Cubic Crystal Symmetry Research Articles

Lattice dynamics of Bi12GeO20 crystal: Polarized Raman scattering and first-principles analysis

Our paper presents a combined experimental and theoretical study of the lattice dynamics of Bi12GeO20 crystal. Polarized Raman spectra were measured in two x-ray oriented single crystals. In a crystal of cubic symmetry, phonon modes of A- and E-types are spectroscopically indistinguishable. Using a special experimental technique, we separated the phonon modes that transform according to A and E irreducible representations. Since the number of E-type lines observed in Raman scattering is higher than theoretically allowed by group theory, we performed a first-principles calculation of the Bi12GeO20 lattice dynamics to accurately identify the symmetry of phonon modes. All experimental lines observed in Raman scattering are classified according to irreducible representations of the cubic I23 group. Using samples of different thicknesses, 10 mm and 0.120 mm, we have demonstrated that the optical activity plays an insignificant role at Raman backscattering in Bi12GeO20 crystal.

Dynamic Properties and Focusing of Phonons in Metallic and Dielectric Crystals of Cubic Symmetry. Review 1

Dynamic Properties and Focusing of Phonons in Metallic and Dielectric Crystals of Cubic Symmetry. Review 1

Formation of pseudo-hexagonal pyramidal structures on (111) crystallographic surfaces of cubic PbTe crystals by ion sputtering

On the example of the PbTe compound we present the results of the study of the interplay between the crystallographic orientation of the surfaces of high reticular density and easy cleaving of the crystals of cubic symmetry and the shape of surface structures induced by prolonged ion sputtering. We found that in case of right combination of the crystallographic orientation of both the sputtering surface and the planes of high reticular density in the aggregate with easy cleaving of the crystals, the pyramidal structures of pseudo-hexagonal shape can be formed on the sputtering surface of cubic crystals. In case of PbTe these are the surfaces of (111) crystallographic orientation subject to the presence of sputter-resistant protective shields on them. When the protective shields are destroyed, the hexagonal structures are transformed into trihedral pyramids. The pseudo-hexagonal pyramidal structures are characterized by indistinct edges and terrace-shaped lateral facets, whereas the facets of the trihedral pyramids, into which they are transformed during the ion bombardment, are mirror-smooth. We prove that: (a) the maximally sharpened pseudo-hexagonal pyramids, ion-induced on PbTe (111) sputtering surfaces, are formed as a result of self-organization of the crystallographic planes of the family {0ī2}, the reticular density of which directly follows the reticular density of the plane of the sputtered surface; (b) the roughness of the lateral facets of pseudo-hexagonal pyramids, as well as the mirroring of the facets of the trihedral pyramids is due to the impact of crystallographic planes of the {100}family, which are the planes of the greatest reticular density and easiest cleaving for PbTe crystals.

Features of the propagation of phonons in graphene nanostructures. Fast high-frequency phonons in a quasi-flexural mode

The character of propagation in graphene nanostructures of quasi-flexural phonons, whose dispersion law differs from that of sound, is analyzed. Based on the calculation of the frequency dependences of the group velocities and the values of the path of quasiparticles for one period of oscillation, the frequency intervals are established at which: i) phonons propagate freely along all directions of reciprocal space—the propagon zone; ii) phonon propagation along some directions is impossible—diffuse zone; iii) phonons are localized at the nodes of the honeycomb lattice—the locon zone. A comparison is made with a similar classification of phonons in a three-dimensional crystal of cubic symmetry.

Deformation of Nickel Titanium Single Crystal under Compressive Pulse

Nickel titanium single crystal is used to show the importance of the volume compressibility anisotropy in analyzing the elastoplastic deformation processes in metal single crystals with the cubic symmetry. The process of uniform bulk deformation corresponds to the process of nonuniform stress-strain state of metal single crystals of cubic symmetry for several orientations of the theoretical coordinate system relative to crystallographic directions of the main axes. The indicative surface of the volume compressibility (or its reciprocal variable of bulk modulus) has an aspherical shape and is the function of the Euler angles. This is shown for the first time based on the solution of the model problem, viz. determination of the stress-strain state of a spherical body made of nickel titanium single crystal during the transmission of a compressive pulse through the tested material. In general case of orientation of the theoretical coordinate system relative to main crystallographic axes, a spherical body made of nickel titanium single crystal undergoes deformation due to the compressive load and acquires the shape of a two-axial ellipsoid.

