Pressure derivatives of shear and bulk moduli from the thermal Grüneisen parameter and volume-pressure data

Geochimica et Cosmochimica Acta, Vol. 67, No. 7, pp. 1207–1227, 2003 Copyright © 2003 Elsevier Science Ltd Printed in the USA. All rights reserved 0016-7037/03 $30.00 ⫹ .00

Pergamon

doi:10.1016/S0016-7037(02)01289-9

Pressure derivatives of shear and bulk moduli from the thermal Gru¨neisen parameter and volume-pressure data A. M. HOFMEISTER1,* and H. K. MAO2 1

Department of Earth and Planetary Sciences, Washington University, St. Louis, MO 63130, USA 2 Geophysical Laboratory, 5251 Broad Branch Rd. NW, Washington D.C. 20015, USA

(Received December 20, 2001; revised 9 October 2002; accepted in revised form October 9, 2002)

Abstract—Gru¨neisen’s parameters are central to studies of Earth’s interior because these link elastic data to thermodynamic properties through the equation of state and can be measured using either microscopic or macroscopic techniques. The original derivation requires that the mode Gru¨neisen parameter (␥i) of the longitudinal acoustic (LA) mode equals the thermodynamic parameter (␥th) for monatomic solids. The success of the Debye model indicates that ␥LA ⫽ ␥th is generally true. Available elasticity data for crystalline solids contain 30 reliable measurements, covering 10 structures, of the pressure derivatives of the bulk (KS) and the shear (G) moduli. For these phases, the measured values of ␥th and ␥LA agree well. Other solids in the database have disparate ␥LA values, suggesting large experimental uncertainties within which ␥LA ⫽ ␥th. This relationship allows inference of the pressure (P) derivative of the shear modulus (⭸G/⭸P ⫽ G') from widely available measurements of ␥th, the isothermal bulk modulus (KT), ⭸KT/⭸P, and G. We predict G' as 1.55 for stishovite, 1.6 to 2.15 for MgSiO3 ilmenite, 1.0 for ␥-Mg1.2Fe0.8SiO4, and 0 for FeS (troilite). Similarly, G' measured for MgSiO3 perovskite suggests that KS' ⫽ 4, corroborating volume-pressure data. For many materials, pairs of G' and KS' ⫽ ⭸KS/⭸P from independent elasticity studies of a given phase define a nearly linear trend, suggesting systematic errors. Non-hydrostatic conditions and/or pressure calibration likely cause the observed variance in KS' and G'. The best values for pressure derivatives can be ascertained because the trend defined by measured pairs of G' with KS' intersects the relationship of G' to K' defined by ␥LA ⫽ ␥th at a steep angle. Our results for isostructural series show linear correlations of KS' with KS and of G' with G. Values of KS' are nearly 4 for high-pressure phases, which is consistent with the harmonic oscillator model, whereas G' has a wide range of ⫺1 to 3. Hence, inference of a detailed mineralogy inside the Earth is best constrained by comparing seismic determinations of shear moduli to laboratory measurements. Copyright © 2003 Elsevier Science Ltd (below and section 1.2) utilize the Gru¨neisen parameter, an important physical property that can be obtained by probing either the macroscopic or the microscopic response of the solid. In correlating elastic properties with various physical properties, the available data are presumed to be reasonably accurate. Accuracy is needed regardless of the various approaches used, such as velocity-density systematics (e.g., Shankland, 1972), structural and chemical trends (e.g., Duffy and Anderson, 1989), or similar behavior of various elastic parameters (e.g., Anderson, 1988; Kung and Rigden, 1999). For metals, Duffy and Wang (1998) showed that good agreement exists between the bulk moduli obtained by completely independent techniques; however, the pressure derivatives from the same studies are inconsistent, and interlaboratory comparisons of ultrasonic data show considerable variation. For insulators, disparities are known to exist between ⭸KT/⭸P obtained from volumetric studies and ⭸KS/⭸P from ultrasonic measurements, such that KS' is generally substantially larger (cf. the compilations of Knittle, 1995, and Bass, 1995), although thermodynamic relationships predict almost indistinguishable values. Similarly, disagreement is seen in the pressure derivatives obtained mostly for ultrasonic studies as tabulated by Isaak (2001). The existence of errors in the pressure derivatives, as suggested by the above comparisons, not only could obscure the true relationships between the pressure derivatives of elastic moduli (or sound speeds) and physical properties, but could also limit the validity of extracting the mineralogy from seismic profiles.

1. INTRODUCTION

The mineralogy of the Earth’s interior is deduced by comparing sound velocities obtained from seismic studies to laboratory measurements of phases that are stable at high pressure and temperature. Elastic moduli are commonly compared due to a close relationship of the adiabatic bulk modulus (KS) to the isothermal bulk modulus (KT) and because KT can be reliably extracted from volumetric data at pressure. Pressure derivatives are especially important because the pressure (P) inside the Earth is known and because the bulk and shear (G) moduli can be extrapolated to great depth by applying an appropriate equation of state (EOS) to the derivatives (KS' ⫽ ⭸KS/⭸P; G' ⫽ ⭸G/⭸P) at ambient conditions. Due to the importance and multivariable nature of the problem (phase, composition, and temperature all affect mineral properties and enter into deductions of Earth’s interior mineralogy from seismologic data) and to the daunting task of acquiring the necessary data on all such relevant phases, much effort has been expended in (1) correlating the above elastic properties with other, well-known physical properties, (2) providing internal consistency in the database by applying thermodynamic relationships (both exact and approximate), and (3) simplifying the EOS to maximize information obtained from the seismic profiles. All three endeavors

* Author to whom correspondence ([email protected]).

should

be

addressed. 1207

1208

A. M. Hofmeister and H. K. Mao

The present report uses a revised definition of the microscopic Gru¨ neisen parameter (Hofmeister and Mao, 2002) to understand the source of the inconsistencies in measurements of KS' and G'. Previous studies examined different relationships (section 1.2) or sought trends among physical parameters (discussed above). Our analysis is timely because the database of elasticity measurements at high pressure has rapidly expanded over the past 5 yr in terms of the numbers of solids examined as well as in the number of repeat measurements on a given solid. Moreover, several new techniques have been developed (e.g., Singh et al., 1998; Dubrovinsky et al., 2000; Olijnyk and Jephcoat, 2000; Hofmeister and Mao, 2001) in addition to adaptation of the Brillouin method to ultrahigh pressures (e.g., Zha et al., 1997; Sinogeikin and Bass, 2000), which provide an independent check against conventional, ultrasonic types of measurements. Our comparison reveals surprising disparities among measurements of the pressure derivatives of the bulk and shear moduli for a given substance. However, reliable data (i.e., reproducible pressure derivatives) do exist, and these data suggest that our revised definition of the microscopic Gru¨ neisen parameter is correct. This result leads to a method for predicting G' from G and thermodynamic properties. For a given substance, the particular dependence of G' on KS' suggests that systematic errors occur: The measured trends and our revised definition together provide clues for the sources of the errors and point to the correct pair of derivatives. The results suggest that K' is close to 4 for almost all solids, whereas G' is variable and structurally dependent. Geophysical implications are discussed.

vibrating and the changes in frequencies across solid solution series. That the vibrational frequencies in isostructural series are explained by a simple harmonic oscillator model also validates the local picture (see Nakamoto, 1978; Hofmeister, 1993a). For monatomic solids, the volume of the unit cell (or the molar volume) is proportional to the atomic volume, allowing direct conversion of Eqn. 4 to Eqn. 3. For cubic diatomic solids (e.g., MgO), only one interatomic distance exists, and the volumes about the cation or anion are each proportional to the molar volume. Eqn. 3 also holds for this case but not necessarily for two-atom structures such as corundum (Al2O3) or stishovite (SiO2) or for diatomics with hexagonal unit cells (e.g., ZnS-wurtzite), as both structure types have two cation– anion bond lengths (e.g., Ross et al., 1990). However, in these structures the volume about the cation is uniquely defined, and the structure can be approximately described in terms an average bond length. This inference is consistent with results from spectroscopic modeling of stishovite using Eqn. 3 (Hofmeister, 1996). The remaining case is that of polyatomic structures, many of which have some type of functional group, e.g., SiO4 tetrahedra. These structures are described by multiple polyhedral volumes (e.g., Hazen and Finger, 1982). Hofmeister and Mao (2002) proposed that for each vibration in a solid, the volume used in Eqn. 3 should not be that of the bulk crystal but instead should be the specific volume which changes during the particular atomic motion that is correlated with any given frequency. This approach upholds the original proposal of Gru¨ neisen (1912, 1926). Eqn. 3 is recast as

␥ i0 ⫽

1.1. Theoretical Background The thermal Gru¨ neisen parameter,

␥ th ⫽ ␣ K TV/C V ⫽ ␣ K SV/C P.

(1)

where ␣ is thermal expansivity and CV is the heat capacity at constant volume (V), relates elastic and volumetric measurements through K S ⫽ K T共1 ⫹ ␣␥ thT兲.

(2)

The macroscopic quantity ␥th is closely related to microscopic behavior of the solid, as mode Gru¨ neisen parameters (␥i) are inferred from spectroscopic measurements of the changes of the acoustic and optic vibrational frequencies (␯i) with pressure. The formula commonly used (e.g., Anderson, 1989) is

␥ i0 ⫽

K T0 ⭸ ␯ i ␯ i0 ⭸P

⭸共ln ␯ 冏 ⫽ ⫺ ⭸共ln 冏. V兲 i

0

(3)

0

However, Gru¨ neisen (1912, 1926) originally considered a monatomic solid, explicitly using the volume about the vibrating atom (Va):

␥i ⫽ ⫺

VA ⭸␯i . ␯ i ⭸V A

(4)

This local picture of the vibrational modes is valid because nearest-neighbor interactions dominate in solids. Examples are use of the vibrational frequency as an indicative of the species

KX ⭸␯i ␯ i0 ⭸P

冏

(5) 0

where KX is the bulk modulus associated with the polyhedral volume involved in the vibration. The longitudinal acoustic (LA) mode expands and contracts the entire unit cell: hence, KX ⫽ KT, and the LA mode Gru¨ neisen parameter is correctly obtained from existing equations, using Debye’s relationship of sound speeds to acoustic frequencies:

␥ LA ⫽

冏

K T ⭸ ␯ LA ␯ LA ⭸P

⫽ ⫺ 0

1 K T 共K⬘S ⫹ 4G'/3兲 ⫹ 6 2 共K S ⫹ 4G/3兲

(6)

(e.g., Sumino and Anderson, 1984). Debye’s theory assumes that the solid is quasi-isotropic. All quantities are measured at ambient conditions, and the primes denote the pressure derivatives. In contrast, the pure shear motion for the transverse acoustic (TA) mode does not change either the unit cell or atomic volumes (e.g., Beyer and Letcher, 1969). Because there is no definable volume change associated with the motion, from Eqn. 4, ␥TA cannot be rigorously defined. At the center of the Brillouin zone, the acoustic frequencies are zero, so that the average Gru¨ neisen parameter only requires knowledge of the optic modes. Because each IR mode interacts with all other vibrational modes in the solid (the damped harmonic oscillator model; see Hofmeister, 2001), the vibrational average should be valid over the entire Brillouin zone and should represent the thermodynamic quantity, i.e.,

Pressure derivatives of shear and bulk moduli

具 ␥ optic典 ⫽ ␥ th

(7)

(e.g., Merkel et al., 2000). In the Debye model of the solid, the acoustic frequencies are taken to represent all vibrations of the solid, and thermal properties such as heat capacity are reasonably represented by this approximation (e.g., Kieffer, 1979). Because the pressure responses of transverse acoustic modes (shear waves) are not relevant to the thermodynamic behavior of the solid,

␥ LA ⫽ ␥ th

(8)

