Pressure and temperature dependence of the single crystal elastic moduli of the cubic perovskite KMgF3

The elastic moduli (c) of single crystal KMgF₃ have been determined by the ultrasonic pulse superposition technique as a function of temperature from T=298?550 K, and as a function of pressure from P=1 bar?2.5 kbar. Room temperature values of the elastic moduli and their temperature derivatives are consistent with Reshchikova's (1969) values. Comparison with the data for SrTiO₃ indicates that, for most of the moduli, 1/c(?c/?T) _P and (?c/?P) _T are very similar for the fluoride-oxide analogue pair, KMgF₃-SrTiO₃. Values of (?c/?P) _T for KMgF₃ are calculated from a simple central force model using parameters determined for KF and are in good agreement with the measured values. The bulk sound velocity-mean atomic weight relationship, v _ф M ^1/2=constant, is well obeyed by the fluoroperovskites; comparison with the perovskite oxide data on a log-log plot of v _ф versus M leads to a value of 70% for the relative effective charge of the oxides with respect to the fluorides.     

Measurement of elastic properties of single-crystal CaO up to 1200 K

The elastic moduli of a single-crystal calcium oxide, CaO, are measured in the temperature range from 300 to 1200 K (1.8 times of the Debye temperature) by the resonant sphere technique (RST). The lowest 18 modes are identified in the frequency range from 0.6 to 1.4 MHz for the vibrating spherical specimen, which is 5.6564 mm in diameter and 3.3493 g/cm³ in density at room temperature, and the resonant frequencies are traced as a function of temperature. The adiabatic elastic moduli are determined in the present temperature range from the observed frequencies by inversion calculations. Most of the elastic moduli, except forC ₁₂ modulus, decrease as temperature increases. The temperature curves ofC _s andC ₄₄ moduli cross at 372 K. This means that the CaO specimen has an isotropic elasticity at the temperature. The temperature derivatives (?C ₁₁/?T) _P and (?C _s/?T) _P become slightly less negative with temperature increase and (?C _s /?T) _P and (?C ₄₄/?T) _P are almost constant. Combining the present elastic data with thermal expansion and specimen heat capacity data of CaO, we present the temperature dependence of thermodynamic parameters important in the studies of earth's interior.     

High-temperature thermal expansion and elasticity of calcium-rich garnets

We present new high temperature elasticity data on two grossular garnet specimens. One specimen is single-crystal, of nearly endmember grossular, the other is polycrystalline with about 22% molar andradite. Our data extend the high temperature regime for which any garnet elasticity data are available from 1000 to 1350 K and the compositional range of temperature data to near endmember grossular. We also present new data on the thermal expansivity of calcium-rich garnet. We find virtually no discernable differences in the temperatureT derivatives at ambient conditions of the isotropic bulkK _S and shearμ moduli when comparing our results between these two specimens. These calcium-rich garnets have the lowest values of ¦(?K _S /?T) _P ¦ = (1.47,1.49) x 10^-2GPa/K, and among the highest values of ¦(?μ/?T) _P ¦ = 1.25 x 10^-2GPa/K, when compared with other garnets. Small, but measurable, nonlinear temperature dependences of most of the elastic moduli are observed. Several dimensionless parameters are computed with the new data and used to illustrate the effects of different assumptions on elastic equations of state extra-polated to high temperatures. We discuss how dimensionless parameters and other systematic considerations can be useful in estimating the temperature dependence of some properties of garnet phases for which temperature data are not yet available. While we believe it is premature to quantitatively predict the temperature variation ofK _S andμ for majorite garnets, our results have bearing on the amount of diopside required to explain the shear velocity gradients in Earth's transition zone.     

