_{1}

^{*}

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
σ
^{if}
_{ αβ} are given and the corresponding angular momentum tensor, using the dynamic equation for the evolution of internal stresses and the correlation between the stresses
σ
^{if}
_{ αβ} in the defect core and the elastic stresses
σ
^{el}
_{ik} 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
Ψ
^{β}
_{⊥}(
β) and cations
, which transforms into linear spectra as
T → 0, 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 disclosed.

Currently, there are several definitions of linear defects in crystalline materials:

1) The phenomenological definition of an edge dislocation [ [

2) In the framework of the theory of an elastic continuum modeling a medium surrounding a dislocation core [

3) To describe the boundaries of tilt and torsion using a model representation—disclination as an element of rotational plasticity [ [

Numerous experimental results using the methods of ion design, X-ray topography, electron microscopy, moire [ [

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 [

It is known [^{−}^{12} s, and the spectral density Ψ ⊥ a t ( β ) on the energy dimensionless scale β = ℏ ω / k T 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 [ [

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.

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 G j k s t , the regular component of which is long-range: G ^ r e g ~ r − 1 (r is the distance from the defect line to the point r ) [ [

In the framework of the discrete model of the nuclei of linear defects [

A μ i f ( x ) = ∫ G μ ν i f ( x − x ′ ) ∑ ν I ν k d 4 x ′ ( ν = e , c a t ; x ≡ r , t ) , (1)

where the components of the tensor of the Green’s function in the generalized space of rectangular momenta have the form

G μ ν i f ( r , t ) = G μ ν + i f ( r ) ⋅ θ ( t ) + G μ ν − i f ( r ) ⋅ θ ( − t ) , (2)

and the functions θ ( t ) and θ ( − t ) are combinations of Heaviside stepwise unit functions of the type [ U ( t ) − U ( t − τ ) ] [

G μ ν ± i f ( r ) = 1 ( 2 π ) 2 ⋅ r 2 ∓ 1 4 π δ ( r 2 ) (3)

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 [

The system of pairs of photoelectrons and cations in the defect nucleus has two degrees of freedom: oscillations and rotations per charged particle, therefore, in addition to T μ ν i f , we must use the angular momentum tensor M μ ν i f of the field E j i f [ [

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 p c a t r e c , 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 σ j k i f and σ l m e l . 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 p c a t r e c from the defect line, which corresponds to the radial elastic deformation of all-round tension, but they can also acquire the tangential components p c a t r e c normal to the defect line and to the field polarization vector E j i f , which in turn corresponds to tangential elastic cation deformation or shear deformation. From this, the correlation between σ j k i f and σ l m e l becomes clear.

Here we can apply the dynamic equation of evolution of internal stresses σ m n e l [

d σ ^ e l d t = − λ e l ⋅ σ ^ e l + α ⋅ ( σ ^ e l ) 2 + γ ⋅ σ ^ e l ⋅ σ ^ i f + η ⋅ σ ^ e l ⋅ σ ^ o u t , (4)

where λ e l = τ r c a t − 1 , τ r c a t 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 relief u ( r , t ) its microscopic component u ′ ( r , t ) , fluctuating over the times t ≪ τ D ( τ D is the Debye time), becomes comparable with its macroscopic component U ( r , t ) ) for t ≫ τ D . The intermittent field reduces the point symmetry of the perfect crystal to axial, thereby creating its dynamic anisotropy in the direction of the field, which essentially forms the corresponding distribution of atom-vacancy displacements. This allows us to draw an analogy between the longitudinal autocorrelation function of the velocity of atoms Ψ ⊥ a t ( t ) on the liquid-gas surface and the same function of atoms, photoelectrons Ψ ⊥ e ( t ) and cations Ψ ⊥ c a t ( t ) in the radial direction from the defect line in the transition region, taking into account the Coulomb attraction.