For primitive unit cells that contain only one atom (most metals) and lack optic modes, Eqn. 8 is strictly true. For all other structure types, the Debye model is an approximation. Should Eqn. 8 prove to be reasonably accurate for a wide range of phases, it would provide an extremely useful connection between thermodynamic (Eqn. 1) and elastic (Eqn. 6) properties. Two pathways exist to test our revised formulas for the mode Gru¨ neisen parameters: (1) detailed examinations of data for certain mineral families or (2) consideration of general trends for diverse phases. Hofmeister and Mao (2002) conducted an in-depth test of Eqn. 5, 7, and 8 against the nesosilicates forsterite, its polymorph, ␥-Mg2SiO4, fayalite, ␥-Fe2SiO4, and spinel (MgAl2O4). These minerals have essentially complete spectroscopic datasets at pressure, band assignments that are well-established by chemical, isotopic, pressure, and temperature studies, and extensive measurements of their structural, thermodynamic, and elastic properties. No other polyatomic substance has the amount of data, which are needed to test Eqn. 5. We found that the average ⬍␥i⬎ calculated separately from each mode type (Raman, IR, or LA) for olivines, and silicate and aluminate spinels equals ␥th within the experimental uncertainties. The good to excellent agreement between each of ␥th, ␥LA, and ⬍␥i⬎ for the spinel and olivine structures suggests that Eqn. 5 through 8 should be generally valid, leading us to test Eqn. 8 in the present paper against substances relevant to the geosciences. 1.2. Relationship to Earlier Studies Gru¨ neisen parameters are tied to elasticity measurements in several different ways because the macroscopic version is proportional to KS (Eqn. 1), whereas microscopic formulations involve KS' and G' (Eqn. 6) and because various approximations are used to simplify the formula. The approach most similar to ours involves comparison of ␥th to the weighted average of the TA and LA Gru¨ neisen parameters [␥ac ⫽ (2␥TA ⫹ ␥LA)/3] (e.g., Sumino and Anderson, 1984). Recently, Anderson (1995) found discrepancies between ␥ac and ␥th for many solids and attributed the lack of correspondence for some solids to these materials departing from the Debye model. This attribution cannot be the underlying cause of the discrepancies because the heat capacity of all minerals departs from a Debye model to some degree (as shown by the Debye temperature being temperature dependent; see tables in Anderson and Isaak, 1995). We explore below whether including the shear waves is the source of the discrepancies. Other researchers derived simple relationships between ␥th and KT', which do not involve G or G', by constructing an EOS

1209

from interatomic potentials (e.g., Dugdale and McDonald, 1953; Irvine and Stacey, 1975). The equations are summarized by Quareni and Mulgaria (1989). For example, Hofmeister (1991a, 1991b) obtained KT'(P) ⫽ 5/3 ⫺4P/9KT ⫹ 2␥th (1 ⫺4P/9KT) by relating vibrational frequencies to KT. Central forces are generally assumed, and ␥th is commonly treated as being independent of temperature and/or pressure. Such relationships are used to approximate the thermal Gru¨ neisen parameter inside Earth’s mantle and core (e.g., Irvine and Stacey, 1975; Quareni and Mulgaria, 1989; Anderson, 1998). Our approach provides a direct link to seismic data with few restrictive assumptions. 2. COMPARISON OF ␥LA TO ␥th

2.1. The Database In comparing thermodynamic and elastic Gru¨ neisen parameters, we strove to create an impartial test by including all crystalline solids for which the bulk and shear moduli were measured at pressure. The compilations of Sumino and Anderson (1984), Bass (1995), Liebermann (2000), Isaak (2001), and Wang and Ji (2001) were utilized and supplemented with data on KS' and G' of metals and recent studies of insulators and minerals, mostly in the materials science literature (Table 1). For convenience, some phases important to geophysics included in the above compilations are repeated in Table 1. Some authors presented the same data in several papers, but we examined only the most recent study (e.g., Abramson et al., 1997), assuming it represents the best analysis. Otherwise, no elasticity measurement at pressure that we encountered during our search of the literature was excluded, although a few substances listed in the above compilations could not be tested because data needed to compute ␥th are unavailable and could not be reliably estimated. Given the size of the database (Table 1), the above compilations, publications on metals cited in Figure 1, and the similar behavior for most materials, the results would not be altered if some studies were missed. Experimental uncertainties vary with the technique (e.g., Every and Sachse, 2001) and are also associated with application of pressure (e.g., Duffy and Wang, 1998). Most authors reported the second decimal place (i.e., K' ⫽ 4.08). Spectroscopic and X-ray experiments calibrate pressure using an internal standard (e.g., the ruby scale), and thus the uncertainties in KS' and G' stem mainly from spectral resolution. Ultrasonic methods are precise, but external calibration is used (e.g., manganin coils), which can contribute significant errors. Curve fitting is not a problem. Most studies, particularly ultrasonic methods, used a linear fit for the pressure dependence of the bulk and shear moduli. The results (KS' and G') represent the average over the range of pressures examined and thus the lower limit for these first derivatives because the second derivatives with pressure are negative. However, for almost all of the ultrasonic studies, the pressure range is narrow, and thus the fit essentially provides the initial slope. When curvature of KS or G with pressure was observed, such as for orthopyroxenes (Table 1), the results were fit to EOS or to polynomials, which also provide the initial slope. Thus, interlaboratory comparisons are valid. Uncertainties also exist in ␥th, mostly arising from thermal

1210

A. M. Hofmeister and H. K. Mao Table 1. Elasticity data for high-pressure phases and other substances of geologic interest. Phase

stishovite quartz andradite “andradite” grossular “grossular” Py52Al32Gr16 Py62Al36Gr2 Py94Al6 pyrope pyrope pyrope pyrope Mj38Py62 Mj50Py50 Mj80Py20 majorite majorite Al2O3 Al2O3 Al2O3 MgSiO3-ilm FeTiO3 MgSiO3-pv ScAlO3-pv CaSnO3-pv CaTiO3-pv CaTiO3-pv SrTiO3-pv BaTiO3-pv BaTiO3-pv orthoenstatite Mg0.8Fe0.2SiO3 Mg0.8Fe0.2SiO3 Mg0.91Fe0.09SiO3 ␣-Mg2SiO4 ␣-Mg2SiO4 ␣-Mg1.8Fe0.2SiO4 ␤-Mg2SiO4 ␤-Mg2SiO4 ␤-Mg1.76Fe0.24SiO4 ␥-Mg2SiO4 ␥-Mg1.82Fe0.18SiO4 ␥-Mg1.2Fe0.8SiO4 ␥-Fe2SiO4 ␥-Fe2SiO4 ␣-Co2SiO4 ␥-Co2SiO4 ␣-Ni2SiO4 ␥-Ni2SiO4 ␥-Mg2GeO4 MgAl2O4 MgAl2O4 MgAl2O4 Y3Al5O12 Y3Al5O12 Y3Fe5O12 MgO MgO MgO MgO

␥th

␥LA

1.34 0.67 1.22

– 1.08 3.09 1.84 1.72 1.94 1.68 1.88 2.08 1.52 1.32 2.00 1.81 2.12 1.58 – 2.51 1.61 1.65 1.53

1.22 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.5 1.5 1.5 1.50d 1.50d 1.57£ 1.57£ 1.63(13)& 1.5 1.42% 1.6¥ n.d. 1.85

KT (GPa) K⬘T KS (GPa)

K⬘S

G (GPa)

303 37.1 157b

2.8a 6.2 4.4b

170b

5.2b

– 171.5

– 4

171b 172.8c –

4.4b 3.8c –

220 49.3 90 86 108.9 98.1 94.7 92 90.2 93.7 92 92

159 159.8 156 161 169.3 253

4.9d 4e 4.4f 4e 4g 5h

168.5 167 162.6 167.3 166 257.4 253.1

5.3 6.4 5.85 4.7 5.46 5.9 4.09 4.93 4.9 4.1 3.22 5.3 4.5 5.3 4.2 – 6.7 4.2 4.4 4.0

210 177 254

4 4 4

212

–

MgO

1.54

1.49@ 1.59 1.56 1.96 1.76 210? 1.46 174.2 5.02 2.4 3.12 105.8 3.17 2.98 – 2.34 – 1.22 128 1.56 128 1.59 – 1.46 171 1.73 174 1.63 173 1.74 182 m 1.39 197.5 1.45 207.6 2.13 200 147 206 164 2.68 227 1.44 – 1.41 194 1.78 1.29 1.57 1.50 1.68 1.58 156 1.61 – 1.61 1.52 1.45 1.58 160.2

MgO CaO MnO

1.54 1.35 1.51

1.525 1.67 1.72

1.63 1.80 1.28% 1.28 1.28 1.28 1.29 1.29 1.31 1.39 1.39 1.39 1.25 1.28 1.33 1.51# 1.51# 1.26 1.41§ 1.44 1.60§ 1.45¶ 1.30 1.30 1.3 1.43⵩ 1.43⵩ 1.35⵩ 1.54 1.54 1.54 1.54

160.5 112 144

5.6? 4.4 8.5i

316 37.8 159.4 162.5 166.8 166.3 170.8 172.8 171 171.2 172.7 171

184.2 120 89.3 105 105.4 112.5 68.4

104 107.8 – 115.4 131.1 130 129.4 170 174 172 184 188 199.5 209.6 201 148 208 165 226 179 196.7 197.9

74.9 75.7 – 77.9 79.4 78.2 78 115 114 106 119 120 112 87.5 79 62 105 80 106 111 108.3 108.5

131.1 80.6 68.3

161.5 160 162.7

4.03

162.5

– 4.2 3.3

162.8 112 149

3.85 4.83 4.7

4n 4

– 4o

1.8 1.9 0.9 1.48 1.44 1.38 5.2

n.d. 3.8 5.0 5.78 5.1 4.3 10.5

4.7 –

– – 3.84 3.84 – 7j 4k 7j 4.2m

89 90 86 88.3 85 162.2 162.9 n.d. 132

246.4 218 167.2 175 177 177.8 139.2

10.9 10.8 9.6 7.82 3.8 4.2 4.29 4.3 4.7 4.6 4.84 4.1 – ⬅4 5.59 – – – 7.61 4.22 4.89 5.7 5.05 4.44 5.16 4.75 4.13 4.19 4.08 3.96 3.83 3.99

180.3

G⬘ n.d. 0.46 4.3 1.7 1.1 1.3 1.76 1.6 2.1 1.3 1.4 1.6 1.73 2.0 1.4 – 1.9 1.4 1.8 1.73 1.94 –

110.8 80.6 130.3 131.1

130.4

References for elasticity data Liu et al. (1999) Average from Bass (1995) Conrad et al. (1999)* Wang and Ji (2001) Conrad et al. (1999)* Wang and Ji (2001) Chai et al. (1997a)‡ Webb (1989) Wang and Ji (2001) Sinogeikin and Bass (2000)* Conrad et al. (1999)* Chen et al. (1999) Chopelas et al. (1996)‡ Rigden et al. (1994) Sinogeikin and Bass (2002)* Sinogeikin et al. (1997) Gwanmesia et al. (1998) Sinogeikin and Bass (2002)* Li et al. (1996), Gieske and Barsch (1968) Kung et al. (2000), Goto et al. (1987) Chopelas et al. (1996)‡ Weidner and Ito (1985)*

