Theory of dislocation tunneling through a pinning point

We have calculated the transition rate for a string unpinning from a point barrier, using a truncated parabolic potential. In this approximation, it was shown that the result for this N-dimensional system has the one-dimensional form R=ν_effexp (−ΔU/kT_eff), where ν_eff is an effective frequency, ΔU the barrier height, T_eff the effective temperature. There is a crossover temperature T* separating the high temperature classical behavior from the low temperature quantum rate and given by T*=ω_eff/2k. The effective temperature is given by the actual temperature above T*, while below it, is given by the ground state energy, calculated using the effective frequency. The important point is that if one knows the transition rate at high T, then the crossover temperature and the low T transition rate may be calculated. The effective frequency has been calculated for dislocations in the classical regime: ν_eff0.26 (U₀/Gb³) ν_D, where U₀ is the binding energy with a pinning atom, G the shear modulus, b the Burgers vector, and ν_D is the Debye frequency. The predicted crossover temperature of a few tenths Kelvin for an Al crystal is in good agreement with our recent experimental results.