Single-crystal Elastic Moduli Research Articles

Pressure derivatives of the elastic moduli of BCC Ti-V-Cr, Nb-Mo and Ta-W alloys

The single-crystal elastic moduli Cij and their pressure derivatives dCij/dP have been measured ultrasonically at 25 degrees C for six Ti-V-Cr, eight Nb-Mo and seven Ta-W solid solutions. The rigid-band model for the alloy electronic structure is known to work fairly well for these BCC metals in predicting the density of states at the Fermi energy and related properties. The results of this study show that changes in the composition dependence of the Cij and dCij/dP appear to be related to specific topological changes in the Fermi surface which occur as the electron population varies in the rigid-band model. The relationship between the bandstructure and the Cij is discussed and a qualitative explanation is given for the correlations between the moduli and the Fermi surface and for some of the differences between the three alloy series.

Effect of nuclear radiation on the elastic moduli of zircon

Zircon crystals that have been damaged by a naturally occurring radioactive contaminant, referred to as metamict zircons, have density values ranging from 4.70 to 3.90 g/cm3. The present work is a study of the effect of the damage on the single-crystal elastic moduli. A systematic and very marked decrease (i.e., up to 69%) of the elastic moduli with density have been observed. All the longitudinal and the shear elastic moduli decrease with radiation damage and approach two common saturation values of 1.5×1012 and 0.49×1012 dyn/cm2, respectively. Present observation of the elastic moduli of the radiation-damaged zircon crystals confirms Warren’s theoretical predictions concerning the effect of different pore geometries on the compressibility. Qualitative correlation of the elastic moduli with the structural changes arising from radiation damage in zircon crystals suggests that the marked decrease of the elastic moduli in the radiation-damaged zircons is caused by the nonspherical defect clusters, microcracks, and pores generated during radiation damage. Thermal annealing, x-ray diffraction, elastic moduli, and density measurements on the zircon samples confirm the ’’three-stage’’ process proposed for the metamict transformation of zircons, but the final product of metamictization does not seem to be a mixture of ZrO2 and SiO2 crystallites.

Temperature dependence of the single-crystal elastic constants of Co-rich Co–Fe alloys

The single-crystal elastic moduli, C11, C12, and C44 of three fcc cobalt-iron alloys (Co–6 at.% Fe, Co–8 at.% Fe, Co–10 at.% Fe) were measured in the range 0–315°C. In addition C11 for the Co–6 at.% Fe alloy, and C′=(1/2)(C11+C12+2C44) for the three alloys are measured over the temperature range 0–1250°C. Plots of the elastic moduli vs temperature exhibit a change in slope and deviation from linearity in the neighborhood of the Curie temperature. The temperature variation of the shear anisotropy in the fcc phase Afcc (≡2C44/C11−C12) differs among the three alloys. Afcc exhibits a highly positive temperature dependence in the Co–10 at.% Fe alloy and a slight negative dependence in the Co–6 at.% Fe and Co–8 at.% Fe alloys. Previous statements in the literature that the hcp⇄fcc transformation in cobalt is preceded by a highly negative temperature dependence of the shear anisotropy ratio A (≡C44/C66) in the hcp phase between 523°K and the transition at about 743°K is not borne out by the present results. Rather it appears that the hcp⇄fcc transformation involves a change from A>1 in the hcp phase to A<1 in the fcc phase.

Pressure-induced elasticity changes in V3Si

The single-crystal elastic moduli of V3Si have been measured for hydrostatic pressures to 2.5 Kbar and temperatures from 13.5 to 296°K to determine the influence of volume decreases on the anomalous C′ = (C11 − C12)/2 shear mode softening in this A-15 material. Moduli values were obtained by measuring the cw resonance frequencies of three thin specimens that did not undergo at room pressure the well-known structural transformation exhibited by some V3Si crystals near 20°K. The measurements show all moduli to be linear in pressure. dBs/dP, which is nearly constant at 4 for temperatures down to 40°K, is near 7 at 13.5°K. dC′/dP varies from unity at 296°K to −5 at 13.5°K, vanishing at near 90°K. These results are compared with those obtained earlier for large C′-type shear loads. The possibility of inducing a structural transformation with hydrostatic pressure is discussed.

