Is “friction” responsible for the reduction of fusion rates far below the Coulomb barrier?

The fusion of two interacting heavy ions traditionally has been interpreted in terms of the penetration of the projectile into the target. Observed rates well below the Coulomb barrier are considerably lower than estimates obtained from penetration factors. One approach in the analysis of the data invokes coupling to nonelastic channels in the scattering as the source of the depletion. Another is to analyze those data in terms of tunneling in semiclassical models, with the observed depletion being taken as evidence of a “friction” under the barrier. A complementary approach is to consider such tunneling in terms of a fully quantal model. We investigate tunneling with both one-dimensional and three-dimensional models in a fully quantal approach to investigate possible sources for such a friction. We find that the observed phenomenon may not be explained by this friction. However, we find that under certain conditions tunneling may be enhanced or diminished by up to $50\%$, which finds analogy with observation, without the invocation of a friction under the barrier.

Figure 1

Modulus of the wave packet $\mid {\Psi }_l(x,t)\mid$ at times $t=0$ (dashed line), 18 (solid line), and 40 (dotted-dashed line). The potential is portrayed by the dotted line.

Figure 2

The average position (solid line) and average momentum (dashed line) of the wave packet as a function of time. The initial momentum is $K=1.06$. To facilitate the display, the momentum has been multiplied by a factor of 10.

Figure 3

$\mid {\Psi }_{+}\mid$ at $t=15.75$ corresponding to maximal presence under the barrier.

Figure 4

Transmitted norm $T\left(t\right)$ for the case $K=1.06$ and for the reference potential.

Figure 5

The potentials used for solving the TDSE. The base potential $\left[W\left(x\right)=0\right]$ is given by the solid line, while the potentials for $\sigma =0.5$ (ears up) and $\sigma =-0.5$ (ears down) are given by the dashed and dotted lines, respectively.

Figure 6

Transmitted norms for a wave packet with $K=1.06$ incident on the three potentials. The curves correspond to the results obtained for the reference (solid line), “ears-up” (dashed line), and “ears-down” (dotted line) potentials.

Figure 7

As for Fig. 6 but for $K=0.6$.

Figure 8

As for Fig. 6 but for $K=0.42$.

Figure 9

As for Fig. 6 but for $K=0.46$ and with the modulating potential $W\left(x\right)$ given by Eq. (9).

Figure 10

Ratio of $T\left(\infty \right)$ against the reference value $\left[T_R\left(\infty \right)=0.0239\right]$ for the runs made using the random pair of $\left\{{\Omega }_c,{\Omega }_s\right\}$ specified in Eq. (10). For these calculations, $K=0.6$. The line is a guide to the eye.

Figure 11

Distribution of values of $T\left(\infty \right)$ for the runs shown in Fig. 10, for which $K=0.6$. The reference value $T_R\left(\infty \right)=0.0239$ is indicated by the dashed line.

Figure 12

As for Fig. 11 but for $K=1.06$ and using a sample of 500 potentials. The value of $T_R\left(\infty \right)$ for the reference potential is 0.2039.

Figure 13

The radial potential $V_{\mathrm{sqC}}\left(r\right)$ in the $3\mathrm{D}$ problem.

Figure 14

Energy variation of the reference action for tunneling under the potential $V_{\mathrm{sqC}\left(r\right)}$.

Figure 15

Ratio $⟨\ln \mathcal{T}⟩∕\mathcal{A}$ as a function of energy. The solid curve was obtained by using the reference potential $V_{\mathrm{sqC}}$. The closed circles depict the result found by using a fluctuation energy of 0.0929 while the open circles correspond to use of an energy three times greater. The crosses are the results found on using a fluctuation set at $10\%$ of the highest strength value.