Constants C11 Research Articles

Per‐Olov Löwdin curriculum vitae

Per‐Olov Löwdin curriculum vitae

Elasticity of single-crystal calcite and rhodochrosite by Brillouin spectroscopy

The single-crystal elastic moduli of natural samples of both calcite (CaCO3) and rhodochrosite (MnCO3) have been measured by Brillouin spectroscopy under ambient condition. Based on the trigonal unit cell, the elastic constants C11, C33, C44, C12, C13, and C14 are 149.4(7), 85.2(18), 34.1(5), 57.9(11), 53.5(9), -20.0(2), and 223.9(15), 132.6(41), 44.5(9), 93.4(21), 76.0(23), -17.3(6) GPa for CaCO3 and MnCO3, respectively. Our data for calcite are in good agreement with earlier data obtained by ultrasonic experiments. The off-diagonal elastic constants (C12, C13, and C14) for rhodochrosite have systematically larger values than the trend defined by other isostructural carbonates, in all of which the divalent cations are alkaline-earth metals. This is a distinctive signature of transition- metal-bearing oxides, which is present in silicates and simple oxides as well.

Thermal expansion of single crystals of the layered compounds TlGaSe2 and TlInS2

Experimental data are reported from studies of the thermal expansion of single crystals of the layered compounds TlGaSe2 and TlInS2 in the plane of the layers and in the direction perpendicular to the layers in the temperature region 20–300 K. Observed features of the thermal expansion are discussed in the light of the available data on the elastic properties of TlGaSe2 and TlInS2. It is found that the differences in the temperature behavior of the thermal expansion coefficients in the plane of the layers α∥(T) in the TlGaSe2 and TlInS2 crystals are due to the different degree of anisotropy of these crystals and to the different value of the “interlayer” elastic constant C13.

Second-order elastic constants for pentaerythritol tetranitrate single crystals

Existing sound speed measurements for various orientations of pentaerythritol tetranitrate single crystals were reanalyzed to obtain corrected values for the second-order elastic constants. Previous analysis in the literature had resulted in an inconsistency in the elastic constant values. The source of the inconsistency is identified. The corrected analysis restores consistency to the results and yields a significantly different value for the elastic constant c13.

Elastic properties of nanocrystalline zirconium–silicon–boron thin films

Thin amorphous films of ZrB3 grown on Si(111) substrates by chemical-vapor deposition are, upon annealing at 960 °C, transformed to films composed of nanocrystallites (6–10 nm extent) with a nominal composition of Zr0.9Si0.3B3. The independent elastic constants of the layers are determined from the dispersion of the surface and pseudo-Sezawa acoustic excitations and reveal large enhancements in the C11 and C44 constants accompanying the conversion of ZrB3 to the nanocrystalline phase. Since the transverse sound velocities of the binary and Zr0.9Si0.3B3 are in near resonance with the sound velocity of Si (VTSi=4.8 km/s), only the Rayleigh surface wave is localized to the film while all higher-order acoustic modes are evanescent. Despite the strong decay channels, high-lying excitations with velocities as large as 25 km/s (≫VTSi) are observed in Brillouin light scattering. Insight into these acoustic properties is provided by evaluating the elastodynamic Green’s functions and associated acoustic-mode densities.

Elastic, electronic and crystal structure of SrH2 by the method of pseudopotentials

The pseudopotentials method is used to calculate some properties of the solid strontium hydride, SrH2, namely the structural, electronic and elastic ones. The validity of these results is tested by comparison with their equivalents calculated in CaH2. The analysis of bands structure, Mulliken's populations and density of states, has shown that SrH2 is an insulator solid, less ionic than CaH2, its band of valence is constituted primarily by the 3s(H) orbital with a weak participation of 1s(H) and 2s(H) orbitals, and its band of conduction is formed essentially by the orbital 2p(H) with a small contribution of both the 2s(H) orbital and the orbital describing the two valence electrons of the Sr atom (5s(Sr) or 5sp(Sr)). The addition of the orbital d(Sr), on the one hand, has a remarkable effect on the distribution of electronic charges and, on the other hand, it participates strongly with the orbital 2p(H) in the formation of the conduction band. The analysis of elastic properties, calculated for the first time, by means of correlations with their equivalents calculated in CaH2, has placed the existence of two series of constants in a prominent position. The first one regroups the constants C11, C12 and C33; the second one is formed by the other constants, including the bulk modulus B0. This separation is confirmed by the analysis of charge densities.

Structural and vibrational properties of GaN

Structural and vibrational properties of device quality pure GaN substrate grown using a lateral epitaxial overgrowth (LEO) technique were studied using x-ray diffraction, Brillouin, Raman, and infrared spectroscopy. Lattice constants were found to be a=3.1896±0.0002 Å and c=5.1855±0.0002 Å. Comparing the results with those on GaN epilayer directly grown on sapphire substrate, it is shown that the GaN substrate is indeed of high quality, i.e., the lattice is relaxed. However the GaN substrate has a small enough but finite residual strain arising from the pileup of the lateral growth front on SiO2 masks in the course of LEO. It was also found that the elastic stiffness constants C13 and C44, are more sensitive to the residual strain than the optical phonon frequencies. The high frequency and static dielectric constants were found to be 5.14 and 9.04. The Born and Callen effective charges were found to be 2.56 and 0.50.

