Phonon dispersion and symmetry of solid-J2

The phonon dispersion curves, spectrum and specific heat of solid J₂ and Br₂ are calculated using a general central force parameter model, which is fitted to the available experimental data. The influence of the intramolecular degrees of freedom on the elastic constants and the compressibility is shown. The usefulness of group theoretical methods in the investigation of normal modes of vibrations in crystals is illustrated by applying them to the decomposition of the dynamical matrix.     

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.     

The second-order elastic constants of AgBr from 77 to 300 K

The three independent second-order elastic constants of AgBr have been measured from room temperature down to liquid nitrogen temperature A single crystal with a (110) axis was used for the measurements The measured longitudinal elastic constant C'₁₁ = (C₁₁ + C₁₂ + 2C₄₄)/2 increases by 16% over this temperature range The elastic shear constant C₄₄) increases by almost 11%, while the elastic shear constant C' = (C₁₁ ? C₁₂)/2 increases by 50% over this range The measured bulk modulus B = (C₁₁) + 2C₁₂)/3 increases by almost 15% as the lattice becomes stiffer The initial temperature derivatives of the elastic constants are similar to those previously measured The changes in the elastic constants are basically linear down to approximately 150 K, where the temperature derivatives begin to decrease in magnitude These results are similar to those previously obtained for AgCl     

Elastic constants of 2H-MoS2 and 2H-NbSe2 extracted from measured dispersion curves and linear compressibilities

Recent neutron data on the dispersion curves and X-ray measurements of the linear compressibilities of the 2H polytypes of MoS₂ and NbSe₂ have been used to obtain approximate values of the five independent elastic constants of these materials. In the case of NbSe₂ sufficient information is available to over-determine the elastic constants and the results are self consistent within estimated uncertainties, although the uncertainties are especially large for c₃₃ and c₁₁. Additional related considerations such as Debye temperatures and model calculations of c₃₃, and c₄₄ are also made. It is found that there is significant and unexplained disagreement between the value of the low temperature specific heat Debye temperature of NbSe₂ and the value determined on the basis of the elastic constants, but that the model predictions of c₃₃ and c₄₄ are in satisfactory agreement with the values extracted from the neutron data for both MoS₂, and NbSe₂.     

Monocrystal elastic constants in the ultrasonic study of welds

For studying welds ultrasonically, the importance of knowing the material's single-crystal elastic constants, the C_ijs, is explained. Where these constants are not known, some guidelines are given for estimating them from polycrystalline elastic constants such as Young's modulus and the shear modulus.The important case of [001] fibre texture is considered. Being transversely isotropic, this case exhibits five macroscopic elastic constants, which are related to the three cubic elastic constants: C₁₁, C₁₂, C₄₄. From these five constants the angular variations of Young's modulus, the torsional modulus, and the sound velocities can be computed. For the same [001] fibre texture, results are given for a standard well-characterized material — copper, where the C_ijs are well known.     

Dimerization in the magnetostrictive one-dimensional XY model

We discuss the one-dimensional first-neighbor $\text{1}\text{2}$-spin magnetostrictive XY model (where the crystalline degrees of freedom are assumed to be three- dimensional), and exhibit that, for all temperatures below T_c, no other contributions to the structural order appear than the pure dimerization one. The influences of temperature and elastic constant on the order parameter and sound velocity are analyzed as well.     

Classical behaviour of the soliton density in Rb2ZnCl4 studied by 87Rb NMR

The temperature dependence of the soliton density n_s(T) which has been studied in Rb₂ZnCl₄ by ⁸⁷Rb NMR could be fitted to a classical formula using only one fit parameter. From our data we cannot confirm recent experimental results which yield a critical exponent $\text{1}\text{2}$ for the soliton density. The results are discussed with respect to the free energy where the soliton density acts as the order parameter. Moreover, couplings to other degrees of freedom, e.g. the elastic strains, are considered.     

