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.     

Single-crystal elastic properties of alunite, KAl3(SO4)2(OH)6

The single-crystal elastic constants of natural alunite (ideally KAl₃(SO₄)₂(OH)₆) were determined by Brillouin spectroscopy. Chemical analysis by electron microprobe gave a formula KAl₃(SO₄)₂(OH)₆. Single crystal X-ray diffraction refinement with R ₁ = 0.0299 for the unique observed reflections (|F _o| > 4σ _F) and wR ₂ = 0.0698 for all data gave a = 6.9741(3) Å, c = 17.190(2) Å, fractional positions and thermal factors for all atoms. The elastic constants (in GPa), obtained by fitting the spectroscopic data, are C ₁₁ = 181.9 ± 0.3, C ₃₃ = 66.8 ± 0.8, C ₄₄ = 42.8 ± 0.2, C ₁₂ = 48.2 ± 0.5, C ₁₃ = 27.1 ± 1.0, C ₁₄ = 5.4 ± 0.5, and C ₆₆ = ½(C ₁₁–C ₁₂) = 66.9 ± 0.3 GPa. The VRH averages of bulk and shear modulus are 63 and 49 GPa, respectively. The aggregate Poisson ratio is 0.19. The high value of the ratio C ₁₁/C ₃₃ = 2.7 and of the ratio C ₆₆/C ₄₄ = 1.6 are characteristic of an anisotropic structure with very weak interlayer interactions along the c-axis. The basal plane (001) is characterized by 0.1% longitudinal acoustic anisotropy and 0.9–1.1% shear acoustic anisotropy, which gives alunite a characteristic pseudo-hexagonal elastic behavior, and is related to the pseudo-hexagonal arrangement of the Al(O,OH)₆ octahedra in the basal layer. The elastic Debye temperature of alunite is 654 K. The large discrepancy between the elastic and heat capacity Debye temperature is also a consequence of the layered structure.     

Composition, lattice parameters, and room temperature elastic constants of natural single crystal xenotime from Novo Horizonte

The composition, lattice parameters, and elastic constants of natural single crystal YPO₄ xenotime from Novo Horizonte (Brazil) were determined using EPMA, XRD, and the pulse-echo technique. The composition indicates a 24% substitution of Y sites with other rare-earth elements. The lattice parameters of the studied crystal deviated only slightly from those reported for synthetic YPO₄ and were in a good agreement with trends obeyed by other orthophosphates with the xenotime structure. The measured elastic constants C ₁₁, C ₃₃, C ₄₄, and C ₆₆ were consistent with synthetic crystals when porosity was accounted for. C ₁₂ and C ₁₃ constants were evaluated based on the comparison with other materials with xenotime structure. The elastic constants could be rationalized using interionic force constants and bond energies.

Anomalous elasticity of talc at high pressures: Implications for subduction systems

Talc is a layered hydrous silicate mineral that plays a vital role in transporting water into Earth’s interior and is crucial for explaining geophysical observations in subduction zone settings. In this study, we explored the structure, equation of state, and elasticity of both triclinic and monoclinic talc under high pressures up to 18 GPa using first principles simulations based on density functional theory corrected for dispersive forces. Our results indicate that principal components of the full elastic constant tensor C₁₁ and C₂₂, shear components C₆₆, and several off-diagonal components show anomalous pressure dependence. This non-monotonic pressure dependence of elastic constant components is likely related to the structural changes and is often manifested in a polytypic transition from a low-pressure polytype talc-I to a high-pressure polytype talc-II. The polytypic transition of talc occurs at pressures within its thermodynamic stability. However, the bulk and shear elastic moduli show no anomalous softening. Our study also shows that talc has low velocity, extremely high anisotropy, and anomalously high V_P/V_S ratio, thus making it a potential candidate mineral phase that could readily explain unusually high V_P/V_S ratio and large shear wave splitting delays as observed from seismological studies in many subduction systems.     

