Molecular orbital study on the onset of the pressure-induced transformations of the metastably compressed germanates

In order to elucidate the onset of the pressure-induced transformations of germanates consisting of GeO₄ tetrahedra at room temperature, we have investigated the stability of the crystal structures near the transition pressure in terms of the stability of the linkage of the tetrahedra. The stability of the linkage near the transition pressure is estimated from the results of the molecular orbital calculations for the model cluster H₆Ge₂O₇, which mimics the linkage of tetrahedra, as functions of the bond length d(Ge–O_br) with bridging oxygen and bond angles Ge–O_br–Ge. The calculation shows that the total energy of the linkage becomes minimum at d(Ge–O_br)=1.758 Å and br–Ge=130.4°, and that it increases with the deviation from the energy minimum geometry. From the compression behavior of framework and chain germanates, we find that the linkage of the tetrahedra becomes unstable with increasing pressure, and that these germanates commonly transform into their high-pressure phases when the linkage of the tetrahedra largely departs from the energy minimum geometry. This suggests that the high-pressure transformations of the metastably compressed germanates are induced by the instability of the linkage of tetrahedra.     

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.     

Composition dependence of elasticity in aluminosilicate glasses

The elastic properties of two types of aluminosilicate (basaltic and rhyolitic) glasses have been studied using both Brillouin and Raman spectroscopy at ambient conditions. It has been found that the elastic moduli of the basaltic glasses decrease with increasing SiO₂ concentration. The shear moduli displayed the least dependence on SiO₂ content. The bulk moduli of the basaltic glasses strongly depend on the sum of the Q ³ and Q ⁴ anionic units. Among the modifiers, iron cations showed the strongest effect on the elastic properties of the rhyolitic glasses. For the elastic moduli of rhyolitic glasses, the major effect of alkaline earth cations is on shear modulus; however, both iron and alkali cations showed stronger effects on bulk modulus and similar relative contribution between bulk and shear moduli (based on the equivalent M⁺ cation). The dependences of elastic moduli on bulk NBO/T observed in both types of glasses suggest that the elastic modulus of an aluminosilicate glass depends on the concentration of effective modifying cations rather than the apparent concentration of all non-network-forming cations. An analysis of data also indicated that the ideal molar mixing model is failed in prediction of the elastic properties of the present multicomponent glasses by using the known parameters.     

Application of empirical ionic models to SiO2 liquid: Potential model approximations and integration of SiO2 polymorph data

Structural and thermodynamic properties of crystalline SiO₂ and SiO₂ liquid have been examined with Monte Carlo (MC), molecular dynamics (MD), and energy minimization (EM) calculations using several ionic potential models obtained from the literature. The MC and MD methods calculate the same structural and thermodynamic properties for liquids when the same potential model is used. The Ewald (1921) method of calculating coulomb interactions reproduced most successfully the structure of liquid silica. Approximating the coulomb interaction by eliminating the inverse lattice sum results in predicted bond distances that are too short and an average 〈Si-O-Si〉 angle of approximately 180°. Introduction of a cut-off in the potential energy function produces irregular tetrahedra and inconsistencies in predicted Si-O coordination in silica liquid. The system internal energies show that liquid structures derived from random starting configurations can be metastable relative to structures calculated from crystalline starting configurations.The static lattice properties of the polymorphs alpha-quartz, coesite, and stishovite were used to evaluate further the accuracy of different sets of repulsive parameters for the full Ewald ionic model. Most of the models studied reproduced poorly the measured structures and elastic constants of the polymorphs. The major weakness of the ionic model is the unreasonably large Si-O bond strength (120 × 10⁻¹² ergs/bond) when formal ionic charges are used. Fractional charge models with a small Si-O bond strength (30 × 10⁻¹² ergs/bond) improve the agreement with experimental data. However, further improvement of the ionic model should include reducing the Si-O bond strength to values in better agreement with published estimates (7 × 10^{− 12} to 13 × 10⁻¹² ergs/bond). By using additional information to constrain the parameterization of the ionic model, such as estimated bond strengths and static properties of the silica polymorphs, a model more representative of the interparticle interactions may be obtained.     