Nondestructive Determination of the Elastic Anisotropy in Aluminum—SiC and —Al 2 O 3 Metal—Matrix Composites

Process control is becoming increasingly important to the high performance applications of aluminum metal—matrix composites. Most processing conditions of metal—matrix composites introduce a degree of anisotropy that results in the directional dependence of material properties. Aluminum alloys reinforced with up to 20 volume % silicon-carbide or alumina particles are examined using ultrasonic velocity measurements. The measurements indicate that these composites are transversely isotropic and thus their elastic properties can be described by the five independent elastic constants C11 , C33 , C44 , C12 , and C23 . Anisotropy parameters are determined assuming either a preferred alignment of the matrix crystallites or a preferred alignment of the reinforcement. Comparing measured and theoretical anisotropy parameters for partially aligned reinforcement or for texture shows that neither is the single contributor to anisotropy in the composite material.

Piezoelectric Resonance Properties of a β Phase Quartz Plate

We have previously reported an artificial twinning (x-axis inversion) technique for rotated Y-cut quartz plates. The twinning is produced by mechanical stress through a thin metal film deposited on the plate. Such a twinning phenomenon occurs at a considerably lower temperature than the α-β transition temperature, Tc, of 573°C. This paper presents the electromechanical properties of quartz at a temperature region higher than the twinning temperature. Degeneration of the piezoelectric resonances corresponding to the x-axis inversion area and the non-inversion area are observed at a temperature between the twinning temperature and Tc. The elastic stiffness constants c55 and c66, estimated from the degeneration mode resonance frequency, agree with these for the β phase. This suggests that the α-β transition occurs by the thin metal film at a considerably lower temperature than Tc of 573°C. An intense piezoelectric response with a high quality factor above 20,000 is observed from the degeneration temperature to above 660°C.

Acoustic softening in ZrTiO4 crystal

The temperature dependence of the elastic constant c 55 of ZrTiO 4 crystal has been studied by ultrasonic method. In the incommensurate phase c 55 is found to soften towards the commensurate–incommensurate transition temperature T c . The transition mechanism is discussed phenomenologically.

Mechanical and photoelastic properties of oriented poly-4-methyl-1-pentene investigated by Brillouin spectroscopy

The elastic constants C11, C13, C33 and C44 of oriented poly-4-methyl-1-pentane (P4M1P) films were measured with Brillouin scattering. The photoelastic constants p11, p13, p31 and p33 were measured by evaluating the integral intensities of the phonon lines. The correlation of the photoelastic constants p13 and p31 with the degree of stretching was determined by evaluating the relative integral intensities of the Brillouin lines with the phonon propagation vector into and perpendicular to the stress direction. The other photoelastic constants were measured by comparing the angle-dependent relative integral intensities of the transverse and longitudinal Brillouin lines. The elastic constants, as well as birefringence, were surprisingly found to be unaffacted by further stretching of the sample above the necking region. The density was constant for all degrees of stretching. The photoelastic constants p11 and p33 showed significant variation during deformation. The results have been compared with former measurements of polypropylene (PP) by Cavanaugh and Wang. Because the elastic constants are unchanged during deformation, the modulation of the intensities is due to the variation of the photoelastic constants for this material.

Temperature dependence of elastic, dielectric, and piezoelectric properties of “single crystalline’’ films of vinylidene fluoride trifluoroethylene copolymer

Elastic, dielectric, and piezoelectric constant matrix elements of a “single crystalline’’ (SC) film of vinylidene fluoride trifluoroethylene copolymer, P(VDF/TrFE), in which the orthorhombic [001] and [110] axes of fully extended chain crystals are preferentially oriented parallel to the stretching axis and normal to the surface, respectively, were measured at temperatures ranging from 10 K to the Curie point (402–404 K) by using a piezoelectric resonance method. All of the electromechanical coupling factors (k31, k32, k33, k24, and k15) are larger than those of conventional lamellar crystalline films. Some of the matrix elements for a P(VDF/TrFE) single crystal are derived from the measured values of constant matrix elements for the SC film. Some features characteristic of the SC film are revealed. The SC film has a large Young’s modulus for the stretching direction (1/s11) (121 GPa at 10 K). The properties related to the molecular motions along the chain axis, such as 1/s11, shear stiffness constant c55, shear piezoelectric constant e15, etc., exhibit strong relaxations around 250 K. The origin of these relaxations in the crystalline phase is discussed.

Similar elastic behaviour in mhseo4crystals near the superionic transition

Abstract Brillouin scattering investigations in NH4-, Rb- and CsHSeO4 are reported near their superionic transition. A similar elastic behaviour is observed, consisting of strong downward jumps of elastic constants C11, C22 and C33, at the transition, and broadenings of the corresponding Brillouin lines above the transition. No pretransitional evolutions occur in both phases.

