On the Relationship of the Discrete Model of the Nuclei of Linear and Planar Defects and the Continuum Models of Defects in Crystalline Materials

A physical and mathematical model of the transition from a discrete model of linear and flat defects nuclei to continuum models of defects such as dislocations and disclinations and their combinations is presented, where the tensors of energy-momentum and angular momentum of an alternating field are considered, for which the type and structure of the Maxwell stress tensor if αβ σ are given and the corresponding angular momentum tensor, using the dynamic equation for the evolution of internal stresses and the correlation between the stresses if αβ σ in the defect core and the elastic stresses el ik σ in its environment, obtains elastic displacement and deformation fields identical to these fields from Burgers and Frank vectors of continuous models. The spectral density of the autocorrelation functions of the velocity of photoelectrons and cations  , which transforms into linear spectra as 0 T → , is considered reflecting the existence of threshold values of oscillation and rotations currents of photoelectrons and cations at all stages of plastic deformation and fracture. The features of the process of sliding linear defects in metals are


Introduction
Currently, there are several definitions of linear defects in crystalline materials: 1) The phenomenological definition of an edge dislocation [[1] [2] p. 235] includes the insertion of an extra half-plane, forming regions of condensation and rarefaction of atoms above and below the slip plane with a normal n . Its edge inside the crystal corresponds to a dislocation line with the unit vector τ , and the Burgers vector is equal to the sum of the increments u, but scr b τ is similar. 2) In the framework of the theory of an elastic continuum modeling a medium surrounding a dislocation core [2] [3], the well-known procedure is applied: the crystal is cut along the slip plane, the dislocation core is removed, the cut surfac- Numerous experimental results using the methods of ion design, X-ray topography, electron microscopy, moire [[4] p. 36; [5] p. 323] basically confirm the phenomenological and continuum definitions of linear defects. At the same time, due to the insufficient resolution of the above methods (not less than 0.6 ÷ 0.8 nm), it has not yet been possible to directly observe the structure of the defect nucleus except refractory metals such as molybdenum and crystals of copper and platinum phtalocyanide.
It is known that the structure of the usual boundaries of tilt and torsion is determined by the method of relaxation tuning of atoms, which form an interlayer with a thickness of 2 -3 atomic layers with a minimum excess energy of the boundary [6]. In the liquid boundary model [7], the crystal structure is completely broken and can be represented as a combination of vacancies and disordered atoms. A number of other works [8] [9], based on a symmetric tilt boundary in the form of a lattice dislocation wall and the Reed-Shockley formula, consider the amorphization process, where the nuclei of these lattice dislocations ( ) z ρ arise [ [11], p. 156] with a spatial period 0 a , the longitudinal autocorrelation function the velocities of atoms for moving along the normal to the surface of the liquid are oscillatory quasicrystalline in time scale with a period equal to the lifetime of surface phonons of the order of 4.5 × 10 −12 s, and the spectral density has two pronounced peaks and three -four less pronounced peak with an exponential envelope for the entire series of peaks above the background, the area of which is also limited by the exponent [ [11], p. 254].
Starting from the works of G. Weingarten, A. Somigliana, and V. Volterra, continuum dislocation models are based on physical and mathematical abstraction in the form of a line endowed with mass, velocity, energy, linear tension and the Burgers vector, where the degree of physicality of the model is not large enough compared to its mathematics; as in the theory of linearly deformed media, only Hooke's physical law holds. A natural question arises: What is the structure of the nuclei of linear and planar defects in the language of charges and their currents, because nothing except them exists in crystalline materials? How does this structure reflect the dynamic processes of generation and slip of such defects?
The aim of this work is to build a transitional model from a discrete model of linear and flat defects nuclei to continuum dislocation-disclination models of crystalline materials.

