < is dominated by the leading term yŽq y 1.ŽgrV . ŽED Eth rET . V in Eq. Ž7. and may be much larger. Thus, for q 4 1, <ŽEK TrET . P < is large because there is a large intrinsic temperature dependence of K T ,

J. Kung et al.r Physics of the Earth and Planetary Interiors 120 (2000) 299–314

312

Table 7 Comparison of the temperature dependence of elastic moduli for Al 2 O 3 and ScAlO 3 Temperature ŽK.

ŽEMrET . P Ž=10y2 .

Intrinsic ŽEMrET . V Ž=10y2 .

Extrinsic ya K T ŽEMrEP . T Ž=10y2 .

ŽEMrET . V rŽEMrET . P Ž%.

Al 2 O 3

300 800 1800

y2.35 y3.16 y3.54

y0.70 y0.62 y0.59

y1.64 y2.54 y2.95

30 20 17

ScAlO 3

300 600 1400 1800

y2.92 y3.80 y4.53 y5.05

y1.40 y1.61 y1.64 y1.71

y1.52 y2.19 y2.89 y3.34

48 42 36 34

300 300

y2.4 a y1.5a

y1.67 y0.75

y0.73 y0.75

70 50

KT

G Al 2 O 3 ScAlO 3

a In the absence of a framework like the MGD EOS for description of the temperature dependence of G, the values of ŽEGrET .P for Al 2 O 3 and ScAlO 3 are simply average values from 300 K to the highest experimental conditions reported in the text for both materials.

leading also to large <ŽEK SrET . P <, through the thermodynamic identity:

Ž EK SrET . P s Ž EK TrET . P Ž 1 q ag T . y ag K T  1 q T Ž EarET . Pra q a q 4 ,

Ž 10 .

Že.g. Jackson and Rigden, 1996.. The strong intrinsic temperature dependence of bulk modulus for ScAlO 3 ŽTable 7. violates the general observation of Anderson et al. Ž1992. that ŽEK TrET . V is typically close to zero. For the MGD model fitted here to the thermoelastic data for ScAlO 3 , the intrinsic contribution, i.e. ŽEK TrET . V to ŽEK TrET . P varies from 48% at 300 K to 34% calculated at 1800 K ŽTable 7.. An even more markedly negative intrinsic temperature derivative of about y0.03 GPa Ky1 is required to explain the very large values of <ŽEK TrET .< P inferred from V Ž P,T . studies of ŽMg, Fe.SiO 3 ŽMao et al., 1991. and Al-bearing magnesium silicate perovskite ŽZhang and Weidner, 1999., for ŽEK TrEP . T ; 4. The temperature dependence of the shear modulus can similarly be expressed as the sum of intrinsic and extrinsic parts: EG EG EG s yaKT . Ž 11 . ET P ET V EP T

ž / ž /

y0.029 GPa Ky1 for MgSiO 3 perovskite ŽSinelnikov et al., 1998. would require a negative intrinsic contribution of much larger magnitude, as for example for Al 2 O 3 ŽTable 7..

ž /

For ScAlO 3 perovskite, these contributions are approximately equal at ambient conditions ŽTable 7., whereas the recent determination of ŽEGrET . P s

5. Conclusion High precision ultrasonic interferometry is used with an internally heated gas apparatus to determine elastic wave speeds up to 800 K at 300 MPa. Lucalox, a well-sintered polycrystalline alumina, was used to demonstrate the precision and accuracy of this technique. The average values of the temperature derivatives of elastic moduli measured for Lucalox alumina are consistent with those calculated from singlecrystal elasticity data within 15%. The capacity to perform high temperature ultrasonic measurements is new to this laboratory. The results of this study demonstrate the feasibility of such measurements, although there remains room for improvement, including extension of the temperature range, optimization of the choice of cup material to surround the specimen and detailed examination of the properties of the bonding layer. The technique has been applied to ScAlO 3 , a close structural analogue for MgSiO 3 perovskite. Measurements to 600 K yield a value of y0.033 GPa Ky1 for ŽEK SrET . P , similar to that predicted by

