Formation and distribution of fragments in the spontaneous fission of ${}^{\mathbf{240}}\mathbf{Pu}$

Background: Fission is a fundamental decay mode of heavy atomic nuclei. The prevalent theoretical approach is based on mean-field theory and its extensions where fission is modeled as a large amplitude motion of a nucleus in a multidimensional collective space. One of the important observables characterizing fission is the charge and mass distribution of fission fragments.

Purpose: The goal of this Rapid Communication is to better understand the structure of fission fragment distributions by investigating the competition between the static structure of the collective manifold and the stochastic dynamics. In particular, we study the characteristics of the tails of yield distributions, which correspond to very asymmetric fission into a very heavy and a very light fragment.

Methods: We use the stochastic Langevin framework to simulate the nuclear evolution after the system tunnels through the multidimensional potential barrier. For a representative sample of different initial configurations along the outer turning-point line, we define effective fission paths by computing a large number of Langevin trajectories. We extract the relative contribution of each such path to the fragment distribution. We then use nucleon localization functions along effective fission pathways to analyze the characteristics of prefragments at prescission configurations.

Results: We find that non-Newtonian Langevin trajectories, strongly impacted by the random force, produce the tails of the fission fragment distribution of ${}^{240}\mathrm{Pu}$. The prefragments deduced from nucleon localizations are formed early and change little as the nucleus evolves towards scission. On the other hand, the system contains many nucleons that are not localized in the prefragments even near the scission point. Such nucleons are distributed rapidly at scission to form the final fragments. Fission prefragments extracted from direct integration of the density and from the localization functions typically differ by more than 30 nucleons even near scission.

Conclusions: Our Rapid Communication shows that only theoretical models of fission that account for some form of stochastic dynamics can give an accurate description of the structure of fragment distributions. In particular, it should be nearly impossible to predict the tails of these distributions within the standard formulation of time-dependent density-functional theory. At the same time, the large number of nonlocalized nucleons during fission suggests that adiabatic approaches where the interplay between intrinsic excitations and collective dynamics is neglected are ill suited to describe fission fragment properties, in particular, their excitation energy.

Figure 1

Top: The density of Langevin trajectories and the corresponding EFPs on the two-dimensional PES on the $(Q_{20},Q_{30})$ plane. The outer turning-point line (OTL) and the scission line are shown by dashed and dashed-dotted lines, respectively. Bottom: Eleven EFPs considered in this Rapid Communication.

Figure 2

Neutron (left) and proton (right) localization functions in ${}^{240}\mathrm{Pu}$ at $(Q_{20}=305\mathrm{b},Q_{30}=32{\mathrm{b}}^{3/2})$ compared to the individual NLFs of the localized prefragments (${}^{80}\mathrm{Ge}$ and ${}^{128}\mathrm{Sn}$). The symmetry axis is marked by the vertical line. The horizontal dotted lines $z_{\mathrm{L}}$ and $z_{\mathrm{H}}$ indicate the position of the maxima of the radial coordinate $r_{\perp }\left(z\right)$ associated with the NLF profile.

Figure 3

Top: Partial mass distributions for different initial configurations shown in Fig. 1. The contribution from the minimum-action fission path EFP 5 is shown by a thick line. Bottom: The potential energy along each EFP. Note the nonmonotonic behavior for EFP 3 and EFP 4.

Figure 4

Neutron NLFs for neutrons for several configurations [indicated by $Q_{20}$ (in barns) and $Q_{30}$ (in ${\mathrm{b}}^{3/2}$)] along the four EFPs indicated. The color legend is same as in Fig. 2. The horizontal dashed lines indicate the position of the maxima of the radial coordinate associated with the NLF profile.

Figure 5

Upper panel: The ranges for the number of localized (a) neutrons and (b) protons for heavier ($N_{\mathrm{H},\mathcal{C}},Z_{\mathrm{H},\mathcal{C}}$) and lighter ($N_{\mathrm{L},\mathcal{C}},Z_{\mathrm{L},\mathcal{C}}$) fragments as a function of the configurations along the EFPs marked in Fig. 1 by circles. The ranges obtained by integrating the nucleon density for the heavier ($N_{\mathrm{H},\rho },Z_{\mathrm{H},\rho }$) and lighter ($N_{\mathrm{L},\rho },Z_{\mathrm{L},\rho }$) fragments from the minimum of the neck. The magic numbers are marked by the horizontal dotted lines.

Figure 6

Number of particles in the neck $q_N=\langle Q_N\rangle$ defined through the expectation value of the neck operator with $a_N=1\mathrm{fm}$ as a function of the length of the EFP.

Figure 7

Average collective momentum of Langevin trajectories for different EFPs (squares) and shortly before reaching the scission point (dots).