A dislocation-dipole in one-dimensional lattice model

A family of equilibria corresponding to a dislocation-dipole, with variable separation between the two dislocations of opposite sign, is constructed in a one-dimensional lattice model. A continuous path is constructed, in the space of lattice configurations, connecting certain members of this family; eventually leading to an energy landscape, using the concept of order-parameter. The energy landscape is found to exhibit the familiar Peierls relief for the variation of energy associated with certain sequential transitions between these equilibria. The results allow an interpretation in terms of quasi-statically separating pair of dislocations of opposite sign from the viewpoint of the original Frenkel-Kontorova model albeit a piecewise-quadratic onsite potential is employed in the paper. The closed form expressions are provided after an application of detailed analytical means wherein an analysis of the effect of an intermediate spinodal region is included.