The extrapolation of elastic moduli to high pressure and temperature

The elastic constants of a crystal under stress, defined as the second derivative of the crystal free energy with respect to strain, require a correction related to the static pressure at non-zero pressures. The corrections required for the elastic constants calculated by the free energy minimisation code PARAPOCS are described and tested by comparison with the elastic constants calculated numerically by applying small stresses in the appropriate orientations to simulated crystals of fluorite, forsterite, α-quartz and albite. The corrected elastic constants are then used to investigate the extrapolation of the bulk and shear moduli (and hence also the seismic wave velocities V _p and V _s) of β-spinel and forsterite to upper mantle pressures. A Murnaghan equation, thirdorder Eulerian finite strain equation, second order polynomial equation and a logistic equation were all fitted to the simulated bulk and shear moduli between 0 and 3 GPa pressure. The parameters derived for these equations are used to extrapolate the bulk and shear moduli to 14 GPa and the results are compared to the simulated high pressure moduli. Over this pressure range, the second order polynomial provides the best extrapolation of the bulk modulus, but the use of the logistic equation results in the best extrapolation of the shear modulus.     

Elasticity of single-crystal MgAl2O4 spinel up to 1273 K by Brillouin spectroscopy

The adiabatic single-crystal elastic constants, C _ij , of stoichiometric magnesium aluminate spinel (MgAl₂O₄) have been measured up to 1273 K by highresolution Brillouin spectroscopy, using a 6-pass tandem Fabry-Pérot interferometer and an argon ion laser (514.5 nm). Two platelet samples were employed for probing the acoustic phonons along [100] and [110] directions by platelet and backscattering geometries. The measured temperature dependences of the elastic moduli show a distinct anomaly at 923 K in the shear modulus C _s = (C₁₁-C₁₂)/2 (along [110] direction) and the longitudinal modulus C₁₁ (along [100] direction). This anomaly is consistent with the order-disorder phase transition, resulting from the atomic exchange between Mg at the tetrahedral site and Al at the octahedral site, which has been well documented recently (Peterson et al. 1991; Millard et al. 1992) by neutron powder diffraction and ²⁷Al magic-angle spinning NMR. The values of the temperature derivatives of v _p , v _s , and K _s , in the temperature range 300–923 K, calculated by the Voigt-Reuss-Hill approximation are -0.40ms^?1 K^?1, -0.26ms^?1 K^?1, and -1.89 x 10^?2GPaK^?1.     

Elasticity of the pyrope (Mg3Al2Si3O12)-majorite (MgSiO3) garnets solid solution

Dense isotropic polycrystalline specimens of majorite-rich garnets (Py₁₀₀, Py₆₂Mj₃₈, Py₅₀Mj₅₀, Py₂₁Mj₇₉ and Mj₁₀₀) along the pyrope (Mg₃Al₂Si₃O₁₂ = Py₁₀₀)-majorite (MgSiO₃ = Mj₁₀₀) join were fabricated in a 2000-ton uniaxial split-sphere anvil apparatus (USSA-2000) at pressures from 10 to 18.5 GPa and temperatures from 1200 to 1850 °C, within their stability fields in runs of 2–4-h duration, using hot-pressing techniques developed by Gwanmesia et al. (1993). These specimens are single-phased, fine-grained (≤5 mm), free of microcracks, and have bulk densities greater than 99% of the corresponding single-crystal X-ray density. Elastic compressional (P) and shear (S) wave velocities were determined at room pressure and temperature for these polycrystalline garnet specimens by phase comparison ultrasonic interferometry. For Mj₁₀₀, the P and S wave velocities are within 1% of the Hashin-Shtrikman averages calculated from the single crystal elastic moduli measured by Brillouin spectroscopy. Both the elastic bulk modulus (K) and the shear modulus (G) decrease continuously with increasing majorite content from pyrope garnet (Py₁₀₀) to pure majorite garnet (Mj₁₀₀). The compositional dependence of K and G are given by K = 172.3 (40) − 0.085X, and G = 91.6 (10) − 0.038X, where X = mol% majorite), respectively, indicating that substitution of Si for Mg and Al decreases both K and G by about 5% along the solid solution series. Received: 25 March 1999 / Accepted: 12 July 1999     