It should be noted here that at absolute temperature T → 0 , all peaks of the spectral density Ψ ⊥ a t ( β ) ( β = ℏ ω / k T ), Ψ ⊥ e ( β ) , Ψ ⊥ c a t ( β ) transform into δ-functions Dirac, while the intervals between neighboring peaks increase by two orders of magnitude with a decrease from room to helium temperatures, the exponential envelope of the peaks tends to the asymptote in the form of a straight line parallel energy scale β , and the background component goes into a narrow band adjacent to the scale β . In other words, Ψ ⊥ a t ( β ) , Ψ ⊥ e ( β ) , Ψ ⊥ c a t ( β ) turn into line spectra, each line of which, apparently, characterizes the time-separated processes of translational, rotational plasticity, fragment formation, microcracks and their merging at all stages of plasticity and fracture.

An analysis of expressions (1)-(4) shows that the dependence of the components σ m n e l ( r ) is long-range and is determined by the shape and size of the vacancy volume, the nature of the distribution of the oscillation and rotations currents of photoelectrons and cations in the nuclei of linear and plane defects. In the generalized space of defect nuclei, the amplitude values of σ m n e l are determined by the amplitude and duration of the rectangular pulses of the alternating field [

We direct the z axis of the rectangular coordinate system along the line of the edge dislocation, the slip plane is compatible with the x 0 z plane with the normal n , and the y axis along n . In the theory of individual dislocations, the elastic stress tensor σ m n e l in the medium surrounding the core of the edge dislocation has a 2 × 2 matrix [ [

σ ^ e d g e l = ( σ x x 0 τ y x σ y y ) , (5)

which corresponds to plane deformation at u z = 0 . Here, the tangent component τ y x creates a shearing in the y 0 z plane in the positive or negative direction of the y axis. In the core of the dislocation, an alternating field forms linear structures along the y axis, moreover, vacancy chains appear in parallel electronic chainseither below the slip plane parallel to the z axis, where cationic chains are located from last at a distance b_{0}, and electronic chains are located at a distance ≤ 2b_{0} from dislocation line, or above the slip plane, etc. The Maxwell stress tensor σ ^ e d g i f preserves the type of matrix (5)

σ ^ e d g i f = ( σ x x i f τ x y i f ( E x o w n ) τ y x i f ( E y o w n ) σ y y i f ) , (6)

where the tangent components τ x y i f and τ y x i f contain only the real parts, depending on the components of the field E j o w n in the x 0 y plane, the imaginary parts of these components [

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

Transverse photons of the alternating field form the LS1, 2, 3, 4 structures. Quantum electrodynamics allows the existence of longitudinal photons [ [_{e}. Hence, the threshold values J e t u r n t h r and J c a t t u r n t h r are connected by a simple relation

J c a t t u r n t h r ≈ 2 m e M c a t J e t u r n t h r . (7)

For J e = J e t u r n t h r , the time dependence of J e ( t ) in the generalized space of rectangular pulses is shown in

σ α β i f = ( σ x x i f ( E x o w n ) 0 τ x z i f ( i H z o w n ) 0 σ y y i f ( E y o w n ) τ y z i f ( i H z o w n ) 0 0 σ z z i f ( E z o w n ) ) , (8)

which reflects the axial symmetry of the field of elastic strains and stresses σ α β e l in a good screw dislocation crystal. In (8) σ x x i f + σ y y i f + σ z z i f 3 = − p , and

σ x x i f = σ y y i f = σ z z i f = − p , which corresponds to all-round extension on the screw dislocation line, and the components vectors H j o w n = ( 0 ; 0 ; H z o w n ) . Here the components of the angular momentum tensor M i k i f represent the antisymmetric tensor of the second rank

M i k i f = ( 0 0 ± τ x z i f ⋅ y 0 0 ± τ y z i f ⋅ x 0 0 0 ) , (9)

where for structures LS1 and LS3 the combination of signs is ± τ x z i f ⋅ y and ∓ τ x z i f ⋅ ( − y ) respectively, and for structures LS2 and LS4: ± τ y z i f ⋅ x and ∓ τ y z i f ⋅ ( − x ) (

Experience shows [

1) Energy criterion. The energy of the currents of photoelectrons and cations from which the photoelectrons were knocked out should correspond to the minimum elastic energy of the crystal in the medium surrounding the defect core;

2) Geometric criterion. The motion paths of the above photoelectrons and cations should be within the layer 3 − 4 a 0 ;

3) The fulfillment of the electroneutrality condition both for the subsystems of photoelectrons and cations, and for conduction electrons and matrix cations;

4) The absence of long-range fields of internal stresses from the boundaries of tilt and torsion.

