Constants C11 Research Articles

Studies of elastic constants of oriented poly(ethylene terephthalate) films using Brillouin scattering

Brillouin scattering is used to study the elastic behavior of oriented poly(ethylene terephthalate) films. The elastic constants C11, C13, C33, and C44 are determined by measurements of the quasilongitudinal and quasitransverse hypersonic velocities at different angles in the film. The total orientational parameter is calculated for the three draw ratios from the Brillouin hypersonic velocities and are found to be similar to the orientational parameter in the amorphous phase obtained with a different technique.

Effects of chain orientation of Raman and Brillouin scattering spectra of hydrostatically extruded polypropylene

Raman and Brillouin scattering techniques have been used to study the effect of chain orientation in uniaxially extruded isotactic polypropylene. The orientation parameters 〈P2〉 and 〈P4〉 are obtained by measuring several Raman scattering intensities associated with the CH2 rocking mode at 841 cm−1. Both 〈P2〉 and 〈P4〉 parameters represent the average of the local orientational distribution of the polypropylene backbone. The Raman scattering result for 〈P2〉 is found in good agreement with the previous study using another vibrational band. The high-frequency elastic constants C11, C33, C44, and C13 of extruded isotactic polypropylene are determined by Brillouin scattering. The elastic constants of deformed polymers have been related to the orientation parameters determined by Brillouin scattering. The result obtained for 〈P2〉 at high extension using Brillouin scattering is found to be smaller than that of Raman. The result suggests that the pair orientation correlation between segments is important in affecting the hypersonic velocity.

Brillouin scattering from GaS under hydrostatic pressure up to 17.5 GPa

Brillouin scattering in GaS crystals had been measured under hydrostatic pressures up to 17.5 GPa. Frequency shifts of the Brillouin lines give the pressure dependence of the elastic constants C11 and C33, and the index birefringence. Exponents derived from Murnaghan’s equation of state are compared with a Lennard-Jones description of interatomic forces in GaS.

Phonon anomalies in intermediate valent TmXSe and TmSe1−yTey

In TmxSe and TmSe1−yTey the degree of valence mixing can be adjusted between nearly 3+ for Tm0.87Se and 2.55+ for TmSe0.7Te0.3. The measurement of sound velocities vL, vTl and vT2 and the evaluation of the Raman effect for various compositions permit the derivation of LA [111] phonon dispersion at critical points in the Brillouin zone. vL decreases with increasing valence mixing. Near the middle of the zone the LA branch gets a dip for intermediate valent compositions, resulting in a characteristic peak in the Ramn spectrum at about 60 cm−1. The elastic constant c12 has been found negative for Tm0.99Se, also at 4.2 K. For uniaxial pressures c12 exhibits strong nonlinearities and even changes sign with pressure in an intermediate valent composition. The optical phonon frequencies, LO (L) also soften proportional with the degree of valence mixing.

Magnetoelastic effects in erbium

A new ultrasonic sing-around system has been used to measure the changes in the elastic moduli of single-crystal specimens of erbium to a much higher precision than has been achieved previously. The variations of the second-order elastic constants C33, C11, C44 and C66 have been measured as a function of temperature over the range 4.2-100K. The variations of the moduli C11 and C33 have also been investigated as a function of magnetic field applied along the c and b crystallographic axes respectively at a series of temperatures in the three magnetically ordered phases of erbium. The results have allowed critical magnetic fields and their dependence upon temperature to be determined.

Brillouin Scattering in Layered Semiconductors using Microprocessor-controlled Fabry-Perot Interferometer

A microprocessor-controlled multipassed Fabry-Perot interferometer has been developed for high-resolution and high-contrast Brillouin scattering spectroscopy. Using this interferometer, Brillouin scattering experiments have been carried out for GaSe1-xSx mixed crystals having layered structures. Specific features of Brillouin scattering in the layered crystals were presented. Anomalous discontinuities of elastic constants of C44 and C33, related to the shearing and stretching force constants, have been found at the compositional ratios x≃0 and x≃0.4, which are related to the ε-γ and γ-β polytype transformations.

Low-temperature Lamb wave propagation in PVF2

Lamb wave propagation in thin films of polyvinylidene fluoride PVF2 is examined as a function of temperature. Results are presented for frequencies from 10 to 70 MHz and temperatures from −50 to −175 °C. Lamb waves were excited in the 28.4-μm-thick films by interdigital transducers with a center frequency wavelength of 92 μm. At the fundamental wavelength, the first five Lamb modes were observed, while at the third harmonic of the transducer only the first two branches of the dispersion curves could be detected. The observed velocities of the seven modes are compared to the theoretical dispersion curves to obtain stiffness constants for the PVF2 film as a function of temperature. The results obtained for constants c33 and c55 are in agreement with previous literature. Values for the constants c11 and c13 are presented for the first time.

