Calculation of the electroelastic Green’s function of the hexagonal infinite medium

The electroelastic 4 × 4 Green’s function of a piezoelectric hexagonal (transversely isotropic) infinitely extended medium is calculated explicitly in closed compact form ((73) ff. and (88) ff., respectively) by using residue calculation. The results can also be derived from Fredholm’s method [2]. In the case of vanishing piezoelectric coupling the derived Green’s function coincides with two well known results: Kröner’s expressions for the elastic Green’s function tensor [4] is reproduced and the electric part then coincides with the electric potential (solution of Poisson equation) which is caused by a unit point charge. The obtained electroelastic Green’s function is useful for the calculation of the electroelastic Eshelby tensor [16].     

Elastic interaction of point defects with an edge dislocation loop within the Green’s function formalism

The universal expressions have been obtained for components of the tensor Green’s function of an elastically anisotropic hexagonal medium. In contrast to the classical expressions (the Lifshitz–Rosenzweig method), they do not contain uncertainties of the type 0/0 upon the transition to the isotropic approximation and hold true for any hexagonal crystal. As an example of their use, the displacement and strain fields created by an edge dislocation loop lying in the basal plane of the crystal have been calculated.     

Interaction élastique de défauts ponctuels dans un cristal hexagonal semi-infini avec ou sans adsorbat

The energy of elastic interaction between a point defect and the (0001) surface of a hexagonal crystal is calculated, as well as the energy of the elastic interaction between two defects near such a surface. The defects are represented by the superposition of three mutually perpendicular double forces without moment. The calculation is done by means of a Green function method. As for an isotropic medium, the energy of a point defect presents a variation in x₃^?3 with the distance x₃ to the surface. On the other hand, the mutual interaction between two defects depends upon different geometric parameters, and not simply on an image factor. We also study the effect of a thin adlayer on these elastic interactions. This is done by showing that the presence of the adlayer is equivalent to effective boundary conditions at the surface of the substrate. We derive these conditions and then the elastic energies to the first order in the thickness h of the layer. Finally we present the mean square displacements of atoms in the presence of a clean or adsorbed surface.     

Green’s tensor for crystals of the hexagonal system

Expressions for components of the Green’s tensor of the basic equation of the elasticity theory for hexagonal system crystals have been obtained using the Lifshitz-Rozentsveig method. A problem is in principle reduced to finding the roots of a sixth-order algebraic equation. They are either complex or purely imaginary for all known hcp metals. In both cases, the desired components of the Green’s tensor are calculated exactly in contrast to metals of the cubic system. A limiting transition to the isotropic approximation is shown.     

Analytical methods for the calculation of the elastic interaction of point defects with dislocation loops in hexagonal crystals

The Green’s function method for hexagonal crystals within the Lifshitz–Rosenzweig (1947) and Kröner (1953) approaches has been used to obtain analytical expressions for the energy of elastic interaction of radiation-induced point defects with dislocation loops of three types: the basal edge dislocation loop (cloop), the basal shear dislocation loop, and the edge a-loop (bedding plane {11 20}, Burgers vector b ^D = 1/3〈11 20〉). In the case of the basal edge dislocation loop, a similar expression has been obtained independently by solving the equilibrium equations using the Elliott method. A numerical comparison of the derived expressions for zirconium has demonstrated a complete identity of the results obtained within the approaches considered in this study.     

Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor,dislocation key-formulas and dislocation loops

The present work provides fundamental quantities in generalized elasticity and dislocation theory of quasicrystals. In a clear and straightforward manner, the three-dimensional Green tensor of generalized elasticity theory and the extended displacement vector for an arbitrary extended force are derived. Next, in the framework of dislocation theory of quasicrystals, the solutions of the field equations for the extended displacement vector and the extended elastic distortion tensor are given; that is, the generalized Burgers equation for arbitrary sources and the generalized Mura–Willis formula, respectively. Moreover, important quantities of the theory of dislocations as the Eshelby stress tensor, Peach–Koehler force, stress function tensor and the interaction energy are derived for general dislocations. The application to dislocation loops gives rise to the generalized Burgers equation, where the displacement vector can be written as a sum of a line integral plus a purely geometric part. Finally, using the Green tensor, all other dislocation key-formulas for loops, known from the theory of anisotropic elasticity, like the Peach–Koehler stress formula, Mura–Willis equation, Volterra equation, stress function tensor and the interaction energy are derived for quasicrystals.     

