Elastic strain and coloration pattern in natural quartz crystals

Very small elastic strains (tilts of the order of 0.25 sec of arc and dilationsΔd/d?10^?6, whered is the spacing of net planes) have been observed in natural quartz crystals. According to their rocking curves which are of almost theoretical width, these crystals were commonly expected to be practically perfect crystals. On double crystal X-ray topographs a pattern was revealed of uniformly strained layers parallel to the rhombohedron faces (10¯11) and with thickness ranging between 20 and 200 microns. This fine structure strain pattern was correlated to the pattern of the optical coloration which could be produced by irradiating the originally transparent crystals with X-rays. A localincrease of coloration matched with a localcontraction of the lattice. Tilts occurred in transition zones between adjacent layers with slightly different lattice spacings. Strain and coloration patterns probably have as a common cause the deposition of layers with different impurity concentrations during the growth process of the crystals. Although no change of the strain as a result of the X irradiation could be detected in the colored layers in particular, experimental evidence has been found for anoverall expansion of the lattice of the order ofΔd/d?5×10^?7 due to X irradiation. This expansion seems to saturate at this low value and is probably caused by electronic processes connected with the coloration. Some aspects concerning the nature of the color centers have been discussed. It seems likely from the results of the strain measurements that both sets of layers, colored and transparent ones, play a combined role in coloration.     

Elastic constants of lead-bismuth alloys

A study of ultrasonic velocities and internal friction has been carried out in Pb-Bi alloys in the concentration range of 0 to 49.5 atomic % Bi using the composite oscillator technique. From the velocity and density data a set of elastic constants namely, Young’s modulus, rigidity modulus, bulk modulus and Poisson’s ratio are estimated. The results are interpreted in terms of the phase changes occurring in the alloy system. Internal friction is found to be more sensitive than the elastic constants to the phase changes.     

Elastic constants of antiferromagnetic chromium-vanadium alloys

Single crystals of chromium alloys containing 0.67 and 1.5% vanadium were prepared by the floating-zone technique. Elastic constants and attenuation were measured by the pulse-echo method between 77 and 300 K. The changes of the elastic properties in the paramagnetic and antiferromagnetic states and in particular at the magnetic transition are described and discussed.     

Elastic constants of nonionic anharmonic crystals

Taking into account the entire anharmonicity of the crystal an exact relation between the long-wavelength limit of the static self-energy and the isothermal elastic constants is established. The result can be reformulated as a sum rule for the displacement correlation function. The diagram technique is also employed to discuss the equation for the elastic constants in the pseudoharmonic approximation as well as to derive the integral equations for the quasiparticle parameters.     

Elastic constants of mixed ammonium halides

In the present communication a method based on the three body forces for the evaluation of second and third order elastic constants of mixed ammonium halides has been described. The Lundqvist three body potential is used along with long range Coulombian and short-range overlap repulsive potentials. The latter is taken of the Born-Mayer type and effective up to first neighbours only. It has been assumed that the charge transfer and short-range parameters depend linearly on the composition of solid solutions. Theoretical results obtained, for the entire range of compositions of mixed NH₄Cl and NH₄Br, have been compared with the recently measured experimental values and the theoretical results of other workers, whenever these are available.The authors are thankful to Dr. J. Shanker, Department of Physics, Agra University, Agra, for helpful discussions.     

Elastic constants from microscopic strain fluctuations

Fluctuations of the instantaneous local Lagrangian strain epsilon(ij)(r,t), measured with respect to a static "reference" lattice, are used to obtain accurate estimates of the elastic constants of model solids from atomistic computer simulations. The measured strains are systematically coarse-grained by averaging them within subsystems (of size L(b)) of a system (of total size L) in the canonical ensemble. Using a simple finite size scaling theory we predict the behavior of the fluctuations as a function of L(b)/L and extract elastic constants of the system in the thermodynamic limit at nonzero temperature. Our method is simple to implement, efficient, and general enough to be able to handle a wide class of model systems, including those with singular potentials without any essential modification. We illustrate the technique by computing isothermal elastic constants of "hard" and "soft" disk triangular solids in two dimensions from Monte Carlo and molecular dynamics simulations. We compare our results with those from earlier simulations and theory.     

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.     

Elastic constants of an aluminum-alumina unidirectional composite

The five independent second-order elastic constants of a transversely isotropic aluminum/alumina fiber composite have been measured for the first time using a resonant ultrasound spectroscopy technique. These data were used to deduce the elastic constants and engineering moduli for off-axis loading conditions.     

Elastic constants of layers in isotropic laminates

The individual laminae elastic constants in multilayer laminates composed of dissimilar isotropic layers were determined using ultrasonic-resonance spectroscopy and the linear theory of elasticity. Ultrasonic resonance allows one to measure the free-vibration response spectrum of a traction-free solid under periodic vibration. These frequencies depend on pointwise density, laminate dimensions, layer thickness, and layer elastic constants. Given a material with known mass but unknown constitution, this method allows one to extract the elastic constants and density of the constituent layers. This is accomplished by measuring the frequencies and then minimizing the differences between these and those calculated using the theory of elasticity for layered media to select the constants that best replicate the frequency-response spectrum. This approach is applied to a three-layer, unsymmetric laminate of WpCu, and very good agreement is found with the elastic constants of the two constituent materials.     

Elastic constants of iron-silicon alloy single crystals

The adiabatic elastic constants of single crystals of Fe-Si alloys in the composition range from 6.3 at.% Si to 25 at.% Si have been determined by the ultrasonic pulse echo method at room temperature. The values of the elastic constants depend significantly on the silicon content, and their composition dependences are influenced by ordering.     

Elastic constants of solids at high pressures

The isothermal and adiabatic nth-order (n ?? 2) elastic constants of a loaded crystal are defined. These constants fully determine the behavior of solids at an arbitrary load and are controlled by both an interatomic interaction and an applied load. Expressions that relate these constants (of the second, third, and fourth order) to Brugger elastic constants of the corresponding order, which are only determined by an inter-atomic interaction, are found for cubic symmetry crystals under hydrostatic pressure. These expressions are used to calculate the equation of state and the second- and third-order elastic constants of bcc tantalum at T = 0 K over a wide pressure range (0?C600 GPa) using an electron density functional method. The results of calculating the equation of state and the second-order elastic constants agree with available experimental data and the calculation results obtained in other works.     

Elastic constants and refractive index of boron phosphide

The refractive index of BP was measured in the visible spectral range by means of the Brewster angle method. The elastic constants were determined by Brillouin scattering. The results confirm the strongly covalent character of the B-P bond.