Elastic constants of parahydrogen determined by Brillouin scattering

Five single crystals of parahydrogen were grown near the triple point, and their Brillouin spectra examined in known crystal directions. An analysis of the measured frequency shifts of longitudinal and transverse components led to the direct determination of the adiabatic elastic constants at 13.2 K: C11 = 3.34 ± 0.05, C13 = 0.56 ± 0.03, C33 = 4.08 ± 0.06, C44 = 1.04 ± 0.03 kbar. The fifth elastic constant C12 = 1.30 ± 0.05 kbar was evaluated from the relation C11 + C12 = C13 + C33. Values of the following constants were calculated: elastic anisotropy, A = 1.02 ± 0.06, adiabatic bulk modulus, Bs = 1.73 ± 0.04 kbar, and Debye temperature, θD = 111.2 K. All of these values were compared with earlier experimental measurements. The elastic constants were found to be in good agreement with a recent self-consistent phonon calculation by Goldman.

Jahn-Teller induced elastic constant changes in TmPO4

Elastic constants C66 and C44 have been measured for TmPO4 from 4.2 to 300K. C66 shows evidence of strong Jahn-Teller coupling of Tm3+ electronic states to lattice strain but there is no cooperative phase transition. Theory previously developed to describe TbVO4 gives a very good description of the data for TmPO4.

The magnetic field dependence of the low-temperature elastic constants of neodymium

The elastic constants c11, c33, c44 and c66 of single-crystal neodymium have been measured at 4.2K as a function of applied magnetic field of up to 8 T. The results are in good agreement with the neutron diffraction and magnetisation data for fields in the (1120) and (0001) directions. There is no comparable neutron work to help understand the effects on the elastic constants of a field in the (1100) direction. The magnetic field produces marked anisotropy in the basal plane of Nd, which is strongly reflected in c44 in particular.

Magnetoelastic interactions in holmium

We report here the experimental results of the change of the elastic constants c22, c33 and c66 as a function of the state of magnetization of single crystal holmium for the temperatures between 130 K and 300 K. We applied magnetic fields up to 75 kOe along the crystallographic a, b and c-axes. To analyze the experimental data we used a thermodynamic treatment of the change of the elastic constants as a function of the state of magnetization similar to Freyne’s. Using a non-linear least squares fitting procedure to the experimental results we obtained a set of magnetoelastic coupling coefficients.

An ultrasonic study of the magnetic phases of dysprosium

The elastic constant C33 and its associated ultrasonic attenuation alpha 33 have been measured for Dy over the temperature range 4.2-300K in magnetic fields of up to 8 T. Anomalies in both the isothermal magnetic field dependence and isofield temperature variations have been used to provide a magnetic phase diagram of the element. Anomalies in the region of 130-140K have been interpreted in terms of changes of the basal plane magnetic easy direction, in fair agreement with anisotropy measurements. It is evident that there exists no simple magnetic state between the low-field antiferromagnetic and high-field ferromagnetic phases but a series of intermediate structures.

The HCP to BCC phase transition in metals

The relative stabilities of HCP, BCC and isotropic melt phases may be estimated from the temperatures at which the HCP phase 'melts' to either of two different anisotropic 'worm' fluids. The transition temperature HCP to BCC, Tc, and the melting point, Tm, in this model depend for any metal only on the elastic constants C66, C44, C11 and C33 and two empirical parameters which are ratios of entropy changes. More direct knowledge of possible pseudo-potentials is not required. An effective anisotropy is defined in the shear elastic constants of the HCP phase A' which is independent of the two empirical parameters. For A' A'max the HCP phase is always expected to be unstable relative to BCC. The absolute values of Tc and Tc/Tm, are calculated with a standard deviation from the experimental values of 20%.

Anisotropic phonon dispersion in GaS

The group theoretical analysis of lattice vibrations in GaS and neutron inelastic scattering measurements of acoustic phonons propagating along the Delta , Sigma and T directions are presented. The elastic constants C11, C33, and C44 are derived and the results for the Delta direction are analysed to determine interlayer force constants. The frequencies of the B1 modes along Sigma and T show a weak quadratic dependence on wavevector. A simple interatomic force model is fitted to the phonon frequencies, and the frequency distribution function is calculated. The temperature dependence of the lattice specific heat and of the corresponding Debye temperature is calculated; over a small temperature range the variation of the specific heat is characteristic of a pseudo-two-dimensional solid.

