Induced nuclear fission viewed as a diffusion process: Transients

Induced nuclear fission is viewed as a diffusion process of the fission degree of freedom over the fission barrier. We describe this process in terms of a Fokker-Planck equation which contains the fission variable and its canonically conjugate momentum. We solve this equation numerically for several energies (temperatures) of the fissioning nucleus neglecting changes of the fission barrier due to the temperature dependence of nuclear shell effects. We pay particular attention to the time $\tau$ needed for the system to build up the quasistationary probability flow over the fission barrier. The rate of the latter is approximated in terms of the Bohr-Wheeler formula or Kramer's transition state expression; the precise value of the quasistationary current depends on the nuclear friction constant $\beta$. Our results for $\tau$ are consistent with those obtained earlier in the framework of a simplified model: As long as $\beta \le {\beta }_0$, the time $\tau$ is proportional to ${\beta }^{-1\mathrm{}}$. This relationship exhibits the fact that with increasing friction $\beta$, the diffusion process is accelerated, so that it takes the system increasingly less time to attain the quasistationary distribution. The constant ${\beta }_0$ is roughly given by $2{\omega }_1$, where ${\omega }_1$ is the frequency of a harmonic oscillator potential which osculates the potential at the minimum corresponding to the initial configuration of the fissioning nucleus. The condition $\beta \le {\beta }_0$ is roughly equivalent with the motion in that minimum being underdamped. The converse relationship—$\tau$ increases with $\beta$—is found for $\beta >{\beta }_0$. We ascribe this to the fact that now the fission variable executes an overdamped motion. Generalizing Kramers's original derivation, we obtain an analytical expression for the time dependence of the probability current over the fission barrier. For $\beta \lesssim {\beta }_0$, this expression agrees well with our numerical results. We use it to calculate the energy dependence of the fission probability $P_f$ and find that $P_f$ grows much less rapidly with increasing excitation energy than would be predicted by the Bohr-Wheeler formula. This is in qualitative agreement with recent experimental findings and suggests that the energy dependence of $P_f$ deserves further investigation and can be used to determine $\beta$ experimentally. Our analysis does not yet include the additional time delay incurred by the system on its way from the saddle to the scission point: Clearly the time needed to establish the quasistationary situation at the scission point will be larger than $\tau$. This would probably lead to additional modifications of the energy dependence of $P_f$.