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 a beryllium silicate (phenacite: Be2SiO4)

The adiabatic single-crystal elastic moduli of a beryllium silicate (phenacite: Be₂SiO₄, trigonal, have been determined at atmospheric pressure and 22° C by Brillouin spectroscopy. The elastic stiffness moduli in gigapascals are: C ₁₁=341.9 C ₃₃=391.0 C ₄₄=91.4 C ₆₆= 96.9 C ₁₂=148.0 C ₁₃=136.0 C ₁₄= 0.1 C ₁₅= 3.5Overall, the elastic stiffness moduli for phenacite parallel and perpendicular to the c axis are comparable (i.e., it is almost cubic in its elastic signature). The elastic moduli can be rationalized in terms of division of the structure into two types of coordination polyhedra (1Si+2Be) with slightly different stiffnesses, which are linked to form a three dimensional framework. Values of the isothermal bulk modulus and the linear compressibilities, as determined from hydrostatic compression experiments of Hazen and Au (1986), are in good agreement with those obtained here. Combining the two studies indicates a low pressure derivative of the bulk modulus for phenacite.     

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.     

High temperature elasticity of rutile-structure MgF2

The elastic moduli of single-crystal MgF₂ have been determined by the ultrasonic pulse superposition technique as a function of temperature from T=298?650° K. These new data are consistent with those obtained by other ultrasonic pulse techniques at and below room temperature and agree favourably with polycrystalline data above room temperature. The elastic moduli (c) are represented by quadratic functions in T over the experimental temperature range with the curvature in the same sense for all the moduli. For the rutile-structure fluorides and oxides, evaluation of the temperature derivatives of the elastic moduli at constant volume indicates that the dominant temperature effect is extrinsic for (?K _S /?T) _P and intrinsic for (?μ/?T) _P , where K _S and μ are the isotropic bulk and shear moduli, respectively. There appears to be no simple relationship between (?c/?T) _P and crystallographic parameters for the rutile structure, and |(?c/?T) _P | for the fluorides is in general very much lower than the corresponding |(?c/?T) _P | for the oxides. For the pair of compounds MgF₂-TiO₂, there is no evident analogue relationship for high-temperature elastic properties.     

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.     

SCF-Xα-SW MO Study of Fe-O and Fe-OH chemical bonds; applications to the mössbauer spectra and magnetochemistry of hydroxyl-bearing Fe3+ oxides and silicates

Results of SCF-Xα-SW molecular orbital calculations on (FeO₄(OH)₂)^7? and (FeO₆)^9? clusters are used to investigate the differences between Fe-O and Fe-OH bonding in hydroxyl-bearing iron oxides and silicates. The Fe³⁺-OH^? bond is more ionic, and has a smaller spinpolarization, then the Fe³⁺-O^2? bond. The smaller spinpolarizability of OH^? ligands explains why superexchange interactions between hydroxo-bridged Fe³⁺ cations are much weaker than those between oxo-bridged Fe³⁺ cations. Replacement of oxygens in the Fe³⁺ coordination environment by OH^? ligands appears to promote the covalency between Fe³⁺ centers and O^2? oxygens. The increased covalency lowers the effective spin of the Fe atom. This, in turn, explains the decreased magnetic hyperfine fields at the Fe nucleus in FeOOH polymorphs relative to those found in Fe³⁺ oxides.     

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.     

Electronic and magnetic structure of vivianite: cluster molecular orbital calculations

The electronic and magnetic structure of the octahydrophosphate vivianite, Fe₃(PO₄)₂·8H₂O, has been investigated by cluster molecular orbital calculations in local spin density approximation. Optical and Mössbauer spectra are well reproduced by the calculations, and the differences between the two iron sites can be correlated with differences in the geometrical structure of the first coordination sphere. The spin structure within the crystallographic ac plane is derived and explained on the basis of different superexchange pathways via edges of the phosphate tetrahedra. The calculations demonstrate that quite large clusters (up to 118 atoms) are necessary to arrive at reliable results.     

Variational stabilization of the ionic charge densities in the electron-gas theory of crystals: Applications to MgO and CaO

The electron-gas theory of crystals is extended to include the effects of many-body forces that arise from both electrostatic and overlap interactions. These effects are incorporated through a self-consistent spherical relaxation of the ionic charge distributions such that the crystal binding energy is minimized. This variational model is used to compute the elastic constants and equations of state of MgO and CaO, and we compare its results to those derived from earlier electron-gas models. In the variational model, the anion charge distributions are markedly more sensitive to the local crystal environment than they are in the PIB or other electron-gas models. We find that for these oxides the variational model gives the best overall agreement with experiment for lattice constants, equations of state, dissociation energies, and elastic moduli.     

