Exact Calculation of the Penetrability Through Two-Peaked Fission Barriers

The penetrability is computed exactly for a fission barrier $V\left(\epsilon \right)$ defined in terms of two parabolic peaks connected smoothly with a third parabola forming the intermediate well. The potential is specified by the peak energies $E_1$ and $E_3$ and the minimum energy $E_2$ of the connecting curve, along with the constants $\hslash {\omega }_1$, $\hslash {\omega }_3$, and $\hslash {\omega }_2$ related to the curvatures of the three parabolas. For an incident wave of unit amplitude, the amplitude of the transmitted wave is determined by requiring that the wave functions (expressed exactly in terms of parabolic-cylinder functions) and their first derivatives match at the points where the parabolas are connected. The penetrability is then obtained from the amplitude of the transmitted wave. The transmission is essentially an increasing exponential function exhibiting narrow resonances at the positions of quasibound states in the intermediate well. The widths of these resonances are extremely small (∼10 eV) for the levels near the bottom of the well but increase dramatically as the energy increases. This trend continues in some cases above the top of the barrier, producing broad peaks in the penetrability function. The exact penetrabilities are accurately reproduced by the WKB approximation for energies well below the barrier tops, but for energies near the barrier tops the WKB approximation is found to overestimate the penetrability for the cases studied.