Geometric Theory Of Defects Research Articles

Combined Screw and Wedge Dislocations

Elastic media with defects are considered manifold with nontrivial Riemann–Cartan geometry in the geometric theory of defects. We obtain the solution of three-dimensional Euclidean general relativity equations with an arbitrary number of linear parallel sources. It describes elastic media with parallel combined wedge and screw dislocations.

Disclinations in the Geometric Theory of Defects

In the geometric theory of defects, media with a spin structure, for example, ferromagnet, is considered as a manifold with given Riemann--Cartan geometry. We consider the case with the Euclidean metric corresponding to the absence of elastic deformations but with nontrivial ${\mathbb S}{\mathbb O}(3)$-connection which produces nontrivial curvature and torsion tensors. We show that the 't Hooft--Polyakov monopole has physical interpretation in solid state physics describing media with continuous distribution of dislocations and disclinations. The Chern--Simons action is used for the description of single disclinations. Two examples of point disclinations are considered: spherically symmetric point "hedgehog" disclination and the point disclination for which the $n$-field has a fixed value at infinity and essential singularity at the origin. The example of linear disclinations with the Franc vector divisible by $2\pi$ is considered.

The ’t Hooft–Polyakov monopole in the geometric theory of defects

The ’t Hooft–Polyakov monopole solution in Yang–Mills theory is given new physical interpretation in the geometric theory of defects. It describes solids with continuous distribution of dislocations and disclinations. The corresponding densities of Burgers and Frank vectors are computed. It means that the ’t Hooft–Polyakov monopole can be seen, probably, in solids.

Chern—Simons Action and Disclinations

We review the main properties of the Chern—Simons and Hilbert—Einstein actions on a three-dimensional manifold with Riemannian metric and torsion. We show a connection between these actions that is based on the gauge model for the inhomogeneous rotation group. The exact solution of the Euler—Lagrange equations is found for the Chern—Simons action with the linear source. This solution is proved to describe one straight linear disclination in the geometric theory of defects.

Chern-Simons term in the geometric theory of defects

The Chern--Simons term is used in the geometric theory of defects. The equilibrium equations with $\delta$-function source are explicitly solved with respect to the $SO(3)$ connection. This solution describes one straight linear disclination and corresponds to the new kind of geometrical defect: it is the defect in the connection but not the metric which is the flat Euclidean metric. This is the first example of a disclination described within the geometric theory of defects. The corresponding angular rotation field is computed.

Optical Anisotropy of Topologically Distorted Semiconductor Nanocrystals.

Engineering nanostructured optical materials via the purposeful distortion of their constituent nanocrystals requires the knowledge of how various distortions affect the nanocrystals' electronic subsystem and its interaction with light. We use the geometric theory of defects in solids to calculate the linear permittivity tensor of semiconductor nanocrystals whose crystal lattice is arbitrarily distorted by imperfections or strains. The result is then employed to systematically analyze the optical properties of nanocrystals with spatial dispersion caused by screw dislocations and Eshelby twists. We demonstrate that Eshelby twists create gyrotropy in nanocrystals made of isotropic semiconductors whereas screw dislocations can produce it only if the nanocrystal material itself is inherently anisotropic. We also show that the dependence of circular dichroism spectrum on the aspect ratio of dislocation-distorted semiconductor nanorods allows resonant enhancing their optical activity (at least by a factor of 2) and creating highly optically active nanomaterials.

Geometric model of a fullerene molecule in the presence of Aharonov–Bohm flux

In this study, we describe a geometric model of a fullerene molecule with Ih symmetry. We combine the well known non-Abelian monopole approach and the geometric theory of defects, where every topological defect is associated with curvature and torsion, to describe a fullerene molecule. The geometric theory of defects in solids is used to consider the topological defects that allow this molecule to form and we apply a continuum formulation to describe this spherical geometry in the presence of an external Aharonov–Bohm flux. We solve a Dirac equation for this model and obtain the eigenvalues and eigenfunction of the Hamiltonian, and we obtain the persistent current for this model and show that it depends on the geometrical and topological properties of the fullerene.

Tube dislocations in gravity

We consider static massive thin cylindrical shells (tubes) as the sources in Einstein’s equations. They correspond to δ- and δ′-function-type energy-momentum tensors. The corresponding metric components are found explicitly. They are not continuous functions, in general, and lead to ambiguous curvature tensor components. Nevertheless all ambiguous terms in Einstein’s equations safely cancel. The interplay between elasticity theory, geometric theory of defects, and general relativity is analyzed. The elasticity theory provides a simple picture for defect creation and a new look on general relativity.

TOPOLOGICAL DEFECT DISTRIBUTIONS AND THE SELF-ENERGY OF A CHARGED PARTICLE

We study a charged particle in a medium with different continuous distributions of topological defects in the approach of the geometric theory of defects. Those distributions allow us to evaluate the self-energy for a single charge in both the internal and external regions of each distribution.

Inside the BTZ black hole

We consider a static circularly symmetric solution of three-dimensional Einstein equations with a negative cosmological constant (the BTZ black hole). The case of the zero cosmological constant corresponding to the interior region of a black hole is analyzed in detail. We prove that the maximally extended BTZ solution with the zero cosmological constant coincides with flat three-dimensional Minkowskian space-time without any singularities and horizons. The Euclidean version of this solution is shown to have a physical interpretation in the geometric theory of defects in solids describing combined wedge and screw dislocations.

AHARONOV–BOHM EFFECT AND DISCLINATIONS IN AN ELASTIC MEDIUM

In this work we investigate quasiparticles in the background of defects in solids using the geometric theory of defects. We use the parallel transport matrix to study the Aharonov–Bohm effect in this background. For quasiparticles moving in this effective medium we demonstrate an effect similar to the gravitational Aharonov–Bohm effect. We analyze this effect in an elastic medium with one and N defects.

Geometricheskaya teoriya defektov

Geometricheskaya teoriya defektov

Wedge dislocation in the geometric theory of defects

We consider a wedge dislocation in the framework of elasticity theory and the geometric theory of defects. We show that the geometric theory quantitatively reproduces all the results of elasticity theory in the linear approximation. The coincidence is achieved by introducing a postulate that the vielbein satisfying the Einstein equations must also satisfy the gauge condition, which in the linear approximation leads to the elasticity equations for the displacement vector field. The gauge condition depends on the Poisson ratio, which can be experimentally measured. This indicates the existence of a privileged reference frame, which denies the relativity principle.

Scattering on Dislocations and Cosmic Strings in the Geometric Theory of Defects

We consider scattering of elastic waves on parallel wedge dislocations in the geometric theory of defects or, equivalently, scattering of point particles and light rays on cosmic strings. Dislocations are described as torsion singularities located on parallel lines, and trajectories of phonons are assumed to be the corresponding extremals. Extremals are found for arbitrary distribution of the dislocations in the monopole, dipole, and quadrupole approximation and the scattering angle is obtained. Examples of continuous distribution of wedge and edge dislocations are considered. We have found that for deficit angles close to −2 πa star located behind a cosmic string may have any even number of images, 2, 4, 6, … The close relationship between dislocations and conformal maps is elucidated in detail.