Effective stiffness tensor of nanocrystalline materials of cubic symmetry: The core-shell model and atomistic estimates

Anisotropic core-shell model of a nano-grained polycrystal, proposed recently for nanocrystalline copper, is applied to estimate elastic effective properties for a set of crystals of cubic symmetry. Materials selected for analysis differ in the lattice geometry (face-centered cubic vs. body-centered cubic) as well as the value of a Zener factor: a ratio of two shear moduli defining elastic anisotropy of a cubic crystal. The predictions are verified by means of the atomistic simulations. The dependence of the overall bulk and shear moduli on the average grain diameter is analysed. In the mean-field approach the thickness of the shell is specified by the cutoff radius of a corresponding atomistic potential, while the grain shell is isotropic and its properties are identified by molecular simulations performed for very small grains with approximately all atoms belonging to the grain boundary zone. It is shown that the core-shell model provides predictions of satisfactory qualitative and quantitative agreement with atomistic simulations. Performed study indicates that the variation of the bulk and shear moduli with the grain size changes qualitatively when the Zener anisotropy factor is smaller or greater than one.

Optical Anisotropy of Photonic Crystals of Cubic Symmetry Induced by Multiple Diffraction of Light

The optical spectra of Bragg reflection from opal-like photonic crystals under conditions of the resonant enhancement of the multiple diffraction of light have been studied experimentally and theoretically using the photonic crystal structures prepared of monodisperse polystyrene globules. It is shown that the reflection signal registered in mutually orthogonal configurations of the polarizer and analyzer is related to the intrinsic optical anisotropy of the crystals and is a specific manifestation of the multiple Bragg diffraction in three-dimensional photonic crystals.

Averaging of elastic constants for polycrystals

Many materials of interest are polycrystals, i.e., aggregates of single crystals. Randomly distributed orientations of single crystals lead to macroscopically isotropic properties. Here, we briefly review strategies of calculating effective isotropic second and third order elastic constants from the single crystal ones. Our main emphasis is on single crystals of cubic symmetry. Specifically, the averaging of third order elastic constants has not been particularly successful in the past, and discrepancies have often been attributed to texturing of polycrystals as well as to uncertainties in the measurement of elastic constants of both poly and single crystals. While this may well be true, we also point out here shortcomings in the theoretical averaging framework.

On the magnetization process and the associated probability in anisotropic cubic crystals

We present a theoretical method to calculate specific magnetic properties, e.g. magnetization curves, magnetic susceptibility and probability landscapes along the [100], [110] and [111] crystallographic directions of a crystal of cubic symmetry. The probability landscape displays the evolution of the most probable angular orientation of the magnetization vector, for selected temperatures and magnetic fields. Our method is based on the premises of classical statistical mechanics. The energy density, used in the partition function, is the sum of magnetic anisotropy and Zeeman energies, however no other energies e.g. elastic or magnetoelastic terms are considered in the present work. Model cubic systems of diverse anisotropies are analyzed first, and subsequently material magnetic systems of cubic symmetry; namely iron, nickel and Cox Fe100−x compounds, are discussed. We highlight a correlation between magnetization curves and the associated probability landscapes. In addition, determination of easiest axes of magnetization, using energy consideration, is done and compared with the results of the present method.

Higher-order elastic constants and megabar pressure effects of bcc tungsten:Ab initiocalculations

The general method for the calculation of $n\mathrm{th}$ ($n\ensuremath{\ge}2$) order elastic constants of the loaded crystal is given in the framework of the nonlinear elasticity theory. For the crystals of cubic symmetry under hydrostatic compression, the two schemes of calculation of the elastic constants of second, third, and fourth order from energy--finite strain relations and stress--finite strain relations are implemented. Both techniques are applied for the calculation of elastic constants of orders from second to fourth to the bcc phase of tungsten at a 0--600 GPa pressure range. The energy and stress at the various pressures and deformations are obtained ab initio in the framework of projector augmented wave+generalized gradient approximation (PAW+GGA) method, as implemented in Vienna Ab initio Simulation Package (VASP) code. Using the obtained results, we found the pressure dependence of Gr\uneisen parameters for long-wave acoustic modes in this interval. The Lam\'e constants of second and third order were estimated for polycrystalline tungsten. The proposed method is applicable for crystals with arbitrary symmetry.