Elastic constant systematics

This papers reviews elastic constant systematics. The bulk modulus of oxides and silicates is generally predictable on the basis of density and mean atomic weight. For constant mean atomic weight, \(\bar M\) , the bulk modulus is inversely proportional to the fourth power of the molar volume, regardless of whether molar volume changes due to temperature,T, pressure,P, or crystal structure. This iso- \(\bar M\) trend has the explanation that the Grüneisen parameter, (?K/?P) _T , and ?(1/αK)(?K/?T) _P , whereK is bulk modulus and α is volumetric thermal expansion, are approximately constant for most materials. For isostructural compounds, the bulk modulus is inversely proportional to the molar volume. This isostructural trend has the explanation that a certain combination of interatomic force parameters are the same for isostructural compounds. Equivalent iso- \(\bar M\) and isostructural trends are discussed for velocity versus density. Exceptions to the systematics exist.     

High-temperature elasticity of magnesioferrite spinel

The elastic moduli of magnesioferrite spinel, MgFe₂O₄, and their temperature dependence have been determined for the first time by ultrasonic measurements on a polycrystalline specimen. The measurements were carried out at 300 MPa and to 700°C in a gas-medium high-pressure apparatus. On heating, both the elastic bulk (K _S) and shear (G) moduli decrease linearly to 350°C. By combining with extant thermal-expansion data, the values for the room-temperature K _S and G, and their temperature derivatives are as follows: K ₀ = 176.3(7) GPa, G ₀ = 80.1(2) GPa, (∂K _S/∂T) _P = −0.032(3) GPa K⁻¹ and (∂G/∂T) _P = −0.012(1) GPa K⁻¹. Between 350 and 400°C, there are abrupt increases of 1.4% in both of the elastic moduli; these closely coincide with the magnetic Curie transition that was observed by thermal analyses at about 360°C.     

A thermodynamic theory of the Grüneisen ratio at extreme conditions: MgO as an example

The Grüneisen ratio, γ, is defined as γy=αK _TV/C_v. The volume dependence of γ(V) is solved for a wide range in temperature. The volume dependence of αK _T is solved from the identity (? ln(αK _T)/? ln V)T ≡ δ _T-K′. α is the thermal expansivity; K _T is the bulk modulus; C _V is specific heat; and δ _Tand K′ are dimensionless thermoelastic constants. The approach is to find values of δ _T and K′, each as functions of T and V. We also solve for q=(? ln γ/? ln V) where q=δ _T -K′+ 1-(? ln C _V/? ln V)T. Calculations are taken down to a compression of 0.6, thus covering all possible values pertaining to the earth's mantle, q=? ln γ/? ln V; δ _T=? ln α/? ln V; and K′= (?K _T/?P)T. New experimental information related to the volume dependence of δ _T, q, K′ and C _V was used. For MgO, as the compression, η=V/V ₀, drops from 1.0 to 0.7 at 2000 K, the results show that q drops from 1.2 to about 0.8; δ _T drops from 5.0 to 3.2; δ _T becomes slightly less than K′; ? ln C _V/? In V→0; and γ drops from 1.5 to about 1. These observations are all in accord with recent laboratory data, seismic observations, and theoretical results.     

Elasticity of diopside

The thirteen single-crystal elastic moduli for diopside as determined by the acoustic technique based on Brillouin scattering are: c₁₁=2.23, c₂₂=1.71, c₃₃=2.35, c₄₄=0.74, c₅₅=0.67, c₆₆=0.66, c₁₂=0.77, c₁₃=0.81, c₁₅=0.17, c₂₃=0.57, c₂₅=0.07, c₃₅=0.43, c₄₆=0.073. The Reuss bound of the adiabatic bulk and shear moduli calculated from these data are K _s=1.08 Mbar and G=0.651 Mbar. The room-pressure isothermal bulk modulus, K _T , and the pressure derivative of the bulk modulus, K′ _T have also been determined on a four-circle diffractometer, from a single crystal mounted in a gasketed opposed-anvil diamond cell, giving values of K _T =1.13 Mbar and K′ _T =4.8. The principal axes of the strain ellipsoid, calculated from the elastic moduli and observed in the static compression data, are identical, and the linear compressibilities are in reasonable agreement. The single-crystal elastic moduli can be correlated with the structural features of diopside.     

