Tunneling of a String Through a Potential Barrier*
D.McKay†, Univ. of Illinois Dept. of Physics, Urbana, IL; T. Kosugi, Hiroshima Univ., Dept. of Physics, Higashi-Hiroshima, JAPAN; and A. V. Granato, Univ. of Illinois Dept. of Physics, Urbana, IL. *Research supported by NSF grant DMR 9705750 †Present Address: Antenna and Microwave Dept., Raytheon Systems Company, Tewksbury, MA.
We have calculated the transition rate for a string through a pinning point potential barrier, using a truncated harmonic oscillator potential for the barrier. In this approximation, it was shown by Link et al.1 that the result for this N–dimensional system has the one–dimensional form
where is an effective frequency, is the barrier height, k is the Boltzmann constant, and is the effective temperature. The frequency factor is the Vineyard2 effective frequency given by the product of the 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 cross–over temperature separating the high temperature classical behavior from the low temperature quantum rate and given by . The effective temperature is given by the actual temperature above this cross–over, while below it, the effective temperature 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 cross–over temperature and the low T transition rate may be calculated.
The case of a continuous string pinned by a single pinning point can be approximated by point masses and springs. For a discrete system of three mass points connected by springs with a pinning force acting on the middle mass point, the potential becomes two–dimensional. Contour plots of the potential landscape provide a graphic visualization of how the effective frequency controls the system behavior. For moderate to strong pinning, the transition becomes quasi–one–dimensional. In this case, the lowest frequencies in the entropy factor cancel in pairs, and therefore the effective frequency is greater than that of the lowest normal mode.
The string model has applications to dislocation unpinning, magnetic flux flow in type II superconductors, and vortex flow in neutron stars. A pinned dislocation a good example of a system governed by a multi–dimensional defect activation process because it is a testable system in which the entropy factor is very large. The effective frequency has been calculated for dislocations in the classical regime by Granato et al.3 They obtain
where is the binding energy, G is the shear modulus, b is the Burgers’ vector, and is the Debye frequency. The predicted cross–over temperature of a few tenths Kelvin for dislocations in relatively pure aluminum differs from earlier estimates given in a review by Startsev4 but is in good agreement with our recent experimental results.5
1B. Link, R.I. Epstein, and G. Baym, Astrophys. J. 403, 285 (1993).
2G.H. Vineyard, J. Phys. Chem. Solids 3, 121 (1957).
3A.V. Granato, K. Lücke, J. Schlipf, and L.J. Teutonico, J. Appl. Phys. 35, 2732 (1964).
4V.I. Startsev, in Dislocations in Solids, F.R.N. Nabarro, ed., (North–Holland Publishing Co., 1983), Chap. 28, and references.
5T. Kosugi, D. McKay, and A.V. Granato, J. de Physique, C–8, 863 (1996).