Electronic contributions to shear constants and susceptibility in cubic transition metals

We have calculated the electronic contributions to the shear constants C = c₄₄ and C' = (c₁₁ ? c₁₂)/2 of cubic transition metals, tight binding Hamiltonians and their derivatives with respect to shear, and thermodynamical perturbation theory. The results obtained for bare susceptibility and temperature derivatives of the shear constants on the basis of a band structure proposed by Posternak and Steinemann are in reasonable agreement with experiment.     

Low temperature elastic constants and Debye temperature of NbO2

The transit times of ultrasonic waves have been measured in single crystal NbO₂ from 295 K down to 1.5 K for quasilongitudinal and shear waves propagating in the [100] direction and down to 160 K for eight other waves. Values are obtained for the C₄₄ elastic constant and for an elastic constant combination which is approximately equal to C₁₁ for temperatures down to 1.5 K and for C₁₁, C₁₂, C₁₃, C₁₆, C₃₃, and C₆₆ down to 160 K. These results are used to deduce 0 K values for the elastic constants and an elastic Debye temperature of 596 ± 7 K at 1.5 K. The acoustic mode heat capacity calculated from the latter is significantly smaller than the heat capacity measured by Wenger and Keesom at low temperatures. Following Wenger and Keesom, the difference is attributed to phasons (excitations involving the phase modulation of charge density waves). An average velocity is deduced for the phasons.     

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.     

Elastic constants of nionium up to the melting point

The elastic constants of a pure niobium single crystal have been measured between 1300 and 2600 K by means of an ultrasonic method and a suitable geometry. We find that the shear constant C₄₄ behaves anomalously up to the melting point and that $\text{1}\text{2}$(C₁₁–C₁₂) decreases monotonically. The anomalous behaviour of C₄₄ associated with the trigonal distortion is explained as an electronic contribution with a (local) degeneracy temperature which is considerably higher than the melting point.     

Stiffness matrix and debye temperature of ReO3 from ultrasonic measurements

We have measured the propagation velocities of bulk acoustic waves in the simple cubic transition-metal oxide ReO₃ by ultrasonic pulse propagation. The elastic stiffness constants at 300 K are: C₁₁ = (47.9 ± 1.4) × 10¹¹ dyne/cm²; C₄₄ = (6.1 ± 0.2) × 10¹¹ dyne/cm²; C₁₂ = (?0.7 ± 2.8) × 10¹¹ dyne/cm². These elastic constants indicate a crystal with highly anisotropic shear propagation. The Debye temperature of the compound from these measurements is 528 K. This value is somewhat higher than previous results from specific heat and resistivity determinations.     

Investigation of elastic moduli and constants of zinc-blende Al<Subscript><Emphasis Type="Italic">y</Emphasis></Subscript>Ga<Subscript>1-<Emphasis Type="Italic">y</Emphasis></Subscript>As alloy by statistical moment method

The elastic moduli and elastic constants of the ternary semiconductor alloy Al _y Ga_1-yAs at finite temperature have been investigated using the statistical moment method. The Young, shear, bulk moduli and elastic constants C₁₁, C₁₂, C₄₄ of the zinc-blende Al _y Ga_1?yAs crystal are calculated as functions of Al composition and temperature. Numerical calculations have been performed and compared with those of the experimental and other theoretical results showing the reasonable agreements. Our study shows that elastic moduli and C₁₁, C₁₂ constants of zinc-blende Al _y Ga_1?yAs alloy are decreasing functions of the temperature and Al composition; C₄₄ constant is a decreasing function of the Al composition.     

Mode softening in CeAl2

Measurements of elastic constants C₁₁ and C₄₄ are reported between 1.5 and 280 K for CeAl₂. With decreasing temperature a softening of the C₄₄ elastic constant of 30% is observed. The softening is discussed in terms of a magnetoelastic interaction.     

