Calculations were made of the transition rate for a string passing through a pinning point potential barrier. The result for an N-dimensional system was of the 1-dimensional form, v = veffexp[-E/kTeff], where veff was the effective frequency, E was the barrier height and Teff was the effective temperature. The frequency factor was a Vineyard effective frequency which was given by the product of the fundamental attack frequency and an entropy factor. The latter was, in turn, equal to the ratio of the product of normal-mode frequencies in the pinned state to those in the activated state. There was found to be a cross-over temperature at Tc = hwo/2πk, where wo was given by the curvature of the potential barrier at the maximum which separated the high-temperature classical behavior from the low-temperature quantum rate. The value of wo depended upon the shape of the pinning potential. The effective temperature was given by the actual temperature above the cross-over. When below it, Teff was given by Tc. An important result was that, if the transition rate at high temperatures was known, the cross-over temperature and the low-temperature transition rate could be calculated.

Tunnelling of a Dislocation through a Pinning Obstacle. A.V.Granato, T.Kosugi, D.McKay: Materials Science and Engineering A, 2001, 309-310, 207-10