The Theoretical Model
First, note that the continual dislocation-disclination models have two important features of the mathematical apparatus: 1) when describing translational plasticity, the Burgers vector is introduced on the defect line, and when describing rotational plasticity, the Frank vector; 2) in these models, in most cases, they consider a static problem using equilibrium equations, where a solution is sought using the Green's function In the framework of the discrete model of the nuclei of linear defects [13], the general structure of the solutions of the wave equation of the alternating field as an analog of the electromagnetic field using the Green's functions [ [14], p. 77] in the form of a vector potential where the components of the tensor of the Green's function in the generalized space of rectangular momenta have the form and describes, respectively, a wave running to the right along the defect line or a wave reflected from its left edge of the defect fixing nodes and a wave running to the left or reflected from the right edge of the defect. According to [13], one of the possible schemes for generating linear defect nuclei includes the photoelectric effect of the electrons of the inner shells of cations as a result of resonant scattering of primary photons of an intermittent (alternating) field and the formation of long-wave secondary photons that are elastically scattered in the region of the defect. The energy of primary photons γ ε is redistributed between the transition energy ij ε of photoelectrons between discrete states i j → in the matrix cations, the kinetic energy of photoelectrons phe ε knocked out of cations, and the energy of secondary photons sph ε , which mainly determine the elastic deformation fields and stresses of a good crystal. The kinetic energy of photoelectrons phe ε is completely determined by the pulse density of the alternating field Let us briefly disclose a scheme of the mechanism of influence of secondary long-wavelength photons of an alternating field on the environment surrounding the core of a defect. According to atomic spectra, these photons cannot produce resonant scattering by cations at lattice sites, but they carry out elastic scattering by them. In addition, they are bosons, obey wave-particle duality, their density decreases from the defect line according to the hyperbolic law. Here, the elastic scattering of photons by cations does not change the internal state of cations, but when they are backscattered by large angles from π/2 to π, within the framework of the momentum conservation law, cations acquire recoil momentum rec cat p , leading to elastic displacements from lattice sites. Note that, in this case, the directions of the polarization vectors of the primary and secondary photons are close or coincide, which leads to the coincidence of the types of tensor matrices if jk σ and el lm σ . Apparently, this is precisely the physical essence of Hooke's law in quantum electrodynamics. In addition, during elastic scattering, the cation nuclei acquire the radial components rec cat p from the defect line, which corresponds to the radial elastic deformation of all-round tension, but they can also acquire the tangential components rec is the fast relaxation time of cations in a good crystal. When considering such a correlation, the structural-phase transition in the region between a good and a bad crystal becomes important. Here in the potential are determined by the amplitude and duration of the rectangular pulses of the alternating field [13].

Edge Dislocation
We direct the z axis of the rectangular coordinate system along the line of the where the tangent components

Screw Dislocation
Let us consider one of the possible schemes for generating linear structures in the core of a screw dislocation. Here, the formation of linear structures occurs in two mutually perpendicular glide planes, while the vacancy chain is formed on the dislocation line by means of four such structures, as shown in Figure 1.     Figure 1). The combinations of signs before the products of stress and coordinate values reflect, respectively, the left-sided and right-sided rotation systems with respect to the unit vector τ . We assume that the probabilities of the appearance of right-handed and left-handed rotation systems are the same and equal to 1/2.

Disclosure Models
Experience shows [19]  The volume vac V for general disclination approaches in shape to a dumbbell axisymmetric with the defect line, and its axial sections are elliptical.  [21] in copper ( ) ( )

Translational Ductility in Metals
where the parameters β = 7.62; A = 0.052; Φ = 0.1015 are taken from [21]; F k is the Fermi wave number, for copper where the change in the kinetic energy of the photoelectrons obtained from the field = π − . The distribution ( ) q ρ is considered in a cylindrical coordinate system, and the photoelectron displacement is in the Cartesian coordinate system, whose z axes coincide. For calculation, we use the standard approach for describing [23] the motion of charged particles in an electromagnetic field for various systems of interacting charges. We assume that the nonrelativistic approximation is applicable here, and the radiation of moving particles is absent. Note that in the calculation, the dielectric constant where p ω is the plasma frequency of the subsystem of conduction electrons . At the first stage of the calculation, we use the fact that the electric field outside the infinitely long circular cylinder uniformly charged in volume or on the surface is identical to the field of the infinite uniformly charged line where the external fields ( ) ( ) where A detailed consideration of the numerical calculation using the Mathcad application, the equations and trajectories of the motion of photoelectrons and donor cations in the nuclei of linear and planar defects is presented in a separate work. We also note that the photoelectron current density phe J using the dimensional method can be represented as

Discussion of the Results
This transition model allows you to identify a number of fundamentally important features: 1) The existence of a correlation between the components of the tensor of the

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.