Sinelikov et al. (1998) Kung et al. (2000) Kung et al. (2001) Kung and Rigden (1999) Fischer et al. (1993) Fischer et al. (1993) Fischer et al. (1993) Ishidate and Sasaki (1989)* 1.60 Flesch et al. (1998), K⬙ ⫽ ⫺0.16/kbar 2.06 Webb and Jackson (1993), K⬙ ⫽ ⫺0.16/kbar 2.38 Frissillo and Barsch (1972) 1.44 Chai et al (1997b)‡ 1.0 Zha et al. (1998)*, K⬙ ⫽ G⬙ ⫽ 0 1.6 Zha et al. (1998)*, K⬙, G⬙ ⫽ 0 1.71 Abramson et al. (1997)‡ 1.4 Zha et al. (1997)* 1.8 Rigden et al. (1992) 1.5 Li and Liebermann (2000) 1.75 Ridgen et al. (1992) 1.3 Sinogeikin et al. (2001)* – Hofmeister and Mao (2001) 0.81 Hofmeister and Mao (2001)“ 1.06 Rigden and Jackson (1991) – Sumino (1979) – G est. by Hofmeister and Mao (2001) – Bass et al. (1984) 1.42 Rigden and Jackson (1991) 1.33 Ridgen and Jackson (1991) 0.51 Chang and Barsch (1973a) 0.70 Yoneda (1990), Voigt limit ⫹ 0.4 0.072 Chopelas et al. (1996)‡ 0.93 Yogurtcu et al. (1980), Voigt limit 0.68 Chopelas et al. (1966)‡ 0.6 Eastman (1996), Voigt limit 2.5 Jackson and Neisler (1982) 2.46 Yoneda (1990) 2.56 Chopelas et al. (1996)‡ 2.35 Sinogeikin and Bass (2000)*, K”, G” not equal to O 2.21 ibid. K⬘ ⫽ G⬙ ⫽ 0, Pmax ⫽ 200 kbar 2.52 Zha et al. (2000)*, K⬙ ⫽ 0, polynomial fit to G, Pmax ⫽ 500 kbar 2.45 Spetzler (1970) 1.78 Oda et al. (1992); Chang and Graham (1977) 1.2 Pacalo and Graham (1991) (continued)

Pressure derivatives of shear and bulk moduli

1211

Table 1. (continued)

␥th

␥LA

MnO Fe0.94O Fe-␣ Fe-␣ Fe-⑀ Fe-⑀ Fe-⑀ Fe-⑀ FeS Au Mo Mo Re Re

1.51 1.36 1.70$ 1.70$ 1.7Þ 1.7Þ 1.7Þ 1.7Þ 0.60 2.90$ 1.62$ 1.62$ 1.8$ 1.8$

2.02 1.98 2.12 1.79 1.80 1.81 2.14 1.7

diamond A1N ceramic A1N ceramic ␤-Si3N4 ceramic TiB2 ceramic NaCl CaF2 CaCO3

0.75$ 1.6$ 1.3$ 1.09 1.1-1.4$ 1.59 1.74 0.55

1.19 1.09 1.18 1.18 1.48 1.72 1.63 0.84

Phase

2.32 1.53 1.64 2.17 1.78

KT (GPa) 154 163.2 167 165.9

K⬘T 3.8 4 4.01 4.92

KS (GPa)

5.28 5.06 5.3

160 167.7 235 84p 171.7 270

82 166.6 266

5.0p 5.5 3.5

360 350

4.5 4q

356.2

1.9

443 200 159.9 276 276 24.9 84.5 77.3

44.4 273 23.8 82.2 77.1

3.8r 4.45s 4.1

K⬘S

152.1 154.8 166.7

G (GPa)

G⬘

4.4 5.39 5.4-G⬘ – 5.33 4.63 4.47 5.4 4.6

66.8 46.4 81.5 78.9 85.9 75 81.9

1.55 0.71 1.8 1.90 1.38 1.67 1.68

31.5 27.6 149

– 0.68 1.1 1.42 1.8 1.4

4.0 4.4 5.2 4.3 3.9 4.72 4.78 5.4

535.7 130 130.8 252 252 14.7 42.8 32

178.7

2.3 0.22 0.2 0.17 2.5 1.75 1.08 ⫺1.7

References for elasticity data Webb et al. (1988) Jackson and Khanna (1990) Average from Bass (1995) Dubrovinsky et al. (2000)† Dubrovinsky et al. (2000)†, P ⬍ 420 kbar Mao et al. (2001)†, initial slope Dubrovinsky et al. (2001)†, assumed ␥ac ⫽ ␥th Fiquet et al. (2001)†, P ⬍ 110 kbar, ␳0 ⬅ 8.3 g/cm3 G Estimated, see Fig. 4 Duffy et al. (1999a)† Duffy et al. (1999a)† Katahara et al. (1979) Manghnani et al. (1974) K⬘(Duffy et al. 1999b)†; c44⬘ (Olijnyk et al. 2001)⬙ McSkimin and Bond (1972) Dodd et al. (2001a) Gerlich et al. (1986) Dodd et al. (2001b) Dodd et al. (2001c) average威 Speziale and Duffy (2002)* Kaga (1968)

The Gru¨ neisen parameter was obtained from Eqn. 1. Except as noted, these values are reported by Sumino and Anderson (1984) and Anderson and Isaak (1995). Elasticity data at ambient conditions are summarized by Bass (1995), Liebermann (2000), Isaak (2001) and Wang and Ji (2001): pressure derivatives are from ultrasonic or pulse-superposition methods, unless noted. Additional substances can be found in these compilations, but not all of these have data needed to compute ␥th. Phase names in quotes are nominal compositions. Isothermal bulk moduli and pressure derivatives are from the summary of Knittle (1995), unless noted. a Ross et al. (1990): Hofmeister (1996) obtained K⬘T ⫽ 4 from applying a semi-empirical model to IR and Raman spectra. b Zhang et al. (1999). c Leger et al. (1996). d Wang et al. (1998). e Yagi et al. (1992). f Morishima et al. (1999). g Hazen et al. (1994). h Richet et al. (1988). i Angel and Jackson (2002). j Hazen et al. (2000); the authors state that K' is not well-constrained. k Fei et al. (1992). m Meng et al. (1993). n Hofmeister and Mao (2001) refit V(P) by Zerr et al. (1993) o Finger et al. (1986). p King and Prewitt (1982). q Liu et al. (1970). r Li et al. (1997). s Hofmeister (1997) * These elasticity data were obtained using Brillouin scattering. ⬙ These elasticity data were obtained from a resonance of the LA mode in the IR for ␥-FeSiO4 or from Raman spectroscopy for metals. † These elasticity data were obtained from diffraction data by various analysis: Debye-Waller temperature factors (Dubrovinsky et al. 2000, 2001), phono density of states (Mao et al. 2001), inelastic scattering (Fiquet et al. 2001), and stress analysis for the other metals. ‡ These elasticity data were obtained from sideband spectrscopy or LIPS spectroscopy for olivine and orthopyroxene. £ Calculated using ␣ of Dubrovinsky et al. (1998). % Composition is MgSiO3; ␥th from Chopelas (2000). Ultrasonic data are for Fe-bearing opx. § Thermal expansivity is taken as 22 ⫻ 10⫺6/K for the Ni and Co silicate spinels; see Hofmeister and Mao (2001). # Thermal expansivity from Mao et al. (1969); see Hofmeister and Mao (2001). 㛳 Computed using ␣ ⫽ 23.4 ⫻ 10⫺6/K of pyrope and the average of the two KT values, which is consistent with KS. Chopelas (2000) obtained a lower ␥th ⫽ 1.28 based on mode Gru¨ neisen parameters obtained from Raman spectroscopy at pressure. This approach typically gives ␥th that is low by 10%, due to use of average bulk moduli in Eqn. 2, rather than polyhedral bulk moduli; see Hofmeister and Mao (2002) for detailed discussion of olivines and spinels. Given the 10% underestimation, Chopelas’ (2000) value supports our determination. ¥ From Hill and Jackson (1990); ␥th is computed from a constant ␣. & Computed using ␣ of Ashida et al. (1988) and CV of Hofmeister and Ito (1991). Data on structural analogues support these values; see text. @ used K⬘S ⫽ K⬘T ⫽ 4. ¶ For the olivine polymorph of Mg2GeO4, ␣ ⫽ 26.4 ⫻ 10⫺6/K (Fiquet et al. 1992), which compares closely to ␣ of various olivines (Fei et al., 1995), suggesting that an average ␣ ⫽ 21 ⫻ 10⫺6/K from silicate spinels (see Fei, 1995; Hofmeister and Mao, 2001) is appropriate for and ␥-Mg2GeO4 and the other germanates with the spinel structure included in Fig. 1b. Heat capacity is known for Mg2geO4 (Ross and Navrotsky, 1987) and was used for the other compounds. ⵩ From Yogurcu et al. (1980), adjusting ␣ of YIG to 22 ⫻ 10⫺6/K, as seen for other garnets. $ Calculated form ␣ and CP compiled by Grigoriev and Meilikhov (1997). Conflicting reports exist for the ceramics, which may be related to whether the measurements were made on single crystals or polycrystalline material. Þ Thermodynamic estimate from Anderson (1998). 威 Average of Spetzler et al. (1972), Chabildas and Ruoff (1976), and Kim et al. (1989).

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Fig. 1. Comparison of ␥LA with ␥th for elements. Nominal experimental uncertainties are about the size of the symbols. Thick line ⫽ one-to-one correspondence. Thin line shows the range of ␥LA observed for materials with multiple measurements (e.g., Fe). All data are individual measurements from Table 1 (Sumino and Anderson, 1984; Bass, 1995) or as noted. Filled symbols ⫽ monatomic metals as labeled. Open symbols ⫽ diatomic metals. Grey symbols ⫽ diatomic C and Ge. Data from Hughes and Maurette (1956), Alers and Neighbors (1957) Schmunk and Smith (1959), Voronov and Vereschagin (1960), Bogardus (1965), Schwartz and Elbaum (1970), van’t Klooster et al. (1979), and Biswas et al. (1981). The cluster of points near the 1:1 line (Mo, Re, ␣- and ␧-Fe) are mostly recent data (Table 1).

expansivity. For many germanate spinels, ␣ and/or CV was estimated (see Table 1). Most values of ␥th were either provided by Sumino and Anderson (1984) or calculated from existing summaries of heat capacity and thermal expansivity data (e.g., Fei, 1995; Grigoriev and Meilikhov, 1997; Chopelas, 2000). Cross-checks (e.g., Shim and Duffy, 2000) were used if available (see notes in Table 1). Determining the accuracy is equivocal, as few duplicate measurements exist. Corundum is a one example, with ␣ ⫽ 16.2 ⫻ 10⫺6/K (White and Roberts, 1983), 19.0 ⫻ 10⫺6/K (Dubrovinsky et al., 1998), or 23.0 ⫻ 10⫺6/K (Aldebert and Traverse, 1984). The most recent ␣ lies near the average of the others, suggesting it is correct. The four measurements of ␣ for forsterite fall between 26.4 ⫻ 10⫺6/K and 30.6 ⫻ 10⫺6/K (Fei, 1995). Soft materials such as metals and well-studied phases (e.g., MgO) have tighter constraints. The average uncertainty for ␥th is probably 5 to 7%. 2.2. General Behavior Inferred from All Available Data At first glance, only a few of the 111 datasets on 65 different crystalline solids at pressure compiled by Sumino and Anderson (1984) apparently adhere to Eqn. 8 (Figs. 1, 2, 3, 4) within the nominal experimental errors of the measurements. Closer inspection reveals that the datapoints for substances with multiple measurements form vertical strips (e.g., Fe, Ag, NaCl, MgO, Al2O3, forsterite, spinel-types, wadsleyite, and pyrope in

Fig. 2. Comparison of ␥LA with ␥th for structures with two kinds of atoms. See Fig. 1 for a general legend. Grey circles ⫽ NaCl; open circles ⫽ other alkali halides; open diamonds ⫽ XF2; square with cross ⫽ CaO; various open squares ⫽ assorted oxides; filled circles ⫽ XS; these data compiled by Sumino and Anderson (1984). The remaining data are from Table 1: filled squares ⫽ MgO. Early ultrasonic data on MgO (Bogardus, 1965; Anderson and Andreatch, 1966) with ␥LA equal to 1.7 are not shown due to overlap with MnO. Filled triangles ⫽ various ceramics. Filled diamond ⫽ CaF2.