The elastic behaviour of the ternary zincblende structure semiconductor CuGe2P3

A single crystal has been grown of CuGe₂P₃, a ternary semiconductor with a zincblende structure in which the copper and germanium atoms are randomly distributed on the cation sites. The second order elastic constants measured by the ultrasonic pulse echo overlap technique (C₁₁ = 13.66, C₁₂ = 6.17, C₄₄ = 6.66 and bulk modulus B = 8.67 in units of 10¹⁰Nm^?2 at 291 K) are closely similar to those of GaP and conform well to Keyes' correlation for zincblende structure crystals. The hydrostatic pressure derivatives of the second order elastic constants ($\text{?C}{}_{11}\text{?P}\text{= 4.40}$, $\text{?C}{}_{12}\text{?P}\text{= 3.9}$, $\text{?C}{}_{44}\text{?P}\text{= 0.93}$ and $\text{?B}\text{?P}\text{= 4.07}$) and the third order elastic constants (C₁₁₁ = ? 8.45, C₁₁₂ = ? 3.49, C₁₂₃ = ? 1.13, C₁₄₄ = ? 2.82, C₁₅₅ = ? 1.44 and C₄₅₆ = ? 0.69 in units of 10¹¹Nm^?2) also resemble those of GaP: even the anharmonicity of the long wavelength acoustic modes of this ternary semiconductor resembles that of its binary “parent” compound. The positive signs of the hydrostatic pressure derivatives and the negative signs of the third order elastic constants show that the acoustic modes at the long wavelength limit stiffen under pressure, an effect which is quantified here by computation of the mode Grüneisen parameters, which are all positive. The harmonic and anharmonic force constants, obtained in the valence force field model, fit well into the pattern shown by related zincblende structure compounds: the ratio ($\text{β}\text{α}$) for bond bending (β) to stretching (α) force constants is greater than 4:1; the dominating anharmonic force constant is γ: most of the anharmonicity is associated with nearest neighbour atoms. Analysis of the temperature dependence of the elastic constants on the basis of a simple anharmonic model again emphasises the similarity between the elastic behaviour of CuGe₂P₃ and GaP. The thermal expansion of CuGe₂P₃ varies almost linearly with temperature between 291 K and 1000 K, the linear coefficient of thermal expansion α being 8.21 ± 0.02 × 10^?6 °C^?1. The similar lattice parameters and elastic behaviour of CuGe₂P₃ and GaP indicate that semiconducting devices made of epitaxial deposits of CuGe₂P₃ on a GaP substrate should prove to be almost strain-free.     

Elastic constants of galena down to liquid helium temperatures

The elastic constants of single crystal galena have been determined from the measured ultrasonic velocities down to liquid helium temperature. A cryostat incorporating an arrangement to inject the liquid bonding material at low temperature is described. At 5 K, the values of elastic constants are C₁₁=14.9₀, C₁₂=3.5₁ and C₄₄=2.9₂×10¹⁰ N/m².     

Study of first principles on anisotropy and elastic constants of Y3Al2 compound

In this work primarily structural parameters of the Y₃Al₂ was optimized. In the calculations, PBE type Exchange-Correlation function was selected in the GGA approximation and ultra-soft pseudopotential was used. Crystalline lattice parameters a = 15.581 Bohr, c / a = 0.923. Total energy was calculated by small deformations in volume. These values were fitted to the equation of state and the bulk modulus was calculated as 58.4 GPa. The working structure has elastic constants as C₁₁, C₁₂, C₁₃, C₃₃, C₄₄ and C₆₆. These constants have been calculated under ambient pressure. Using the elastic constants, bulk, shear, Young modulus and Poisson ratio (in Voigt approach) were obtained as 58.12, 34.31, 86.01 GPa and 0.25, respectively. Melting temperature of the material was guessed from the elastic constant. The results were compared with the current experimental and theoretical data.     

Hole contribution to the elastic constants of SnTe

Ultrasonic wave velocities have been measured in SnTe single crystals with hole concentrations of 1.0 and 4.5 × 10²⁰/cm³. The shear elastic stiffness constant C₄₄ is sensitive to the hole concentration but $\text{1}\text{2}\text{(C}{}_{11}\text{? C}{}_{12}\text{)}$ is not, a result which is consistent with the valence band pockets being sited at the L points. The non-ellipsoidal, non-parabolic multivalley band model has been used to calculate the hole contribution to the elastic constants. The calculated difference between the shear constant C₄₄ (2.78 × 10¹⁰ dyne cm^-2) for the two crystals is in agreement with that measured experimentally (2.67 × 10¹⁰ dyne cm^-2). The shear deformation potential constant E_u for the SnTe valence band is 7.8 eV at 293°K.     

Anomalous temperature behaviour of the shear elastic constant C44 in vanadium

The adiabatic elastic constants of a pure vanadium single crystal have been measured between 300 and 2100 K in a suitable geometry by means of an ultrasonic method. We find that the shear constant C₄₄ has an anomalous temperature behaviour very similar to that found in niobium.     

Ultrasound studies of single crystal thulium in an applied magnetic field

The paper reports the first measurements of the single crystal elastic constants of the heavy rare earth metal thulium as a function of temperature and magnetic field. The constants were obtained from ultrasonic velocity measurements over a temperature range of 4.2–296 K and in applied magnetic fields of up to 5 T. The elastic constants; C₁₁, C₃₃, C₄₄ and C₆₆=(C₁₁–C₁₂)/2 were determined from the ultrasonic velocities. Anomalies in the elastic constants were observed at 58 K from the c-axis propagated shear wave measurements and at 55 K from the c-axis propagated longitudinal wave measurements. Significant softening of the elastic constants C₃₃ and C₄₄ was observed close to T_N. Application of a magnetic field (>2 T) along the c-axis direction induced further softening of the material. Electromagnetic acoustic transducers (EMATs) were also employed in addition to conventional piezoelectric quartz transducers. A marked increase in the EMATs acoustic coupling efficiency (generation and detection efficiency) occurred close to T_N.     

