Geometric theory of defects

Abstract

A description of dislocation and disclination defects interms of the Riemann–Cartan geometry is given, with thecurvature 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 ispresented and equations of nonlinear elasticity theory are usedto specify the coordinate system. Application of the Lorentzgauge leads to equations for the principal chiral SO(3) field.In the defect-free case, the geometric model reduces to elasticitytheory for the displacement vector field and to a principal chiralSO(3)-field model for the spin structure. As illustrated by theexample of a wedge dislocation, elasticity theory reproducesonly the linear approximation of the geometric theory of defects. It is shown that the equations of asymmetric elasticitytheory for Cosserat media can also be naturally incorporatedinto the geometric theory as gauge conditions. As an applicationof the theory, phonon scattering on a wedge dislocation isconsidered. The energy spectrum of impurities in the field of awedge dislocation is also discussed.