Values Of Third Order Elastic Constants Research Articles

Diamond’s third-order elastic constants: ab initio calculations and experimental investigation

In order to obtain more reliable values of diamond’s third-order elastic constants, experiments on bulk acoustic wave propagation under uniaxial stress application (up to 450 MPa) in pure IIa-type synthetic single-crystal diamond were carried out, and values of third-order elastic constants were calculated. We have also provided theoretical analysis using ab initio density functional theory approach which has shown close correspondence with the experimentally measured data. From ab initio calculations, the values of third-order elastic constants are (GPa): C 111 = −7611, C 112 = −1637, C 144 = −199, C 155 = −2799, C 123 = 604, C 456 = −1148, while experimental values are C 111 = −7750 ± 750, C 112 = −2220 ± 500, C 144 = −1780 ± 440, C 155 = −2800 ± 220, C 123 = 2100 ± 200, C 456 = −30 ± 150. The estimated values on diamond’s fourth-order elastic constants were obtained. The calculated stress–strain curves for different crystal orientations were investigated, including shear stress for [111] direction. From calculations for [100], [110], and [111] directions, the values of critical stress in case of the pure shear were estimated as 222, 113, and 84 GPa, respectively.

Higher-order elastic constants of La1.84Sr0.16CuO4 high-temperature superconductor

Finite deformation theory is used to obtain the strain energy density ϕ of a tetragonal 2–1–4-type single crystal of the high-temperature superconductor La2−xSrxCuO4. The complete set of second and third-order elastic constants of the high-temperature superconductor La2−xSrxCuO4 (x=0.16) is calculated by taking into account the interactions between nine nearest-neighbour atoms in the lattice and using Mie–Grüneisen interatomic potential. For the sake of comparison we have also computed the values of these constants for x=0.13–0.20. The values of third-order elastic constants of La2−xSrxCuO4 (x=0.13–0.20) are negative and their absolute magnitudes are one order higher than those of the second-order elastic constants.

Comment on “Reappraisal of experimental values of third-order elastic constants of some cubic semiconductors and metals”

In a recent paper A. S. Johal and D. J. Dunstan [Phys. Rev. B 73, 024106 (2006)] have applied multivariate linear regression analysis to the published data of the change in ultrasonic velocity with applied stress. The aim is to obtain the best estimates for the third-order elastic constants in cubic materials. From such an analysis they conclude that uniaxial stress data on metals turns out to be nearly useless by itself. The purpose of this comment is to point out that by a proper analysis of uniaxial stress data it is possible to obtain reliable values of third-order elastic constants in cubic metals and alloys. Cu-based shape memory alloys are used as an illustrative example.

Reply to “Comment on ‘Reappraisal of experimental values of third-order elastic constants of some cubic semiconductors and metals’ ”

In his Comment [Phys. Rev. B 74, 146101 (2006)], Ma\~nosa claims that other methods of calculation can obtain reliable values of third-order elastic constants from data for acoustic measurement under uniaxial stress, from data sets which fail under multivariate linear regression analysis (MVLA). His analysis does not consider errors. We show that his method leaves errors on the derived values, orders of magnitude larger than those obtained by MVLA, and that his values are therefore completely unreliable.

Reappraisal of experimental values of third-order elastic constants of some cubic semiconductors and metals

The third-order elastic constants of many cubic materials have been determined by measurements of the changes in acoustic velocities under hydrostatic and uniaxial stress. These measurements give, typically, 14 data to determine the six material parameters ${c}_{\mathrm{IJK}}$, related by linear equations. Not least because of the unreliability of uniaxial stress experiments, the resulting ${c}_{\mathrm{IJK}}$ has been considered to be dubious. Multivariate linear regression analysis is applied to the published data for some semiconductors, Si, Ge, GaAs, GaP, and InSb, some elemental metals, Cu, Ag, Au, Al, and Ni, and some miscellaneous materials, to obtain the best estimates of the material parameters and of the errors in them. We find that the published data are remarkably good and many of the published ${c}_{\mathrm{IJK}}$ and errors on them need little adjustment. Some values and errors do require significant adjustment. Uniaxial stress data on metals turns out to be nearly useless, by itself, but we show that in combination with hydrostatic data it gives essential and reliable input.

Third-order elastic constants and Grüneisen parameters of silicon and germanium between 3 and 300 °K

We present a method to determine all six independent third-order elastic (TOE) constants of diamond-like solids as a function of temperature from measured values of the ultrasonic nonlinearity parameters. Ultrasonic harmonic generation experiments along the three principal directions of a cubic crystal lead to the determination of the nonlinearity parameters, which contain three linear combinations of TOE constants. These data are sufficient to determine the three anharmonic first and second neighbor force constants in the Keating model for diamond-like solids. Ultrasonic velocity data are used to determine the two harmonic force constants in this model. Since our experiments have given data on silicon and germanium between liquid helium temperature and room temperature, we are able to evaluate and plot the Keating model force constants and hence all the six third-order elastic constants between 3 and 300 °K. Most of the TOE constants of silicon and germanium are smoothly varying functions of temperature; however, C123 and C144 go through zero and assume positive values at low temperatures. To determine whether the behavior of the calculated TOE constants is reasonable, we have used them to calculate the generalized Grüneisen parameters γ in the quasiharmonic approximation and have plotted γ as a function of temperature. These curves are in better agreement with curves obtained from thermal expansion experiments than the curves plotted by earlier workers using room temperature values of third-order elastic constants. For completeness we also have calculated the pressure dependence of the SOE constants and the Anderson Grüneisen parameters over the same temperature range.

Morse-Potential Evaluation of Second- and Third-Order Elastic Constants of Some Cubic Metals

The second- and third-order elastic constants of Al, Cu, Ag, Au, Na, and K have been calculated in the framework of a central-force model. The interatomic potential was expressed in terms of the Morse potential. The parameters in the potential were determined by using the lattice parameter, bulk modulus, and cohesive energy. The range of the interatomic potential was cut off after 176 and 168 neighbors in the case of face-centered and body-centered cubic metals, respectively. The evaluated second- and third-order elastic constants were used to calculate the pressure derivatives of the second-order elastic constants. These pressure derivatives compare fairly well with the experimental values. The agreement between experimental values and theoretical values of third-order elastic constants is less satisfactory. Theory predicts ${\mathrm{C}}_{123}$ to be positive for all the fcc metals and negative for bcc metals. The available experimental data on the third-order elastic constants of Cu, Ag, and Au show that ${\mathrm{C}}_{123}$ is positive for Ag and negative for Cu and Au. The possible implications of this result are discussed.