Geometric theory of defects

Abstract

A description of dislocations and disclinations defects in terms of RiemannCartan geometry is given, with the curvature and torsion tensors being interpreted as the surface densities of the Frank and Burgers vectors, respectively. A new free energy expression describing the static distribution of defects is presented, and equations of nonlinear elasticity theory are used to specify the coordinate system. Application of the Lorentz gauge leads to equations for the principal chiral SO(3)-field. In the defect-free case, the geometric model reduces to elasticity theory for the displacement vector field and to a principal chiral SO(3)-field model for the spin structure. As illustrated by the example of a wedge dislocation, elasticity theory reproduces only the linear approximation of the geometric theory of defects. It is shown that the equations of asymmetric elasticity theory for the Cosserat media can also be naturally incorporated into the geometric theory as the gauge conditions. As an application of the theory, phonon scattering on a wedge dislocation is considered. The energy spectrum of impurity in the field of a wedge dislocation is also discussed.

Microkinetic model of nucleation and its application in electrochemistry

“…Note, it is not occasional that T(r) coincides in its form with the kinetic energy of the quasiparticle in spherical coordinates. This is associated with the fact that the nonlinear model under consideration admits solutions in the form of quasiparticles (nonlinear waves) called instantons [8][9][10][11][12][13]. Also note that the quasiparticles (instantons) are not dynamic particles in our case but topological…”

“…The variation of the thermodynamic potential should be equal to zero (the extremum condition). If we omit the variations of the external forces, which we consider hereafter, the obtained equation takes the form (8) this leads to the equation S '' = 0. Let us pass to a cylindrical coordinate system.…”

“…Then, we write the specified part of the free energy in these variables: (9) where θ ( r ) is the angle between the cylinder axis and the spin vector (the magnetization vector in the theory of micromagnetizm (see [3][4][5][6][7][8][9][10][11][12] for more details)), and r is the radial coordinate. The model kinetic energy T(r) is a classical analogue of the exchange (potential) energy in the Heisenberg model for the two-dimensional space (see [3][4][5][7][8][9][10][11][12][13] for more details). Note, it is not occasional that T(r) coincides in its form with the kinetic energy of the quasiparticle in spherical coordinates.…”

“…In [3][4][5], an instanton model was proposed, which involves, for example, microscopic parameters of the system, such as the energy of interaction of atoms in the nanoparticle. It is expedient to compare the results of these approaches, especially taking into account that an instanton model of disclinations already exists [7][8].…”

“…Then, from (8) and (9), a known nonlinear equation follows (10) Here, it is convenient to introduce the relative coordinate ρ = r/r cr [3][4][5][7][8][9][10][11][12][13] (because the theory should be scale-invariant [9][10][11][12][13]), which changes as 0 < ρ < 1.…”