Construction of a dynamical matrix for an HCP crystal using phonon data at the symmetry points of the Brillouin zone and elastic constants

General analytical expressions are obtained for the dynamical matrix D(k) and the elastic constants C _ik in an HCP crystal in terms of the Born-von Karman (BvK) parameters. An analytical method is proposed for constructing D(k) on the basis of data about the phonon frequencies ω _i (N) at the symmetry points of the Brillouin zone and the elastic constants C _ik . A number of relations between the values of ω _i (N) and C _ik are presented for conventional interaction models. It is shown that the standard method for determining BvK parameters by fitting them to experimental phonon spectra in HCP lattices is, as a rule, ambiguous, whereas the analytical method proposed allows one to find all the solutions of the problem. The methods developed are illustrated by the construction of dynamical matrices for Tb, Sc, Ti, and Co.     

Longitudinal-phonon relaxation mechanisms in cubic crystals of germanium,silicon, and diamond

The relaxation rates of thermal and high-frequency longitudinal phonons are calculated using an anisotropic-continuum model. Three-phonon scattering mechanisms (L ? L + L, L ? T + L) for the phonon relaxation are considered. Anisotropic anharmonic phonon scattering in cubic crystals is described in terms of the second-and third-order elastic moduli. The parameters determining the longitudinal-phonon relaxation rates are found for germanium, silicon, and diamond crystals. The long-wavelength limit and the transition to the isotropic-medium model are considered, and the dependences of the relaxation rates of thermal and high-frequency phonons on temperature and phonon wave vector are analyzed for these crystals.     

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).     

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.     

Nonadiabatic effects in the lattice dynamics of compressed rare-gas crystals

Electron-ion contributions to the energy of rare-gas crystals are discussed from first principles in the framework of the Tolpygo model and its variants. The frequencies of phonons in a neon crystal at pressures p ≠ 0 are calculated in terms of models that go beyond the scope of the adiabatic approximation. Analysis of the contributions from different interactions to the lattice dynamics of the crystals demonstrates that the phonon frequencies calculated in the framework of the simplest model (allowing only for the nearest neighbors) and the most complex model (with the inclusion of the nearest neighbors, next-nearest neighbors, nonadiabatic effects, etc.) for small wave vectors are close to each other. The difference between the phonon frequencies calculated within the above models is most pronounced at the Brillouin zone boundary. Under strong compression, the phonon spectrum along the Δ direction is distorted and the longitudinal mode is softened as a result of the electron-phonon interaction. The contribution from terms of higher orders in the overlap integral S at p ≠ 0 to the phonon frequencies is more significant than that obtained in the band-structure calculations of the neon crystal.     

Molecular dynamics calculations of anharmonic properties of the vibrational spectrum of BCC zirconium under pressure

The structural stability and lattice dynamics of the high-pressure bcc phase of Zr at a constant temperature T = 500 K are studied for various volumes using molecular dynamics simulation with the Animalu pair pseudopotential. Dispersion curves of the vibrational spectrum calculated by the molecular dynamics method for various volumes are compared to the phonon spectrum obtained in the harmonic approximation. It is demonstrated that, as the volume decreases, all frequencies of the vibrational spectrum increase gradually and bcc zirconium remains strongly anharmonic along all high-symmetry directions of the Brillouin zone over the entire range of volumes studied. The strongly anharmonic N _T1 phonon is significantly softened near the point of structural instability of bcc-Zr at T = 500 K and V = 0.87V ₀. As the volume decreases to V = 0.73V ₀ under pressure, the anharmonic corrections for this phonon decrease by almost an order of magnitude and the phonons near the H point of the Brillouin zone become anharmonic. The damping of the T ₁ phonon mode along the [110] direction is calculated as a function of pressure.     

Messungen des van der Waals-Potentials zwischen Alkaliatomen

Total collision cross sections for the interaction between alkali atoms have been measured using a modulated atomic beam technique. Since the determination of the density in the scattering chamber is the main source of error in absolute scattering measurements, particular attention is given to this problem: The cross section for partnersA andB is measured first withA as beam particles anB as target particles, and then withB as beam particles andA as target particles. The data are used to deduce the long rangevan der Waals (inverse sixth power) potential constants, which can be compared with theoretically calculated values of different authors. The interaction constants resulting from the present measurements are in good agreement with those calculated byFontana, while other calculations give much larger values.     

Mechanical and Dynamic Stability of Complete and Nonstoichiometric 3<Emphasis Type="Italic">C</Emphasis>-Si<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>C<Subscript><Emphasis Type="Italic">y</Emphasis></Subscript> from Ab Initio Calculations

The energies of formation of vacancies in the carbon and silicon sublattices, the independent elastic constants, the all-round compression, shear and Young’s moduli, and the anisotropy coefficients are determined for the complete and nonstoichiometric cubic phases of 3C-Si_xC_y (x, y = 1.0–0.75) by ab initio methods of the band theory. In the formalism of the density functional perturbation theory (DFPT), the phonon dispersion dependences are obtained for these phases (the comparison with the experiment is given for the complete phase). It is shown that the mechanical characteristics of the phases become strongly anisotropic upon the transition from 3C-SiC_0.875 to 3C-SiC_0.75. It is established from the analysis of the phonon dispersion curves that the 3C-SiC_0.875 and 3C-SiC_0.75 phases, in contrast to the complete 3C-SiC phase, are dynamically unstable at T = 0 K.     

