Dislocation pinning/depinning transition: Fixed-point analysis

Abstract

We present a fixed-point analyis for the equations describing the motion of a dislocation in the presence of thermally activated cross-slip pinning events. Previously, a stochastic, finite-difference model was proposed [Chrzan and Daw, Phys. Rev. B 55, 798 (1997)] in relation to the yield strength anomaly observed in ${L1}_{2}$ intermetallic compounds. (In such materials, the strength increases with increasing temperature.) In addition to numerical solutions, Chrzan and Daw also derived deterministic, mean-field equations to describe the dynamics. In this paper, we use a fixed-point stability analysis to solve analytically those mean-field equations. We compare the analytic solutions to the previously obtained numerical solutions. As with the (numerical) exact solutions, the mean-field solutions exhibit a pinning/depinning transition. Mean-field critical exponents and scaling relations are determined analytically.