Third-order elastic constants and pressure derivatives of the second-order elastic constants of β-tin

The second- and third-order elastic constants and pressure derivatives of second-order elastic constants of tetragonal β-tin have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second- and third-order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The 12 non-vanishing third-order elastic constants and the six pressure derivatives of the second-order elastic constants in tetragonal β-tin are obtained in the present work and are compared with the available experimental values. The second-order elastic constant C₃₃ obtained in the present study is in reasonable agreement with the experimental values. The third-order elastic constants are generally one order of magnitude greater than the second-order elastic constants as expected of a crystalline solid. The third-order elastic constant C₃₃₃ is higher in magnitude than all other values. This shows a greater anharmonicity of β-tin along the c-axis direction of the crystal.     

High pressure of solid noble gases in metals

The fact that noble gas ions implanted in metals with concentrations of a few percent precipitate at room temperature as solid bubbles under high pressure with a structure epitaxial with the host matrix (fcc or hcp) has been investigated recently. A brief survey of the results measured by transmission electron microscopy, glancing angle X-ray diffraction, energy-loss spectroscopy and positron annihilation is given. The main part of this paper deals with a study on⁸³Kr⁺ implanted in Al using Mössbauer spectroscopy. The results are as follows: the precipitates grow from 1.5 nm after implantation to 3.7 nm after 700 K annealing. The recoilless fractions can be fitted very well to a Debye model. On annealing. the characteristic Mössbauer temperature gradually decreases from 89 K to 64 K, the value for solid Kr at 1 bar. At the same time a decrease in the isomer shift is observed. Even after annealing at 700 K our experiments do not show melting in the temperature range 0–230 K.     

Equation of state of LiCoO_2 under 30 GPa pressure

LiCoO_2 is one of the most important cathode materials for high energy density lithium ion batteries. The compressed behavior of LiCoO_2 under high pressure has been investigated using synchrotron radiation x-ray diffraction. It is found that LiCoO_2 maintains hexagonal symmetry up to the maximum pressure of 30.1 GPa without phase transition. The elastic modulus at ambient pressure is 159.5(2.2) GPa and its first derivative is 3.92(0.23). In addition, the high-pressure compression behavior of LiCoO_2 has been studied by first principles calculations. The derived bulk modulus of LiCoO_2 is 141.6 GPa.     

The second and third order elastic constants of alkali metals

The second and third order elastic constants of the alkali metals have been calculated on the long wave method using the Heine-Abarenkov lacal model potential with different exchange-correlation corrections. It is found that the use of an exchange correlation correction which satisfies the compressibility sum rule leads to a good agreement between the calculated and measured second order elastic constants of the alkali metals Na, K, Rb and Cs. The shear elastic constants however come out correct even if the compressibility sum rule is violated by the exchange-correlation correction. The third order elastic constants and the pressure derivatives of the second order elastic constants and the pressure derivatives of the second order elastic constants calculated on the HA local potential are lower than the experimental values at room temperature. The discrepancy is pronounced for the heavier alkali metals. Similar calculations using the Wallace potential for Li, Na and K and the Schneider-Stoll potential for Rb give the pressure derivative in good agreement with experiment. In view of the important role by the exchange correlation correction, Suzuki's results calculated without taking this correction into account can only be accepted with some reservation.     

Equation of state and electronic structure of solid CaO under high pressure

The equations of state (EOSs) and the electronic structures of solid CaO both in the B1 and in the B2 structure have been calculated by using the local-density functional theory with the APW method in the muffin-tin approximation. The EOS of the B1 phase is in excellent agreement with the diamond-anvil-cell experiments. The metallization at T = 0 K is shown to occur in the B2 phase at a pressure of 480 GPa. By using the Debye temperature and the Grüneisen constant, both of which are theoretically determined, the hugoniot has also been calculated. The agreement with experiment is good for the B1 phase and is moderate for the B2 phase.     

Study of the elastic stiffness constants of thiourea by Brillouin scattering

All the elastic stiffness constants of thiourea have been measured at room temperature. The value of C₆₆(0.7 × 10¹⁰dynes/cm²) is very low. The C₁₁ and C₃₃ constants exhibit discontinuities at 169 and 202 K. The longitudinal acoustic phonons propagating in the c direction are very strongly damped in the vicinity of 202 K.     