Hydrostatic pressure derivatives of the single crystal elastic moduli of Gd, Dy, and Er

Hydrostatic pressure derivatives of the single crystal elastic moduli of Gd, Dy, and Er

Contribution to the study of the metal-insulator transition in the [formula omitted] system I. Crystallographic and transport properties

Measurements of single-crystal elastic moduli under ambient conditions are reported for nine calcium to calcium-sodium amphiboles that lie in the composition range of common crustal constituents. Velocities of body and surface acoustic waves measured by Impulsive Stimulated Light Scattering (ISLS) were inverted to determine the 13 moduli characterizing these monoclinic samples. Moduli show a consistent pattern: C33 > C22 > C11 and C23 > C12 > C13 and C44 > C55 ∼ C66 and for the uniquely monoclinic moduli, |C35| ≫ C46 ∼ |C25| > |C15| ∼ 0. Most of the compositionally-induced variance of moduli is associated with aluminum and iron content. Seven moduli (C11 C12 C13 C22 C44 C55 C66) increase with increasing aluminum while all diagonal moduli decrease with increasing iron. Three moduli (C11, C13 and C44) increase with increasing sodium and potassium occupancy in A-sites. The uniquely monoclinic moduli (C15 C25 and C35) have no significant compositional dependence. Moduli associated with the a∗ direction (C11 C12 C13 C55 and C66) are substantially smaller than values associated with structurally and chemically related clinopyroxenes. Other moduli are more similar for both inosilicates. The isotropically averaged adiabatic bulk modulus does not vary with iron content but increases with aluminum content from 85 GPa for tremolite to 99 GPa for pargasite. Increasing iron reduces while increasing aluminum increases the isotropic shear modulus which ranges from 47 GPa for ferro-actinolite to 64 GPa for pargasite. These results exhibit far greater anisotropy and higher velocities than apparent in earlier work. Quasi-longitudinal velocities are as fast as ∼9 km/s and (intermediate between the a∗- and c-axes) are as slow as ∼6 km/s. Voigt-Reuss-Hill averaging based on prior single crystal moduli resulted in calculated rock velocities lower than laboratory measurements, leading to adoption of the (higher velocity) Voigt bound. Thus, former uses of the upper Voigt bound can be understood as an ad hoc decision that compensated for inaccurate data. Furthermore, because properties of the end-member amphiboles deviate substantially from prior estimates, all predictions of rock velocities as a function of modal mineralogy and elemental partitioning require reconsideration.

Variation of Ultrasonic Grain-Scattering Factors with Velocity

This paper explores the dependence of ultrasonic grain-scattering attenuation in a polycrystalline aggregate upon the various methods of determining the elastic-wave velocities in the aggregate. High powers of the velocities appear in the denominators of the grain-scattering factors, so small errors in velocity will produce disproportionately large variations in the scattering factors. The elastic-wave velocities in the aggregate may be measured directly on the attenuation specimen or may be computed from the elastic moduli of single crystals of the material by the methods of Voigt or Reuss. These two methods are approximate and yield different values for the velocities. Usually the experimental velocities will fall between the Voigt and Reuss velocities. Because of the velocity dependence of the scattering factors, the attenuation predicted from Voigt and Reuss velocities will differ considerably in most materials. The variability of the scattering factors with velocity will be explored. In this paper, the ultrasonic grain-scattering factors are computed and compared for several materials using experimental velocities, theoretical Voigt and Reuss velocities, and the average theoretical velocity (VOIGT+REUSS)/2. It is shown that the scattering factors computed from the experimental velocities often agree with those from the average theoretical velocities to within 20% to 50%. Measured attenuation usually agrees with predictions made in this manner to within a factor of 2.

Elastic Moduli of the Lead-Thallium Alloys

The adiabatic elastic moduli of single crystals of lead containing 1%–17% thallium have been measured using the ultrasonic pulse echo technique. Data at room temperature, 77°, and 4.2°K were obtained. It was observed that increasing the thallium content decreased all the moduli in spite of the fact that the interatomic distance decreases with thallium additions. Values of the Debye θ0 parameter were calculated from the 4.2°K modulus values.

Single-Crystal Elastic Moduli and the hcp → bcc Transformation in Ti, Zr, and Hf

The variations with temperature of the elastic moduli in single crystals of hcp Ti, Zr, and Hf have been experimentally determined from 4\ifmmode^\circ\else\textdegree\fi{}K up to 1155\ifmmode^\circ\else\textdegree\fi{}K, using an ultrasonic wave interference technique. The ${c}_{11}$, ${c}_{33}$, and ${c}_{44}$ differ considerably among the three metals, whereas the ${c}_{66}$ shear moduli for Ti and Zr are similar to within 1% between 4 and 300\ifmmode^\circ\else\textdegree\fi{}K and less than 10% at 1100\ifmmode^\circ\else\textdegree\fi{}K. There is very marked anisotropy in the temperature dependence of the shear moduli in Ti and Zr, ${c}_{66}$ decreasing by 75% between 4 and 1100\ifmmode^\circ\else\textdegree\fi{}K whereas ${c}_{44}$ decreases by 40%. A very pronounced deviation from a linear temperature dependence of ${c}_{66}$ occurs at a temperature which coincides with the anomalous curvatures in the published electrical resistivity data. The relatively small ${c}_{66}$ shear moduli at the hcp \ensuremath{\rightarrow} bcc phase transformation temperatures are as expected from the transformation shear mechanisms suggested by Burger. A survey of existing elastic modulus and thermal expansion data in hcp metals leads to the conclusion that a relatively high temperature dependence in ${c}_{66}$ and the occurrence of hcp \ensuremath{\rightarrow} bcc transformations are characteristic of metals with $\frac{c}{a}<1.60$. It is suggested that these characteristics due to the transverse thermal vibrations in the basal planes, and that the transformation results from anharmonic contributions to the free energy which are indicated by specific heat data. Contrary to comparison in other hcp metals, the Debye $\ensuremath{\theta}$ from 4\ifmmode^\circ\else\textdegree\fi{}K elastic moduli in Ti, Zr, and Hf are in excellent agreement with published specific heat data. A slight anomaly at 18\ifmmode^\circ\else\textdegree\fi{}K in Ti is indicated which may be associated with published resistivity minima in Ti alloys.