Temperature variation of elastic constants of quartz across the α - β transition

The α − β transition of quartz was successfully observed with using a single sample by means of the rectangular parallelepiped resonance (RPR) method. An oriented rectangular parallelepiped of α-quartz single crystal was prepared and the resonant frequencies of 30–11 vibrational modes were measured from room temperature to 700°C. The softening of quartz crystal was observed as the significant reduction of resonant frequencies near the α–β transition. The present study is the first application of the RPR method to the study of phase transition. The complete set of elastic constants of α- and β-quartz were determined as a function of temperature by the least-squares inversion of the measured frequency data obtained by a single run. This is a merit yielded by the RPR method. It is shown near the α − β transition in both α- and β-quartz that the elastic parameters decrease proportionally to |T−T ₀|⁻ⁿ , where T is temperature and T ₀ is the transition temperature, 573.0°C for α-quartz and 574.3°C for β-quartz. It was also seen that linear incompressibilities K ₁ = (C ₁₁ +C ₁₂ +C ₁₃)/3 and K ₃ = (C ₃₃ +2C ₁₃)/3 decrease rapidly toward the transition, whereas, shear moduli C ₄₄, C _S1 = (C ₁₁ +C ₃₃ -2C ₁₃)/4 and C _S3 = (C ₁₁ -C ₁₂)/2 = C ₆₆ decrease only slightly. The shear modulus C _S3 = C ₆₆ increased slightly in α-quartz. The elastic properties of isotropic aggregate of quartz were calculated, and it is shown that the longitudinal wave velocity significantly decreases at the α − β transition, whereas, the shear wave velocity decreases only slightly.     

Elasticity of single-crystal aragonite by Brillouin spectroscopy

The elastic constants of natural single-crystal aragonite (CaCO₃) have been measured by Brillouin spectroscopy at ambient conditions. The elastic constants C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃ and C₂₃ are 171.1±1.0, 110.1±0.9, 98.4±1.2, 39.3±0.6, 24.2±0.4, 40.2±0.6, 60.3±1.0, 27.8±1.6 and 41.9±2.0 GPa, respectively, for aragonite. The linear compressibilities of the a-, b- and c-axis for aragonite at ambient conditions were derived from our measured data to be 3.0±0.2, 4.2±0.2 and 7.3±0.6×10^–3 GPa^–1, respectively. The aggregate bulk and shear moduli for aragonite using the Voigt-Reuss-Hill (VRH) scheme are thus calculated to be 68.9±1.4 and 35.8±0.2 GPa, respectively. The value of bulk modulus is in remarkable contrast to the literature value of 46.9 GPa measured almost a century ago. Our new datum, however, is closer to that derived from recent atomistic simulation and static compression studies.     

Structural states of Mg-cordierite II: Landau theory

Mg-cordierite undergoes a ferroelestic phase transitionP6/mmc-Cccm. The order parameterQ is proportional to the spontaneous strain as reflected by changes of the lattice parametersa andb during the phase transition. The order parameter,Q _od, which describes the Al, Si ordering, isnot directly involved in the phase transition and only “triggers” the structural collapse. Landau theory predicts that cordierite can exist in stable or metastable states with hexagonal, orthorhombic or monoclinic symmetry. Hexagonal cordierite can develop modulated structures which have been found by Putnis et al. (1987). The phase transition is predicted to be accompanied by singularities of the elastic constantsC ₁₁,C ₂₂ andC ₁₂ leading to an elastic softening of the crystal structure.     

