Symmetric and Anti-Symmetric Lamb Waves in a Two-Dimensional Phononic Crystal Plate

It is weft known that Lamb waves in a plate with a mirror plane can be separated into two uncoupled sets： symmetric and anti-symmetric modes. Based on this property, we present a revised plane wave expansion method （PWE） to calculate the band structure of a phononie crystal （PC） plate with a mirror plane. The developed PWE method can be used to calculate the band structure of symmetric and anti-symmetric modes separately, by which the depending relationship between the partial acoustic band gap （PABG）, which belongs to the symmetric and anti-symmetric modes alternatively, and the position of the scatterers can be determined. As an example of its application, the band structure of the Lamb modes in a two-dimensional PC plate with two layers of void circular inclusions is investigated. The results show that the band structure for the symmetric and anti-symmetric modes can be changed by the position of the scatterers drastically, and larger PABGs will be opened when the scatterers are inserted into the area of the plate, where the elastic potential energy is concentrated.     

Tetrad formulation of the motion of an elastic medium in the special theory of relativity

The tetrad formalism is used to construct the four-dimensional covariant kinematics for an elastic medium. The deformation tensors which are generalizations of the Almansi, Green, and Cauchy tensors are determined. Kinematic equations are derived to relate the stress-propagation tensor to the rate of change of the deformation tensor.Translated from Izvestiya VUZ. Fizika, No. 4, pp. 45–51, April, 1970.In conclusion the author thanks professor V. I. Rodichev for valuable comments on and fruitful discussion of this study.     

An algorithm for the calculation of the net induction energy and induced multipole tensors in a set of charge distributions with non-linear contributions of all potentials of external fields and both permanent and induced multipoles

This algorithm for the calculation of the induced multipole tensors in a set of charge distributions includes the contributions of partial derivatives of arbitrary order of the potentials defined by both induced and permanent multipoles as well as non-uniform external fields of arbitrary strength. Specific equations are given for both systems with and without translational order. The algorithm, which is based on the Maxwell invariant form, uses direct extensions of algorithms previously developed and tested for the calculation of permanent multipole energies and induced dipole vectors (when the non-uniformity of the field is neglected). The induced tensor components are calculated iteratively. The first approximation, which gives the components as the solution of a set of simultaneous linear equations, includes all nonlinear, non-uniform contributions of permanent multipoles and external fields as well as all contributions linear in derivatives of the induced potentials. The induced tensor components are then used to calculate the net induction energy. The general relations between the polarizability tensors with respect to a centre and the moments of the polarizability densities about the centre are derived.     

Mechanism of Plastic Collapse of Nanosized Crystals with BCC Lattice under Uniaxial Compression

Within the dislocation–kinetic approach, based on the nonlinear kinetic equation for dislocation density, an attempt is made to consider the problem of a catastrophic plastic collapse of defect-free nanocrystals of metals with bcc lattice under their uniaxial compression with a constant deformation rate. Solutions of this equation were found in the form of moving waves, describing the dislocation multiplication process as the wave moves along the crystal from a local dislocation source. Comparison of the theory with the results of experiments on defect-free Mo nanocrystals showed that their ultrahigh strength at the initial stage of deformation is associated with a low rate of rise of crystal plastic deformation in comparison with the growth of its elastic component. The subsequent plastic collapse of crystal is caused by a sharp increasing the plastic component, ending with reaching the equality of elastic and plastic deformation rates.

     

The proton nuclear magnetic shielding tensors in biphenyl: experiment and theory

Line-narrowing multiple pulse techniques are applied to a spherical sample crystal of biphenyl. The 10 different proton shielding tensors in this compound are determined. The accuracy level for the tensor components is 0.3 ppm. The assignment of the measured tensors to the corresponding proton sites is given careful attention. Intermolecular shielding contributions are calculated by the induced magnetic point dipole model with empirical atom and bond susceptibilities (distant neighbours) and by a new quantum chemical method (near neighbours). Subtracting the intermolecular contributions from the (correctly assigned) measured shielding tensors leads to isolated-molecule shielding tensors for which there are symmetry relations. Compliance to these relations is the criterion for the correct assignment. The success of this program indicates that intermolecular proton shielding contributions can be calculated to better than 0.5 ppm. The isolated-molecule shielding tensors obtained from experiment and calculated intermolecular contributions are compared with isolated-molecule quantum chemical results. Expressed in the icosahedral tensor representation, the rms differences of the respective tensor components are below 0.5 ppm for all proton sites in biphenyl. In the isolated molecule, the least shielded direction of all protons is the perpendicular to the molecular plane. For the para proton, the intermediate principal direction is along the C-H bond. It is argued that these relations also hold for the protons in the isolated benzene molecule.     