Elastic Properties of Diamond Like Carbon Thin Films: a Brillouin Scattering Study

AbstractDiamondlike carbon (DLC) thin films have been widely used as hard coatings in a variety of applications where increased wear resistance and hardness are required. Vacuum arc DLC films are among the hardest, with measured hardness values of up to 68 GPa. In our deposition process a repetitively pulsed bias voltage is applied to the substrate, which controls the energy of the incoming C ions. DLC chemical and mechanical properties are strongly affected by the energy of the depositing ions. In this paper, we relate the mechanical properties of these films to the deposition parameters, and describe our initial Brillouin scattering measurements of the elastic constants of monolithic DLC films. Evidence for bulk longitudinal and surface Rayleigh excitation in films with thickness of 50 and 500 nm has been observed. Since the DLC films are amorphous, they are modeled as isotropic solids and the elastic constants CII and C44 are derived.

Timoshenko's Shear Coefficient for Transverse Vibrations of Trigonal Crystals of Rectangular Cross Section

In this paper, we derive the shear coefficient K for trigonal crystals of rectangular cross section in Timoshenko's beam theory of three dimensional (3-D) elasticity. The theoretical result gives K as a function of the transformed elastic stiffness constants c22, c44, c66. For an isotropic material it agrees completely with that derived by G. R. Cowper. In addition, values of K versus cut angle for quartz crystal, lithium tantalate, and lithium niobate which belong to the trigonal system are calculated, so that K has a value of 0.796-0.867 for quartz crystal, 0.834- 0.857 for lithium tantalate, and 0.831-0.863 for lithium niobate, when Z-plates of the crystals are rotated with a cut angle θof 0-180°about the x-axis. The values of K are then compared with those for isotropic materials obtained by other authors.

The elastic behaviour of polycarbonate in the glassy state determined by Brillouin scattering

Measurements of the temperature and hydrostatic pressure dependence of the elastic constants of amorphous polycarbonate in the glassy state using high-resolution Brillouin spectroscopy are reported. It was found that the temperature and the pressure dependence of the longitudinal and the transverse sound velocity are strictly linear, in contrast to the behaviour found in polymethylmethacrylate (PMMA). A simple equation for the behaviour of the elastic constants c11 and c44, and the shear, Young's and bulk modulus, can be given, which fits the experimental data well.

Ultrasonic observation of the [formula omitted] lock-in in holmium

The elastic constant C33 and attenuation α33 of Ho measured in b-axis directed applied fields indicate marked anomalies which correlated with the temperature extent of the lock-in at wave vector τ = 518c∗ as observed with neutron diffraction. A phase diagram is presented.

Elastic behaviour of RbHSeO4at the superionic phase transition investigated by brillouin scattering

Abstract Brillouin investigations have been performed over the temperature range 160–200°C, which includes the superionic phase transition near ~178°C. Strong discontinuities are obtained for the three elastic constants Cll, C22 and C33 at the superionic transition. In the superionic phase, two longitudinal Brillouin doublets can be observed for every direction of the acoustic wave vector. This fact can be attributed to the presence of two kinds of regions above the transition. Besides, systematic broadenings of these longitudinal lines are observed in the superionic phase. The results are qualitatively discussed in connection with theoretical expectations.

New criterion for measurement of stiffness constants c44, c55, c66 and c33 of lithium niobate and quartz

In this paper a new face shear resonance criterion to determine the shear stiffness constants is established. The criterion states that during longitudinal vibrations of a rectangular plate face shear resonance can occur if the face diagonal end points have effective phases of pi /2. The resulting shear wavelength lambda becomes equal to 2 square root (ab) where a and b denote the plate face edge dimensions. The shear stiffness constants c44, c55 and c66 for lithium niobate and quartz crystals were determined using specimens in the form of bars, square or rectangular plates with edges along the crystallographic X, Y, Z axes. The axial stiffness constant c33 for quartz was determined from the arithmetic harmonics of c33/2 generated due to shear vibrations of YZ or ZX faces. This criterion provides a much simplified method for the measurement of these constants without loss of accuracy.

Elastic and gyrotropic anomalies and acoustic activity in lead germanate

The paper discusses elastic and gyrotropic anomalies as well as the acoustic activity of lead germanate during the phase transition from the space group C3h to C3. Expressions are derived for all the anomalies of the elastic constants, and relations among them have been established. We show that the elastic constants C11, C33, C12, and C13 suffer a dip at the transition temperature, in agreement with experimental results. Several invariants involving the order parameter, strain variables, and their derivatives are constructed and from these the contribution to the coupling energy of the gyrotropic energy has been calculated. Expressions are given for the anomalies of the various gyrotropic constants. We show that the system will exhibit anomalies in acoustic activity in the low symmetry phase, and an expression for the polarization angle has been obtained.