Hydrostatic Pressure Derivatives Research Articles

Effects of changes in volume and c/a ratio on the pressure derivatives of the elastic moduli of H.C.P. Ti and Zr

The hydrostatic pressure derivatives of the single crystal elastic moduli of Ti have been measured to 5·5 Kbar. The pressure derivatives are dC 11IdP = 5·01, dC 33/ dP = 4·88, d C 44/d P = 0·52, d C 66/d P = 0-45, d C 12 = 4·11, and d C 13/d P = 4·05. The positive value for dC 44/d P is in sharp contrast to d C 44/d P < 0 for Zr. This difference is assumed to arise from the difference between d( c/a)/d P in the two h.c.p. crystals and quantitative values of d C 44/d V and d C IJ/d( c/a) are calculated. It is then shown that the large differences between the average Grüneisen mode γ H calculated from d C IJ/d P and that obtained from thermal expansion data for both Ti and Zr can be explained by the differences between d( c/a)/d V under hydrostatic pressure and during thermal expansion, respectively. The relatively large negative value for d C 44/d( c/a) is quantitatively consistent with Cousins' calculations of the dependence of the electrostatic contribution to C4, on the cla ratio in any h.c.p. metal lattice.

Effect of Axial Ratio Changes on the Elastic Moduli and Grüneisen γ for Lower Symmetry Crystals

Gerlich has shown that Sheard's model for calculating mode γ's from hydrostatic pressure derivatives of the elastic moduli of hcp Mg and Cd yields Gruneisen γ's at both high and low temperatures that are in good agreement with the γ's derived from thermal-expansion measurements. For hcp Ti and Zr, however, large differences arise, primarily from very small values for dC44/dP. It is proposed that these small values are caused by the changes in c/a ratio with hydrostatic pressure because of a large dependence of C44 on the c/a ratio. The disagreement with thermal-expansion data can be removed by taking into account the difference in d(c/a)/dV between hydrostatic-pressure and thermal-expansion conditions. The effect of Δ(c/a) is not found in tetragonal TiO2, rutile, where γ̄H is in excellent agreement with the thermal expansion γ∞.

Hydrostatic Pressure Derivatives of the Single-Crystal Elastic Moduli of Zirconium

The adiabatic elastic moduli of single-crystal Zr vary linearly with hydrostatic pressures up to 4.7 kbar. The pressure derivatives are dC11/dP=3.93, dC33/dP=5.49, dC44/dP=−0.22, dC66/dP=0.26, dC12/dP=3.42, and dC13/dP=4.25. The average high-temperature Gruneisen mode γ̄H calculated from this data is in wide disagreement with that calculated from the volume thermal expansion coefficient. It is proposed that this disagreement arises because the negative dC44/dP is caused by the change with pressure in the c/a ratio, rather than the volume change.

Third-Order Elastic Constants of Aluminum,

The complete set of six third-order elastic constants of single-crystal Al has been experimentally determined by measuring both hydrostatic-pressure and uniaxial-stress derivatives of the natural sound velocities using a two-specimen interferometric technique. The specimens were neutron-irradiated to eliminate dislocation effects from the uniaxial experiments. A self-consistent set of hydrostatic-pressure derivatives of the second-order elastic constants has been calculated from the measured third-order constants. The third-order elastic constants have also been used to calculate the thermal expansion in the anisotropic-continuum model at both high and low temperatures, and a comparison has been made with the directly measured expansion coefficients.

Third-Order Elastic Constants of Aluminum

The complete set of six third-order elastic constants of single-crystal Al has been experimentally determined by measuring both hydrostatic-pressure and uniaxial-stress derivatives of the natural sound velocities using a two-specimen interferometric technique. The specimens were neutron-irradiated to eliminate dislocation effects from the uniaxial experiments. A self-consistent set of hydrostatic-pressure derivatives of the second-order elastic constants has been calculated from the measured third-order constants. The third-order elastic constants have also been used to calculate the thermal expansion in the anisotropic-continuum model at both high and low temperatures, and a comparison has been made with the directly measured expansion coefficients.

Elastic Moduli of Gallium Antimonide under Pressure and the Evaluation of Compression to 80 kbar

Ultrasonic wave velocities in GaSb have been measured at 25°C as a function of hydrostatic pressure to 2 kbar. The adiabatic ``zero-field'' stiffness moduli calculated from these velocities (for 1 atm) are as follows: c11 = 8.839±0.008, c12 = 4.033±0.010, c44 = 4.316±0.004, and Bs (bulk modulus) = 5.635±.012. All moduli are in units of 1011 dyn/cm2 and are based on a density of 5.6137 g/cm3 taken as exact with respect to estimates of errors. Hydrostatic pressure derivatives for c11, c12, c44, and Bs are, respectively, 4.96±0.10, 4.64±0.15, 1.01±0.02, and 4.75±0.15. The static compression of GaSb to 80 kbar has been evaluated using the isothermal bulk modulus (BT) of 5.614×1011 dyn/cm2 and (∂BT/∂P) of 4.78.

Elastic Moduli of GaAs at Moderate Pressures and the Evaluation of Compression to 250 kbar

Ultrasonic wave velocities in GaAs have been measured at 25°C as a function of hydrostatic pressure. The adiabatic ``zero-electric-field'' stiffness moduli calculated from these velocities are as follows: c11=1.1877± 0.0006, c12=0.5372±0.0009, c44=0.5944±0.0003. All moduli are in units of 1012 dyn/cm2 and are based on a density of 5.3169 g/cm3 taken as exact with respect to estimates of errors. Hydrostatic pressure derivatives for c11, c12, and c44, are, respectively, 4.63±0.03, 4.42±0.05, and 1.10±0.02. From the isothermal bulk modulus (BT) of 746.6 kbar and its pressure derivative (∂BT/∂P) of 4.67 calculated using the above data, the static compression of GaAs has been computed to a pressure of 250 kbar.