Hydrostatic compression of γ-Mg2SiO4 to mantle pressures and 700 K: Thermal equation of state and related thermoelastic properties

P-V-T equations of state for the γ phase of Mg₂SiO₄ have been fitted to unit cell volumes measured under simultaneous high pressure (up 30 GPa) and high temperature (up to 700 K) conditions. The measurements were conducted in an externally heated diamond anvil cell using synchrotron x-ray diffraction. Neon was used as a pressure medium to provide a more hydrostatic pressure environment. The P-V-T data include 300 K-isothermal compression to 30 GPa, 700 K-compression to 25 GPa and some additional data in P-T space in the region 15 to 30 GPa and 300 to 700 K. The isothermal bulk modulus and its pressure derivative, determined from the isothermal compression data, are 182(3) GPa and 4.2(0.3) at T=300 K, and 171(4) GPa and 4.4(0.5) at T=700 K. Fitting all the P-V-T data to a high-temperature Murnaghan equation of state yields: K _TO=182(3.0) GPa, K _TO=4.0(0.3), ?K _T /?T)₀=?2.7(0.5)×10^?2 GPa/K and (?² K _T /?P?T)₀=5.5(5.2)×10^?4/K at the ambient condition.     

Elasticity of TiO2 rutile to 1800 K

The ambient pressure elastic properties of single-crystal TiO₂ rutile are reported from room temperature (RT) to 1800 K, extending by more than 1200 oK the maximum temperature for which rutile elasticity data are available. The magnitudes of the temperature derivatives decrease with increasing temperature for five of the six adiabatic elastic moduli (C _ij ). At RT, we find (units, GPa): C ₁₁=268(1); C ₃₃=484(2); C ₄₄=123.8(2); C ₆₆=190.2(5); C ₂₃=147(1); and C ₁₂=175(1). The temperature derivatives (units, GPa K⁻¹) at RT are: (∂C ₁₁/∂T) _P =−0.042(5); (∂C ₃₃/∂T) _P =−0.087(6); (∂C ₄₄/∂T) _P =−0.0187(2); (∂C ₆₆/∂T) _P =−0.067(2); (∂C ₂₃/∂T) _P =−0.025; and (∂C ₁₂/∂T) _P −0.048(5). The values for K _S (adiabatic bulk modulus) and μ (isotropic shear modulus) and their temperature derivatives are K _S =212(1) GPa; μ=113(1) GPa; (∂K _S /∂T) _P =−0.040(4) GPa K⁻¹; and (∂μ/∂T) _P =−0.018(1) GPa K⁻¹. We calculate several dimensionless parameters over a large temperature range using our new data. The unusually high values for the Anderson-Gròneisen parameters at room temperature decrease with increasing temperature. At high T, however, these parameters are still well above those for most other oxides. We also find that for TiO₂, anharmonicity, as evidenced by a non-zero value of [∂ln (K _T )/∂lnV] _T , is insignificant at high T, implying that for the TiO₂ analogue of stishovite, thermal pressure is independent of volume (or pressure). Systematic relations indicate that ∂² K _S /∂T∂P is as high as 7×10⁻⁴ K⁻¹ for rutile, whereas ∂²μ/∂T∂P is an order of magnitude less. Received: 19 September 1997 / Revised, accepted: 27 February 1998     

High-temperature and high-pressure equation of state for the hexagonal phase in the system NaAlSiO4 – MgAl2O4