Constitutive relations in multidimensional isotropic elasticity and their restrictions to subspaces of lower dimensions

The mechanical meaning and the relationships among material constants in an n-dimensional isotropic elastic medium are discussed. The restrictions of the constitutive relations (Hooke’s law) to subspaces of lower dimension caused by the conditions that an m-dimensional strain state or an m-dimensional stress state (1 ≤ m < n) is realized in the medium. Both the terminology and the general idea of the mathematical construction are chosen by analogy with the case n = 3 and m = 2, which is well known in the classical plane problem of elasticity theory. The quintuples of elastic constants of the same medium that enter both the n-dimensional relations and the relations written out for any m-dimensional restriction are expressed in terms of one another. These expressions in terms of the known constants, for example, of a three-dimensional medium, i.e., the classical elastic constants, enable us to judge the material properties of this medium immersed in a space of larger dimension.     

Ultrasonic attenuation in insulating crystals

A theory for the dampingΓ of ultrasonic waves due to three-phonon processes is developed by using a Green's function method. The imaginary part of the self-energy of the impressed ultrasound phonons interacting with thermal phonons is calculated. In the limits ofω τ very large and very small the known results are rederived, whereω is the frequency of the ultrasonic wave andτ the thermal phonon relaxation time. The intermediate range ofω τ values is discussed in detail for the case of longitudinal phonon attenuation. It is found, that forω τ>1 a Landau-Rumer type law applies also for longitudinal phonons,Γ～ωT ⁴. But it is shown that dispersion effects and large third-order elastic anisotropy can lead to a stronger temperature dependence thanT ⁴ and a weaker dependence on frequency thanω. These results are compared with recent experiments.     

Determination of the effective nuclear charge for free ions of transition metals from experimental spectra

The refined set of values of the effective nuclear charge (\(\bar Z_{ef} \)) and the set of one-electron spin-orbital constants (ξ _3d ^teor ) are obtained for ions of transition metals with the help of a semiempirical method of calculation of the spectrum of free ions with allowance for the spin-orbital interaction. The effective nuclear charge, which is a variable theoretical parameter, was determined from a comparison of theoretically calculated ion absorption spectra with experimental ones. The form of the potential for the calculation of the spin-orbital constant was chosen such that the calculated value ξ _3d ^teor would coincide with the experimental one ξ _3d ^exp . The calculated set of values \(\bar Z_{ef} \), ξ _3d ^teor } can be used in semiempirical calculations of energy levels of 3d ions (ions with an incomplete 3d shell).     

Shape,size and phonon scattering effect on the thermal conductivity of nanostructures

A phenomological model is described here to study the effect of size, shape and phonon scattering on the thermal conductivity of nanostructures. Using the classical model proposed by Guisbiers et al (Phys. Chem. Chem. Phys. 12, 7203 (2010), J. Phys. Chem. C 112, 4097 (2008)) in terms of the melting temperature of nanostructures, the expression for variation of thermal conductivity is obtained in terms of shape and size parameter. An additional term is included in the expression of thermal conductivity to consider the impact of phonon scattering due to the surface roughness with a decrease in size. The expression of thermal conductivity is obtained for spherical nanosolids, nanowires and nanofilms. The thermal conductivity is found to decrease in nanostructures in comparison with the counterpart bulk material. The values of thermal conductivity obtained from the present model are found to be close to the available experimental data for different values of roughness parameter which verifies the suitability of the model.     

Stoßionisierungskoeffizient α, mittlere Elektronenenergien und die Beweglichkeit von Elektronen in Gasen

From the analysis of oscillograms of electron avalanches values of the ionization coefficient α for electrons were determined for oxygen, methane, carbon dioxide, and some organic vapors atE/p-values between some 100 and several 1000 volts/cm Torr. Over a certainE/p-region the values satisfy the relation α/p=A exp(-Bp/E)for which the constants are given. — Values of the mean diffusion energyD/b of the electrons were determined for oxygen, nitrogen, hydrogen, and some other gases and vapors ranging from 4 evolts atE/p?50–100 to some 20 evolts atE/p of several 1000 volts/cm Torr. These values of the diffusion energy are in agreement with those obtained from values of the ionization coefficient α. — The mobility of the electrons is calculated in dependence on the energyD/b and compared with experimental values. It is shown that satisfactory agreement is obtained only when the first two terms of the expansion of the velocity distribution are taken into account.     

Structure and lattice dynamics of rare-earth ferroborate crystals: Ab initio calculation

The ab initio calculation of the crystal structure and the phonon spectrum of crystals RFe₃(BO₃)₄ (R = Pr, Nd, Sm) has been performed in the framework of the density functional theory. The ion coordinates in the unit cell, the lattice parameters, the frequencies and the types of fundamental vibrations, and also the intensities of lines in the Raman spectrum and infrared reflection spectra have been found. The elastic constants of the crystals have been calculated. For low-frequency A₂ mode in PrFe₃(BO₃)₄, a “seed” vibration frequency that strongly interacts with the electronic excitation on a praseodymium ion was found. The calculation results satisfactory agree with the experimental data.     