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 elastic dielectric

Macroscopic field equations, boundary conditions and equations of state are derived for the non-linear, macroscopic elastic and dielectric response of an insulator. A centrosymmetric polynomial representation of order four is introduced for the energy density; the equations of state for the electric field and stress tensor are then deduced as polynomials of degree three in the displacement gradients and electric displacement field. The results are applied to the special case of m3m material symmetry.

A finite, point-charge model of a centrosymmetric ionic crystal is introduced and used to determine 0°K microscopic expressions for the electric field and stress tensor equation of state coefficients introduced in the macroscopic analysis. The results are used to calculate the full set of second and third-order non-linear coefficients for NaI, based on a Born-Mayer potential and the 4·2°K elastic stiffness data of Claytor and Marshall.     

Averaging out the Einstein equations

A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.     

Anharmonic properties of rocksalt structure solids

An effort has been made for obtaining the anharmonic properties of rocksalt structure solids starting from primary physical parameters viz. nearest-neighbor distance and hardness parameter assuming long- and short-range potentials at elevated temperatures. The elastic energy density for a deformed crystal can be expanded as power series of strains for obtaining coefficients of quadratic, cubic and quartic terms which are known as the second-, third- and fourth-order elastic constants, respectively. When the values of the higher-order elastic constants are known for a crystal, many of the anharmonic properties of the crystal can be treated within the limit of the continuum approximation in a quantitative manner. In this study, higher-order elastic constants are computed up to their melting temperature for rocksalt structure solids, which are alkali cyanides, sodium and potassium halides. The first order pressure derivatives of second- and third-order elastic constants, the second-order pressure derivatives of second-order elastic constants and partial contractions are also evaluated at different temperatures for these substances. The results thus obtained are compared with experimental data and found in well agreement with present values.     

Slow propagation of plasmonic modes in a symmetrical waveguide containing anisotropic metamaterial

We study the slow propagation of plasmonic modes in a three-layer symmetric waveguide with anisotropic metamaterial claddings and a dielectric core. The dispersion equations for symmetric and anti-symmetric plasmonic modes are derived, and a graphic method is used to verify the zero-group-velocity point on the dispersion curves. Simulation results show the influences of signs of tensors and anisotropy of metamaterial on trapped plasmonic modes.     

Einfluß von Drehungen auf die piezoelektrische und photoelastische Wechselwirkung im kristallinen Medium

Influence of Electric Polarization on the Piezoelectric and Photoelastic Interaction in Crystalline Medium Because of the moments of force induced by electric polarization in solids the stresstensor is no longer symmetric, so that the antisymmetric parts of the distortion tensor is a thermodynamic state function too. The corresponding antisymmetric parts of the piezoelectric and photoelastic tensors are expressed by the spontaneous electric polarization and the linear electric susceptibility. Some consequences, also with regard to numerical values, are discussed.     

Method for estimating local lattice distortions near a magnetic ion based on the parameters of the ligand hyperfine interaction: Ce3+ in the fluorite homologous series

A method is suggested for estimating local lattice distortions near a paramagnetic center by using the parameters determined from experimental tensors of the ligand hyperfine interaction with surrounding nuclei through the use of a simple mathematical procedure. In general, a tensor with nine independent components can be unambiguously reduced to nine independent parameters characterizing the hyperfine interaction: the isotropic part, deviations from the symmetric and axially symmetric form, the axially symmetric component (containing dipole and pseudodipole contributions), and the angles defining the bond direction. Under certain physical assumptions, these parameters are used to estimate distortions of the first and second coordination shells of Ce³⁺ in the homologous series CaF₂, SrF₂, and BaF₂.     

Spherically symmetric free fall collapse

The general dynamical equations for spherical gravitational collapse are derived by introducing the eigenvalue of the conformal Weyl tensor in the 2-2 component of the Einstein tensor and assuming the material content of the models to be a perfect fluid. Since this eigenvalue is coupled always with the material energy density, it has been interpreted as theenergy density of the free gravitational field whose presence is related with anisotropy and inhomogeneity. As a particular case, the collapse of a spherically symmetric dust (zero pressure) with vanishing radial acceleration (free fall collapse) is discussed. It is shown that the model is inhomogeneous with non-vanishing shear of the congruence of world lines of the dust particles. The model contains gravitational radiation by Szekere’s criterion since both shear invariant and the spatial gradient of density are non-vanishing. This is in contrast to the Oppenheimer-Synder model for which both the above mentioned characteristics are absent. A particular solution which is anisotropic and inhomogeneous has been given to prove the emission of gravitational radiation by the freely falling dust and in this case the energy density of the free gravitational field contains a typeN term superposed on the coulombian field.     