Consider the form of vacancy volumes V v a c for various types of disclinations. Here we will use the disclinations formation schemes in the elastic continuum model [ [

A detailed quantitative description of the transition model is possible by numerically calculating the Vlasov system of equations, the wave equation of the alternating field, and the equation of evolution of internal stresses, which requires separate work. In this paper, we restrict ourselves to the consideration of translational plasticity of metallic crystals with face centered lattice and estimate the threshold values of the photoelectron oscillation current J e o s c t h r of the onset of slip of a screw dislocation in copper. Here, the process of knocking out and moving the photoelectrons is essentially an internal photoelectric effect, where the kinetic energy received from the field E j i f must not be lower than the sum of the work of the returning forces F j r e s t of the cation from which it was knocked out, of three pairs of its nearest neighbors along the cation chain and the forces of electrostatic interaction from the side of the excess charge distribution q ( ρ ) [cm^{−}^{3}] conduction electrons [

q ( ρ ) = β ⋅ 8 π 3 k F π ⋅ A ⋅ cos ( 2 k F ρ + Φ ) ρ 5 / 2 , (10)

where the parameters β = 7.62; A = 0.052; Φ = 0.1015 are taken from [

m e ⋅ v 0 x 2 2 − m e ⋅ v f x 2 2 = ∑ i 3 A i = ∑ i 3 ∑ j 3 F j ⋅ l j ⋅ cos γ i ⋅ cos α , (11)

where the change in the kinetic energy of the photoelectrons obtained from the field E j i f is equal to the work of the sum of the Coulomb interaction forces from the point charges of the nuclei of the donor cations and the three pairs of nearest neighbors along the cation chain and the distribution q ( ρ ) ; l j is the photoelectron displacement vector along the x axis; γ i is angle between the directions of the vectors F j and l j ; α is the angle between the direction of the Coulomb force and the x axis, α = π − γ . 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 [

ε m ( k , ω ) = 1 − ω p 2 ω r e p 2 , (12)

where ω p is the plasma frequency of the subsystem of conduction electrons [^{−3}], where for copper n_{0} = 8.45 × 10^{22} сm^{−3} [_{e} = 9.11 × 10^{−}^{28} g; e = 4.8 × 10^{−}^{10} CGSE_{q} [^{14} ÷ 10^{15} s^{−}^{1}, and ω p = 1.64 × 10 16 сек − 1 , then according to (12) | ε m ( k , ω ) | = 10 2 ÷ 10 4 . 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 E l i n e ~ χ ⋅ ρ − 1 , where χ is the linear charge density, and the field inside hollow cylinder is zero [ [

E ( x ) = ∑ n = 1 n [ E n + 1 ( x ) + Δ E n l ( x ) ] , (13)

where the external fields E n ( x ) and Δ E n l ( x ) , induced respectively by the entire n-layer and the lower tube of the n-layer, have the form

E n ( x ) = χ n 2 π k ε m ⋅ x ; Δ E n l ( x ) = χ n l ( x ) 2 π k ε m ⋅ x , (14)

where the linear charge densities χ n and χ n l ( x ) are equal

χ n = ∫ ρ n ρ n + 1 q ( ρ ) ρ d ρ ∫ 0 1 d z ∫ 0 2 π d θ ; χ n l ( x ) = ∫ ρ n x − ρ n q ( ρ ) ρ d ρ ∫ 0 1 d z ∫ 0 2 π d θ ; (15)

k = 1 in CGSE_{q}; k = 4 π ε 0 in MKSA. It should be noted here that the fields E n ( x ) and Δ E n l ( x ) are directed from the dislocation line at q ( ρ ) − n 0 > 0 , and toward the dislocation line at q ( ρ ) − n 0 < 0 with respect to the photoelectron motion.