Thermal equation of state of an Al-rich phase with Na_1.13Mg_1.51Al_4.47Si_1.62O₁₂ composition has been derived from in situ X-ray diffraction experiments using synchrotron radiation and a multianvil apparatus at pressures up to 24 GPa and temperatures up to 1,900 K. The Al-rich phase exhibited a hexagonal symmetry throughout the present pressure–temperature conditions and the refined unit-cell parameters at ambient condition were: a=8.729(1) Å, c=2.7695(5) Å, V ₀=182.77(6) Å³ (Z=1; formula weight=420.78 g/mol), yielding the zero-pressure density ρ₀=3.823(1) g/cm³ . A least-square fitting of the pressure-volume-temperature data based on Anderson’s pressure scale of gold (Anderson et al. in J Appl Phys 65:1534–543, 1989) to high-temperature Birch-Murnaghan equation of state yielded the isothermal bulk modulus K ₀=176(2) GPa, its pressure derivative K ₀ ^′ =4.9(3), temperature derivative (?K _T /?T) _P =?0.030(3) GPa K^?1 and thermal expansivity α(T)=3.36(6)×10^?5+7.2(1.9)×10^?9 T, while those values of K ₀=181.7(4) GPa, (?K _T /?T) _P =?0.020(2) GPa K^?1 and α(T)=3.28(7)×10^?5+3.0(9)×10^?9 T were obtained when K ₀ ^′ was assumed to be 4.0. The estimated bulk density of subducting MORB becomes denser with increasing depth as compared with earlier estimates (Ono et al. in Phys Chem Miner 29:527–531 2002; Vanpeteghem et al. in Phys Earth Planet Inter 138:223–230 2003; Guignot and Andrault in Phys Earth Planet Inter 143–44:107–128 2004), although the difference is insignificant (<0.6%) when the proportions of the hexagonal phase in the MORB compositions (～20%) are taken into account.     

Thermal properties of forsterite at mantle pressures derived from vibrational spectroscopy

The pressure dependence of the Raman spectrum of forsterite was measured over its entire frequency range to over 200 kbar. The shifts of the Raman modes were used to calculate the pressure dependence of the heat capacity, C _v, and entropy, S, by using statistical thermodynamics of the lattice vibrations. Using the pressure dependence of C _v and other previously measured thermodynamic parameters, the thermal expansion coefficient, α, at room temperature was calculated from α = K _S (?T/?P) _S C _V/TVK _T, which yields a constant value of (? ln α/? ln V)_T= 6.1(5) for forsterite to 10% compression. This value is in agreement with (? ln α/? ln V)_T for a large variety of materials. At 91 kbar, the compression mechanism of the forsterite lattice abruptly changes causing a strong decrease of the pressure derivative of 6 Raman modes accompanied by large reductions in the intensities of all of the modes. This observation is in agreement with single crystal x-ray diffraction studies to 150 kbar and is interpreted as a second order phase transition.     

P-V-T equation of state of magnesiowüstite (Mg0.6Fe0.4)O

The lattice parameter of magnesiowüstite (Mg_0.6Fe_0.4)O has been measured up to a pressure of 30 GPa and a temperature of 800 K, using an external heated diamond anvil cell and diffraction using X-rays from a synchrotron source. The experiments were conducted under quasi-hydrostatic condition, using neon as a pressure transmitting medium. The experimental P-V-T data were fitted to a thermal-pressure model with the isothermal bulk modulus at room temperature K _T0 = 157 GPa, (?K _TO /?P) _T =4, (?K _T /?T) _P =-2.7(3) × 10^-2 GPa/K, (?K _T /?T) _v =-0.2(2) × 10^-2 GPa/K and the Anderson-Grüneisen parameter δ _T =4.3(5) above the Debye temperature. The data were also fitted to the Mie-Grüneisen thermal equation of state. The least-squares fit yields the Debye temperature θ _DO = 500(20) K, the Grüneisen parameter γ ₀=1.50(5), and the volume dependence q=1.1(5). Both thermal-pressure models give consistent P-V-T relations for magnesiowüstite to 140 GPa and 4000 K. The P-V-T relations for magnesiowüstite were also calculate by using a modified high-temperature Birch-Murnaghan equation of state with a δ _t of 4.3. The results are consistent with those calculated by using the thermal-pressure model and the Mie-Grüneisen relation to 140 GPa and 3000 K.     