Elastic properties of the incommensurate phase of akermanite,Ca2MgSi2O7

The crystal structure of akermanite, Ca₂Mg-Si₂O₇, consists of mixed tetrahedral sheets formed by [MgO₄] tetrahedra and [Si₂O₇] groups interleaved along the c axis with Ca²⁺ ions in eight-fold coordination. Above 358 K, the structure is tetragonal , and below it is incommensurate with modulations parallel to [110] and . The elastic stiffness moduli, C ^ij of the incommensurate phase at room temperature were measured from wave velocities in the 20–75 MHz carrier frequency range by the ultrasonic phase comparison method using optically clear synthetic single crystal plates (3×3×2 mm) oriented parallel to (100), (001), (110) and (101) planes. The C _ij values (GPa) are: C ₁₁ 159.40, C ₃₃ 149.43, C ₄₄ 30.26, C ₆₆ 58.10, C ₁₂ 76.58 and C ₁₃ 57.80. In (010) and (001) planes, the compressional modulus, V ²(L) from the longitudinal wave, L is considerably larger than the shear moduli, V ²(T₁, T ₂) both from the in-plane and perpendicular-to-plane shear waves, T ₁ and T ₂. The relatively small values of the shear moduli indicate the ease of tetrahedral rotations in response to in-plane and perpendicular-to-plane shears and may provide preconditions for structural changes involving shear-type atomic movements.     

Elasticity of an upper mantle clinopyroxene

The ambient pressure elastic properties of a natural clinopyroxene (C2/c symmetry) from Kilbourne Hole, NM have been determined. In terms of end-members, diopside (CaMgSi₂O₆), hedenbergite (CaFeSi₂O₆), jadeite (NaAlSi₂O₆), cosmochlor (NaCrSi₂O₆), and Mg-Tschermak (MgAl(AlSi)O₆), its composition is Di₇₂He₉Jd₃Cr₃Ts₁₂. The analytic density, based on chemistry and cell parameters is 3.327 (0.003) g/cm³. The elastic constants [c₁₁, c₁₂, c₁₃, c₁₅, c₂₂, c₂₃, c₂₅, c₃₃, c₃₅, c₄₄, c₄₆, c₅₅, c₆₆] are [273.8 (0.9), 83.5 (1.3), 80.0 (1.1), 9.0 (0.6), 183.6 (0.9), 59.9 (1.6), 9.5 (1.0), 229.5 (0.9), 48.1 (0.6), 76.5 (0.9), 8.4 (0.8), 73.0 (0.4), 81.6 (1.0)] GPa where uncertainties are reported at the 95% confidence level. The aggregate (mean of Hashin-Strikman bounds) adiabatic bulk modulus is 117.2 (0.7) GPa, and the shear modulus is 72.2 (0.2) GPa. Although measured moduli are broadly consistent with trends in elasticity versus atomic volume, deviations from the systematics would produce significant (percent level) changes in calculated velocities for candidate mantle mineral assemblages. The compositional dependence of elasticity for several clinopyroxenes is explored on the basis of just the Di+He and Jd+Ts mole fractions. The bulk modulus lies within experimental uncertainties of the linear mixture of end-member properties while the shear modulus deviates by 3%. Received: 29 September 1997 / Revised, accepted: 4 March 1998     

Elasticity of superhydrous B

The adiabatic single-crystal elastic moduli of superhydrous B, Mg₁₀Si₃O₁₄(OH)₄, have been measured at ambient conditions using Brillouin spectroscopy. This material is the first hydrous phase found to be stable at the extreme conditions of 20 GPa and 1400 °C. The single-crystal moduli, in GPa, are: C ₁₁=280.0±1.5, C ₂₂=307.4±1.6, C ₃₃=293.4±1.4, C ₄₄=90.0±1.1, C ₅₅=99.2±0.8, C ₆₆=89.6±0.6, C ₁₂=66.1±2.2, C ₁₃=105.6±2.6, C ₂₃=81.8±2.6. With aggregate elastic properties of K _VRH =154.0±4.2 and μ _VRH =97.0±0.7 GPa, superhydrous B is approximately 16% suffer than forsterite and 20% softer than magnesium silicate spinel; it is also considerably more elastically isotropic than forsterite. The single-crystal moduli are compared to those of forsterite, magnesium silicate spinel and periclase, materials that are both structurally and compositionally similar to superhydrous B. The longitudinal moduli of superhydrous B and forsterite follow similar trends and appear to be dominated by the incompressibility and rotation of silicon tetrahedra. The shear and off-diagonal moduli more closely resemble those of periclase and spinel and may reflect the properties inherent to layers of magnesium octahedra.     