Figs. 1, 2, 3, 4), showing that large errors may be present in any individual measurement. The spread in ␥LA for most of these substances is 0.6, which is 40% of the mean value of 1.5. Most ultrasonic studies have ␥LA greater than or equal to ␥th, but the noble metals and a few other “soft” substances straddle the line. The inconsistency noticed for the older ultrasonic data on metals by Duffy and Wang (1998) exists also for insulators and includes recent measurements, although the newer techniques tend to provide ␥LA that lies close to ␥th (Table 1). Potential errors in ␥th do not alter this conclusion. The inconsistencies among the literature values of KS' and G' imply that the majority of the measurements are considerably less accurate than the stated uncertainties. If the spreads observed in ␥LA for ␣-Fe, Au, NaCl, forsterite, ␥-Fe2SiO4, pyrope, and majorite represent the precision, then Eqn. 8 holds for all solids, given the uncertainties in the data. This comparison is not satisfying, in that other relationships could be supported due to range of possible values for solids with multiple measurements. Therefore, we develop an independent criterion for ascertaining the reliability of the elastic data and test Eqn. 8 against this subset of the data. 2.3. A Test Based on Reproducible and Consistent Elasticity Data We eliminate categories of experiments from the comparison based on reproducibility, consistency, and reliable pressure calibrations. In removing suspicious studies from consider-

Pressure derivatives of shear and bulk moduli

Fig. 3. Comparison of ␥LA with ␥th for polyatomics other than garnet or perovskite. See Fig. 1 for a general legend. X ⫽ hydrated sulfates. These data, quartz, calcite, and beryl, were compiled by Sumino and Anderson (1984). Square with cross ⫽ germanate spinels (data from Ridgen and Jackson, 1991). The remaining data are from Table 1 or Bass (1995). Dot ⫽ forsteric olivine. Large dark grey circle ⫽ fayalite; see text. Open square ⫽ ␥-Mg2SiO4. Open circle ⫽ wadsleyite. Open cross ⫽ aluminous spinels. Cross ⫽ enstatite (the highest cross actually has ␥LA ⫽ 3.17 but was offset to best present the other data).

ation, we necessarily eliminate some good data, but given the large number of measurements, these stringent criteria are met by a sufficient number of phases to constitute a test. 2.3.1. Discussion of inconsistencies and possible problems with various techniques Because the early (ca. 1960) ultrasonic studies on metals are inconsistent with each other and with shock-wave data (Duffy and Wang, 1998), the old studies of insulators are also suspect. Figure 1 and Table 1 show that many recent ultrasonic pressure derivatives are inconsistent with each other. The potential problems in the data are discussed in sections 4 and 5.1. For the comparison of ␥LA with ␥th, we disallow inconsistent data as well as measurements on phases that have not been independently confirmed. The present study used only the ultrasonic data from phases for which KS' and G' from several studies fall within 10% of each other for all of the post 1968 measurements. Even this relatively low level of reproducibility is seen only for Al2O3, MgO, and fluorite (Table 1). Wadsleyite and CaTiO3 are included, although only two measurements were made. AlN is included because the disparity in the K' values are due to tradeoffs in determining KS and K'; utilizing the tradeoffs leads to well-constrained ␥LA values. Data obtained using Brillouin spectroscopy should not have significant uncertainties in pressure due to the use of an internal standard. Thirteen measurements on 11 different substances have been obtained so far (Table 1). This type of measurement

1213

Fig. 4. Comparison of ␥LA with ␥th for garnet and perovskite structured polyatomics. See Fig. 1 for a general legend. Open triangles ⫽ flourite perovskites from Sumino and Anderson (1984). Filled triangle ⫽ various oxides with perovskite structure. Filled square ⫽ nearly pure pyrope. Grey squares ⫽ majorite. Open squares ⫽ yttrium garnets. Square with X ⫽ pyrope-almandines. Square with cross ⫽ grossular. Half-filled square ⫽ andradite.

is used for the test because the pressure derivatives so derived in two studies of MgO are consistent not only with each other but also with the reproducible derivatives from three ultrasonic studies (Table 1, Fig. 2). Two substances examined by Brillouin spectroscopy clearly have suspect data and are omitted from the comparison. Soft-mode behavior is observed for BaTiO3 (Ishidate and Sasaki, 1989). The transition via tilting of the octahedra likely affects the elastic and thermodynamic properties in different ways, as it occurs gradually over a range of pressures. For andradite, G' (Conrad et al., 1999) is substantially higher than the values not only for the garnets but also for all of the other substances in Table 1. Several vibrational spectroscopic techniques have been recently developed. The sideband technique (Chopelas et al., 1996) requires that small amounts of impurity ions (such as Cr3⫹) be substituted in the lattice. The results generally agree with ultrasonic studies (Table 1), but KS' is higher than results from Brillouin scattering, and G' results seem scattered to higher and lower values (Table 1). Two potential problems exist. First, mode Gru¨ neisen parameters depend on the change of the volume vibrating and thus on the polyhedral bulk modulus (Eqn. 4). Because the sideband spectra are intimately connected with the local modes of Cr3⫹ in the lattice, the extracted bulk moduli are close to that of the lattice but are not exactly the same. For several phases, the larger Cr ion is stuffed in the smaller Al sites, causing the polyhedral bulk modulus for the local mode to be higher. The shear moduli would be affected by the substitutions, but whether an increase is expected cannot be deduced simply. Second, sideband spectroscopy samples a section of the Brillouin zone, rather than a

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A. M. Hofmeister and H. K. Mao

point. Because the frequency changes with reciprocal lattice (dispersion), the Gru¨ neisen parameters may depend on the size and location of the section in the Brillouin zone, and thus K and G and their pressure derivatives could vary from their zone center values measured in Brillouin scattering techniques. Thus, sideband spectroscopy provides a reasonable approximation to the pressure derivatives of the elastic moduli or for the acoustic velocities but not necessarily the true values. Laser-induced phonon spectroscopy (LIPS) for garnet (Chai et al., 1997a) and olivine (Abramson et al., 1997) gives shear moduli considerably higher than the Brillouin technique for pyrope and forsterite (Table 1; see also Hofmeister and Mao, 2002). The results also differ from the ultrasonic data. It is doubtful that the substitution of 10% Fe in the olivine sample has much of an effect owing to the similar values for forsterite and fayalite. The ultrasonic values for pyrope and pyropealmandine-grossular garnets similarly cluster. The source of the discrepancy is attributed to the fact that pure transverse (shear) waves are not observed in LIPS (Abramson et al., 1997), which is consistent with shear velocities and their pressure derivatives varying more than the bulk moduli and K' do from the Brillouin scattering results. Quasi-transverse modes are induced in LIPS by probing sections that depart from crystallographic directions by a known amount. These modes apparently include contributions of the longitudinal mode, leading to higher G and G' in LIPS than in Brillouin scattering, simply because K and K' are larger than G and G', respectively. Sideband and LIPS data are not considered further. In general, neither are new X-ray techniques such as stress analysis (e.g., Duffy et al., 1999a) demonstrably reproducible, except for ␧-Fe, which was studied many times (Table 1) and which is included. Acoustic modes in IR spectra have been investigated once (Hofmeister and Mao, 2001). The LA mode appears in ␥-Fe2SiO4 because of an accidental degeneracy with an IR mode, and thus its mode Gru¨ neisen parameter should be as accurate as that measured for the IR modes. 2.3.2. The general equality of the acoustic and thermal Gru¨neisen parameters The 30 measurements on 13 different chemical compositions (10 different structures) show that ␥LA equals ␥th within the experimental uncertainty of ⬃7% (Fig. 5) for all but four samples. Some problems are associated with these four measurements. The value KS' of 3.2 for pyrope (Conrad et al., 1999) is surprisingly low (see Table 1). Grossular measured in the same study has an unusually high KS' of 5.5. The two ultrasonic determinations of wadsleyite (Rigden et al., 1992; Li and Liebermann, 2000) agree with each other but not with Brillouin results (Zha et al., 1997). Least-squares fitting omitting the four suspect measurements recapitulates the 1:1 correspondence. Least-squares fitting to all data gave a line that is slightly offset from the 1:1 correspondence (Fig. 5), probably due to larger experimental uncertainties. From the good agreement in Figure 5, which involves all categories of substances with optic modes (hexagonal elements, diatomics, non-cubic diatomics, and polyatomics) and all reliable elasticity measurements, the Debye model and our revised definition (Eqn. 4, 5) reasonably represent the Gru¨ neisen parameters of most geologic materials. The Appendix discusses

Fig. 5. Comparison of ␥LA with ␥th for substances with reproducible or consistent pressure derivatives. Sources of data are in Table 1. Filled symbols ⫽ Brillouin scattering experiments. Open symbols ⫽ ultrasonic. Half-filled symbols ⫽ acoustic mode in IR spectra. X ⫽ X-ray studies of ␧-Fe. Square ⫽ olivine polymorphs. Circle ⫽ pyropemajorite solid solutions or grossular. Triangle ⫽ MgO. Diamond ⫽ fluorite. Cross ⫽ corundum. Right-triangle ⫽ AlN ceramic. Square with cross ⫽ CaTiO3. Dotted line ⫽ least-squares fit to all data. Solid line ⫽ fit omitting grossular or the ultrasonic data on the wadsleyite (open squares). Conrad et al. (1999) data are labeled.

these structure types in detail. It also appears that ␥LA does not equal ␥th for pyroxenes, which initially compress through kinking of the tetrahedral chains (Webb and Jackon, 1990). Other likely exceptions are phases with structural transitions at low pressure (e.g., BaTiO3) because the various physical properties can be affected to different degrees. These cases may be tied together in that the kinking can be regarded as a gradual phase transition. Alternatively, solids that compress dominantly through bond-bending may not follow Eqn. 8 because the underlying assumption of quasi-isotropic compression is violated. Accurate determinations of ␣, KS', and G' are needed for more phases to differentiate these hypotheses. 2.4. Can ␥LA be Distinguished from ␥ac? Previously, the transverse acoustic mode was averaged with the longitudinal mode to compute the average acoustic Gru¨ neisen parameters (section 1.2). We examine the subset of reliable data as well as ␣-Fe (Table 2) to ascertain whether including the TA mode is warranted. For the cubic element Fe and the diatomic oxide MgO, ␥LA is nearly the same as ␥ac, whereas for the remaining phases, ␥LA is roughly 20% larger than ␥ac. Given the uncertainties in ␥LA of ⬃10% (as indicated by Fig. 5), ␥th equals ␥LA but differs from ␥ac. The smaller difference for MgO (Table 2) is still larger than experimental uncertainty in the Zha et al. (2000) ultrahigh-pressure study (see Appendix), and thus Eqn. 8 is supported by the reproducible data. The bcc form of Fe may be unusual by having equivalent pressure

Pressure derivatives of shear and bulk moduli

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Table 2. Comparison of LA (Eqn. 6) and acoustic ((2␥TA ⫹ ␥LA)/3) Gru¨ neisen parameters. Phase

Structure

␣-Fe ⑀-Fe MgO A1N CaF2 Al2O3 CaTiO3 forsterite wadsleyite ringwoodite majorite pyrope

cubic hexagonal cubic hexagonal cubic hexagonal orthorhombic orthorhombic orthorhombic cubic cubic cubic

␥th

␥LA

␥ac

Reference

1.7 1.7 1.54 1.17 1.74 1.57 1.85 1.29 1.39 1.28 1.5 1.50

1.8 1.8 1.52 1.09 1.63 1.59 1.86 1.39 1.46 1.39 1.61 1.52

1.8 1.4 1.41 0.37 1.12 1.34 1.29 1.06 1.07 1.03 1.34 1.19

Dubrovinsky et al. (2000)† Dubrovinsky et al. (2000)†; Mao et al. (2001)‡ Zha et al. (2000)*; Sinogeikin and Bass (2000)* Dodd et al. (2001a)§ Speziale and Duffy (2002)* Li et al. (1996)§; Kung et al. (2000)§ Fischer et al. (1993)§; Kung et al. (2000)§ Zha et al. (1998)*; average of two fits Zha et al. (1997)* Sinogeikin et al. (2001)* Sinogeikin and Bass (2002)* Sinogeikin and Bass (2000)*