Theoretical foundations of resonant ultrasound spectroscopy at high pressure

The theory of free-vibration acoustic resonance (FVAR) of solids at high pressure is developed within the framework of nonlinear elasticity and the calculus of variations. The FVAR state is formulated as a generalization of the conventional theory that was originally developed by Rayleigh and Ritz in the sense that it includes geometrical and material nonlinearities as well as the potential energy of external pressure. Magnetic point groups and quasi-harmonic approximation are used so as to obtain a natural extension of normal mode phonons in the high-pressure regime. The numerical analysis of eight different cubic-symmetry crystals reveals that FVAR frequencies depend linearly on the pressure, and the slopes vary with the FVAR modes, including the sign. We estimated the mode Grüneisen parameter up to N =2400 and proved that the high-frequency limit γ ∞ is equivalent to the conventional Grüneisen parameter γ . Quantitative agreement of the parameters demonstrates that nearly the entire third-order elastic constants tensor can be determined from high-pressure ultrasound spectroscopy experiments.

Variation of spectral properties of dielectric ionic crystal in the terahertz range due to the polariton absorption.

The dispersion equations for polariton waves in dielectric ionic crystal with the absorption are obtained. The self-consistent solutions of the system of Maxwell electromagnetic field equations and the equations of motion of ions have been used. The elastic and absorption properties of the crystal are taken into account in the ion equations of motion. It is shown that the separated equations of motion for positive and negative ions allow obtaining all branches of phonon and polariton spectrum by the example of the ionic crystal of cubic symmetry at the terahertz range. It has been shown that the variation of absorption in the crystal leads to changing of the character of spectrum branch and the polariton velocities.

A model for finite-deformation nonlinear thermomechanical response of single crystal copper under shock conditions

A physically consistent framework for combining pressure–volume–temperature equations of state with crystal plasticity models is developed for the application of modeling the response of single and polycrystals under shock conditions. The particular model is developed for copper, thus the approach focuses on crystals of cubic symmetry although many of the concepts in the approach are applicable to crystals of lower symmetry. We employ a multiplicative decomposition of the deformation gradient into isochoric elastic, thermoelastic dilation, and plastic parts leading to a definition of isochoric elastic Green-Lagrange strain. This finite deformation kinematic decomposition enables a decomposition of Helmholtz free-energy into terms reflecting dilatational thermoelasticity, strain energy due to long-range isochoric elastic deformation of the lattice and a term reflecting energy stored in short range elastic lattice deformation due to evolving defect structures. A model for the single crystal response of copper is implemented consistent with the framework into a three-dimensional Lagrangian finite element code. Simulations exhibit favorable agreement with single and bicrystal experimental data for shock pressures ranging from 3 to 110GPa.

Potentials of acousto-optical spectrum analysis on a basis of a novel algorithm of the collinear wave heterodyning in a large-aperture KRS–5 crystalline cell

The technique under proposal for a precise spectrum analysis within an algorithm of the collinear wave heterodyning implies a two-stage integrated processing, namely, the wave heterodyning of a signal in a square-law nonlinear medium and then the optical processing in the same solid state cell. The technical advantage of this approach lies in providing a direct multichannel parallel processing of ultra-high-frequency radio-wave signals with essentially improved frequency resolution. This technique imposes specific requirements on the cell's material. We focus our attention on the solid solutions of thallium chalcogenides and take the TlBr-TlI (thallium bromine-thallium iodine) solution, which forms KRS-5 cubic-symmetry crystals with the mass-ratio 58% of TlBr to 42% of TlI. Analysis shows that the acousto-optical cell made of a KRS-5 crystal oriented along the [111]-axis and the corresponding longitudinal elastic mode for producing the dynamic diffractive grating can be exploited. With the acoustic velocity of about 1.92 × 105 cm/s and attenuation of ∼10 dB/(cm GHz2), a similar cell is capable of providing an optical aperture of ∼5.0 cm and one of the highest figures of acousto-optical merit in solid states in the visible range. Such a cell is rather desirable for the application to direct 5000-channel parallel spectrum analysis with an improved up to 10−5 relative frequency resolution.

Photorefractive vectorial wave mixing in different geometries

We analyze vectorial wave mixing in a photorefractive crystal of cubic symmetry in different geometries of beam interactions--reflection, transmission, and orthogonal. It is shown that orthogonal geometry in contrast with others supports an efficient phase demodulation of a depolarized object wave in linear mode without using any polarization-filtering elements. As a result adaptive interferometers based on the orthogonal geometry can provide a higher signal-to-noise ratio due to lower noise and lower optical losses.