Single-crystal elastic properties of chondrodite: implications for water in the upper mantle

Chondrodite, a member of the humite group of minerals, forms by hydration of olivine and is stable over a range of temperatures and pressures that includes a portion of the uppermost mantle. We have measured the single crystal elastic properties of a natural chondrodite specimen at ambient conditions using Brillouin spectroscopy. The isotropic aggregate bulk (K) and shear (μ) moduli calculated from the single-crystal elastic moduli, C_ij, are: K_S=118.4(16) GPa and μ=75.6(7) GPa. A comparison of the structures and elasticity of olivine and chondrodite indicate that the replacement of O with (OH,F) in M²⁺O₆ octahedra has a small effect on the elasticity of humite-group minerals. The slightly diminished elastic moduli of humite-group minerals (as compared to olivine) are likely caused by a smaller ratio of strong structural elements (SiO₄ tetrahedra) to weaker octahedra, and perhaps a more flexible geometry of edge-sharing MO₄(O,OH,F)₂ octahedra. In contrast to the humite-olivine group minerals, the incorporation of water into garnets and spineloids leads to a more substantial decrease in the elastic properties of these minerals. This contrasting behavior is due to formation of O₄H₄ tetrahedra and vacant hydroxyl-bearing octahedra in the garnets and spineloids, respectively. Therefore, the mechanism of incorporation of H/OH into mineral phases, not only degree of hydration, should be taken into account when estimating the effect of water on the elastic properties of minerals. The bulk elastic wave velocities of chondrodite and olivine are very similar. If humite-like incorporation of OH is predominant in the upper mantle, then the reaction of OH with olivine will have a minor or possibly no detectable effect on seismic velocities. Thus, it may be difficult to distinguish chondrodite-bearing rocks from “anhydrous” mantle on the basis of seismically determined velocities for the Earth. Received: 25 February 1998 / Revised, accepted: 18 August 1998     

The nonlinear elasticity and related properties of anhydrous and hydrous γ-Mg2SiO4: a theoretical study

The second-order elastic constants up to 30 GPa, which encompass the stability field of the spinel forms, their pressure derivatives and the third-order elastic constants of both hydrous and anhydrous -Mg₂SiO₄ have been obtained theoretically. A combination of deformation theory and finite strain elasticity theory has been employed to arrive at the expressions for second-order and third-order elastic constants from the strain energy of the lattice. The strain energy is calculated by taking into account the interactions up to second nearest neighbours in the -Mg₂SiO₄ lattice. This is then compared with the strain-dependent lattice energy from continuum model approximation to obtain the expression of elastic constants. The second-order elastic constants C_ij compare favourably with the measurements in the case of anhydrous as well as hydrous -Mg₂SiO₄ and with other calculations on the anhydrous phase. All the third-order elastic constants of both the compounds are negative. The third-order elastic constant C₁₄₄(–52.41 and –45.07 GPa for anhydrous and hydrous -Mg₂SiO₄, respectively) representing the anisotropy of shear mode has a smaller value than C₁₁₁ (–2443.94 and –2101.25 GPa for anhydrous and hydrous phases, respectively), which corresponds to the longitudinal mode. The pressure-induced variations in the longitudinal elastic constants (i.e.,dC₁₁/dp) are relatively large (4.08 and 4.09 for dry and hydrous ringwoodite, respectively) compared with those for the shear (0.22 and 0.32 for dry and hydrous ringwoodite, respectively) and off-diagonal constants (1.40 and 1.41 for dry and hydrous ringwoodite, respectively). The variation of the shear moduli C_s and anisotropy factor A with pressure have also been studied. The average value of elastic anisotropy is 0.835 in the case of anhydrous -Mg₂SiO₄ and 0.830 in the hydrous phase. The reversal of sign of the Cauchy pressure C₁₂ – C₄₄, which describes the angular character of atomic bonding in metals and other compounds, at around 21 GPa for both the compounds may be a precursor to the phase transition from ringwoodite to periclase and perovskite at an elevated temperature. The aggregate elastic properties like the adiabatic bulk modulus K (175.4 and 150.2 GPa for anhydrous and hydrous phases, respectively), and the isotropic compressional (P) and shear (S) wave velocities were calculated and the mode Grüneisen Parameters (GPs) of the acoustic waves were determined based on the quasi-harmonic approximation. The low temperature limit of both hydrous and anhydrous phases of -Mg₂SiO₄ are positive (1.69 and 1.78, respectively, for hydrous and anhydrous phases) and hence we expect the thermal expansion to be positive down to absolute zero. The Anderson–Grüneisen parameter obtained for hydrous as well as anhydrous phases of -Mg₂SiO₄ from the second-order and third-order elastic constants are 2.30 and 2.29, respectively.     

