Exact calculation of the penetrability for a simple two-dimensional heavy-ion fusion barrier

In a study of the effect of the quantal zero-point oscillations of nuclei on their low-energy fusion cross section, we calculate exactly the penetrability for a simple two-dimensional barrier $V(r,\sigma )$. The coordinate $r$ is the distance between the centers of mass of the two nuclei, and $\sigma$ is the sum of the root-mean-square extensions along the symmetry axis of the matter distribution of each nucleus about its center of mass. The potential $V(r,\sigma )$ is a parabolic peak in $r$ and is one or the other of two harmonic oscillators in $\sigma$, depending upon whether $r$ is greater than or less than a critical value $r_1$. The oscillators differ both in the locations of their minima and in their curvatures. This simulates the dominant feature in the two-dimensional nuclear potential-energy surface of two misaligned valleys (the fission and fusion valleys) separated by a ridge between them. When an incident wave that is localized in the fusion valley encounters the potential-energy ridge, it is partially transmitted and partially reflected in waves that correspond to different excited states in the transverse direction and hence to different amounts of energy in the fusion direction. The amplitudes of these waves are determined by requiring that the wave functions (expressed exactly in terms of parabolic-cylinder functions) and their first derivatives be continuous at $r_1$. The penetrability is then obtained from the amplitudes of the transmitted waves. As a specific example, we use this formalism to calculate the penetrability for a two-dimensional potential-energy surface appropriate to the reaction ${}_{}{}^{100}{}_{}{}^{}\mathrm{Mo}$ + ${}_{}{}^{100}{}_{}{}^{}\mathrm{Mo}$ → ${}_{}{}^{200}{}_{}{}^{}\mathrm{Po}$. The calculated penetrability is substantially different from the result for a one-dimensional calculation. In particular, 10 MeV below the maximum in the one-dimensional fusion barrier the two-dimensional penetrability is ${10}^{10}$ times as large as the one-dimensional result. Also, for equal penetrability the slopes of the two curves are very different.