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

Grüneisen’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 Grüneisen parameter (γ_i) of the longitudinal acoustic (LA) mode equals the thermodynamic parameter (γ_th) for monatomic solids. The success of the Debye model indicates that γ_LA = γ_th is generally true. Available elasticity data for crystalline solids contain 30 reliable measurements, covering 10 structures, of the pressure derivatives of the bulk (K_S) and the shear (G) moduli. For these phases, the measured values of γ_th and γ_LA agree well. Other solids in the database have disparate γ_LA values, suggesting large experimental uncertainties within which γ_LA = γ_th. This relationship allows inference of the pressure (P) derivative of the shear modulus (∂G/∂P = G′) from widely available measurements of γ_th, the isothermal bulk modulus (K_T), ∂K_T/∂P, and G. We predict G′ as 1.55 for stishovite, 1.6 to 2.15 for MgSiO₃ ilmenite, 1.0 for γ-Mg_1.2Fe_0.8SiO₄, and 0 for FeS (troilite). Similarly, G′ measured for MgSiO₃ perovskite suggests that K_S′ = 4, corroborating volume-pressure data. For many materials, pairs of G′ and K_S′ = ∂K_S/∂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 K_S′ and G′. The best values for pressure derivatives can be ascertained because the trend defined by measured pairs of G′ with K_S′ intersects the relationship of G′ to K′ defined by γ_LA = γ_th at a steep angle. Our results for isostructural series show linear correlations of K_S′ with K_S and of G′ with G. Values of K_S′ 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.     

Pressure dependence of the elastic constants of nonmetamict zircon

The single crystal elastic constants of nonmetamict zircons have been measured as a function of pressure to 12 kb at room temperature and also as a function of temperature between 25 and 300° C at atmospheric pressure. The pressure derivatives of the elastic constants are: C ₁₁=10.78, C ₃₃=5.88, C ₄₄=0.99, C ₆₆=?0.31, C ₁₂=3.24, C ₁₃=6.20. The anomalous negative behaviour of C ₆₆ versus pressure could be associated with a high pressure phase transition. The pressure and temperature derivatives of the isotropic elastic wave velocities and elastic moduli for nonmetamict zircon are calculated from the present single crystal data by the Voigt, Ruess, and Hill approximations and compared with the values of some other oxides and silicates. The pressure derivative of the isotropic adiabatic bulk modulus is relatively high (dK _S/dP=6.50), and the pressure derivative of the shear modulus is relatively low, (dG/dP=0.78), compared to the corresponding values for some other oxides and silicates. The Debye temperature, ?_D, and the high temperature limit of the Grüneisen parameter, γ_Ht, calculated from the elastic constants and their pressure derivatives, agrees well with the Debye temperature and the thermal Grüneisen parameter, γ_th, calculated from the thermal expansion, heat capacity, and compressibility data.     

In search of the mixed derivative ?2 M /? P ? T ( M = G , K ): joint analysis of ultrasonic data for polycrystalline pyrope from gas- and solid-medium apparatus

Elastic wave velocities for dense (99.8% of theoretical density) isotropic polycrystalline specimens of synthetic pyrope (Mg₃Al₂Si₃O₁₂) were measured to 1,000 K at 300 MPa by the phase comparison method of ultrasonic interferometry in an internally heated gas-medium apparatus. The temperature derivatives of the elastic moduli [(∂Ks/∂T) _P = −19.3(4); (∂G/∂T) _P = −10.4(2) MPa K⁻¹] measured in this study are consistent with previous acoustic measurements on both synthetic polycrystalline pyrope in a DIA-type cubic anvil apparatus (Gwanmesia et al. in Phys Earth Planet Inter 155:179–190, 2006) and on a natural single crystal by the rectangular parallelepiped resonance (RPR; Suzuki and Anderson in J Phys Earth 31:125–138, 1983) method but |(∂Ks/∂T) _P | is significantly larger than from a Brillouin spectroscopy study of single-crystal pyrope (Sinogeikin and Bass in Phys Earth Planet Inter 203:549–555, 2002). Alternative approaches to the retrieval of mixed derivatives of the elastic moduli from joint analysis of data from this study and from the solid-medium data of Gwanmesia et al. in Phys Earth Planet Inter 155:179–190 (2006) yield ∂² G/∂P∂T = [0.07(12), 0.20(14)] × 10⁻³ K⁻¹ and ∂² K _S /∂P∂T = [−0.20(24), 0.22(26)] × 10⁻³ K⁻¹, both of order 10⁻⁴ K⁻¹ and not significantly different from zero. More robust inference of the mixed derivatives will require solid-medium acoustic measurements of precision significantly better than 1%.     