Multicritical behavior in ferroelectrics

Solid state systems exhibit besides usual second order phase transitions a rich variety of multicritical phenomena like Lifshitz points (or lines), tricritical points (or lines) and even tricritical Lifshitz points. Realizations of such points are numerous and were also verified in the family of ferroelectrics of the type (Pb_ySn_1y)₂P₂(Se_xS_1-x)₆. A review of the critical behavior at such points is presented here. Because of the importance of the uniaxial dipolar interaction in ferroelectrics the critical behavior is different from systems with short range interaction only. Moreover the coupling to the elastic degrees of freedom may not be neglected, and leads under certain conditions to a critical temperature dependence in certain elastic constants. Crossover phenomena, which are expected in the experimental accessible region of experiments are also considered.     

Effect of intrinsic degrees of freedom on the quantum tunneling of a collective variable

The effect of intrinsic degrees of freedom on tunneling through a potential barrier is discussed using a BKW-like approximation. In the present work intrinsic degrees of freedom are represented by a single harmonic oscillator. The theory leads to a formula for the effect of the coupling on the decay width Γ of a metastable state. When the frequency ω of the intrinsic degree of freedom is large, then Γ → Γ_ad where Γ_ad is the decay width calculated with the adiabatic barrier. An inequality Γ ≤ Γ_ad is proved for any form of the coupling hamitonian. Corrections are discussed and are shown to be of order $\text{1}\text{M}$ where M is the mass of the tunneling coordinate. An application to fission is considered. The generalization of our formula for Γ to the case of many intrinsic degrees of freedom is given explicitly.     

Third-order elastic constants of ZnS and ZnSe

The complete set of non-vanishing third-order elastic constants of the semiconductors ZnS and ZnSe is obtained theoretically. The strain energy density is estimated using finite strain elasticity theory by considering the interactions up to two nearest neighbours of each atom in the unit cell of these compounds. This energy density is compared with the strain dependent lattice energy density from the continuum model approximation. The second-order parameter of the potential function φ is obtained from the measured principal axis C_ij. The third-order potential parameter is estimated by assuming a Lennard-Jones type of interatomic potential. The interlattice displacements as well as the second-order elastic constants are evaluated along with the six third-order elastic constants of ZnS and ZnSe. Using these second- and third-order elastic constants of ZnS, the pressure derivatives of second-order elastic constants are evaluated. The second- and third-order elastic constants of ZnSe are compared with the available experimental values. The third-order elastic constants show anisotropy in different directions.     

Theoretical study of the nuclear degrees of freedom in the isotropic and anisotropic hyperfine coupling constants of the C2H5 radical: a Feynman path integral–density functional approach

The isotropic and anisotropic hyperfine coupling (hfs) constants of the C₂H₅ radical have been theoretically studied under the conditions of thermal equilibrium, i.e. under the explicit consideration of the nuclear degrees of freedom. For this purpose the Feynman path integral quantum Monte Carlo (PIMC) formalism has been combined with an electronic Hamiltonian of the B3LYP–EPRIII type. The density functional operator has been used to derive both the distribution functions for the isotropic and anisotropic hfs constants of the ethyl radical as well as the thermal mean values. The electron paramagnetic resonance (EPR) timescale enables only the measurement of the thermal averages. The underlying distribution functions of these mean values, however, offer insight into the nature and strength of the nuclear degrees of freedom contributing to the observable thermal averages. The EPR parameters of C₂H₅ have been studied between 25 and 1000?K. This temperature (T?) window is large enough to consider nuclear fluctuations beyond zero-point effects. The deviations between the thermally averaged hfs constants and the values at the minimum of the potential energy surface (PES) are caused by (i) enlargements of the bond lengths in thermal equilibrium under the influence of anharmonicities in the internuclear potential, and (ii) by the intramolecular methylene rotation. The latter degree of freedom leads to a planar CH₂ unit in thermal equilibrium. At the minimum of the PES the methylene fragment exhibits a certain pyramidalization. The ensemble corrections as well as the T dependence of the isotropic hfs constants are larger than the ensemble shifts and T dependence of the anisotropic parameters. The non-validity of the crude Born–Oppenheimer approximation for the theoretical evaluation of physically meaningful isotropic hfs constants of the ethyl radical has been explained on the basis of specific nuclear degrees of freedom. Theoretical results of the ensemble averaged Monte Carlo type as well as single-nuclear configuration data are compared with experiment whenever available.     

