Geometric Theory of Defects

TL;DR

This paper develops a geometric framework using Riemann--Cartan geometry to describe defects like dislocations and disclinations in materials, linking elasticity, gauge theory, and defect dynamics.

Contribution

It introduces a novel free energy model for defect distribution and integrates nonlinear elasticity and gauge conditions into a unified geometric theory.

Findings

Reproduces linear elasticity as a special case of the geometric model.

Shows how asymmetric elasticity theory fits as gauge conditions.

Analyzes phonon scattering and impurity spectra near defects.

Abstract

A description of dislocations and disclinations defects in terms of Riemann--Cartan 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.