Microscopic theory of second-order and third-order elastic constants for an NaCl crystal

Relations between the second-order and third-order symmetry-independent elastic constants and the energy of interatomic interactions dependent on the mutual arrangement of pairs and triplets of atoms are obtained for crystals belonging to the crystal class O _h. The derived relations and experimental data on the elastic constants are used to calculate four third-order elastic constants and the temperature dependence of the elastic anisotropy factor a(T) for an NaCl crystal. The calculated dependence a(T) is in qualitative agreement with the experimental dependence a _exp(T).     

Experimental determination of third-order elastic constants of diamond

To determine the nonlinear elastic response of diamond, single crystals were shock compressed along the [100], [110], and [111] orientations to 120 GPa peak elastic stresses. Particle velocity histories and elastic wave velocities were measured by using laser interferometry. The measured elastic wave profiles were used, in combination with published acoustic measurements, to determine the complete set of third-order elastic constants. These constants represent the first experimental determination, and several differ significantly from those calculated by using theoretical models.     

Factorization of quantities arising in second- and third-order perturbation theory for atoms

A closed expression is found for the real representation of the Coulomb Green's function for negative energy. Equations are derived to take into account the atomic characteristics in second- and third-order perturbation theory for atoms of the first and second groups in the periodic table.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 54–58, December, 1969.     

Second-order and third-order elastic constants of B4C ceramics

The linear and nonlinear elastic properties of B₄C boron carbide ceramics have been studied. The second-order elastic constants and other parameters of the theory of elasticity in the linear approximation have been calculated based on the experimentally measured density and velocity values of longitudinal and shear bulk acoustic waves in the samples. The Thurston-Brugger method has been used to determine the third-order elastic constants of B₄C. To achieve this, we have measured the relative changes of the longitudinal and shear bulk acoustic wave velocities depending on the uniaxial compression applied to the sample.     

Superintegrability with second- and third-order integrals of motion

Superintegrable systems in two-dimensional Euclidean space E ₂ with scalar potentials and integrals of motion that are second- or third-order polynomials in the momenta are discussed. We review old results, obtained in collaboration with Ya.A. Smorodinsky more than 40 years ago and present new ones that are their continuation. The text was submitted by the author in English.     

Statistical theory of the elastic constants of nematic liquid crystals

A statistical mechanical theory of the Frank elastic constants is formulated. The free energy functional is constructed for the deformed sample and the free energy density is defined for the case of small spatial gradients. The Frank constants are expressed in terms of the direct correlation function c(1, 2) and the orientational single particle distribution function. For the example of Onsager spherocylinders three constants K ₁, K ₂ and K ₃ are calculated. The results of these calculations are similar to those given by Priest and by Straley.     

Effect of concentration gradient on the first-, second- and third-order polarizabilities

The effect of concentration gradients on the polarizabilities of chemical systems of mixed composition is examined theoretically. It is found that the first-, second- and third-order polarizabilities all exhibit linear dependences on concentration gradient but nonlinear dependences on dielectric gradient. Furthermore, sample calculations imply that under selected conditions, concentration and/or dielectric gradient effects should be readily observable experimentally and indeed, under some conditions could dominate the experimental response. Examples of systems that might display the predicted effects include compositionally graded polymer, semiconductor, composite films, as well as thermally graded liquid mixtures and colloidal suspensions.     

Second-order and third-order elastic constants of B95 aluminum alloy and B95/nanodiamond composite

All independent second-order and third-order elastic constants in B95 aluminum alloy and B95/nanodiamond composite have been determined. To determine the second-order elastic constants, the densities and velocities of longitudinal and shear bulk acoustic waves in the materials under study have been measured. To quantitatively characterize the nonlinear elastic properties, the third-order elastic constants (TOECs) of B95 alloy and B95/nanodiamond composite have been determined. The Thurston-Brugger method has been used to experimentally determine the TOECs. For this purpose, the relative changes in the bulk wave velocity have been experimentally measured depending on the uniaxial compression applied to the samples under study and all independent TOECs have been calculated. The elastic wave velocities have been measured by the ultrasonic pulse method at a frequency of 10 MHz. The results obtained have been discussed.     

Investigation of the Kerr effect and third-order susceptibility constants in a nematic liquid crystal

In this paper, the static Kerr effect was investigated for the W-1680 at temperatures above the nematic-isotropic transition temperature and compared with 9CHBT. We experimentally measured the electro-optical Kerr constant and the pre-transitional behavior. The Kerr law has been confirmed and the variation of the inverse Kerr constant, with a temperature above the smectic-isotropic transition temperatures, were determined. The linear dependence of (T − T*)^− 1 on the Kerr constant is found to be in good agreement with the predications of the Landau-De Gennes model. Additionally, the temperature dependence of third order susceptibility has been studied. This value for these samples will reduce when the temperature increases.     

Third-order elastic constants and pressure derivatives of the second-order elastic constants of β-tin

The second- and third-order elastic constants and pressure derivatives of second-order elastic constants of tetragonal β-tin have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second- and third-order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The 12 non-vanishing third-order elastic constants and the six pressure derivatives of the second-order elastic constants in tetragonal β-tin are obtained in the present work and are compared with the available experimental values. The second-order elastic constant C₃₃ obtained in the present study is in reasonable agreement with the experimental values. The third-order elastic constants are generally one order of magnitude greater than the second-order elastic constants as expected of a crystalline solid. The third-order elastic constant C₃₃₃ is higher in magnitude than all other values. This shows a greater anharmonicity of β-tin along the c-axis direction of the crystal.     

Dispersive correction for the second- and the third-order nonlinear terms of the electrical energy density for a light-dielectric system

Optical and Quantum Electronics - Classification of hyperspectral image (HSI) is widely used for the study of remotely sensed images. Convolutional Neural Networks (CNNs) are one of the most...     