See Tables 1 and Appendix Table A for notes. These data are deemed reliable due to the reproducibility of pressure deriviatives. Au and Mo have ␥ac that is 0.3 to 0.4 lower than ␥LA, like the silicates and hexagonal metals. * Brillouin spectroscopy. † Debye-Waller factors or stress analysis. ‡ Nuclear resonant inelastic X-ray scattering. § Ultrasonic measurement.

dependence of its transverse and longitudinal acoustic modes. However, the acoustic parameters in Table 2 for ␣-Fe are obtained from averaging various datasets that are inconsistent (Fig. 1) and thus may not be accurate. From the comparison in Table 2, the Gru¨ neisen parameter of TA mode is not physically meaningful. 3. ESTIMATING THE PRESSURE DERIVATIVE OF THE SHEAR MODULUS FROM VOLUMETRIC DATA

The connection established above between the thermodynamic and LA Gru¨ neisen parameters allows limits to be set on G' in lieu of elasticity data at pressure. Combining Eqn. 6 and 8 gives G' ⫽

冉

3 1 ␥ th ⫹ 2 6

冊冉

冊

K S ⫹ 4G/3 3 ⫺ K⬘S. KT 4

(9)

If replacing KS' by KT' is valid, Eqn. 9 allows prediction of the pressure derivative of the shear modulus from physical properties that are widely available. From Eqn. 2, K⬘S ⫽ K⬘T共1 ⫹ ␣␥ thT兲 ⫹ ␣␦ th␥ thT ⫺ ␣ q th␥ thT

(10)

where ␦T ⫽ ⫺(1/␣KT) ⭸KT/⭸T兩P is the Anderson-Gru¨ neisen parameter, and the second thermal Gru¨ neisen parameter is: q th ⫽ 1 ⫺ K⬘T共1 ⫹ T ␣␥ th兲 ⫺ 共T ␥ th/ ␣ 兲⭸ ␣ /⭸T ⫺ 共2T ␥ th/K T兲⭸K T/⭸T兩 P ⫺ 共1/ ␣ K T兲⭸K T/⭸T兩 P.

(11)

The term T␣␥th is much smaller than unity and can be ignored. The term (2T␥th/KT) ⭸KT/⭸T is about ⫹0.1, whereas (T␥th/␣) ⭸␣/⭸T is about ⫺0.2. These terms are both small and nearly cancel. A useful approximation is: q th ⫽ 1 ⫺ K⬘T ⫹ ␦ T.

(12)

Because KT' is near 4 and ␦T is near 6, qth is 2 for many substances (for examples, see Anderson and Isaak, 1995; Shim and Duffy, 2000). Because the physical parameters in Eqn. 10 do not vary much among minerals, the difference KS' ⫺ KT' is estimated (from ␣ ⬃20 ⫻ 10⫺6/K, ␥th ⬃1.5, T ⬃298 K) as ⫹

0.04 ⫹ 0.04 ⫺ 0.02 ⫽ 0.06. Thus, using KT' will underestimate G' by ⬃0.045. This amount is within the uncertainty of the individual parameters in Eqn. 9. Obviously, errors in KT' exist, and the ambiguity in a given volumetric determination needs consideration. In contrast to elasticity data, which are almost always fit linearly or perhaps to a quadratic polynomial (see section 2.1), V(P) data are fit to an equation of state (EOS) to extract KT(P). The results depend somewhat on the choice of the equation of state, but the majority of studies use a Birch-Murnaghan EOS and assume that d2KT/dP2 is negligible. Although this commonly used approach provides KT0 and KT', it is not exactly equivalent to direct measurement of KS in elasticity studies because tradeoffs exist between KT0 and KT' through fitting V(P). Using measurements of elastic constants at ambient conditions to constrain KT0 in the EOS gives KT' that is equivalent to the elastic determinations (Bass et al., 1981). Not all studies use this approach, as V(P) studies frequently predate elasticity measurements. However, most of the data in Table 1 have consistent initial bulk moduli; i.e., KT0 is ⬃3% (⬃5 GPa) lower than KS0, and, thus, the isothermal pressure derivatives are appropriate for use in Eqn. 9. Small uncertainties exist in ␥th, which arise mainly from the thermal expansivity (section 2.1). With these caveats, predictions of G' (Table 1) are made for various high-pressure phases lacking such determinations. For stishovite, KS' of 5.3 (Liu et al., 1999) is suspect because polycrystalline material was examined. Spectroscopic data to pressures exceeding 36 GPa (Kingma et al., 1994) suggest that KT' ⫽ 4 (Hofmeister, 1996), whereas X-ray diffraction up to 16 GPa gives an unusually low value of 2.8 (Ross et al., 1990). Using KT' ⫽ 4 in Eqn. 9 gives G' ⫽ 1.55, which is comparable to silicate perovskite (Sinelikov et al., 1998). For akimotoite (MgSiO3 with the ilmenite structure), the thermal expansivity is uncertain. Lack of discernable curvature in V(T) leads to ␣ that decreases slightly across this temperature range rather than the expected increase (Fig. 6). The range of possible values for ␣ and ␥th are obtained as follows. The

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A. M. Hofmeister and H. K. Mao

Fig. 6. Thermal expansivity (right y-axis) and volumes (left y-axis) of akimotoite as a function of temperature. Circles ⫽ V(T) data from Ashida et al. (1988). Solid lines ⫽ linear fit to V(T) and corresponding ␣. The linear fit best describes the data and intersects the single-crystal volume at ambient conditions (square) from Horiuchi et al. (1982). Dashed line ⫽ ␣ of Al2O3 from Dubrovinski et al. (1998). Dash-dotted line ⫽ ␣ of FeTiO3 from Wechsler and Prewitt (1984).

Ashida et al. (1988) ␣ at 298K is close to the average value for the structurally related compounds FeTiO3 and Al2O3 (Fig. 6). The bulk moduli follow the same relations (Table 1). Because these structures consist of O atoms in a close-packed array with cations occupying the voids, contraction through squeezing or freezing should have the same effect. Supporting evidence is that both ␣ and measured KT values of akimotoite closely resemble those of Fe2O3 (e.g., the compilations of Fei, 1995; Knittle, 1995). We conclude that ␣ ⫽ 25 ⫻ 10⫺6/K at 298 K (Ashida et al., 1988) is near the true value, although it appears high from the T dependence of ␣ (Fig. 6). This gives an upper limit of 1.75 for ␥th. One alternative is simply averaging ␥th of Al2O3 and FeTiO3, giving a slightly lower value 1.62. If the problem in ␣ for akimotoite is lack of precision in V, then ␣ at the middle of the temperature range should be correct. This value matches ␣ at 600 K of corundum measured by Dubrovinsky et al. (1998), suggesting a lower limit of ␥th as 1.3. Chopelas (2000) used Raman data at pressure to determine ␥th. Her value of 1.3 for ␥th, obtained by averaging mode Gru¨ neisen parameters, is underestimated because Eqn. 2 was used. The present lack of knowledge of KX of octahedral Si prohibits accurate determination of ␥i from Eqn. 4. Typically, using Eqn. 2 leads to ⬍ ␥i ⬎ that is 10 to 15% lower than ␥th (Chopelas et al., 1994), suggesting that ␥th of akimotoite is 1.4 to 1.5. However larger discrepancies of 25% with ⬍ ␥i ⬎ from Eqn. 2 have been observed, and thus we suggest that ␥th is 1.5 to 1.75. Consequently, G' is 1.6 to 2.15 for KT' of 4 (using Table 1). The lower limit is close to the values for stishovite and MgSiO3-perovskite, all of which have Si in octahedral coordination. For ␥-Mg1.2Fe0.8SiO4, the nominal value of the isothermal K' gives G' ⫽ 0.96. This value is consistent with Fe-rich spinels having smaller G' values than Mg-rich spinels (Table 1). For FeS (troilite), data on G do not exist. Comparison of the

Fig. 7. Estimation of the shear modulus of FeS by comparing KS and G for simple compounds. Data from Bass (1995) and Grigoriev and Meilikhov (1997). Squares ⫽ Fe oxides. Triangles ⫽ XO compounds. Diamonds ⫽ metals. Circle ⫽ sulfides. Selected samples are labeled. Dotted line ⫽ KS of FeS calculated from KT of King and Prewitt (1982). Solid line connects the hexagonal compounds ZnS and ZnO. Dashed line connects Zn with Fe. The trends show that the anions are the dominant factor for diatomic compounds. The shear modulus of FeS should be like that of ZnS and have consistent Zn, Fe pairs. The polymorph is unimportant. The XS compounds cluster closely together, suggesting that the true value should not differ significantly from the prediction of G ⫽ 31.5 GPa.

bulk to the shear modulus for monatomic and diatomic substances suggests the troilite should have G ⫽ 31.5 GPa (Fig. 7). From this value, existing volumetric data and a nominal value of 4 for K', G' could range from ⫺0.62 to ⫹0.13. Two other minerals have G' ⱗ 0: calcite (Kaga, 1968) and ZnS (Chang and Barsch, 1973b). All three minerals have polymorphic phase transformations (calcite-aragonite, wurtzite-sphalerite, troiliteMnP type), and the decrease in G as P increases may be related to soft mode behavior. For the sulfides, a shearing motion of the anion layers relates the ccp and hcp structures. 4. DETERMINING G' and KS' FROM THE EXPERIMENTAL TRENDS IN ELASTICITY DATA AND BY REQUIRING THAT ␥LA EQUALS ␥th

Figures 1, 2, 3, and 4, and indicate that problems exist with a significant fraction of the elasticity determinations at pressure. In particular, the older datasets almost always have ␥LA that is large compared to recent Brillouin measurements. The discrepancies could have many sources, e.g., transducer bonding or non-hydrostatic conditions (for discussion, see Every and Sachse, 2001; Schilling and Ramelow, 2001). Likely causes are pressure errors, which could arise from use of external pressure standards, whereas the recent DAC studies use internal standards such as ruby fluorescence. This type of error is revealed by comparison: If the variations in ⭸KS/⭸P and ⭸G/⭸P among different studies of the same substance are solely due to sys-

Pressure derivatives of shear and bulk moduli

1217

Fig. 8. Graphical illustration of a method to constrain KS' and G' by coupling existing measurements of metals with Eqn. 9. Heavy lines ⫽ least-squares linear fits to the available elasticity data; fits are in Table 2. Light lines ⫽ Eqn. 9 with KS' taken as a variable. The point of intersection determines the preferred values of KS' and G', labeled by pairs of numbers for each element or compound. Sources of data are Table 1, Bass (1995), Sumino and Anderson (1984). (a) Soft metals and Re. Data from van’t Klooster et al. (1979), Biswas et al. (1981), and Voronov and Vereschagin (1960). G' values represent the Voigt limit for some older studies. Filled diamonds and solid lines ⫽ Re. Grey ⫽ Al. Half-filled squares and dotted lines ⫽ Cu. Squares and dashed lines ⫽ Ag. Triangles and long-dashed lines ⫽ Au. The outlying point for Ag is from Hiki and Granato (1966). The oldest Al datum from Hughes and Maurette (1956) is suspect and is excluded from the fitting. (b) Fe and Mo. Filled circles and dashed lines ⫽ ␣-Fe. Triangles and solid lines ⫽ Mo. Grey ⫽ ␧-Fe. Fiquet et al. (2001) provides the compressional velocity, so possible K' values are shown (dot-dashed line). Open circle ⫽ XRD data on ␣-Fe (Dubrovinsky et al., 2000). For ␣-Fe, the suspect high G' datum of Hughes and Maurette (1956) is omitted from the fit. An alternate fit to recent data is shown.

tematic errors in the pressure calibration, then a plot of G' as a function of KS' will provide a straight line, with the slope equal to the ratio of true values, G'(true)/KS'(true). This relationship arises because ⭸G/⭸P exactly equals ⌬G/⌬P when G linearly depends on P and because curvature is small enough that it is not detected in most studies. The same holds for the bulk modulus. If instead the errors are random, a cluster of points would be seen on a G' vs. KS' plot. If the errors are indeed random, our approach cannot reveal the cause. Independent of either type of trend in the elastic data, Eqn. 9 shows that G' linearly depends on KS', with a slope of ⫺3/4 and an intercept dependent on the thermodynamic Gru¨ neisen parameter and on values of the elastic moduli at ambient conditions. The best representation for the pressure derivatives of the elastic moduli can be ascertained from the intersection of Eqn. 9 with the elasticity trend, be it a line or a cluster. Several substances have been studied multiple times, which allows us to test the methodology proposed above and to try and differentiate pressure errors from other possible sources of discrepancies among elasticity measurements. Problems in thermal parameters are considered as well. Figures 8, 9, 10, and 11 show the available data on the derivatives of the elastic moduli for metals, alkali halides, oxides, and silicates. Because the data on alkali halides are old and plentiful, only substances with at least three measurements are examined. Table 3 lists the fits—the pairs of preferred values and the slopes expected on a G' vs. KS' plot for those values.