Low-Temperature Elastic Moduli of Aluminum

The adiabatic elastic moduli of single crystals of aluminum have been measured from 4.2° to 300°K by using the ultrasonic pulse-echo technique. The values obtained by extrapolation to 0°K are C11=11.430, C12=6.192, and C44=3.162 in units of 1011 dyn/cm2. The Debye temperature obtained from the 0°K modulus values is 430.3°K, in excellent agreement with the value from heat-capacity measurements.

Low-Temperature Phase Transition in Alpha Uranium

A phase transition in alpha uranium at 42\ifmmode^\circ\else\textdegree\fi{}K\ifmmode\pm\else\textpm\fi{}1\ifmmode^\circ\else\textdegree\fi{}K is indicated by measurements of the single-crystal elastic moduli using ultrasonic pulse techniques. The indicated transition coincides closely to anomalies noted in the thermal expansion, Hall coefficient, electrical resistivity, and thermoelectric power. An examination of the literature specific heat data shows that the transition is associated with an inflection point in the entropy vs temperature curve. On the basis of the elasticity and available lattice parameter data it is postulated that the temperature dependence of the $y$ positional parameter becomes relatively large in this range and the phase instability is a result of a change in the nature of certain interatomic bonds.

Determination of the elastic moduli of tin single crystals, and their variation with temperature

An apparatus is described for the measurement of the frequencies of resonance of longitudinal and torsional vibrations of circular cylindrical rods at temperatures in the range -180 to +195° C. The values obtained have been used to determine the elastic moduli of single crystals of tin over this range of temperature. The equations relating the elastic moduli to these frequencies of resonance for tetragonal crystal symmetry are given. The values of the elastic moduli at 15° C are given as S11 = 41.6, S33 = 14.9, S44 = 45.6, S66 = 42.8, S12 = -31.2 and S13 = -4.8 in units of 10-13 cm2 dyn-1.

Measurement of Ultrasonic Wave Velocities and Elastic Moduli for Small Solid Specimens at High Temperatures

An improved buffer rod-specimen unit is described for measuring longitudinal wave velocities in solids at temperatures up to 350°C. The method is particularly useful in determining elastic moduli of single crystals since very small path lengths in any prescribed direction can be employed. The use of molten metals to provide coupling at still higher temperatures is discussed. Several high-temperature techniques involving mode conversion at an impedance discontinuity for generating shear waves will be detailed, with particular interest in application to crystalline media. The various methods are illustrated with data for fused silica and single crystal germanium.

Measurements of the compressibility of solids

In high pressure research, static megabar pressures are typically produced by compression of a thin sample by two diamonds in various types of diamond anvil cells. This process is accompanied by large plastic deformation (sample thickness is reduced by a factor of 30), and finite elastic deformation of a sample and even the diamond. A thermodynamically consistent system of equations for large elastic and plastic deformation of an isotropic material obeying nonlinear elasticity and pressure dependent yield condition is formulated. The Murnaghan elasticity law and pressure-dependent J2 plasticity are utilized. The finite-strain third-order elasticity law for cubic crystals is utilized for diamond. A computational algorithm is presented with emphasis on the stress update procedure and derivation of the consistent tangent moduli. It is implemented as a user material subroutine in the finite element code ABAQUS. Material parameters for a rhenium sample, as an example, and a diamond are calibrated based on the experimental and atomistic simulation results in the literature. The evolution of the stress and strain tensor fields in the sample and diamond is studied up to a pressure of 300 GPa. Good correspondence between numerical and experimental pressure distributions at the diamond-sample contact surface is obtained. Because there is a significant scatter of the magnitude of reported third-order single-crystal elastic moduli for diamond, their effect on strains and stresses is studied in detail. With the smaller third-order elastic moduli, the phenomenon of cupping of the diamond-sample contact surface is reproduced, which plays an important role in increasing maximum pressures for a given anvil geometry. The results provide important insight into the mechanical response in diamond anvil cells, interpretation of materials properties under extreme conditions from heterogeneous fields, and optimum design of cells for reaching the maximum static pressure in a volume sufficient for the desired measurements.