Hydrogen bonding effects on the (15)N and (1)H shielding tensors in nucleic acid base pairs

The results of systematic ab initio calculations of (15)N and (1)H chemical shielding tensors in the GC base pair as a function of hydrogen bond length are presented for the first time. The hydrogen bond length characterized by the distance r(N...N) between purine N1 and pyrimidine N3 was varied between 2.57 and 3.50 A and the chemical shift tensors were calculated by the sum-over-states density functional perturbation theory. It is shown that the hydrogen bond length has a strong effect on the chemical shielding tensor of both imino proton and nitrogen, on their orientation, and, as a consequence, on the relaxation properties of both nuclei. For a nitrogen nucleus not involved in hydrogen bonding, the shielding tensor is nearly axially symmetric and almost collinear with the bond vector. As the length of the hydrogen bond decreases, the least shielding component sigma(11) deflects from the N-H vector and the shielding tensor becomes increasingly asymmetric. The significance of the presented results for the analysis of relaxation data and the efficiency of TROSY effects together with a summary of the relevant shielding parameters are presented and discussed.     

Determination of elastic strain fields and geometrically necessary dislocation distributions near nanoindents using electron back scatter diffraction

The deformation around a 500-nm deep Berkovich indent in a large grained Fe sample has been studied using high resolution electron back scatter diffraction (EBSD). EBSD patterns were obtained in a two-dimensional map around the indent on the free surface. A cross-correlation-based analysis of small shifts in many sub-regions of the EBSD patterns was used to determine the variation of elastic strain and lattice rotations across the map at a sensitivity of ～±10^?4. Elastic strains were smaller than lattice rotations, with radial strains found to be compressive and hoop strains tensile as expected. Several analyses based on Nye's dislocation tensor were used to estimate the distribution of geometrically necessary dislocations (GNDs) around the indent. The results obtained using different assumed dislocation geometries, optimisation routines and different contributions from the measured lattice rotation and strain fields are compared. Our favoured approach is to seek a combination of GND types which support the six measurable (of a possible nine) gradients of the lattice rotations after correction for the 10 measurable elastic strain gradients, and minimise the total GND line energy using an L¹ optimisation method. A lower bound estimate for the noise on the GND density determination is ～±10¹² m^?2 for a 200-nm step size, and near the indent densities as high as 10¹⁵ m^?2 were measured. For comparison, a Hough-based analysis of the EBSD patterns has a much higher noise level of ～±10¹⁴m^?2 for the GND density.     

Some Properties of the Bel and Bel-Robinson Tensors

The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum. We present some new properties of these tensors which might throw some light onto this relationship. First, for any spacetime we find a decomposition of the Bel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the pure matter super-energy tensor and the matter-gravity coupling super-energy tensor. We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the square of the usual energy-momentum tensor. This, together with the fact that the Bel-Robinson tensor has dimensions of energy density square, leads us to the definition of square root for the Bel-Robinson tensor: a two-covariant symmetric traceless tensor with dimensions of energy density and such that its square gives the Bel-Robinson tensor. We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented. The properties of this new tensor are examined and some interesting explicit examples are analyzed. Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves. We also examine the decomposition of the whole Bel tensor for Vaidya's radiating metric and Kerr-Newman's solution. Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types.     

Supersymmetric quantum mechanics and the index theorem

We consider the classical mechanics of the spinning particle and investigate which Abelian interactions can be added without breaking supersymmetry. A quantum theory is presented. The well known index theorem for the Dirac operator is extended to take into account the effect of anti-symmetric Abelian tensor fields. Furthermore interactions with non-Abelian anti-symmetric tensor fields are investigated. It turns out in both cases that these fields do not give any non-trivial contributions to the index.     

On the incorporation of cubic and hexagonal interfacial energy anisotropy in phase field models using higher order tensor terms

In this paper, we show how to incorporate cubic and hexagonal anisotropies in interfacial energies in phase field models; this incorporation is achieved by including up to sixth rank tensor terms in the free energy expansion, assuming that the free energy is only a function of coarse-grained composition, its gradient, curvature and aberration. We derive the number of non-zero and independent components of these tensors. Further, by demanding that the resultant interfacial energy is positive definite for inclusion of each of the tensor terms individually, we identify the constraints imposed on the independent components of these tensors. The existing results in the invariant group theory literature can be used to simplify the process of construction of some (but not all) of the higher order tensors. Finally, we derive the relevant phase field evolution equations and describe some preliminary results from our 1D simulations.     

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.