Self-consistent theory of second order elastic constants for nonionic anharmonic crystals

Elastic constants of strongly anharmonic nonionic dielectric crystals are studied within the pseudoharmonic approximation by means of two-time Green functions. The relation between the method of homogeneous deformation by Leibfried and Ludwig and that of long waves is investigated. These considerations lead to a generalization of the method of homogeneous deformation for the case of strongly anharmonic crystals. A comparison between the results of the two methods shows that the pseudoharmonic approximation satisfies the exact elastic sum rule. On leave from the Institute of Theoretical Physics University of Wrocław, Wrocław, Poland.     

A cobalt (II) complex with 6-methylpicolinate: Synthesis,characterization, second- and third-order nonlinear optical properties,and DFT calculations

A cobalt(II) complex of 6-methylpicolinic acid, [Co(6-Mepic)₂(H₂O)₂]·2H₂O, was prepared and fully determined by single crystal X-ray crystal structure analysis as well as FT-IR, FT-Raman. UV–vis spectra were recorded within different solvents, to illustrate electronic transitions and molecular charge transfer within complex 1. The coordination sphere of complex 1 is a distorted octahedron according to single crystal X-ray results. Moreover, DFT (density functional theory) calculations with HSEH1PBE/6-311 G(d,p) level were carried out to back up the experimental results, and form base for future work in advanced level. Hyperconjugative interactions, intramolecular charge transfer (ICT), molecular stability and bond strength were researched by the using natural bond orbital (NBO) analysis. X-ray and NBO analysis results demonsrate that O−H···O hydrogen bonds between the water molecules and carboxylate oxygen atoms form a 2D supramolecular network, and also adjacent 2D networks connected by C−H···π and π···π interactions to form a 3D supramolecular network. Additionally, the second− and third−order nonlinear optical parameters of complex 1 were computed at DFT/HSEH1PBE/6-311 G(d,p) level. The refractive index (n) was calculated by using the Lorentz−Lorenz equation in order to investigate polarization behavior of complex 1 in different solvent polarities. The first−order static hyperpolarizability (β) value is found to be lower than pNA value because of the inversion symmetry around Co (II). But the second−order static hyperpolarizability (γ) value is 2.45 times greater than pNA value (15×10⁻³⁰ esu). According to these results, Co(II) complex can be considered as a candidate to NLO material. Lastly molecular electrostatic potential (MEP), frontier molecular orbital energies and related molecular parameters for complex 1 were evaluated.     

Individual and joint influence of second- and third-order dispersion on the transmission quality in the presence of coherent interference

The shape of a resulting pulse (coherent interference superimposed to Gaussian pulse) along the linear optical fiber is determined by using attained analytical expressions. These analytical expressions enable the investigation of the influence of arbitrary dispersion order on pulse propagation. Individual and joint influence of second- and third-order dispersion on the transmission quality is examined in this paper. Pulse shape, eye diagram and bit error rate (BER) are used to represent in the best manner the above-mentioned influences.     

Second- and third-order elastic moduli of Co I

Elastic moduli of Co with A ₁-type structure are calculated using models in which the total energy is presented in the form of the energy of clusters consisting of pairs, triplets, and quadruplets of identical atoms. The irreducible energies of the clusters are assumed to contain two terms: E ∝ ?τ^?6, corresponding to the mutual attraction of the particles, and E ∝ ?τ^?6, corresponding to their mutual repulsion. Here, τ stands for the half-edge of an elementary cell. All 27 types of models expected in accordance with symmetry theory are analyzed. Of these, the 13 that yielded elastic modulus values satisfying the structural stability criteria are selected. The potentials of the selected models are used to predict the rigidity moduli of Co films with an A2-type structure.     

The elastic constants of the alkali-metals

Experimental results of the elastic constants c₁₁, c₄₄ and c are compared with theoretical predictions. It is shown that the alkali-metals obey a law of corresponding states rather closely, if lithium is excluded to some extent. The absolute magnitudes of the reduced elastic constants and their temperature dependence are finally discussed in the light of recent more sophisticated theories and other experimental evidence.     

Anomalous temperature dependence of third-order elastic constants in NH4Cl and NH4Br near the λ point

Second order elastic constants of NH₄Cl and NH₄Br crystals were measured under hydrostatic and uniaxial pressures near the λ-type phase transitions. Third order elastic constants were calculated from the pressure derivatives of the second order elastic constants. Results show that the temperature dependences of the third order elastic constants of NH₄Cl differ from those of NH₄Br dramatically. In addition, it is found that the coefficients $\text{1}\text{54}\text{(C}{}_{111}\text{+ 6C}{}_{112}\text{+ 2C}{}_{123}\text{)}$ and $\text{(}\text{1}\text{24}\text{√3)(C}{}_{111}\text{? 3C}{}_{112}\text{+ 2C}{}_{123}\text{)}$ of strains in the free energy of NH₄Cl show quite different temperature dependences from those of NH₄Br.     

The elastic constants of the alkali-metals

Experimental results of the elastic constants c₁₁, c₄₄ and c are compared with theoretical predictions. It is shown that the alkali-metals obey a law of corresponding states rather closely, if lithium is excluded to some extent. The absolute magnitudes of the reduced elastic constants and their temperature dependence are finally discussed in the light of recent more sophisticated theories and other experimental evidence.     

Macroscopic theory of elasticity of fourth order elastic constants of cubic structure solids

The macroscopic theory of elasticity is extended to determine the fourth order elastic constants. The expressions for the small amplitude sound wave velocity and for a natural velocity, in statically stressed media, are derived in terms of second, third and fourth order elastic constants.