Kinetics of the Reactions of Water, Hydroxide Ion and Sulfide Species with CO2, OCS and CS2: Frontier Molecular Orbital Considerations

The kinetics of the reactions of water, hydroxide ion and sulfide species with CO₂, OCS and CS₂ are investigated using the molecular orbital approach and available kinetic data. Although these reactions are symmetry allowed, the lowest unoccupied molecular orbital (LUMO) for CO₂ is a poor electron accepting orbital as it has a positive potential energy. At low pH, hydration of CO₂ requires that the waters interact with CO₂ via hydrogen bonding for subsequent formation of H₂CO₃ in an effort to overcome the high energy of activation. These factors are significant for the slow kinetics of hydration and the persistence of CO₂ in water. The reaction of hydroxide ion with CO₂ has a much smaller energy of activation. For the isoelectronic species OCS and CS₂, their LUMO orbitals are good electron acceptor orbitals, and the energy of activation is less than that for the corresponding CO₂ reactions. The LUMO orbitals for OCS and CS₂ have less carbon character whereas the LUMO for CO₂ has more carbon character. The relative rates of these reactions (CO₂ > OCS > CS₂) reflect the increased carbon character of the π^* LUMO orbital for CO₂ over CS₂ and the fact that the LUMO for OCS is σ^*, which when filled can readily break the C—S bond leading to sulfide (even though the C character of the LUMO is less than those for CO₂ and CS₂). Also, the higher hydrogen bonding interactions with nearest water molecules is in the order CO₂ > OCS > CS₂ indicating that hydrolysis via water catalysis is retarded as the number of S atoms increases. Solid phase FeS has a highest occupied molecular orbital (HOMO) with a potential energy similar to that of CO₂ and can activate (or bond with) the carbon atom in CO₂ so that organic compounds can be produced under hydrothermal vent conditions.     

Polymerization of silicate and aluminate tetrahedra in glasses,melts and aqueous solutions—II. The network modifying effects of Mg2+ , K+, Na+, Li+, H+, OH−, F−, Cl−, H2O,CO2 and H3O+ on silicate polymers

The effect of the group IA and VIIA ions, as well as Mg²⁺, and the molecules H₂O, CO₂, H₃O⁺ and OH^? on the energy of the Si-O bond in a H₆Si₂O₇ cluster has been calculated using semiempirical molecular orbital calculations (CNDO/2). Three types of elementary processes, i.e. substitution, addition, and polymerization reactions have been used to interpret data on the dynamic viscosity, surface tension and surface charge, hydrolytic weakening, diffusivity, conductivity, freezing point depression, and degree of polymerization of silicates in melts, glasses, and aqueous solutions. As a test of our calculational procedure, observed X-ray emission spectra of binary alkali silicate glasses were compared with calculated electronic spectra. The well known bondlength variations between the bridging bond [Si-O(br)] and the non-bridging bond [Si-O(nbr)] in alkali silicates are shown to be due to the propagation of oscillating bond-energy patterns through the silica framework. A kinetic interpretation of some results of our calculations is given in terms of the Bell-Evans-Polanyi reaction principle.     

Compressibility and crystal structure of sillimanite, Al2SiO5, at high pressure

The unit-cell dimensions and crystal structure of sillimanite at various pressures up to 5.29 GPa have been refined from single-crystal X-ray diffraction data. As pressure increases, a and b decrease linearly, whereas c decreases nonlinearly with a slightly positive curvature. The axial compression ratios at room pressure are β_a:β_b:β_c=1.22:1.63:1.00. Sillimanite exhibits the least compressibility along c, but the least thermal expansivity along a (Skinner et al. 1961; Winter and Ghose 1979). The bulk modulus of sillimanite is 171(1) GPa with K′=4 (3), larger than that of andalusite (151 GPa), but smaller than that of kyanite (193 GPa). The bulk moduli of the [Al1O₆], [Al2O₄], and [SiO₄] polyhedra are 162(8), 269(33), and 367(89) GPa, respectively. Comparison of high-pressure data for Al₂SiO₅ polymorphs reveals that the [SiO₄] tetrahedra are the most rigid units in all these polymorphic structures, whereas the [AlO₆] octahedra are most compressible. Furthermore, [AlO₆] octahedral compressibilities decrease from kyanite to sillimanite, to andalusite, the same order as their bulk moduli, suggesting that [AlO₆] octahedra control the compression of the Al₂SiO₅ polymorphs. The compression of the [Al1O₆] octahedron in sillimanite is anisotropic with the longest Al1-OD bond shortening by ～1.9% between room pressure and 5.29 GPa and the shortest Al1-OB bond by only 0.3%. The compression anisotropy of sillimanite is primarily a consequence of its topological anisotropy, coupled with the compression anisotropy of the Al-O bonds within the [Al1O₆] octahedron.     

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.