4.1. Metals Except for the XRD study of Au (Duffy et al., 1999a), elasticity measurements on the noble metals involve ultrasonic techniques (see summaries by van’t Klooster et al., 1979; Biswas et al., 1981). Cu and Au have well-defined trends (Fig. 8a) and a ratio of the intercept values that is consistent with the slope defined by the elasticity measurements (Table 3). The study by Hiki and Granato (1966) gave KS' and G' values that seem inconsistent with other measurements, with the greatest disparity occurring for silver (Fig. 8a). For Cu, the Hiki and Granato (1966) datum (with the largest G' value) and Lazarus (1949) datum (with the smallest G' value) affect the slope in fitting but have little affect on the intercept (Table 3). The well-defined slope for Au and the average of the two fits for Cu are in agreement with the predicted slopes, suggesting that the errors for the ultrasonic data on noble metals, which predate 1981, are mainly due to problems in pressure calibration. Therefore, G' for Ag is probably 1.24 like Cu and Au. Silver should be reexamined. For Al, the data of Hughes and Marretti (1956) were excluded from the fitting, as this study yielded inconsistent data for ␣-Fe as well. The remaining data on Al define a steep slope similar to that of Cu, Au, and Re (Fig. 8a). Pressure calibration errors are the likely cause of most discrepancies. The oldest studies (ca. 1960) likely have additional problems that were overcome as the techniques improved. In contrast, errors for Fe and Mo appear to be random (Fig. 8b). For ␣-Fe, recent XRD measurements (Table 1) provide

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A. M. Hofmeister and H. K. Mao

Fig. 9. Plots of G' vs. KS' for alkali halides. See Fig. 8 for a general description. (a) Chlorides. Square and dotted lines ⫽ NaCl. Triangle and dashed lines ⫽ KCl. Diamond and solid line ⫽ RbCl. The fit to KCl is estimated. (b) Bromides, iodides, and fluorite. Grey dots and lines ⫽ fluorite. Open circle and dotted lines ⫽ RbBr. Diamonds and dashed lines ⫽ KI. Triangles and black line ⫽ RbI. Very flat trends are seen for these older data on soft solids at low pressure.

pressure derivatives that are consistent with Eqn. 8 and 9, whereas the ultrasonic data on ␣-Fe are highly scattered (Fig. 8b). If one discounts the earliest studies (Hughes and Maurette, 1956; Voronov and Vereschagin, 1960), then two linear trends are suggested, depending on whether the Dubrovinsky et al. (2000) XRD datapoint is included in one of the fits involving the ultrasonic data or not included (Fig. 8b). The intersections

are similar, suggesting that the fitting approach is robust. All data on hexagonal ␧-Fe are from diffraction methods and have some uncertainties due to the extrapolation needed to ambient conditions. Mao et al. (2001) use several different methods providing KS' and G' in the middle of the cluster and ␥LA close to ␥th. 4.2. Alkali Halides NaCl provides a stringent test of the equations and methodology due to a large number of measurements. The pressure derivatives for the elastic moduli of NaCl define a linear trend (Fig. 9a), which steeply intersects Eqn. 8. The ratio of the preferred derivatives equals the slope obtained from leastsquares fitting (Table 3), which suggests that errors in pressure calibration cause discrepancies among the pressure derivatives for NaCl. The trend is robust because it does not change if one or two studies are omitted from the fitting. The preferred value for KS' equals KT', calculated from IR spectral data at pressure (Hofmeister, 1997), and describes existing V(P) data well. Values for G' for the other diatomic halides are roughly constant (Fig. 9b). These data were obtained over narrow pressure ranges. Fits for KCl and RbI are ill-defined. The data as a whole suggest that K' ⬃4.5 and G' ⬃0.9 represent the potassium and rubidium halides. Fluorite has three consistent studies, including the Brillouin data of Speziale and Duffy (2002). The comparison suggests that ␥th is ⬃5% high, probably due to the uncertainty in ␣.

Fig. 10. Plots of G' vs. KS' for diatomic oxides. Small filled squares and solid lines ⫽ MgO. Triangles and dashed lines ⫽ CaO. Square with circle and dotted lines ⫽ MnO. Grey squares and grey lines ⫽ SrO. Circles with dot and dot-dashed lines ⫽ aluminous spinels. The arrow connects the two Brillouin scattering measurements for MgO.

4.3. Diatomic Oxides For MgO, the ultrasonic data define a shallow trend, whereas the Brillouin scattering measurements provide a steep trend (Fig. 10). The fit to the Brillouin scattering measurements of

Pressure derivatives of shear and bulk moduli

1219

Fig. 11. Plots of G' vs. KS' for polyatomics. (a) X2SiO4 polymorphs. Triangles and dotted line ⫽ wadsleyite. Square and long dashed line ⫽ ␥-Fe2SiO4. Square with cross ⫽ ␥-Mg2SiO4. Circles ⫽ Fo90. Grey ⫽ forsterite. The dot-dashed line between two points for forsterite connects the two fits of Zha et al. (1998). (b) Garnets and complex oxides. Diamonds and dotted lines ⫽ CaTiO3. Circles and dashed lines ⫽ MgAl2O4. Squares and solid line ⫽ pyrope. Square with cross ⫽ pyrope-almandines. Grey ⫽ majorite. Thick long-dashed line is a fit to elasticity data for the pure spinel samples, although these can be disordered. The other samples have Fe substitution or are non-stoichiometric (see Wang and Simmons, 1972; Hibert and Graham, 1989).

Sinogeikin and Bass (2000), which allows for non-zero second pressure derivatives, has ␥LA ⫽ ␥th within the uncertainty of the measurements. The Brillouin scattering study of Zha et al. (2000), which reached pressures a factor or two larger (55 GPa), provides nearly the same values for KS' and shows that K'' is negligible. The tie line between the Brillouin data intersects Eqn. 9 near the same location as the fit to all elasticity data (Fig. 10). The ultrasonic data were gathered at significantly lower pressures and were fit to polynomials for the most part with K'' ⫽ G'' ⫽ 0. The Brillouin data show that the ultrasonic results represent the initial value for K' but should provide slightly less than the initial value for G'. High values are instead observed (Fig. 10), suggesting that errors exist other than in pressure determinations. Simple oxides other than MgO (Fig. 10) occupy trends that are less rigorously defined due to the small number of elasticity studies. The results for SrO are suspect. 4.4. Polyatomics Most of the polyatomic solids with functional groups for which elastic properties have been multiply determined at pressure are orthosilicates. Brillouin scattering data of forsterite (Zha et al., 1996) are equally well described as first- or secondorder polynomials in pressure. The linear fit comes closest to Eqn. 9 (Fig. 11a). This conclusion is consistent with the low amount of curvature in the data and the tendency of the highest pressure points to influence the fit. Also, a least-squares fit to all elastic data on ␣-Mg2SiO4 provides comparable values. In contrast, the data on Mg-rich olivines are scattered (Fig. 11a). Similarly, for wadsleyite, ␥-Fe2SiO4, ringwoodite, majorite, and pyrope, the results from Brillouin scattering experiments

and IR spectroscopy come closest to Eqn. 9 (Fig. 11; Tables 1, 3). Additional ultrasonic data on pyrope-almandines lie on the same trend, and the slope is consistent with errors in pressure, as is seen for ␥-Fe2SiO4 (Table 3). For pyrope, the Brillouin scattering result of Conrad et al. (1999) affects the fit due to the very low KS' value. The ultrasonic values are scattered and do not provide a well-defined trend by themselves. For this reason, we suggest that the Brillouin study of Sinogeikin and Bass (2000) best represents pyrope. Spinel has varying amounts of Mg-Al disorder (Millard et al., 1992), non-stoichiometry, and chemical substitutions, all of which affect its physical properties. The elasticity data on the two pure spinels (see Bass, 1995) point to KS' ⫽ 4.5 and G' ⫽ 0.5, which is in agreement with the fit to all data. These trends and ␥th do not support results from sideband spectroscopy with G' ⫽ 0.07 and KS' ⫽ 5.05 (Chopelas, 1996). The ultrasonic data for perovskite (Fischer et al., 1993; Kung and Rigden, 1999) are recent and in good agreement with each other and with ␥th (Fig. 11b). 5. DISCUSSION AND CONCLUSIONS

Measurement of the pressure derivatives of elastic moduli has been the subject of numerous papers in the physics, materials science, and geologic literature. Due to improvements in techniques, new methodologies, and achievement of increasingly higher pressure, recent studies are generally assumed to provide the most accurate values of KS' and G'. However, disparities exist among recent datasets as well as the older data (Table 1; Figs. 1, 2, 3, 4, 8, 9, 10, 11). Using a Debye model (i.e., ␥th ⫽ ␥LA) constrains pairs of KS' and G' through Eqn. 9. Multiple studies of a given material should provide a second,

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A. M. Hofmeister and H. K. Mao Table 3. Pressure derivatives of the shear and bulk moduli from the intercept of experimental values with Eqn. 9. Elasticity fit

Phase

␣-Fe ⑀-Fe Al Re Cu Ag Au Mo CaF2 NaCl KCl RbCl RbBr KI RbI MgO MnO CaO SrO MgAl2O4 forsterite Olv⫹For wadsleyite ␥-Mg2SiO4 ␥-Fe2SiO4 majorite pyrope pyr-alm CaTiO3

Intercept

G' ⫽

R

No.