Hubbard-type solution in the planar approximation

The Hubbard solution to the Hubbard model showed a non-trivial metal-insulator transition. The value of the one-particle density of states at the Fermi energy in that solution decreased continuously with increasing value of the Hubbard interaction and vanished at a critical value of the interaction. Such a solution is derived from a planar model, as an approximation to the exact construction of the model's one-particle Green function.     

The tensor force and collective excitations in nuclear matter

The formalism for the particle-hole Green function is developed to handle the nucleon-nucleon tensor potential in order to investigate both the ring correlation energy and collective excitations in nuclear matter. The formalism is exact for direct ring diagrams. An approximation for exchange ring diagrams is also introduced. The one-pion-exchange potential with cut-off radius is used throughout. No collective excitations are found coming from the tensor force.     

The thermodynamic one-particle Green function in the renormalized spin wave approximation for isotropic cubic ferromagnets

The thermodynamic one-particle Green function in the renormalized spin wave approximation for isotropic cubic ferromagnetic insulators with Dyson's spin wave theory as a base is derived. In quantitative respect, dynamic and kinematic effects of spin waves are approximated by the graphs deficient in the energy denominators, wherefore at low temperature kinematic interaction turns out to be too strong. As against the one-particle Green function for independent spin waves, dynamic interaction of ferromagnons is shown to effect the renormalization of the spin wave energy, whereas kinematic interaction directly modifies the average ferromagnon population numbers. In the matter of magnetization, its formula based on the Green function assumes a similar form as in the spin wave theory without interactions on the understanding that it remains valid within the entire range of temperatures from absolute zero up to the critical point.     

Field of a point source in a semi-infinite elastic medium

An exact solution for the tensor Green's function of a harmonic field for a semi-infinite elastic medium is presented in an easy-to-use form in the theory of wave scattering. The solution is derived in the form of a sum of the Green's functions for an infinite medium and the term satisfying the homogeneous wave equation for a semi-infinite elastic medium. The results reproduce the known far-field asymptotics containing longitudinal, transversal and surface Rayleigh-type wave modes. The near-field asymptotic is essentially different for the regions far and near the boundary.     

Prediction study of the structural and elastic properties for the cubic skutterudites LaFe4A12 (A = P,As and Sb) under pressure effect

We have performed accurate ab initio total energy calculations using the full-potential linear augmented plane wave plus local orbitals method with the local density approximation for the exchange–correlation potential to investigate the systematic trends for structural and elastic properties of the cubic LaFe₄A₁₂ skutterudites’ family depending on the type of A pnicogen atom (A stands for P, As and Sb). The calculated equilibrium lattice constants and internal free parameters are in good agreement with the experimental results. For the first time, the numerical estimates of the independent elastic constants and their pressure dependence are performed using the total energy variation as function of strain technique. Isotropic elastic parameters and related properties, namely bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, Lamé’s coefficients, average sound velocity and Debye temperature, are estimated in the framework of the Voigt–Reuss–Hill approximation for ideal polycrystalline LaFe₄A₁₂ aggregates.     

Punktdefekte im kubischen elastischen Kontinuum

The determination of the displacement and strain fields of a point defect in a cubic crystal requires even in the framework of continuum elasticity theory numerical calculation. These fields of elastic dipoles are expanded in suitable vector and tensor fields. The coefficients of this expansion are calculated up to polynomials of 5th and 4th order in the direction cosines using the ratios of elastic constants as parameters. With this expansion the interaction of elastic dipoles in a cubic medium can be calculated. The results have been applied to the interaction of F-centres and of O₂ ^?-centres in alkali halides.     

