Pure Mode Directions Research Articles

Interactions of collinear acoustic waves propagating along pure mode directions of crystals

Previous studies on the interaction of collinear acoustic waves have been devoted to waves propagating along pure modes directions of cubic crystals. In this paper, we show that the calculations can be readily extended to all crystal point groups. Nonlinearity parameters characterizing the nonlinear interactions are defined here. The effective third order elastic constants involved in the parameters can be calculated by using the method presented in this paper. Our results are very useful for the study of elastic nonlinearity of crystals with any given symmetry.

Two-finger (TF) SPUDT cells [Correspondence

SPUDT cells including two fingers are only known thus far for so-called NSPUDT directions. In that case, usual solid-finger cells are used. The purpose of the present paper is to find SPUDT cell types consisting of two fingers only for pure mode directions. Two-finger (TF) cells for pure mode directions on substrates like 128°YX LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> and YZ LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> were found by means of an optimization procedure. The forward direction of a TF-cell SPUDT on 128°YX LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> was determined experimentally. The properties of the new cells are compared with those of conventional SPUDT cells. The reflectivity of TF cells on 128°YX LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> turns out to be two to three times larger than that of distributed acoustic reflection transducer (DART) and Hanma-Hunsinger cells at the same metal layer thickness.

Nonlinearity parameter, nonlinearity constant, and frequency dependence of ultrasonic attenuation in GaAs

Mason’s nonlinearity constants D [W. P. Mason, in Physical Acoustics, edited by W. P. Mason (Academic, New York, 1965), Vol. IIIB, Chap. VI] in GaAs at 298 K have been estimated from second- and third-order elastic constants along the [100], [110], and [111] directions for longitudinal and shear waves. These values are further used to estimate the frequency dependence of the ultrasonic attenuation in the range 40–640 MHz and are found to be in good agreement when compared with experimental values available in the literature. Previously, it was established that phonon-phonon interactions produce a drag on moving dislocations in a solid. In the present work, the drag coefficients for edge and screw dislocations along the [100] and [110] directions in single-crystal GaAs have been estimated from the evaluated nonlinearity constant D values at 298 K. We have also estimated the values of Breazeale’s nonlinearity parameters β [M. A. Breazeale and J. Philip, in Physical Acoustics, edited by W. P. Mason and R. N. Thurston (Academic, New York, 1984), Vol. XVII, Chap. I] along the [100], [110], and [111] directions and found them to be consistent for the three principal orientations, compared with the results for other samples. It is pertinent to note that the mode Grüneisen number γ ji involved in the estimation of Mason’s nonlinearity constant D is deceptively similar to Breazeale’s nonlinearity parameter β. The nonlinearity constant D is evaluated from knowledge of the Grüneisen number γ ji for various modes and directions, whereas Breazeale’s nonlinearity parameter β is the negative ratio of coefficients of nonlinear term to the linear term of the (dissipationless) nonlinear wave equation for pure mode directions. Both D and β, in general, are temperature dependent.

Sound speed anisotropy of V 2O 5 single crystals determined by laser generated acoustic waves

The application of laser-light induced acoustic waves for the determination of sound velocities for pure mode directions in V 2O 5 single crystals is reported. The longitudinal sound speeds along the crystallographic c and a axes have been measured for the first time and along the b axis has been measured and compared with results of earlier measurements.

Elastic nonlinearity parameters of tetragonal crystals

The equations of motion for the propagation of finite amplitude elastic waves in crystals of tetragonal symmetry have been derived starting from the expression for the elastic strain energy. The equations have been solved for a finite amplitude sinusoidal wave propagating along the pure mode directions which are [100], [110] and [001] for the tetragonal group TI. The solutions corresponding to longitudinal wave propagation yield expressions for the amplitudes of the fundamental and generated second harmonic for these directions in terms of certain combinations of second and third order elastic constants of the medium. The results will aid the experimenter to determine these constants using ultrasonic harmonic generation technique.

Acoustic radiation-induced static strains in solids.