High pressure elasticity and phase transformation in brucite, Mg(OH)2

The first pressure derivatives of the second-order elastic constants have been calculated for brucite, Mg(OH)₂ from the second- and third-order elastic constants. The deformation theory and finite strain elasticity theory have been used to obtain the second- and third-order elastic constants of Mg(OH)₂ from the strain energy of the lattice. The strain energy ϕ is calculated by taking into account the interactions up to third nearest neighbors in the Mg(OH)₂ lattice. ϕ is then compared with the strain dependent lattice energy from continuum model approximation to obtain the expressions of elastic constants. The complete set of six second-order elastic constants C _IJ of brucite exhibits large anisotropy. Since C ₃₃ (= 21.6 GPa), which corresponds to the strength of the material along the c-axis direction, is less than the longitudinal mode C ₁₁ (= 156.7 GPa), the interlayer binding forces are weaker than the binding forces along the basal plane of Mg(OH)₂. The 14 nonvanishing components of the third-order elastic constants, C _IJK , of brucite have been obtained. All the C _IJK of brucite are negative except the values of C ₁₁₄ (= 230.36 GPa), C ₁₂₄ (= 75.45 GPa) and C ₁₃₄ (= 36.98 GPa). The absolute values of the C _IJK are, in general, one order of magnitude greater than the C _IJ ’s in the Mg(OH)₂ system as usually expected for a crystalline material. To our knowledge, no previous data are available to compare the pressure derivatives of brucite. The pressure derivatives of the two components viz., C ₁₄ and C ₃₃ become negative indicating an elastic instability in brucite while under pressure. This may be related to the phase transition of brucite largely involving rearrangements of H atoms revealed in the Raman spectroscopic, powder neutron diffraction and synchrotron X-ray diffraction studies.     

Modelling of structural and elastic changes of forsterite (Mg2SiO4) under stress

The method of crystal static deformation, including inner strain effects, was applied to calculate the structure configuration and the elastic constants of forsterite under anisotropic and isotropic pressure. A Born type interatomic potential is used, with optimized atomic charges and repulsive radii; SiO₄ tetrahedra are approximated as rigid units. Computations were carried out in the range 1–8 GPa, with steps of 1 GPa, for the three uniaxial stresses τ₁, τ₂, τ₃ and for pressure p. By interpolation of results, interatomic distances and elastic tensor components are shown to depend quadratically on stress. A non-linear behaviour generally appears above 4 GPa; the importance of inner strain and non-linear effects is analyzed. Mg-O bond lengths and O-O edges of coordination polyhedra respond differently to anisotropic and to isotropic stresses, according to the topological features of the structure. Elastic and structural results for hydrostatic pressure are compared to experimental literature data, discussing the range of validity of the rigid body approximation for SiO₄ groups.     