Calculated elastic constants of dilute alloys based on bcc metals

The elastic constants of dilute alloys based on bcc metals have been calculated using the Green’s function method obtaining explicit expressions for change in elastic constants in terms oft-matrix. The crystal impurity problem is discussed within an impurity model containing central and non-central force constant changes extended up to second neighbours of the impurity. The effect of volume change on elastic constants and a contribution from electron pressure term are considered. Numerical results for changes in three elastic moduli have been presented for a number of dilute alloys based on Mo, Nb, W, Ta and V.     

Condensed state physics elastic constants of alkali metal alloys

The second-and third-order Brøgger elastic constants of the disordered Cs-K, Cs-Rb, and Rb-K alloys with different concentrations of the second component are calculated for the pseudopotential model with the use of the Krasko-Gurskii model potential. Results of calculations are in good qualitative agreement with the available experimental data on the elastic constants of alloys.     

A model analysis of the elastic constants of metals and alloy phases

A lattice energy function using a Mie inverse power interatomic potential and a quadrupole distortion term was fitted to the cohesive energy, lattice parameter, bulk modulus and C₄₄ of pure metals. Using this function and the fitted parameters, values were then predicted for C₄₄ and other elastic constants of alloy phases. It was found that the maximum discrepancy in C₄₄ for B2, C15, and L1₂ type phases was 40%, with the RMS value equal to 25%.It was found that the predicted nearest-neighbor bond angle dependent elastic constants were always smaller than the observed values. It is believed that this result is due to the formation of eovalent bonds in the alloy phases, a phenomenon that is not included in the model. An empirical procedure for correcting the elastic constants to account for eovalent bonding reduces the maximum discrepancy in both shear constants to less than 20%. Evidence is presented in support of this hypothesis. One of the consequences of this work is that ground state superstructures, of heteroelectronic type alloy components that are deduced on the basis of constancy of pair or many body interactions, need not necessarily correspond to the most stable superstructures.     

Equation of state and elastic properties of lithium: Isotope effects

The equations of state and elastic properties of the lithium isotopes Li⁶ and Li⁷ at high pressures are investigated at a temperature of 76 K by a pulsed ultrasonic method. Proofs of the existence of appreciable quantum contributions to the pressure and elastic moduli of lithium are given. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 1, 36–40 (10 January 1999)     

Temperature and pressure derivatives of the elastic constants of cubic rubidium cyanide

The temperature and pressure derivatives of all elastic constants of rubidium cyanide have been measured from 133 to 380 K respectively from 0 to 1500 × 10⁵ Nm^-2 by ultrasonic methods. The thermoelastic behaviour resembles that of KCN and NaCN. A strong softening of the shear resistance c₄₄ approaching the transition temperature from higher temperatures is observed. The pressure derivatives are also quite similar to those of KCN and NaCN, but fully different from those of normal alkali halides of rocksalt-type. This behaviour confirms a rule already observed in other isotypic crystal groups: the quasi-invariant pressure derivatives are shifted in a characteristic way for a certain structure type, if rare-gas-like ions are replaced by asymmetric ions. The nonlinear elastic behaviour is qualitatively interpreted by interactions of volume-conserving type as existing in fluids.     

Equation of state for solids under high pressure,with application to alkali halides

A new equation of state has been derived from an analysis of the relations between energy, pressure and volume combined with a Padé approximation for the pressure. This modified equation of state is first tested by comparison with the exact expressions that are obtained for four-potential models, the Lennard-Jones, the Morse, the Born-Mayer, and the Varshni-Shukla potentials, with excellent results. It is also compared with experimental high-pressure data for four alkali halide crystals with quite good results.     

The elastic constants and their temperature and pressure derivatives of AgBr-AgCl mixed crystals

The three independent second-order elastic constants and their temperature and pressure derivatives have been measured for four AgBr-AgCl mixed crystals, with 19.5, 39.1, 56.6 and 78.7 mole % AgCl, using the ultrasonic pulse-echo technique at room temperature. The explicit temperature dependence of the elastic constants is calculated and is found to be much larger than that of other NaCl structure crystals. The violation of the Cauchy relation C₁₂ = C₄₄ is found to be significant and increases between AgBr and AgCl. The high temperature limit of the Gruneisen parameter is calculated from the elastic data. A comparison is made between the elastic properties of the silver halides and the alkali halides.