Number of particles in fission fragments

Background: In current simulations of fission, the number of protons and neutrons in a given fission fragment is almost always obtained by integrating the total density of particles in the sector of space that contains the fragment. The semiclassical nature of this procedure and the antisymmetry of the many-body wave function of the whole nucleus systematically leads to noninteger numbers of particles in the fragment.

Purpose: We seek to estimate rigorously the probability of finding $Z$ protons and $N$ neutrons in a fission fragment, i.e., the dispersion in particle number (both charge and mass). Knowing the dispersion for any possible fragmentation of the fissioning nucleus will improve the accuracy of predictions of fission fragment distributions and the simulation of the fission spectrum with reaction models.

Methods: Given a partition of the full space ${\mathbb{R}}^3$ in two sectors corresponding to the two prefragments, we discuss two different methods. The first one is based on standard projection techniques extended to arbitrary partitions of space. We also introduce a novel sampling method that depends only on a relevant single-particle basis for the whole nucleus and the occupation probability of each basis function in each of the two sectors. We estimate the number of particles $A$ in the left (right) fragment by statistical sampling of the joint probability of having $A$ single-particle states in the left (right) sector of space.

Results: We use both methods to estimate the charge and mass number dispersion of several scission configurations in ${}^{240}\mathrm{Pu}$ using either a macroscopic-microscopic approach or full Hartree-Fock-Bogoliubov calculations. We show that restoring particle-number symmetry naturally produces odd-even effects in the charge probability, which could explain the well-known odd-even staggering effects of charge distributions.

Conclusions: We discuss two methods to estimate particle-number dispersion in fission fragments. In the limit of well-separated fragments, the two methods give identical results. It can then be advantageous to use the sampling method since it provides a $N$-body basis for each prefragment, which can be used to estimate fragment properties at scission. When the two fragments are still substantially entangled, the sampling method breaks down, and one has to use projector techniques, which gives the exact particle-number dispersion even in that limit. Note that, in this paper, we have assumed that scission configurations could be described well by a static Bogoliubov vacuum: the strong odd-even staggering in the charge distributions could be somewhat attenuated when going beyond this hypothesis.

Figure 1

Illustration of the partition of the space $\mathcal{V}$ into ${\mathcal{V}}_0$ and ${\mathcal{V}}_1$ using the local density. The color scale corresponds to the value of the local density associated with a Slater determinant $|\mathrm{\Phi }\rangle$. The red line separates ${\mathcal{V}}_0$ (to the left of the line) and ${\mathcal{V}}_1$ (to the right of the line).

Figure 2

Mass fragmentation probabilities (light fragment) for all the configurations listed in Table 2.

Figure 3

Charge fragmentation probabilities (light fragment) for all the configurations listed in Table 2.

Figure 4

Standard deviations associated with the probabilities to measure the light fragment with a mass $A$ for the shape II. Top: For different values of $n_{\mathrm{pair}}$ and with $n_{\mathrm{loc}}=10000$. Bottom: For different values of $n_{\mathrm{loc}}$ and with $n_{\mathrm{pair}}=10000$.

Figure 5

Same as Fig. 4 for the charge fragmentations.

Figure 6

Probabilities to measure the light fragment with a mass $A$ for the shape II for different number of shells $N_{\mathrm{sh}}$ in the basis.

Figure 7

Same as Fig. 6 for the charge of the light fragment.

Figure 8

Probabilities to measure the light fragment with a charge $Z$ for the shape II different values $v_{\mathrm{thresh}}^2$ of the occupation numbers' threshold.

Figure 9

Geometric shapes associated with the scission configurations of Table 3.

Figure 10

Upper panel: Neutron fragmentation probabilities (light fragment) for all the configurations listed in Table 3 obtained with our method. Lower panel: same probabilities obtained with the method presented in Ref. [24].

Figure 11

Same as Fig. 10 for the charge fragmentation probabilities.

Figure 12

Left panel: Comparison of the neutron fragmentation probabilities obtained with our Monte Carlo sampling (full lines) and the projection method (dashed black lines) for different values of $\langle {\widehat{Q}}_{\mathrm{neck}}\rangle$. Right panel: the same for the proton fragmentations.