Geometric theory of defects

A description of dislocation and disclination defects in terms of the Riemann – Cartan geometry is given, with the curvature and torsion tensors 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 Cosserat media can also be naturally incorporated into the geometric theory as gauge conditions. As an application of the theory, phonon scattering on a wedge dislocation is considered. The energy spectrum of impurities in the field of a wedge dislocation is also discussed.