Elastic and inelastic scattering of 800-MeV protons on <Superscript>16</Superscript>O and <Superscript>20</Superscript>Ne nuclei

Differential cross sections and polarization observables for the elastic and inelastic scattering of 800-MeV protons on ¹⁶O and ²⁰Ne nuclei are calculated on the basis of the theory of multiple diffractive scattering and the α-cluster model involving dispersion. The single-particle nucleon-density distributions obtained within the α-cluster model involving dispersion are used in the calculations. The differential cross sections and polarization calculated for elastic and inelastic p¹⁶O and p²⁰Ne scattering are compatible with available experimental data. The spin-rotation functions calculated for elastic p¹⁶O and p²⁰Ne scattering within the independent-nucleon model differ qualitatively from their counterparts calculated within the α-cluster model involving dispersion.     

Resonant scattering of nonequilibrium phonons (<Emphasis Type="Italic">λ</Emphasis><Subscript>ph</Subscript> = 10–50 nm) in nanostructured ceramics based on YSZ + Al<Subscript>2</Subscript>O<Subscript>3</Subscript> composites

Specific features of the transport of weakly nonequilibrium thermal phonons (λ _ph = 10–50 nm) in nanoscale ceramics at a transition from micro-to nanosizes have been investigated. On the basis of the model of spherical shells randomly distributed in space and modeling grain boundaries, whose elastic properties differ from the elastic properties of grains, features of the phonon spectrum in the wavelength range λ _ph ～ R _g have been studied. The conditions leading to the occurrence of a gap in the phonon spectrum of nanoscale materials are analyzed. It is shown that the position of the top gap edge in the phonon spectrum is determined to a large extent by the structure of phase boundaries, while the presence of inclusions (pores, other phases) with characteristic sizes smaller than that of grains of the main ceramic material shifts the gap to high frequencies in the phonon spectrum. Temperature dependences of the diffusion coefficient of nonequilibrium phonons near the top gap edge in the phonon spectrum have been measured for multiphase ceramics based on YSZ + 14.3% Al₂O₃ composites.     

Light-field-induced spatially periodic freedericksz transition in a planar nematic cell

It is shown that a threshold spatially periodic reorientation of the director by a light field is possible in a planar nematic liquid crystal cell if the ratio of the Frank elastic constants, K ₂/K ₁, exceeds a critical value. The periodic director structure arising in the cell leads to a self-diffraction of the incident light wave. The dependences obtained for this phenomenon make it possible to determine the values of the elastic constants K ₁ and K ₂, the director reorientation threshold, and the period of the director structure from the experimental values of the self-diffraction angle.     

Discrete breathers in biatomic crystals of <Emphasis Type="Italic">AB</Emphasis> and <Emphasis Type="Italic">A</Emphasis><Subscript>3</Subscript><Emphasis Type="Italic">B</Emphasis> composition

The conditions for the existence of discrete breathers (DBs) in biatomic crystals of AB and A ₃ B composition are established, and their properties are studied by means of molecular mechanics using the examples of CuAu and Pt₃Al, respectively. The phonon spectra of the crystals are analyzed, and a gap in the phonon spectrum of CuAu is obtained via considerable homogeneous elastic strain. There is a gap in the phonon spectrum of the Pt₃Al crystal at zero strain, due to the considerable difference between the atomic weights of its components. The frequencies at which discrete breathers can exist in the considered crystals are determined. The energy localized on different types of DBs is estimated. The propagation of a current pulse through Pt₃Al resulting in the excitation of DBs with mild nonlinearity is simulated.     

QCD sum rules for the coupling constants <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">KYN</Emphasis></Subscript> and <Emphasis Type="Italic">g</Emphasis><Subscript>KYΞ</Subscript>

New relations between Borel QCD sum rules for the strong couplings of baryons to kaons are derived. It is shown that, on the basis of the sum rules for the coupling constants g _πΣΣ and g _πΣΛ, the corresponding sum rules can be directly obtained for the coupling constants g _KYN and g _KYΞ, Y = Σ,Λ. The values of these coupling constants are calculated in the fiducial interval 1.0 ≤ M ² ≤ 1.4 GeV² at t = ?1.     

Bound-exciton emission bands in ZnSe single crystals and mixed plasmon-phonon modes

The paper reports a cathodoluminescence study of ZnSe single crystals annealed in a Bi melt at 1200 K for 120 h. It is found that the distance between the phonon structure satellites in the bound-exciton series I ₁ ^s -nLO and I ₁ ^d -nLO and the relative satellite intensity are different in samples with different conduction electron concentrations. It is shown that this difference is due to the mixing of the plasmon and phonon modes. The shape of the bound-exciton emission spectrum in ZnSe crystals in the 450–470 nm region is calculated, and a satisfactory agreement with the experiment calculations is obtained.