K⬘S

G'

1.12 ⫹ 0.132 KS⬘ 2.12 ⫺ 0.131 KS⬘ 1.67 ⫹ 0.066 KS⬘ ⫺0.9 ⫹ 0.5 KS⬘ 0.386 ⫹ 0.276 KS⬘

1 0.27 1 1 0.56

0.17 ⫹ 0.173 KS⬘ 10.36 ⫺ 2.0 KS⬘ 1.27 ⫺ 0.023 KS⬘ 0.527 ⫹ 0.242 KS⬘ 1.0 1.23 ⫺ 0.0613 KS⬘ 0.98 ⫺ 0.142 KS⬘ 1.339 ⫺ 0.024 KS⬘ 2.19 ⫺ 0.237 KS⬘ 1.998 ⫹ 0.114 KS⬘ ⫺1.636 ⫺ 0.603 KS⬘ 1.98 ⫺ 0.487 KS⬘ 2.30 ⫺ 0.158 KS⬘ 0.605 ⫹ 0.032 KS⬘ ⫺0.582 ⫹ 0.466 KS⬘ ⫺0.216 ⫹ 0.406 KS ⫺2.80 ⫹ 0.977 KS⬘ ⫺1.19 ⫹ 0.61 KS⬘ 0.181 ⫹ 0.157 KS⬘ ⫺0.89 ⫹ 0.545 KS⬘ 0.61 ⫹ 0.23 KS⬘ 1.386 ⫹ 0.0236 KS⬘ 1.14 ⫹ 0.059 KS⬘

0.95 1 0.27 0.96

5 4 2 2 5 2 3 2 3 7 3 4 4 3 4 9 2 3 2 5 3 6 2 2 2 2 5 6 2

3.92† 4.2 3.86 4.67† 4.7† 5.4 6.20† 4.53† 4.8‡ 4.42† 4.2 4.37 4.37 3.6 4.2 3.97† 4.20 4.73† 5.04† 4.51 3.90†

1.79† 1.57 1.98 1.46† 1.25† 1.4 1.24† 1.23† 1.14‡ 1.60† 1.0 0.96 0.91 1.2 0.9 2.441† 0.92 1.74† 1.51† 0.47 1.13†

4.20† 3.88 4.13† 4.2† 4.10† 5.34†

0.63 0.67 0.58 0.29 0.24 1 0.40 1 0.06 0.87 0.75 1 1 1 1 0.6 0.12 1

G'/K⬘S

KS (GPa)

G (GPa)

0.45 0.37 0.51 0.31 0.26* ⬅0.23 0.20* 0.28 0.24 0.23* 0.25 0.21 0.20 0.33 0.21 0.62 0.22 0.37 0.30 0.10* 0.28

166.7 165 76 356.2 137.3 102 171.7 270 86.3 24.9 18.1 16.4 13.7 12.1 11 163.2 149 112 87.3 197 131.1

81.5 80 26 178.7 46.9 29.2 27.6 149 42.4 14.7 9.4 7.6 6.5 6 5 130.2 68.3 80.6 58.1 108.4 79.4

1.30† 1.15 0.84† 1.4† 1.30†

0.30 0.3 0.20* 0.34 0.32*

170 184 209.6 167.3 171.2

115 119 87.5 88.3 93.7

1.45†

0.27

176

105.2

R is the residual, and equals unity where the fit is defined by two points. No. is the number of different elasticity studies. The intercept values are the averages of the possible fits to the elasticity measurements for the various substances. The slope is determined from their ratio. References for bulk and shear moduli are in Tables 1, Figs. 1–5 and 8 –11, and the Appendix. For majorite, we assume that the Brillouin study (Sinogeikin and Bass, 2002) is correct. * Agreement occurring between predicted slope and slope from least-squares fits suggests that the disparities among elasticity measurements are due to pressure calibration. Most datapoints for these cases are from older ultrasonic studies. † Denotes values that are well-constrained by thermodynamic data and multiple, consistent elastic measurements. ‡ The cluster of points probably represents KS⬘ and G⬘: thermal expansivity may err.

independent relationship of G' with KS' that allows us to distinguish the most reliable values. For many substances, three different elasticity studies are not sufficient to establish the values of KS' and G'. However, only a few different types of trends are seen in plotting G' as a function of KS', which allows inference of potential problems in the data. The cubic metals, for which Eqn. 9 is exact, behave similarly to the other phases (Figs. 8, 9, 10, 11), implying that the disparities are due to experimental uncertainties and not to limited applicability of Eqn. 8 or 9. This inference is corroborated by the well-defined trends for NaCl and MgO, which have a large number of independent measurements. As a corollary, trustworthy elasticity data (i.e., three consistent measurements) can point to inconsistent thermal expansivity data. For calcic garnets and fluorite, additional measurements of ␣ are warranted. 5.1. Possible Sources of Experimental Uncertainties in Elasticity Data Problems with pressure calibration can account for the trends in K' and G' for many cases (Cu, Au, Re, NaCl, pure MgAl2O4, ␥-Fe2SiO4, and pyrope) because the slope from fitting nearly

equals G'/KS'. These substances generally have been studied many times (Table 3; Figs. 8, 9, 10, 11). For other substances, G' values are roughly constant, whereas KS' varies. Substances with flat trends or slightly negative slopes in Figures 8, 9, 10, and 11 are relatively soft (CaO, SrO, CaF2, and K and Rb halides), having KT below 115 GPa. A few hard substances (Fe, CaTiO3, and MgO) also have fairly flat trends. The trend for Fe involves combining crystallographic and ultrasonic techniques and thus reflects factors in addition to pressure calibration. For MgO, the oldest studies have the highest ␥LA, suggesting that improvement in technique over time may be relevant. A few cases (MnO and wadsleyite) have steep trends (positive and negative slopes), which probably reflect random rather than systematic errors. Problems with deviatoric stress are discussed by Wang and Duffy (1998), Duffy et al. (1999a,b), and Dubrovinsky et al. (2000). Uniaxial compression may cause divergence of the pressure derivatives from their true values. Soft substances deform easily: A small amount of length shortening could alter the pressure derivative for the longitudinal waves although it should not affect the transverse waves. Flat trends associated

Pressure derivatives of shear and bulk moduli

1221

Fig. 12. Pressure derivatives of elastic moduli as a function of the elastic moduli at ambient conditions. (a) Bulk modulus. (b) Shear modulus. Various lines are least-squares fit to the data except for KS' of the chlorides, which is a guide to the eye. Up-pointing triangles ⫽ Al, Fe, Mo, Re metals. Grey triangles ⫽ noble metals. Right triangle ⫽ chlorides. Square with cross ⫽ CaF2. Filled circles ⫽ XO compounds. Grey circle ⫽ MgAl2O4, A12O3, and stishovite. Diamonds ⫽ silicates. Square ⫽ oxide perovskites. Data are from Tables 1 and 4 Appendix Table A. LiCl and AgCl data from Sumino and Anderson (1984) and overestimate KS' and probably G'. The G' curves for oxides and perovskites are quadratic fits.

with the alkali halides, Al, Ag, CaO, and SrO, all of which have KS ⬍ 110 GPa, could be connected to such behavior. Varying degrees of non-hydrostaticity among the different experiments could lead to disparities as well. Many of the ultrasonic measurements were preformed on polycrystalline samples (e.g., Wang and Ji, 2001). Porosity has been considered to be a problem, and substantial effort has been expended to make compact samples (e.g., Gwanmesia et al., 1998). Compaction of pore space appears no longer to be a consideration. However, a polycrystalline sample can deform in a manner not possible for a single crystal, and such deformation could depend on grain size and non-hydrostaticity. Spectroscopic experiments should be little affected by nonhydrostatic conditions because vibrational modes arise through nearest-neighbor interactions. Deviatoric stress (i.e., a uniaxial component in compression) is probably the underlying cause of discrepancies in the pressure derivatives of the elastic constants and moduli obtained by methods other than spectroscopic. 5.2. Disparities among Spectroscopic Studies Acoustic modes are directly probed by four different types of spectroscopy. Brillouin scattering is used most often. This technique provides KS' and G' that are in agreement with ␥th and with the trends established by the ultrasonic studies except for pyrope. Brillouin scattering results for pyrope (Conrad et al., 1999) provide the only value of K' below 3.5. One possibility is that a ruby grain used to determine pressure was not surrounded by the pressure medium (i.e., bridging occurred), giving a higher pressure than the sample was exposed to. Acoustic modes in IR spectra have been investigated once (Hofmeister and Mao, 2001). The results are consistent with

existing data (Fig. 11a, Table 4), in that G'/K' equals the slope defined by the IR and the ultrasonic experiments. Potential problems with sideband spectroscopy and LIPS were discussed in section 2.3.1. Studies using LIPS provide G' values larger than the “best” values obtained from either Brillouin spectroscopy or from the intersection of the experimental trends with the Debye relationship (Tables 1, 3), which is consistent with probing impure shear waves. The sideband method tends to give values of either G' or KS' that are high but not both, which seems consistent with the modes being local in origin. 5.3. Systematic Behavior of KS' and G' Linear trends of KS' with KS are seen for various chemical series crystallizing in the rocksalt structures (Fig. 12a). The least-squares fits for the oxide series is K⬘S ⫽ 6.361 ⫺ 0.001485 K S

(13)

Diatomic oxides that are stiff at ambient pressure are less affected by compression as P increases. Eqn. 13 is intimately connected with the cation radius, which decreases (radius order Sr ⬎ Ca ⬎ Mn ⬎ Fe ⬎ Mg) in the same order as KS increases. For the rocksalt structure, the O anion is bigger than any of the cations investigated (see Jaffe, 1988), and the behavior upon compression depends on the repulsion forces between the O atoms such that the smallest cation leads to the strongest repulsion and the largest KS. The perovskite series behaves similarly (Fig. 12) as is expected but is complicated as two cations are involved. For the series XCl, the KCl point is uncertain, so data on LiCl and AgCl were included, which overestimate KS' to an

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A. M. Hofmeister and H. K. Mao

unknown degree. Metals form a trend but the number of samples is limited, and noble metals, which do have chemical similarities, do not define a trend. Generally, G' increases with G for various chemical series (Fig. 12b). The least-squares fit for the simple oxides is: G' ⫽ ⫺ 2.284 ⫹ 0.0071G ⫺ 2.676 G 2/10 6

(14)

Perovskites behave similarly to the oxides, suggesting the O sublattice is the controlling factor. The trends can be rationalized in that the more resistant to shear a structure is at ambient conditions, the more difficult it is to shear as P increases. Figure 12 and Table 3 show that KS' clusters near 4 for Fe, MgO, and many silicates expected to occur inside the Earth, but the variation in G' is fairly large. Comparison of shear moduli (and probably shear velocities) from seismic observations to laboratory data will thus be a much more sensitive indicator of mantle mineralogy. Fortunately, the pressure derivatives of shear moduli appear to be much less affected by experimental techniques than dKS/dP. Theoretical reasons exist for expecting that K' will be near 4. For the second- and third-order Eularian finite strain equations, K' below 4 leads to the structure being mathematically unstable at sufficiently high compression; i.e., at some very high pressure, the structure collapses (Hofmeister, 1993b). (Thermodynamic instability may occur before such a high pressure is reached, but this is not germane.) For K' larger than 4, the structure is mathematically unstable at sufficiently high dilation (see the potential curves in Fig. 2, Table 1, and Eqn. 16 of Hofmeister, 1993). Thus, the closer K' is to 4, the wider the range of mathematical stability; i.e., the solid has the right combination of repulsive and attractive terms in the potential to be maximally stable against increases as well as decreases in volume. Because of this stability problem, use of the universal equation of state (Vinet et al., 1987) is recommended, as this formulation is mathematically stable for all positive values of K', having been derived from the interatomic potential for a harmonic oscillator. Our finding that K' is near 4 for a wide range of minerals largely recapitulates the validity of the harmonic oscillator model and its corollaries. The quasi-harmonic oscillator model, which accounts for pressure affecting the vibrational frequencies, predicts thermodynamic properties with reasonable accuracy (e.g., Kieffer, 1979), whereas the damped harmonic oscillator model, which allows for interactions between modes, provides thermal conductivity values in good agreement with available measurements (e.g., Hofmeister, 1999, 2001; Geisting and Hofmeister, 2002). Acknowledgments—Support was provided by NSF grants EAR-9712311 and 00125883 to AMH and EAR-0001173 to HKM. We thank C. S. Zha for sharing data on MgO. Comments by L. Dubrovinsky, B. Mysen, and anonymous reviewers led to substantial improvements. Associate editor: B. Mysen REFERENCES Abramson E. H., Brown J. M., Slutsky L. J., and Zaug J. (1997) The elastic constants of San Carlos olivine to 17 GPa. J. Geophys. Res 102, 12253–12263. Aldebert P. and Traverse J. P. (1984) A high-temperature thermal expansion standard. High Temp.-High Pres 16, 127–135.

