Slow Transverse Research Articles

Simple quantum mechanics and complicated ultrasonic attenuation

The ultrasonic attenuation of Al2O3 along the a axis at low temperatures is explained by taking into account the finite lifetimes of the thermal phonons. The longitudinal attenuation proceeds via unallowed transitions. The slow transverse is normal attenuation. The fast transverse is a mixed case.

On interference effects in the elastodynamic Green’s functions G33( x, ω) of Si and GaAs

In this paper, we analyze interference effects present in the elastodynamic Green’s functions G 33( x, ω) of the cubic crystals Si and GaAs, which are associated with folded portions of the wave surface of the slow transverse (ST) acoustic mode. G 33( x, ω) represents the three dimensional extension of the amplitude distribution imaged in the transmission acoustic microscopy of these crystals. The intensity contrast for oscillations of a particular wave vector k in the interference pattern is determined essentially by the 3D Fourier transform of G 33 ( x , ω ) G 33 ∗ ( x , ω ) . According to the Fourier autocorrelation theorem, that transform is equivalent to the autocorrelation function of the corresponding distribution G 33( k, ω) in k-space. We show that due to the linear mapping between k-space and the slowness vector s-space, the interference phenomena discussed here are related to geometrical features of the slowness surface of the ST mode. We present calculations of these effects based on the angular spectrum technique.

Interference of non-aligned Airy diffraction patterns in transmission acoustic microscopy images of crystals

This paper describes interference effects present in narrow band transmission acoustic microscopy images of crystals that arise from the superposition of pairs of non-aligned but symmetry equivalent Airy diffraction patterns. In the cases we examine, these Airy patterns are associated with folded portions of the slow transverse (ST) sheet of the acoustic wave surface of cubic crystals intersecting in the {110} symmetry planes. Each set of Airy fringes is inclined in the (001) observation plane to its overlapping counterpart, giving rise to curved Moiré-like fringes. Also, because of tilting of the two portions of the wave surface in opposite directions out of the (001) plane, each composite Airy fringe breaks up along its length into a large number of narrow, regularly spaced fringes, in a manner akin to wedge diffraction. We present calculations of these effects in GaAs and Ge, based on the elastodynamic Green's function, , for the infinitely extended anisotropic elastic continuum. A measured time-gated diffraction image is presented for GaAs, which is in good agreement with calculation.

Determination of all elastic constants of transversely isotropic media with a cusp around the symmetric axis by use of elastic pulses propagating in two principal directions.

This paper describes a method of determining all five elastic constants of a hexagonal or transversely isotropic medium of type II which has the cuspidal region of group velocity around the highestsymmetry axis [001] perpendicular to the transversely isotropic basal plane. The elastic constants are obtained from wave-speed data of broad-band ultrasonic signals propagating in two principal directions, namely the [001] and the other parallel to the basal plane. The method uses specific cuspidal features of the slow transverse (ST) mode propagating in the [001] direction with two distinct group velocities. An analytic formula that relates the slower group velocity of the ST mode with the elastic constant C 13 is derived

Focusing of fast transverse modes in silicon and slow transverse modes in zinc

This paper presents focusing of fast transverse (FT) modes in silicon and slow transverse (ST) modes in zinc, both observed at utrasonic frequencies. Using a (001)-oriented silicon disk, the FT modes are generated and detected by two tiny shear PZT transducers, both polarized in the [100] direction. One transducer fixed on the bottom surface acts as a source and the other serves as a detector scanning the top surface along the [100] direction across various points on the [010] axis at some distances away from the epicenter. The observed focusing pattern indicates strong concentration of FT modes on the narrow band about the (100) plane containing the source. For a (001)-oriented zinc crystal, strong focusing of ST or quasi-transverse (QT) modes toward the symmetry direction is observed: first in theoretical and experimental, epicentral Green’s functions; second with very small PZT source and PZT detector; finally with focused pulsed laser excitation and small PZT detector. The focusing patterns observed both in silicon and in zinc are consistent with the results predicted by the theory of phonon focusing. [Work supported by ONR.]