J. Kung et al.r Physics of the Earth and Planetary Interiors 120 (2000) 299–314

Duffy and Anderson Ž1989.. ŽEGrET . P for ScAlO 3 at y0.015 GPa Ky1 is comparable to the corresponding derivatives for many other mantle phases Žolivine, garnet and orthopyroxene.. In order to extrapolate the thermoelastic properties beyond the experimental conditions, the framework provided by the MGD EOS is preferred. It has been demonstrated here that this EOS describes the measured thermal expansivity and temperature dependence of bulk modulus for Al 2 O 3 very well, especially, in the high temperature regime. Experience with fitting of such experimental data for Al 2 O 3 and ScAlO 3 to the MGD EOS shows that the values of g 0 and q, in particular, can be tightly constrained by acoustic measurements of the temperature dependence of the bulk modulus. The value of g 0 Ž1.3. constrained most directly for ScAlO 3 by the V ŽT . data of Hill and Jackson Ž1990. is unexceptional, whereas the strong temperature dependence of the bulk modulus requires a substantial intrinsic contribution manifest in an unusually high value Ž3.6. for q. Based on the similarities of crystal structure, molar volume, thermal expansion and compressional behaviour, ScAlO 3 is a good analogue for MgSiO 3 perovskite. In addition, the recently determined pressure derivatives of elastic moduli for ScAlO 3 ŽKung et al., 2000. are comparable to those for MgSiO 3 perovskite. Close similarities are also expected between the temperature derivatives of the elastic moduli for these two materials. The value Žy0.039 GPa Ky1 . of ŽEK TrET . P for ScAlO 3 derived here by direct acoustic measurements of ŽEK SrET . P is intermediate between those inferred less directly from V Ž P,T . studies of Fe-free and Fe- or Al-bearing MgSiO 3 perovskite, whereas the value of ŽEGrET . P Žy0.015 GPa Ky1 . is markedly smaller in magnitude than that Žy0.029 GPa Ky1 . measured in a recent ultrasonic study of MgSiO 3 ŽSinelnikov et al., 1998.. It seems that substantial intrinsic contributions to both ŽEK TrET . P and ŽEGrET . P are characteristic of both of these perovskite phases, but possible differences in degree are not yet understood. Pending the completion of definitive acoustic measurements on the silicate perovskite, the pressure and temperature dependence of the elastic moduli for ScAlO 3 provide useful guidance concerning its EOS. The implications of the ScAlO 3 –MgSiO 3 analogy

313

for the inferred composition and temperature of the lower mantle will be explored in a forthcoming paper ŽKung et al., 2000.. Acknowledgements We thank G. Fischer for design of the modifications to the part of the pressure vessel and G. Horwood and machine shop for the technical support. The improvements to the manuscript resulted from the comments and suggestions of two reviewers ŽN.L. Ross and D.G. Isaak.. References Anderson, O.L., Isaak, D.G., 1995. In: Ahrens, T.J. ŽEd.., Elastic Constants of Mantle Minerals at High Temperature vol. 64 American Geophysical Union, p. 70. Anderson, O.L., Isaak, D.G., Oda, H., 1992. High temperature elastic constant data on minerals relevant to geophysics. Rev. Geophys. 30, 57–90. Bass, J.D., 1984. Elasticity of single-crystal SmAlO 3 , GdAlO 3 and ScAlO 3 perovskites. Phys. Earth Planet. Inter. 36, 145– 156. Duffy, T.S., Anderson, D.L., 1989. Seismic velocities in mantle minerals and the mineralogy of the upper mantle. J. Geophys. Res. 94, 1895–1912. Gieske, J.H., Barsch, G.R., 1968. Pressure dependence of the elastic constants single crystalline aluminium oxide. Phys. Status Solidi 29, 121–131. Goto, T.S., Yamamoto, I.O., Anderson, O.L., 1989. Elastic constants of corundum up to 1825 K. J. Geophys. Res. 74, 5949–5960. Hemley, R.J., Stixrude, L., Fei, Y., Mao, H.K., 1992. Constraints on lower mantle composition from P – V – T measurements of ŽFe,Mg.SiO 3 –perovskite and ŽFe,Mg.O. In: Syono, Y., Manghnani, M.H. ŽEds.., High-Pressure Research: Application to Earth and Planetary Sciences vol. f67pp. 191–196, TerrapubrAGU, TokyorWashington. Hill, R.J., Jackson, I., 1990. The thermal expansion of ScAlO 3 — a silicate perovskite analogue. Phys. Chem. Miner. 17, 89–96. Jackson, I., 1998. Elasticity, composition and temperature of the Earth’s lower mantle: a reappraisal. Geophys. J. Int. 134, 291–311. Jackson, I., Rigden, S.M., 1996. Analysis of P – V – T data: constraints on the thermoelastic properties of high-pressure minerals. Phys. Earth Planet. Inter. 96, 85–112. Jones, L.E.A., 1979. Pressure and temperature dependence of the single crystal elastic moduli of the cubic perovskite KMgF3 . Phys. Chem. Miner. 4, 23–42. Knittle, E., Jeanloz, R., 1987. Synthesis and equation-of-state of ŽMg, Fe.SiO 3 perovskite to over 100 GPa. Science 235, 669–670.

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