Triclinic elastic constants for low albite

The full set of elastic constants for plagioclase end-member phase albite (NaAlSi₃O₈) is reported for the first time. Velocities of surface acoustic waves (both Rayleigh and pseudo-surface waves) were measured using impulsively stimulated light scattering on polished surfaces having six different orientations (three normal to the Cartesian axes and three lying on diagonals). Data were inverted and results tested using several non-linear optimization techniques. Compliance moduli determined under hydrostatic compression provided additional constraints and reduced covariance in the reported constants. The Cartesian coordinate system associated with the constants (using the unit cell) has the y-axis parallel to the crystal b axis, the x-axis parallel to a* (perpendicular to b and c) and the z-axis consistent with a right-handed coordinate system. The values of the moduli C₁₁, C₁₂, C₁₃, C₁₄, C₁₅, C₁₆, C₂₂, C₂₃, C₂₄, C₂₅, C₂₆, C₃₃, C₃₄, C₃₅, C₃₆, C₄₄, C₄₅, C₄₆, C₅₅, C₅₆, C₆₆ and their 2σ uncertainties (in parentheses) are, respectively, 69.1(0.6), 34.0(0.7), 30.8(0.5), 5.1(0.1), −2.4(0.1), −0.9(0.1), 183.5(2.7), 5.5(2.2), −3.9(0.5), −7.7(0.7), −5.8(0.7), 179.5(2.3), −8.7(0.4), 7.1(0.6), −9.8(0.6), 24.9(0.1), −2.4(0.1), −7.2(0.1), 26.8 (0.2), 0.5(0.1), 33.5(0.2). These constants differ significantly from the previously reported pseudo-monoclinic constants that were based on velocity measurements on polysynthetic twinned crystal aggregates. Differences are consistent with systematic errors in the earlier study associated with sparse data and the presence of cracks and other imperfections.     

Effects of carnallite content on mechanical properties of Maha Sarakham rock salt

Mechanical characterization tests are performed to determine the effects of carnallite contents on the strength, elasticity, and time-dependent parameters of rock salt specimens obtained from the Lower Member of the Maha Sarakham formation. The specimens are prepared with carnallite content (C _%) varying from 0% (pure halite) to 100% (pure carnallite). The compressive and tensile strengths and elastic moduli of the specimens exponentially decrease with increasing C _%. Specimens with higher C _% tend to dilate more than those with lower C _%, as evidenced by the increasing of the Poisson’s ratio. The strength reduction due to the carnallite content decreases as the confining pressures increase. The elastic, visco-elastic, and visco-plastic parameters of the creep test specimens are defined as a function of C _%. They rapidly decrease with increasing C _%. Pure halite tends to behave as the Burgers material while pure carnallite behaves as the Maxwell material. The different creep rates and deformation mechanisms between halite and carnallite explain the occurrences and structures of potash ore deposit at the shelves and flanks of the salt domes and anticlines in the salt basin.     

Elasticity of CaTiO3, SrTiO3 and BaTiO3 perovskites up to 3.0 Gpa: the effect of crystallographic structure

Elasticity of CaTiO₃, SrTiO₃ and BaTiO₃ perovskites has been experimentally investigated as a function of pressure up to 3.0 GPa in a liquid-medium piston cylinder apparatus using a high precision ultrasonic interferometric technique. Specimens used are hot-pressed fine-grained (3–10 μm) polycrystalline aggregates with low porosity (<1.5%). Compressional and shear wave velocities and their pressure derivatives have been measured. The results are compared with previous studies on other perovskites and the role of structural transitions is examined. We find that the role of Ti-O₆ polyhedral tilting (such as observed in CaTiO₃) is small in the sense that a single well-defined general trend exists in perovskites with a wide range of tilting angles, although there is suggestion that cubic perovskites have slightly higher bulk modulus than orthorhombic perovskites. In contrast, cation-anion displacement that changes crystal symmetry from cubic to tetragonal in BaTiO₃ has very large effects on elasticity. This distortion significantly reduces the bulk modulus (but not much the shear modulus) and results in an unusually large pressure derivative of bulk modulus (dK/dP～10). A large change in elasticity in BaTiO₃ associated with the structural transition (without a significant volume change) is a clear example of the breakdown of the Birch's law between densities and elastic wave velocities.