Pressure dependence of the elastic moduli of semimetallic(Ti1−xVx)2O3at 296 K. Softening of a slow shear mode and two other elastic eigenvalues whenx≈0.04

Ultrasonic 30-MHz-wave travel times have been measured at 296 K for semimetallic, single-crystal ${({\mathrm{Ti}}_{1\ensuremath{-}x}{\mathrm{V}}_{x})}_{2}{\mathrm{O}}_{3}$ samples with $0.02\ensuremath{\le}x\ensuremath{\le}0.09$ as a function of hydrostatic pressure up to 4 kbar. The moduli of all waves except the slow transverse (ST) mode propagating in the [100] direction in the $x=0.04$ sample stiffen with increasing pressure. This ST mode is a symmetry-breaking one, and its modulus, ${\mathrm{C}}_{\mathrm{ST}}$, is an eigenvalue of the elastic constant matrix. Linear extrapolation of ${\mathrm{C}}_{\mathrm{ST}}$ implies a structural transition at very high pressure (1000 kbar using the Born criterion ${\mathrm{C}}_{\mathrm{ST}}$=0, and 275 kbar using a Demarest et al.-type criterion $\frac{{\mathrm{C}}_{\mathrm{ST}}}{B}=0.2$, where $B$ is the bulk modulus). The pressure dependences of all six independent elastic constants, ${C}_{11}$, ${C}_{12}$, ${C}_{13}$, ${C}_{14}$, ${C}_{33}$, and ${C}_{44}$ are determined for samples with $x=0.02$ and $0.04$ or $0.05$ and of all ${C}_{\mathrm{ij}}$'s except ${C}_{13}$ for $x=0.09$ samples. Comparison with data for semiconducting ${\mathrm{Ti}}_{2}$${\mathrm{O}}_{3}$ does not reveal any simple correlation between the $\frac{{\ensuremath{\Delta}C}_{\mathrm{ij}}}{\ensuremath{\Delta}P}$ values with electrical resistivity, lattice parameters, interatomic spacings, or the values of the ${C}_{\mathrm{ij}}$'s. It is found that the other eigenvalues of the ${C}_{\mathrm{ij}}$ matrix usually increase with pressure. Samples with $x=0.04$ or $0.05$ are exceptional again in that two other eigenvalues are decreased by pressure. One of them is associated with a symmetry-breaking mode and is approximately equal to ${C}_{11}\ensuremath{-}{C}_{12}$ because ${C}_{14}$ is very small.

Analysis of Resonant Brillouin Scattering in GaAs

AbstractThe dispersion curves of the resonant Brillouin scattering cross sections in GaAs measured by Garrod and Bray are analyzed on the basis of the resonant Brillouin scattering theory and piezobirefringence theory. The resonant enhancement and cancellation are found to be well interpreted in terms of the sign reversal of the Brillouin amplitude (or photoelastic constant). A good agreement is found between the experimental data and the present analysis. In addition the present analysis predicts a cancellation of the Brillouin scattering at Λ = 0.976 μm for the scattering by the slow transverse (Tl) acoustic phonons. This prediction is supported by the analysis of the photoelastic constant made by using piezobirefringence theory.

Hypersonic attenuation at low temperatures in Al2O3

Abstract A numerical calculation is undertaken to determine the anharmonic attenuation of both transverse modes and the longitudinal acoustic mode propagating along the a axis in Al2O3. This calculation takes full account of crystal anisotropy and the anharmonic coupling term and, in addition, finite thermal phonon lifetimes are used throughout. It is shown that the attenuation of the longitudinal mode by interactions with longitudinally polarized thermal phonons (Fitzgerald and Silverman 1967) dominates by a large factor the attenuation due to interactions with transversely polarized thermal phonons (Kalejs, Maris and Truell 1966). Good agreement with experimental attenuation data is found for the slow transverse and longitudinal modes, but considerable discrepancies are found to exist for the fast transverse mode. The effect of the wave intensity on the energy-conserving Landau and Rumer (1937) type of process is investigated and found to be of significance for acoustic intensities ∼10−3 watts cm−2 at t...

Microwave Phonon-Attenuation Measurements in Quartz

Measurements have been made of the temperature dependence of the attenuation at 9000 Mc/sec of two slow transverse, two fast transverse, and three longitudinal phonon modes in "perfect" natural-quartz specimens. The attenuation of the slow transverse waves follows the theory of Landau and Rumer. The attenuation of the longitudinal and fast transverse waves follow the theories of Maris and Shiren, which take into account the finite lifetime of the thermal phonons with which the microwave phonons interact, but some discrepancies still exist in the magnitude of the attenuation. The steps observed in the attenuation of imperfect quartz are identified as two peaks at 16\ifmmode\pm\else\textpm\fi{}3\ifmmode^\circ\else\textdegree\fi{}K and 24\ifmmode\pm\else\textpm\fi{}3\ifmmode^\circ\else\textdegree\fi{}K, sumperimposed on a phonon-phonon background. Possible mechanisms for these attenuation peaks are discussed.