One-Dimensional Atomic Model for a Dislocation Line

In a previous paper the author used a two-dimensional lattice model to examine the equilibrium properties of a dislocation kink. The width, energy, and Peierls stress of the kink were computed for a range of values of the shear stress of a perfect lattice. In the present work a one-dimensional approximation to this two-dimensional model is developed, through the use of single-chain localized modes, and the same properties computed. This approximation is shown to be equivalent to a "discrete string" model of the dislocation line, in which the string tension and Peierls potential appear naturally, and with which the kink Peierls stress can be determined. All the equilibrium properties are in excellent agreement with the properties of the two-dimensional model, and the computation time is about $\frac{1}{3000}$ as long. It appears that the same one-dimensional model could be used for more extensive investigations of dislocation behavior, including dynamic effects.