Mode softening in CeAl2

Measurements of elastic constants C₁₁ and C₄₄ are reported between 1.5 and 280 K for CeAl₂. With decreasing temperature a softening of the C₄₄ elastic constant of 30% is observed. The softening is discussed in terms of a magnetoelastic interaction.     

Study of elastic properties of CeO2 by statistical moment method

The purpose of the present paper is to investigate the temperature and pressure dependences of the elastic properties of cerium dioxide using the statistical moment method (SMM). The equation of states of bulk CeO₂ is derived from the Helmholtz free energy, and the pressure dependences of the elastic moduli like the bulk modulus, B_T, shear modulus, G, Young’s modulus, E, and elastic constants (C11, C12, and C44) are presented taking into account the anharmonicity effects of the thermal lattice vibrations. In the present study, the influence of temperature and pressure on the elastic moduli and elastic constants of CeO₂ has also been studied, using three different interatomic potentials. We compare the results of the present calculations with those of the previous theoretical calculations as well as with the available experiments.     

Third-order elastic constants for 2:2 chalcide crystals possessing the sodium chloride structure

Third-order elastic constants of 45 chalcide crystals having the sodium chloride structure are reported using Born-Mayer potential model. We have considered repulsive interaction up to second nearest neighbours. The temperature coefficients of the third-order elastic constants have also been computed for these crystals. As is the case for NaCl-type alkali halides we find that C₁₁₁, C₁₁₂, C₁₆₆ are negative and C₁₄₄ are positive for 2:2 chalcide crystals possessing the NaCl-type structure. We have found that a₁₂₃, a₄₅₆ and a₁₄₄ are negative whereas a₁₁₂ and a₁₆₆ are positive, once again in agreement with the situation found for the alkali halides a₁₁₁ values are positive for alkali halides whereas they are both positive and negative depending upon the interionic distance for the chalcide crystals. We have found that the nature of the variation of C⁰_αβγ with interionic separation is similar for alkali halides and for the 2:2 chalcides having the NaCl-structure. We have also computed the values of the pressure derivatives of second-order elastic constants for MgO, CaO, and SrO which agree well with the experimental values indicating the satisfactory nature of our computed data for C_αβγ.     

The elastic constants of sodium from 20 to 95°C

We report ultrasonic pulse echo measurements of the elastic constants of sodium in the high temperature region. At room temperature the results agree with those of earlier work. No pronounced effects of anharmonicity are found contrary to theoretical predictions. The behaviour of the elastic constants is further analysed in terms of compressibilities, of Grüneisen parameters and of effects caused by lattice defects. The quantity[?lnω_i(q)/?T]_v is calculated from the temperature and pressure variation of the elastic constants.     

Ab initio calculations of elastic constants and thermodynamic properties of Li2O for high temperatures and pressures

The elastic constants and thermodynamic properties of Li₂O for high temperatures and pressures are calculated by the ab initio unrestricted Hartree-Fock (HF) linear combination of atomic orbital (LCAO) periodic approach. The lattice constant, elastic constants, Debye temperature, and thermal expansion coefficient obtained are in good agreement with the available experimental data and other theoretical results. It is found that at zero pressure the elastic constants C₁₁, C₁₂ and C₄₄, bulk modulus B and Debye temperature Θ_D decrease monotonically over the wide range of temperatures from 0 to 1100 K. When the temperature , C₁₂ approaches zero, consistently with the transition temperature 1200 K. However, with increasing pressure, they all increase monotonically and the anisotropy will weaken.     

Effect of pressure on the elastic properties of silicon carbide

The pressure dependences of the second-order elastic constants C _ij and the velocity of sound in 3C-SiC and 2H-SiC crystals are calculated in the framework of the Keating model. The third-order elastic constants C _ijk for 3C-SiC are determined from the dependences of the second-order elastic constants C _ij on the pressure p.     

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).