The controversy surrounding the magnitude of the radiation-induced static strain accompanying acoustic wave propagation in solids is resolved by a consideration of the associated Boussinesq radiation stress. Experimental verification of the results is presented for waves propagating along the pure mode directions of crystalline silicon.

Pure and Semipure Mode Directions in the Piezoelectric Crystal

The contribution of piezoelectricity was taken into consideration in the elastic wave equation referred to the rotated coordinates, and investigated for all of the pure and semipure mode directions in the crystals of the piezoelectric point groups. In the point groups belonging to cubic, tetragonal I, trigonal I, orthorhombic, monoclinic and triclinic systems, pure and semipure mode directions are determined similarly to the case of neglecting the piezoelectric effect. In the points groups belonging to trigonal II, tetragonal II and hexagonal systems, it is found that the piezoelectric effect shows a remarkable contribution to pure and semipure mode directions. The pure and semipure mode directions can be derived taking account of the crystal symmetry elements.

The elastic stiffness constants and their hydrostatic pressure derivatives for TGS

The thirteen adiabatic elastic stiffness moduli of triglycine sulphate (TGS) have been determined at room temperature from measurements of the ultrasound wave velocities. Results are used to describe the elastic behaviour of TGS. Accidental pure mode directions have been found at 9.4° and 106.5°. A method has been developed for obtaining the hydrostatic pressure derivatives (∂CIJ/∂P) of the elastic constants of a monoclinic crystal from the hydrostatic pressure dependences of ultrasound wave velocities. This method has been applied to find the ∂CIJ/∂P for TGS. There is no evidence for acoustic phonon mode softening under pressure.

Relationship between solid nonlinearity parameters and thermodynamic Grüneisen parameters

The relationship between the ultrasonic nonlinearity parameter for solids and the acoustic Grüneisen number has been derived for longitudinal ultrasonic wave propagation in the pure mode directions of cubic crystals and isotropic solids. Agreement between the acoustic Grüneisen number and the thermodynamic Grüneisen parameter is best for ultrasonic harmonic generation measurements along the [100] direction of a cubic crystal. Comparison of the temperature curves of the acoustic Grüneisen number of copper shows that the acoustic Grüneisen number generally follows the temperature dependence of the lattice contribution to the thermodynamic Grüneisen parameter.

Grüneisen gamma from solid nonlinearity parameters: A generalized approach

Generalized Grüneisen parameters which measure the strain derivatives of the lattice vibrational frequencies are often used to test atomic theories of crystalline solids and to model their anharmonic properties. The relationship between these parameters and the solid nonlinearity parameters measured directly in ultrasonic harmonic generation experiments is derived using a generalized approach valid for normal mode acoustic wave propagation in any crystalline direction. The resulting Grüneisen parameters are purely isentropic in contrast to the Brugger‐Grüneisen parameters which are of a mixed thermodynamic state. The equations are specialized to the case of longitudinal wave propagation in the three pure mode directions of cubic crystals in order to compare available experimental data.

Ultrasonic wave propagation in human cortical bone—I. Theoretical considerations for hexagonal symmetry

From the theory of wave propagation in an anisotropic elastic medium are derived the basic equations relating the five independent second-order elastic stiffness constants (fourth-rank tensor) to the ultrasonic wave speeds in a hexagonal medium, with special emphasis on determining the microtextural symmetry of human cortical bone. In addition, the three pure mode directions of high symmetry in a hexagonal medium are explicitly shown. Finally, expressions relating the ‘technical moduli’ such as Young's modulus, shear modulus and bulk modulus to the elastic compliances are presented for the most general case (triclinic symmetry) and then are specialized for the hexagonal system.

Single crystal growth and the elastic behaviour of In 2Bi

Ultrasound velocities have been measured in In 2Bi crystals grown by zone-levelling. The elastic constants are reported. The elastic behaviour is remarkably isotropic: a pure shear and two almost pure waves propagate in all directions other than pure mode directions.