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A.2. Hexagonal Metals The newer data support Eqn. 8. Equivalence of ␥LA and ␥th is seen for various types of XRD measurements of ␧-Fe (Dubrovinsky et al., 2000; Fiquet et al., 2001; Mao et al., 2001). These studies use different approaches but provide equal values for ␥LA (Table 1). Dubrovinsky et al. (2001) obtained a higher value (Table 1), but these authors derived the results assuming that ␥th ⫽ ␥ac. For Re, Raman scattering (Olijnyk and Jephcoat, 2000; Olijnyk et al., 2001) results on ⭸c44/⭸P were used to compute ⭸G/⭸P because inconsistencies exist in the lattice strain data on c44. The basal plane elastic moduli (c11 and c12) respond reasonably to pressure (Duffy et al., 1999b) and provide ⭸KS/⭸P that is reasonably close to the isothermal value (Liu et al., 1970). Ultrasonics study of Re (Katahara et al., 1979) gave larger derivatives (Fig. 1). Given the variation in ␥LA seen for Re and the above results for monatomic metals, the ultrasonic determinations of C, Ge, Mg, and Zn (Fig. 1a), which predate 1973, are uncertain (see also Duffy and Wang, 1998). Elements with hexagonal unit cells also have one and only one Raman-active mode, which thus provides the average ␥i. The available data (Table A) on ␧-Fe, Re, Mg, and Ru have ␥i consistent with ␥th. Other diatomic metals studied by Raman spectroscopy show soft-mode behavior (i.e., d␯/dp ⬍ 0) or gradual phase transitions (Olijnyk and Jephcoat, 2000). Their pressure data are thus problematic for comparison because the acoustic, optic, and thermodynamic properties can be affected in different ways. A.3. Cubic Diatomics Eqn. 8 appears to be valid for MgO and NaCl within the experimental uncertainty (Fig. 2). These substances are the only diatomics for which many independent measurements have been made of KS' and G' (Table 1). The Birch-Murnaghan fit to G for MgO by Zha et al. (2000) implicitly requires finite G''' ⫽ d3G/dP3. The value of the third derivative ⫽ 0.00158/GPa2 is large, as indicated by the large curvature in G(P) resulting from utilizing the fitting parameters in a quadratic equation (Fig. A). Least-squares fitting of the data leads to lower G' and G'' (Table 1, Fig. 2). The fit is not improved by including higher order terms (not shown), suggesting that the form of the fourth-order BirchMurnaghan equation of state for G includes terms that the data cannot constrain. Through tradeoffs in the equation of state, the result is large values of G'. To compare directly the initial slope G' from the existing

Table A. Gru¨ neisen parameters of hexagonal metals

␥th

␥optic

␥LA

K T (GPa)

Mg ⑀-Fe

1.56* 1.7† 1.74¶

1.48§ 1.6‡ 1.68㛳

Re Ru

1.78* 1.53*

1.8‡ 1.5‡

1.76# 1.81& 2.14¶ 1.7¥ 1.84@ n.d.

36.3# not used 167.7¶ 235¥ 356.2@ 220$

* From Grigoriev and Meilikhov (1997). † Anderson (1997). ‡ Raman data on optical modes of Olijnik et al. (2001). § Raman data of Oliynk and Jephcoat (2000). Data are also available on Zr and Gd, but these appear to have phase transitions at low pressure (soft modes). For Zn, a gradual phase transition occurs. Also, the KT0 data appear inconsistent. 㛳 Raman data of Merkel et al. (2000), who calculated ␥i from ␯(V). All other studies calculated ␥i from ␯(P). We used the KT data listed above. # Schmunk and Smith (1959). & Dubrovinsky et al. (2000); Mao et al. (2001). @ c⬘11 and c⬘12 from Duffy et al. (1999b); c⬘44 from Olijnyk et al. (2001); initial values from Manghnani et al. (1974). $ Compiled by Winter (2001). ¶ Dubrovinsky et al. (2001), assuming ␥th ⫽ ␥ac. ¥ Fiquet et al. (2001); data on compressional sound speed, assuming ␳0 ⫽ 8.3 g/cm3.

Fig. A. Shear modulus of MgO as a function of pressure. Dots ⫽ Brillouin data from Zha et al. (2000). Solid line ⫽ quadratic fit with R ⫽ 0.99. Dotted line ⫽ quadratic fit using the derivatives obtained from the Birch-Murnaghan fit of Zha et al. (2000). Dashed line shows the third derivative implicit in the Birch-Murnaghan fit.

datasets, we use equal order (quadratic) fits to G(P) measurements of both Zha et al. (2000) in Figure A and of Sinoeikin and Bass (2000). The quadratic fits to the two different Brillouin scattering measurements on MgO (Sinogeikin and Bass, 2000; Zha et al., 2000) provide the points closest the 1:1 correspondence (Fig. 2). Data from recent ultrasonic studies (Jackson and Niesler, 1982; Yoneda, 1990) lie above the 1:1 line. The linear fits to the Brillouin scattering experiment of Sinogeikin and Bass (2000) has the lowest values of ␥LA (not shown). Early ultrasonic studies (Bogardus, 1965; Anderson and Andreatch, 1966) have ␥LA ⫽ 1.7. The average from the recent determinations of ␥LA (Table 1) ⫽ ␥th within the experimental uncertainty. Fitting of the length change measurements for NaCl (Chabildas and Ruoff, 1976) and elasticity data (Spetzler et al., 1972) to a universal equation of state provides a point very close to the 1:1 correspondence (not shown). The result agrees with recent ultrasonic data (Kim et al., 1989). Because the V(P) data are highly curved, a second order term is important (K'' ⫽ d2K/dP2). Hence, tradeoffs exist between K' and K'' through the equation of state. The V(P) measurements are fit equally well by K' ⫽ 4.45, which is derived from a phenomenological model that uses infrared spectroscopic measurements at pressure (Hofmeister, 1997). This slightly lower K' gives ␥LA as 1.55, which equals ␥th (Table 1). Similarly, K', predicted for KCl (Hofmeister, 1997), provides adherence to Eqn. 8 for this substance. All the other alkali halides, both of the sulfides and many of the oxides, have ␥LA ⫽ ␥th, if ␥LA is as uncertain as suggested by the spread in ␥LA of ⬃0.6 for Fe, Ag, Au, and NaCl. Many of these data predate 1970 and could have errors in pressure, as calibration relied on external standards. The MgO data, which are more recent than the NaCl data (Table 1), are better constrained (Fig. 2). Recent ultrasonic measurements of ␥LA for ceramics (AlN, ␤-Si3N4, and TiB2; Dodd et al., 2001a, 2001b, 2001c) also occupy a narrow range and lie close to ␥th (Fig. 2). A.4. Non-Cubic Structures with Two Kinds of Atoms The corundum and rutile structures lack the distinct functional groups of the silicates but have more than one interatomic distance. Only corundum has many independent measurements of KS' and G' (Figs. 2, 5). The spread in ␥LA compares closely to the complete dataset on MgO (Figs. 2, 5). Older studies of GeO2, SnO2, and Cu2O (Fig. 2) have ␥LA marginally larger than the spread in ␣-Fe and NaCl values,

Pressure derivatives of shear and bulk moduli which probably is not significant. Except for Fe2O3, the measurements of ␥LA are higher than predicted, which is consistent with pressure being generally underestimated in the older datasets A.5. Polyatomic Substances Individual measurements of ␥LA from ca. 1970 (F-bearing perovskites, yttrium garnets, calcite, hydrated sulfates) correspond poorly with ␥th (Figs. 3, 4), whereas substances with multiple measurements (e.g., spinel) cluster near the 1:1 correspondence. Recent Brillouin scattering measurements at pressure (forsterite, wadsleyite, pyrope, and majorite; Table 1, Fig. 5) have ␥LA ⫽ ␥th. Ultrasonic study of MgSiO3 perovskite (Sinelnikov et al., 1998) reported G' only. Volumetric studies of MgSiO3 perovskite give KT' ⫽ 4 (e.g., Yagi et al., 1982; Kudoh et al., 1987) and KT in agreement with KS (Yaganeh-Haeri et al., 1990). Shim and Duffy (2000) summarize recent results, which provide KT' of 4 and show that the various thermodynamic quantities obtained from P-V-T data are consistent. The resulting ␥LA ⫽ ␥th (Table 1). Ultrasonic measurements generally provide ␥LA that is larger than values obtained using Brillouin scattering. Such behavior is seen for pyrope, majorite, forsterite, and wadsleyite. Systematic errors may exist in G' and KS' although the shear modulus is less affected. Data on aluminous spinel are ultrasonic or from sideband spectroscopy. Three different compositions were examined (Table 1), which may cause the divergent pressure derivatives. For wadsleyite, majorite, and pyrope, the ultrasonic data were recently obtained from hot-pressed polycrystalline aggregates, and the pressure calibration should be valid; yet the values for ␥LA seem high (Table 1). For example, the polycrystalline data on ␥-Mg2SiO4 has high G' and KS' (Rigden et al., 1992), whereas low-pressure derivatives were obtained for ␥-Fe2SiO4 through appearance of the LA mode in infrared spectra at pressure through a resonance (Hofmeister and Mao, 2001) and for ringwoodite using Brillouin spectroscopy (Sinogeikin et al., 2001). The dependence of KT' on composition across the ringwoodite binary (Hazen, 1993) also suggests the ultrasonic study of polycrystals overestimates G' and KS'. Ultrasonic data obtained from aggregates of majorite and the majorite-pyrope solid solution provide large KS' of 6.7 (Rigden et al., 1994; Gwanmesia et al., 1998), whereas Brillouin spectroscopy provides KS' near the nominal value of 4 (Sinogeikin and Bass, 2002). Given the similar values for the other physical properties of majorite and pyrope (Table 1), KS' and G' should be comparable. The above information shows to varying degrees of confidence that ultrasonic measurements from polycrystalline aggregates overestimate KS' and G', as discussed further in section 5. The equivalence of ␥LA to ␥th (and to the average of the optic mode

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␥i) for olivines and spinels is discussed in detail by Hofmeister and Mao (2002). We noted that the ultrasonic study of fayalite has a large KS' ⫽ 5.2 (Graham et al., 1988) that is not supported by measurements of KT' from volume-pressure data (Kudoh and Takeda, 1986), and KS is smaller than a recent elasticity study at temperature (Isaak et al., 1993). Using KT' ⫽ 4 gives KT that is in agreement with KS at ambient conditions of Isaak et al. (1993). For fayalite, KS' ⫽ 4 gives ␥LA equal to ␥th. The largest values of KS' occur for orthopyroxenes (Table 1). Although the discrepancy between ␥LA and ␥th is extreme at ambient conditions, with KS' initially equaling 7.8 (Chai et al., 1997b) to ⬃10 (Frissillo and Barsch, 1972; Webb and Jackson, 1990; Flesch et al., 1998), as pressure increases, KS' decreases rapidly and approaches the more common values of 4 to 5 (e.g., Webb and Jackson, 1990). Ultrasonic and transient grating techniques give pressure derivatives that are larger than those obtained by Brillouin scattering (compare the olivine and garnet data in Table 1). Thus, it appears that at slightly elevated pressures, agreement between ␥LA and ␥th is obtained. The chains of SiO4 tetrahedra in this structure initially kink during compression, but the behavior suggests that at moderate pressures (⬃0.9 GPa), kinking is no longer significant and compression occurs via bond shortening (Webb and Jackson, 1990). Similarly, ␥LA for quartz does not equal ␥th. A bond-bending mechanism violates the underlying assumptions of quasi-isotropic compression. Recent studies of the thermal expansivity for nearly pure grossular (Isaak et al., 1992) and a solid solution of pyrope-almandine (Suzuki and Anderson, 1983) gave ␣ at 298 K as 20 ⫻ 10⫺6/K and as 23.6 ⫻ 10⫺6/K, respectively, which are typical of silicates. The corresponding values of ␥th (Table 1) supports the Brillouin scattering measurement of pyrope by Sinogeikin and Bass (2000) but not that of Conrad et al. (1999). These two studies differ in the pressure derivatives of the bulk but not of the shear, modulus (Table 1). Also, KS' near 4 (Sinogeikin and Bass, 2000) agrees with recent volumetric data (Leger et al., 1990; Zhang et al., 1999). This finding raises some questions regarding Brillouin scattering data on calcic garnets (Conrad et al., 1999). Andradite has G' ⫽ 4.3 determined through Brillouin scattering (Conrad et al., 1999). This value is unusually high (Table 1). Wang and Ji (2001) obtained lower G' for their andradite and lower K', which is more in line with volumetric data. Grossular may have high K' (Table 1), but the agreement between the volumetric and elastic measurements does not convince one of the accuracy of the pressure derivatives. Independent confirmation is needed.