A multichannel adaptive interferometry system

A multichannel adaptive interferometer based on dynamic reflecting holograms multiplexing in a photorefractive crystal of cubic symmetry is proposed. The efficiency of hologram multiplexing is studied theoretically and experimentally. For the recording geometry used, a crosstalk between channels is shown to be less than the single-channel inherent noise for a wide angular range of beams entering the crystal, and the sensitivity depletion in one channel is insignificant for a large number of channels.

The calculation of magnetic induction in grain orientated ultra-thin silicon steel sheets

In this work the magnetic induction in grain orientated ultra-thin silicon steel sheets (with a thickness of 0.08 mm) was calculated by employing the crystal orientation distribution function and the formula for the anisotropy of energy for a single crystal of cubic symmetry. The incomplete pole figures of {110}, {200} and {112} were measured and the corresponding orientation distribution function was determined. On the basis of the texture data and the corresponding magnetic anisotropy energy, the magnetic induction in the ultra-thin silicon steel sheet was calculated. The calculation results are in good agreement with the experimental measurement.

Elastic constants of equiatomic L1 0-ordered FePd single crystals

The elastic properties of a γ 1-FePd single crystal have been measured using the pulse–echo overlap method. The anisotropic elastic constants of this tetragonal L1 0 structure are explained in part by the different bond strength between Fe–Pd neighbors compared with Fe–Fe and Pd–Pd neighbors. The tetragonal elastic anisotropy as measured for the single crystal appears to depend on the remaining fraction of misoriented order variants. The anisotropy parameters for γ 1-FePd are ( c 44/ c 66) = 0.93 ± 0.04, A 1 = 2 c 66/( c 11 − c 12) = 2.78 ± 0.14 and A 2 = 4 c 44/( c 11 + c 33 − 2 c 13) = 2.53 ± 0.40, implying a shear stiffness along [1 0 0] that is smaller in (0 0 1) than in (0 1 0). The magnitude of the elastic anisotropy (e.g., A 1 and A 2) of γ 1-FePd is similar to that of some face-centered cubic metals (e.g., Au with A = 2 c 44/( c 11 − c 12) = 2.90 and Ni with A = 2.51) and considerably larger than that of γ-TiAl (e.g., A 1 = 1.44 and A 2 = 1.98). However, the deviations of the considerably anisotropic elastic properties of the tetragonal γ 1-FePd from that expected for an equivalent elastically anisotropic crystal of cubic symmetry are less pronounced than for γ-TiAl. In contrast to γ-TiAl, the elastic constants indicate that shearing in the (1 1 1) plane is easiest along [−1 1 0] and increases in difficulty as the shear direction changes via [−1 0 1] towards [−1 −1 2] for γ 1-FePd.

Poisson’s ratio of anisotropic systems

The Poisson’s ratio of anisotropic materials depends, in general, both on a “longitudinal” direction along which the stress is changed and on a “transverse” direction in which the transverse deformation is measured. For cubic media there exist “longitudinal” directions, parallel to the 4-fold and 3-fold axes, for which the Poisson’s ratio does not depend on the “transverse” direction. Depending on the tensor of elastic compliances (or elastic constants), crystals of cubic symmetry can exhibit negative Poisson’s ratio in both these directions (they are called strongly auxetics), in one of them (i.e. either along the 4-fold axis or along the 3-fold one; they are called partially auxetic) or in none of them. For crystals exhibiting 3-fold symmetry axis the Poisson’s ratio along this axis does not depend on the “transverse” direction. For other “longitudinal” directions the Poisson’s ratio depends, in general, on the “transverse” direction. The Poisson’s ratio averaged with respect to the “transverse” direction depends only on the “longitudinal” direction and can be conveniently presented graphically. As an example the f.c.c. hard sphere crystal is considered. It is shown that the average (with respect to “transverse” direction) Poisson’s ratio of the hard sphere crystal is positive for all “longitudinal” directions. One should add, however, that there exist directions for which the (not averaged) Poisson’s ratio of hard spheres is negative.

Electric polarization induced by optical orientation of dipolar centers in non-polar piezoelectrics

We predict new nonlinear optical phenomenon, static electric polarization induced in non-polar piezoelectric by linearly polarized light owing to orientation of the impurity centers with permanent dipole moment. Microscopic model of this effect for the crystals of cubic symmetry is proposed. Estimations of the model show that the induced polarization might be much higher than that caused by optical rectification. It is shown that a transient current Jd generated under non-steady-state illumination could be observed at extremely high modulation frequency. We expect that the effect could be applied for ultrafast optical signal processing.