Development of the self-consistent approximation and its application to the problem of magnetoelastic resonance in an inhomogeneous medium

Our previously proposed approximation involving both the first and second terms of the expansion of the vertex function is generalized to the system of two interacting wavefields of different physical nature. A system of self-consistent equations for the matrix Green’s function and matrix vertex function is derived. On the basis of this matrix generalization of the new self-consistent approximation, a theory of magnetoelastic resonance is developed for a ferromagnetic model, where the magnetoelastic coupling parameter ε(x) is inhomogeneous. Equations for magnetoelastic resonance are analyzed for one-dimensional inhomogeneities of the coupling parameter. The diagonal and off-diagonal elements of the matrix Green’s function of the system of coupled spin and elastic waves are calculated with the change in the ratio between the average value ε and rms fluctuation Δε of the coupling parameter between waves from the homogeneous case (ε ≠ 0, Δε = 0) to the extremely randomized case (ε = 0, Δε ≠ 0) at various correlation wavenumbers of inhomogeneities k _c. For the limiting case of infinite correlation radius (k _c = 0), in addition to approximate expressions, exact analytical expressions corresponding to the summation of all diagrams of elements of the matrix Green’s function are obtained. The results calculated for an arbitrary k _c value in the new self-consistent approximation are compared to the results obtained in the standard self-consistent approximation, where only the first term of the expansion of the vertex function is taken into account. It is shown that the new approximation corrects disadvantages of the Green’s functions calculated in the standard approximation such as the dome shape of resonances and bends on the sides of resonance peaks. The appearance of a fine structure of the spectrum in the form of a narrow resonance on the Green’s function of spin waves and a narrow antiresonance on the Green’s function of elastic waves, which was previously predicted in the standard self-consistent approximation, is confirmed. With an increase in the parameter k _c, the Green’s functions calculated in the standard and new approximations approach each other and almost coincide with each other at k _c/k ≥ 0.5. At the same time, the results of this work indicate that the new self-consistent approximation has a certain advantage for studying the problems of stochastic radiophysics in media with long-wavelength inhomogeneities (small k _c values), because it describes both the shape and width of peaks much better than the standard approximation.     

Post-Hartree-Fock methods and dynamic correlation in atoms and molecules

The methods of calculation of the matrix of the exchange-correlation interaction are considered within the framework of one post-Hartree-Fock one-electron method of investigation of the properties of many-electron systems. Such post-Hartree-Fock methods are based on two-step variational self-consistent calculations of the spin orbitals and superposition coefficients of configurations in the multiconfiguration approximation. The post-Hartree-Fock method used involves an approach related to the extended Koopmans’ theorem, which, in turn, proves to be a high-energy approximation for quantum Green’s functions. Obvious application areas of the calculations of the exchange-correlation interaction within the framework of the method proposed are the multiparticle perturbation theory, the parameterization of the energy representation as a functional of the single-particle density matrix, and the theory of Green’s functions in the multiconfiguration approximation. A relativistic generalization of the method with the aim of calculating the radiative corrections for many-electron atoms and for problems of interaction with an external field in the nonstationary Floquet theory is possible.     

Electroweak measurements of neutron densities in CREX and PREX at JLab,USA

We analyze microscopic many-body calculations of the nuclear symmetry energy and its density dependence. The calculations are performed in the framework of the Brueckner-Hartree-Fock and the self-consistent Green’s functions methods. Within Brueckner-Hartree-Fock, the Hellmann-Feynman theorem gives access to the kinetic energy contribution as well as the contributions of the different components of the nucleon-nucleon interaction. The tensor component gives the largest contribution to the symmetry energy. The decomposition of the symmetry energy in a kinetic part and a potential energy part provides physical insight on the correlated nature of the system, indicating that neutron matter is less correlated than symmetric nuclear matter. Within the self-consistent Green’s function approach, we compute the momentum distributions and we identify the effects of the high momentum components in the symmetry energy. The results are obtained for the realistic interaction Argonne V18 potential, supplemented by the Urbana IX three-body force in the Brueckner-Hartree-Fock calculations.     