The elastic behaviour of tetrahedral materials with vacant sites

As part of a study of defect tetrahedral structure compounds, the elastic constants of single crystal specimens of Hg₃Ga₂□Te₆ and HgIn₂□Te₄ have been measured between 77 K and room temperature using the pulse superposition technique. These compounds are included in the series HgTe, Hg₅In₂□Te₈, Hg₃In₂□Te₆ and HgIn₂□Te₄ and HgTe, Hg₅Ga₂□Te₈, Hg₃Ga₂□Te₆ where there is a progressive increase in the concentration of vacant sites. While the other compounds studied are cubic, HgIn₂□Te₄ has a tetragonal structure with a $\text{c}\text{a}$ ratio of 2.0. The components of the elastic stiffness tensor of this material at 77 K are (in units of 10¹¹ dyne cm^?2) C₁₁ = 4.31 C₁₂ = 2.54 C₄₄ = 2.14 C₃₃ = 4.47 C₁₃ = 2.18 C₆₆ = 2.41. In a cubic material C₁₁ = C₃₃, C₄₄ = C₆₆ and C₁₂ = C₁₃: the elastic behaviour of this tetragonal compound closely resembles that of a cubic material, as might be anticipated from its structure. This similarity is further illustrated by reference to the symmetry of phase velocity and Young's modulus surfaces. Examination of the elastic constants and reduced elastic constants of these compounds shows a regular trend, the elastic stiffness decreases as the number of vacant sites increases. There is an approximately linear relationship between the reduced bulk modulus and the number of sited vacancies.     

Study of elastic properties of CeO2 by statistical moment method

The purpose of the present paper is to investigate the temperature and pressure dependences of the elastic properties of cerium dioxide using the statistical moment method (SMM). The equation of states of bulk CeO₂ is derived from the Helmholtz free energy, and the pressure dependences of the elastic moduli like the bulk modulus, B_T, shear modulus, G, Young’s modulus, E, and elastic constants (C11, C12, and C44) are presented taking into account the anharmonicity effects of the thermal lattice vibrations. In the present study, the influence of temperature and pressure on the elastic moduli and elastic constants of CeO₂ has also been studied, using three different interatomic potentials. We compare the results of the present calculations with those of the previous theoretical calculations as well as with the available experiments.     

Brillouin scattering in cubic potassium cyanide

The temperature dependence of the elastic shear constant C₄₄ of cubic potassium cyanide has been measured by optical Brillouin spectroscopy in the temperature range from 300-168 K. A strong softening of the constant C₄₄ was found in agreement with the results of ultrasonic and neutron measurements.     

Elastic and elasto-optic constants of ammonium perchlorate

Using Brillouin scattering, the elastic and elasto-optic constants of ammonium perchlorate have been measured at room temperature. The polarized Brillouin scattering spectra yielded the nine elastic constants (in the units of $\text{10}{}^{11}\text{dynes}\text{cm}{}^2\text{)}$ as follows: C₁₁ = 2.51, C₂₂ = 2.46, C₃₃ = 3.15, C₄₄ = 0.66, C₅₅ = 0.47, C₆₆ = l.03, C₁₂ = 1.63, C₂₃ = 1.15, C₂₃ = 0.76; as well as the absolute value of nine elasto-optic constants. In the light of these results, some thermodynamic properties of the crystal are discussed.     

Velocity of hypersounds in lithium hydride by spontaneous brillouin scattering

The elastic constants of lithium hydride have been measured, at room temperature, by means of Brillouin scattering for frequencies of about 60 GHz. The results are C₁₁=6.71, C₁₂=1.7 andC₄₄=4.60 (units 10¹⁰Nm^-2). The values of C₁₁ and C₁₂ are slightly higher than that obtained for ultrasonic frequencies. The strong departure from the Cauchy relation is corroborated.     

Adiabatic elastic moduli in ZnSe : Mn<Superscript>2+</Superscript> and ZnSe : V<Superscript>2+</Superscript> crystals

The temperature dependences of the elastic moduli C ₄₄ (C ₁₁ ? C ₁₂)/2 and C _l = (C ₁₁ + C ₁₂ + 2C ₄₄)/2 of ZnSe : V²⁺ (impurity concentration, 6 × 10¹⁸ cm^?3) and ZnSe : Mn²⁺ (9.4 × 10²⁰ cm^?3) are measured in the temperature range 1.4–100.0 K at frequencies of 52 and 156 MHz. The temperature dependences of the adiabatic elastic moduli are derived. It is established that softening of the symmetry modules is observed only in the crystal with an impurity having orbitally degenerate states.