Fabrication of New Piezoelectric Ceramic and Its Surface Wave Velocity

Strong piezoelectric ceramic consisting of 0.02Sr(Li2/5W3/5)O3-0.46PbTiO3-0.52PbZrO3 solid solution has been developed and its dielectric, piezoelectric and elastic properties are determined by the resonant technique of IRE standards. Phase velocities of surface waves propagating in the pure mode directions on the piezoelectric ceramic substrate are measured by means of the pulse-echo-overlap method of bulk wave technique and determined with precisions better than 5 parts in 104.

Anomalous ultrasonic attenuation at the 105°K transition in strontium titanate

At the 105°K transition a strong peak in the attenuation and a step in the sound velocity were observed for all polarizations in all pure mode directions of propagation. The attenuation anomaly is due to the interaction of the sound wave with soft optical lattice vibrations having a wavevector →q = π/α (1, 1, 1)). The experimental results can be quantitatively explained in terms of an Akhiezer type process with strongly temperature dependent Grüneisen parameters.

Ultrasonic Phase Velocity by the Pulse-Echo-Overlap Method Incorporating Diffraction Phase Corrections

The pulse-echo-overlap method for ultrasonic time-delay measurements is reviewed. In this method, pairs of echoes are compared by driving the x axis of a viewing oscilloscope at a frequency equal to the reciprocal of the travel time between the echoes. A method for choosing the correct cyclic overlap for the rf within the echoes is given. Utilization of the proper cyclic overlap permits the accurate measurement of ultrasonic phase velocity. When corrections for the phase advance due to ultrasonic diffraction are applied to the travel times between various pairs of echoes, the accuracy of the average round-trip travel time is improved. Experimental verification of this is presented for longitudinal waves in isotropic materials and in the pure mode directions in cubic crystals. Delay times are accurate to 0.2 nsec or better.

Elastic Constants of and Wave Propagation in Antimony and Bismuth

Abstract : Ultrasonic wave velocities for 14 different modes were obtained on two differently oriented single crystal antimony cubes from the time between successive unrectified radio frequency pulse echoes. This redundant set of data was fitted by a least squares technique to Voigt theory to yield the six room temperature adiabatic elastic stiffness constants. In units of 10 to the 10th power dynes/sq cm, c sub 11 = 99.4(1), c sub 33 = 44.5(9), c sub 44 = 39.5(5), c sub 66 = 34.2(3), c sub 13 = 26.4(4), c sub 14 = +21.6(4), the positive sign for c sub 14 following from our choice of positive Cartesian axes. When similarly treated, Eckstein, Lawson, and Reneker's bismuth data yield in these same units c sub 11 = 63.22, c sub 33 = 38.11, c sub 44 = 11.30, c sub 66 = 19.40, c sub 13 = 24.40 =.09, c sub 14 = +7.20. Also included are a visual method of fixing the laboratory coordinate system in antimony by means of an imperfect cleavage plane, a calculation of the pure mode directions in the mirror plane, a simple formula for choosing the nonextraneous value of c sub 13 for trigonal crystals having six independent elastic constants without resorting to lattice stability criteria, and a calculation of the deviation of elastic wave particle displacement and energy flux directions from the propagation direction. For waves propagating in the (0,1,1) and (0,1,1) directions, the particle displacement deviations for antimony and bismuth do not exceed 15 degrees and 13 degrees respectively, and corresponding energy flux deviations up to 45 degrees and 27 degrees are obtained. (Author)

Pure Modes for Elastic Waves in Crystals

In an anisotropic medium there are only certain directions along which elastic waves can propagate in pure longitudinal and transverse modes. For the determination of third-order elastic coefficients from sound speed measurements in stressed crystals it is desirable to know these modes. Using a method due to Borgnis the pure mode directions are determined for all crystal point groups belonging to the orthorhombic, tetragonal, cubic, rhombohedral, and hexagonal systems. The eigenvalue problem is solved for each of these directions, and the polarization vectors and the wave speeds are tabulated.