Oxide perovskites: pressure derivatives of the bulk and shear moduli

The pressure derivatives of elastic moduli (∂M/∂P; M=K_S and G) for a suite of polycrystalline oxide perovskites (2 titanates, 1 stannate and 2 aluminates) have been measured up to 3 GPa using the ultrasonic interferometry method combined with a buffer rod technique. Two empirical systematic relationships (∂G/∂P vs K_S/G and ∂K_S/∂P vs K_S (/ρ)^1/3) have been used to investigate the elasticity systematics of this suite of perovskites and to estimate ∂M/∂P of MgSiO₃ perovskite. The pressure derivatives ∂G/∂P and ∂K_S/∂P for this suite of perovskites scatter between well-defined linear trends for the rutile, rocksalt and spinel structures. The more diffuse trends observed for the perovskites might reflect greater flexibility in the response of its corner-connected octahedral framework structure to changing pressure. The pressure derivatives of the elastic moduli for MgSiO₃ perovskite estimated by the “perovskite bands” are ∂G/∂P=1.6–2.2 and ∂K_S/∂P=3.9–4.2. Received: 13 November 1997 / Revised, accepted: 31 August 1998     

Measured elastic moduli of single-crystal MgO up to 1800 K

Using the rectangular parallelepiped resonance method we measured the temperature dependence of the adiabatic elastic moduli of single-crystal MgO over the temperature range 300–1800 K. The high temperature limit of our measurements extends by 500 K the upper limit over which elasticity data on MgO are now available. Although our measured temperature dependence of C _ij ^s are generally in good agreement with previous measurements over a more narrow range in temperature, we found that C ₄₄ ^s decreases more rapidly with temperature, for T > 1000 K, than previous studies suggest. We also found that each of the slopes (?C ₁₁ ^s /?T)_p, (?K_s/?T)_p, and (C ₄₄ ^s /?T)_p become less negative with increasing temperature for T > 1400 K. From our measurements on elasticity we are able to confirm that the Grüneisen parameter at zero pressure is nearly constant with temperature up to 1800 K, with only a slight decrease above 1000 K. Utilizing our new data we present calculations showing the temperature dependence of thermodynamic parameters important in studies of earth's interior.     

The pressure and temperature dependence of the elastic properties of single-crystal fayalite Fe2SiO4

The nine adiabatic elastic stiffness constants of synthetic single-crystal fayalite, Fe₂SiO₄, were measured as functions of pressure (range, 0 to 1.0 GPa) and temperature (range, 0 to 40° C) using the pulse superposition ultrasonic method. Summary calculated results for a dense fayalite polycrystalline aggregate, based on the HS average of our single-crystal data, are as follows: V_p = 6.67 km/s; V_s = 3.39km/s; K= 127.9 GPa; μ = 50.3 GPa; (?K/?P)_T = 5.2; (?μ/?P)_T=1.5;(?K/?T)_P= ?0.030 GPa/K;and,(?/?T)_P =-0.013 GPa/K (the pressure and temperature data are referred to 25° C and 1 atm, respectively). Accuracy of the single-crystal results was maintained by numerous cross and redundancy checks. Compared to the single-crystal elastic properties of forsterite, Mg₂SiO₄, the fayalite stiffness constants, as well as their pressure derivatives, are lower for each of the on-diagonal (C _ij for which i=j) values, and generally higher for the off-diagonal (C _ij for which i≠j) data. As a result, the bulk moduli (K) and dK/dP for forsterite and fayalite are very similar, but the rigidity modulus (μ) and dμ/dP for polycrystalline fayalite are much lower than their forsterite counterparts. The bulk compression properties derived from this study are very consistent with the static-compression x-ray results of Yagi et al. (1975). The temperature dependence of the bulk modulus of fayalite is somewhat greater (in a negative sense) than that of forsterite. The rigidity dependencies are almost equivalent. Over the temperature range relevant to this study, the elastic property results are generally consistent with the data of Sumino (1978), which were obtained using the RPR technique. However, some of the compressional modes are clearly discrepant. The elastic constants of fayalite appear to be less consistent with a theoretical HCP model (Leibfried 1955) than forsterite, reflecting the more covalent character of the Fe-O bonding in the former.