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Israel Journal of Chemistry

Collapse of transmissivity in a triple barrier

Roland Lefebvre, Osman Atabek, R. Lefèbvre, Manuel Martín

Published Dec 1, 2003• 0 citations

Abstract

Abstract One generally expects that the transmissivity through a symmetric arrangement of barriers and wells should go to unity when the energy of the incident particle coincides with a resonance energy of the system. It has been noticed before that for a symmetric triple barrier, there are conditions that lead to a merging of the two lowest unit transmissivity peaks, accompanied by a collapse of the transmissivity profile. This feature has been interpreted as meaning that there is a reduction of the number of resonances under these conditions. We reanalyze this situation with the help of a model that involves the same number of bound and continuum states, but which possesses an analytic solution. Because the transition probability in this case is expressed entirely in terms of resonance parameters (positions and widths), it is possible to calculate with the expression derived from the model a profile that agrees completely with that calculated with the transfer matrix method. The resonance parameters are obtained from the poles of the transition amplitude. This shows that the merging of the peaks and the collapse of the transmissivity occurs without a reduction in the number of resonances.

DOI: 10.1560/f8jp-3dd7-0an4-harr

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Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr\"odinger equation with a cutoff wave initial condition. We derive a two-level formula for incidence energies E near the first resonance doublet of the system. Based on that expression we find that the probability density along the internal region of the potential is governed by three oscillation frequencies: one of them refers to the well known Bohr frequency, given in terms of the first and second resonance energies of the doublet, and the two others represent a coupling with the incidence energy E. This allows us to manipulate the above frequencies to control the tunneling transient behavior of the probability density in the short-time regime.

A system constituted of three bosons interacting via two-body separable potentials with fixed two-boson binding is known to lead to bound-state collapse in the case where the potential parameters allow two-boson S-matrix poles close to (resonance) and on (continuum bound state) the real momentum axis. The collapse is shown to be accompanied by an increase in the average kinetic energy of the two-body bound state, which signals a decrease in the range of the two-body interaction for fixed two-body binding. The collapse is claimed to be a manifestation of the well-known Thomas effect which leads to a collapse of the three-body system when the range of the two-body interaction goes to zero for a fixed two-body binding.

We study a diatomic homonuclear molecule incident upon a potential barrier. We allow transitions between binding energy states of the molecule during the process of tunneling and consider both single and double barriers. By applying an elegant method to solve this system we show that transmission resonances do occur in both cases and that expressions for the locations of the resonances can be found. Moreover, we find that the addition of binding energy levels tends to decrease the probability of tunneling. We also show that by relaxing rigor and treating the length of the molecule as though it is fixed we can obtain exact analytical expressions for the transmission probability that capture important features of the full solution.