Anisotropic lattice distortions in random alloys from first-principles theory

Within the framework of the exact muffin-tin orbitals (EMTO) theory we have developed a new method to calculate the total energy for random substitutional alloys. The problem of disorder is treated within the coherent potential approximation (CPA), and the total energy is obtained using the full charge density (FCD) technique. The FCD-EMTO-CPA method is suitable for determination of energy changes due to anisotropic lattice distortions in random alloys. In particular, we calculate the elastic constants of the Cu-rich face centered cubic Cu-Zn alloys ( alpha-brass) and optimize the c/a ratio for the hexagonal Zn-rich alloys for both the epsilon and eta phases.     

Tensor-Optimized Few-Body Model for s-Shell Nuclei

We propose a tensor-optimized few-body model (TOFM) in the few-body framework with bare nucleon–nucleon interaction. The physical concept of TOFM comes from the tensor-optimized shell-model (TOSM), which is applicable for the study of medium and heavy nuclei. The TOSM wave function describes the deuteron-like tensor correlation and provides a good reproduction of the binding energy with the bare nucleon–nucleon interaction. Using the spirit of the TOSM approximation, we show the performance of TOFM for s-shell nuclei. It is found that the TOFM can account for the contribution of the tensor interaction very well and almost reproduces the total energy and various energy components as compared with rigorous few-body calculations. The relative correlation function is also provided to improve the performance of TOSM for the study of heavy nuclei.     

Greens tensor of a weakly anisotropic cubic crystal: Efficiency of point defect absorption by spherical pore

Using the Lifshitz-Rozentsveig method, an improved expression for the Green tensor in the case of a weakly anisotropic cubic crystal has been derived. It contains a number of additional terms disregard previously. An expression for the energy of elastic interaction of a point defect with a gas-filled pore, which differs from its analogue (commonly used referring to Eshelby) by an additional factor in the radial part, has been derived. In the case of weak interaction (in comparison with thermal energy), a correction to the efficiency of point defect absorption by a gas-filled pore is obtained, which appears quadratic in the anisotropy parameter and negative. As a result, the pore exhibits preference for absorption of vacancies in comparison with absorption of interstitials.     

Diffusivity and derivatives for interstitial solutes: activation energy,volume, and elastodiffusion tensors

Computational atomic-scale methods continue to provide new information about geometry, energetics and transition states for interstitial elements in crystalline lattices. This data can be used to determine the diffusivity of interstitials by finding steady-state solutions to the master equation. In addition, atomic-scale computations can provide not just the site energy, but also the stress in the cell due to the introduction of the defect to compute the elastic dipole. We derive a general expression for the fully anistropic diffusivity tensor from site and transition state energies, and three derivatives of the diffusivity: the elastodiffusion tensor (derivative of diffusivity with respect to strain), the activation barrier tensor (logarithmic derivative of diffusivity with respect to inverse temperature) and activation volume tensor (logarithmic derivative of diffusivity with respect to pressure). Computation of these quantities takes advantage of crystalline symmetry, and we provide an open-source implementation of the algorithm. We provide analytic results for octahedral–tetrahedral networks in face-centred cubic, body-centred cubic, hexagonal closed-packed lattices, and conclude with numerical results for C in Fe.     

An exact solution for polaron in the nonlinear lattice in the Su–Schrieffer–Heeger approximation

The exact solution for a polaron in a lattice with cubic nonlinearity is obtained. The electron–phonon interaction is taken into account in the Su–Schrieffer–Heeger approximation. The system of nonlinear differential equations in partial derivatives, obtained in the continuum approximation, is exactly integrable at a certain ratio between the parameters of nonlinearity, α, and the electron–phonon interaction, χ. An approximate solution is obtained for arbitrary values of α and χ. A good agreement between the analytical results with the numerical simulation is observed at not too large values of parameters α and χ, where the continuum approximation is valid. Stable solutions also exist at higher values of these parameters.