Elastic constants of the bcc phase in niobium-zirconium alloys between 4.2 and 300 K

The adiabatic elastic constants have been determined for quenched single crystals of niobium-zirconium alloys of the following nominal compositions: 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, and 70 at.% Zr. The c11, c12, and c44 constants as well as the shear constant C′= (1/2) (c11−c12) and the bulk modulus are given between 4.2 and 300 K. We observe for the whole range of compositions an anomalous temperature behavior for c44 similar to that found in pure Nb, but here the amplitude of the effect disappears progressively with increasing Zr concentration. The Debye temperatures were computed from the constants at 4.2 K.

Hypersound propagation in oriented polymethylmethacrylate

AbstractThe hypersonic elastic properties of oriented polymethylmethacrylate (PMMA) have been studied using high contrast (>108) multipass Fabry‐Perot techniques. Four samples of different degrees of orientation (birefringence Δn = −4, −9, −15, −24 × 10−4) were prepared by drawing at 130°C. Measurements of longitudinal sound velocity, VL, as a function of ϕ, the angle between the draw and propagation directions, show an anisotropy which scales with Δ and amounts to 6% in the sample having Δn = 24 × 10−4. At ϕ = 0, VL = 2.80 kilometers per second, (KMS−1), the same as the value in an isotropic sample. The elastic constants C11 and C33 are measured as a function of Δn. Data as a function of sound wavelength, Λ, show dispersive effects in both VL and attenuation αΛ, for Λ > 0.7 μ. Over the range 0.7 μ < Λ < 1.3 μ, αΛ increases from 0.1 to 0.5 and VL increases by 8%. Attenuation is consistently higher for ϕ = 0. The data are interpreted in terms of densification (up to 15%) accompanied by void formation on orientation, the structure having characteristic dimensions of approximately 1.5 μ. The structure may be asymmetric with the longer dimension perpendicular to the draw direction. Additional Brillouin peaks may result from sound propagation in the voids (VL = 5.8 KMS−1) in which case the density is approximately 0.3 g cm−3. Measurements of the Landau‐Placzek ratio support these ideas, but evidence from electron micrographs is inconclusive. Dispersion measurements should in principle help determine details of the structure shape.

Frequency Change and Elastic Constants of Quartz Irradiated by 1 MeV Electrons

The fractions of frequency change Δf/f for quartz resonator plates exposed to 1 MeV electrons are measured as a function of electron fluence up to 2×1017 electrons/cm2, from which the adiabatic elastic constants of c44, c14 and c66 are determined. The ratio of these fractional changes is found to have a correlation of Δc44/c44: Δ|c14|/|c14|: Δc66/c66=1.3: 3.1: 1.0. The Δf/f values against fluence are calculated for various cut angles. The result indicates that the frequency in AT-cut resonators decreases a little and then increases (∼10-5), but in BT- and Y-cut resonators a drastic increase and then a decrease (10-3∼10-2) are expected.

Brillouin scattering in ferroelectrical Pb3MgNb2O9

Brillouin spectra of transverse and longitudinal phonons which correspond to the elastic constants C11, c12 and c44 have been measured in a wide temperature region from 77 to 650°K. The observed phonons show softening in the phase transition region.

Structure and dynamics of microcrystalline graphite, graphon, by neutron scattering

The structure and dynamics of “graphon”, a carbon black, have been characterised by a combination of neutron diffraction and inelastic scattering experiments. The techniques employed were those of neutron diffraction, time-of-flight and beryllium-filter spectroscopy.The dimensions of the unit cell were found to be: a= 2.47 ±(0.01)A, c= 6.93 ±(0.01)A, and the average dimensions of the layer lattice were estimated from high angle diffraction as La= 30 A and Lc= 85 A.Well-defined average phonon groups were observed by inelastic scattering and used to construct an almost sinusoidal dispersion curve for longitudinal acoustic lattice vibrations along the c-axis. The stiffness constant C33, found from the dispersion curve, was 3.00 ±(0.02)× 1011 dyn cm–2. The frequency distribution of the lattice vibrations shows singularities at energies qualitatively in agreement with those calculated by Young and Koppel from specific heat measurements on graphite.

Elastic constants of niobium-rich zirconium alloys between 4.2 K and room temperature

The adiabatic elastic constants were determined for niobium and for its alloys containing 1.4, 3.6, and 6.0 at.% zirconium. The c11, c12, and c44 constants as well as the shear constant C′ = (1/2)(c11 − c12) and the bulk modulus were tabulated between 4.2 and 290 K. The unusual temperature dependence of the shear constant c44 found in niobium was also observed in the alloys. The Debye temperatures were computed from the constants at 4.2 K.