Tunneling of a dislocation through a pinning obstacle

We have calculated the transition rate for a string through a pinning point potential barrier. The result for this N-dimensional system has the one-dimensional form τ−1=νeff exp (−ΔU/kTeff), where νeff is an effective frequency, ΔU the barrier height, and Teff an effective temperature. The frequency factor is the Vineyard effective frequency given by the product of the fundamental attack frequency and an entropy factor, which is the ratio of the product of normal mode frequencies in the pinned state to those in the activated state. There is a crossover temperature T∗=ℏωB/2πk, where ωB is given by the curvature of the potential barrier at the maximum, separating the high temperature classical behavior from the low temperature quantum rate, ωeff and ωB are closely related and have the same order of magnitude, which depends in detail on the shape of the pinning potential. The effective temperature Teff is given by the actual temperature above this crossover, while below it, Teff is given by T∗. An important point is that if one knows the transition rate at high T, then the cross-over temperature and the low T transition rate may be calculated.