Motion of a Frenkel—Kontorowa Dislocation in a One-Dimensional Crystal

An analytic solution is given for the equations of a linearized Frenkel---Kontorowa one-dimensional dislocation. It is shown that the motion of the defect with velocity $v$ excites lattice vibrations with $k$ given by $\ensuremath{\omega}(k)=vk$, where $\ensuremath{\omega}(k)$ is the dispersion of the lattice waves. At high velocity there is only one $k$ excited (in one dimension) appearing at the back of the defect, so that the frictional force is very small, one order of magnitude smaller than the Peierls force. At low velocities, however, there are many waves excited appearing both ahead and behind the moving defect, and the frictional force increases to the point of making steady-state nonthermally activated motion impossible.