An analysis of expressions (10)-(15) shows that, in contrast to the tabulated values of n 0 [^{−}^{8} N], which allows us to estimate the height of the potential barrier from the first peak q ( ρ ) ( n = 2 ) , which does not exceed 1.2 eV, and the depth of the first potential valley q ( ρ ) ( n = 3 ) of the order of (0.2 ÷ 0.3) eV. The barrier is due to the action of the fields E 1 ( x ) and Δ E 2 l ( x ) from the inside of the peak and the electron interaction F e e , whose effective radius is 0.1 nm [

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 J p h e using the dimensional method can be represented as

J p h e = 2 ⋅ n p h e ⋅ e ⋅ v p h e = ρ d ⋅ 2 b 0 e ⋅ v p h e , (16)

where n p h e is the density of donor cations in the nuclei of defects; ρ d is the dislocation density [m^{−2}]; b 0 = 0.71 a 0 ; v p h e is the speed of the photoelectron.

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 Maxwell stresses of the alternating field σ α β i f in the core of the defect and the internal elastic stresses σ i k e l in its environment.

2) The conversion of the spectral densities Ψ ⊥ a t ( β ) , Ψ ⊥ e ( β ) , Ψ ⊥ c a t ( β ) as T → 0 into linear spectra leads to the appearance of threshold values of oscillation currents and rotations of photoelectrons and cations, of which they were knocked out at all stages of plastic deformation and fracture. In this case, intermittent deformation takes place near absolute zero [

3) The expression of the force acting on the dislocation in the form of σ i k b k in the framework of the continuum models is not applicable in the framework of the continuum models in this transition model, because it reflects the change in the elastic energy of the crystal Δ ε e l when the dislocation of unit length is displaced by one a 0 , and Δ ε e l is several orders of magnitude smaller than the changes energies of the alternating field in the core of a dislocation at similar displacements.

4) At first glance, the Coulomb’s static law is not applicable to the description of the defect nucleus, but at an arbitrary time t ± Δ t ≥ ℏ / 2 Δ ε p h e with v p h e ≪ v F , the distribution q ( ρ ) creates an additional potential relief u q ( r , t ) , which is adjacent to the main u ( r , t ) from the Coulomb attraction of the initially immobile cationic chain and essentially “stitches” with it. If the photoelectron moves with velocity v p h e ( t ) along the x axis, then the expression for the Coulomb force contains the Lorentz correction ( 1 − v p h e 2 ) − 1 / 2 times

[ ( x − v p h e ⋅ t ) 2 1 − v p h e 2 + y 2 + z 2 ] − 1 [ [

a purely quantum phenomenon takes place: a tunnel transition through the first peak q ( ρ ) , where quasistable states in the form of oscillations and rotations arise in the first potential valley q ( ρ ) , which goes from zero to 2π in the range of angles θ from zero to 2π. Hence, translational plasticity within the framework of discrete and transitional models takes on a new meaning.

5) Generation of nuclei of linear defects involves the formation of linear structures under the influence of an alternating field in the generalized space of rectangular pulses and due to the secondary long-wavelength photons of the field E j i f and the correlation σ j k i f and σ m n e l , elastic deformation and stress fields arise in the surrounding defect core medium, and at a distance from the defect line ρ 1 = ( 0.071 ÷ 0.087 ) nm equal to the cation radius [

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

Busov, V.L. (2020) On the Relationship of the Discrete Model of the Nuclei of Linear and Planar Defects and the Continuum Models of Defects in Crystalline Materials. Applied Mathematics, 11, 862-875. https://doi.org/10.4236/am.2020.119056