Diabatic paths through the scission point in nuclear fission

An outstanding problem in the theory of nuclear fission is to understand the Hamiltonian dynamics at the scission point. In this work the fissioning nucleus is modeled in self-consistent mean-field theory as a set of generator coordinate method (GCM) configurations passing through the scission point. In contrast to previous methods, the configurations are constructed in the Hartree-Fock approximation with axially symmetric mean fields and using the $K$-partition numbers as additional constraints. The goal of this work is to find paths through the scission point where the overlaps between neighboring configurations are large. A measure of distance along the path is proposed that is insensitive to the division of the path into short segments. For most of the tested $K$ partitions two shape degrees of freedom are adequate to define smooth paths. However, some of the configurations and candidate paths have sticking points where there are substantial changes in the many-body wave function, especially if quasiparticle excitations are present. The excitation energy deposited in fission fragments arising from thermal excitations in the pre-scission configurations is determined by tracking orbital occupation numbers along the scission paths. This allows us to assess the validity of the well-known scission-point statistical model, in which the scission process is assumed to be fully equilibrated up to the separated fission fragments. The nucleus ${}^{236}\mathrm{U}$ is taken as a representative example in the calculations.

Figure 1

Energy of Glider with the D1S energy functional and constrained by the quadrupole field $Q_2$.

Figure 2

Density distributions of the Glider configurations at two HF minima at the scission point. Top panel: configuration $|168p\rangle$; bottom panel: $|166\rangle$. Contour lines of the mass density are spaced by $\mathrm{\Delta }\rho =0.016{\mathrm{fm}}^{-3}$.

Figure 3

Neck parameter $N_{\mathrm{neck}}$ as a function of deformation for Glider. The circles starting from the upper left are the HF minima constrained by $Q_2$ stepping from the previous solution at $Q_2-2$ b. The minima starting from the lower right are similarly constrained stepping from the previous solution at $Q_2+2$ b. There are two distinct minima at $Q_2=168$ and 170 b.

Figure 4

Orbital locations and their energies as a function their expectation value ${\langle z\rangle }_{\alpha }$. Panel (a): $|168p\rangle$; panel (b): $|166\rangle$. Proton and neutron orbitals are shown as circles and diamonds, respectively. The markers are red for $K=1/2$ orbitals and otherwise blue.

Figure 5

$K=1/2$ orbital energies and location parameters $\langle z\rangle$ as the Glider neck parameter changes. Panels (a) and (b) show proton and neutron orbitals respectively, for $|\varepsilon -{\varepsilon }_f|<4$ MeV. Red diamonds: initial configuration $|166\rangle$; blue diamonds: final configuration $|168p\rangle$. See text for comments on the marked orbitals.

Figure 6

Energy of the A configuration with the D1S energy functional and constrained by the quadrupole field $Q_2$. Black circles: iteration from the left; red diamonds: iteration from the right.

Figure 7

Overlap distance $\zeta$ from $N_{\mathrm{neck}}=3.0$ to 0.7 for the four configurations Glider, A, B, and C with the D1S energy functional. The A configuration is distinguished by red lines and markers.

Figure 8

Hartree-Fock potential energy surfaces for ${}^{236}\mathrm{U}$ along the fission valley constrained by $Q_2$. The energies functionals are the D1S (black lines) and the BCPM (red lines). The cusps in the curves mark the positions where the orbital occupancies and $K$ partitions change. While the $Q_2$ is continuous by construction, the other moments of the shape distribution have discontinuities at these points. For example, the $Q_3$ moment jumps by $\approx 2{\mathrm{b}}^{3/2}$ at $Q_2=96$ b where a neutron pair in a $K=3/2$ orbital jumps to a $K=1/2$ orbital.

Figure 9

Glider neck size with the BCPM energy functional and constrained by the quadrupole field $Q_2$. As in Fig. 6, iterations from the left and right are shown as black circles and red diamonds, respectively.

Figure 10

Overlap distance $\zeta$ from $N_{\mathrm{neck}}=3.0$ to 0.7 for the four configurations Glider, A, B, and C calculated with the BCPM energy functional.

Figure 11

Single-particle energies, relative to the Fermi energy, plotted as a function of neck size for the two lowest $K=5/2$ proton orbitals for the configuration $A$ at $Q_2=149\mathrm{b}$. Energy units are MeV.

Figure 12

Single-particle densities for the two $K=5/2$ proton orbitals closest to the Fermi energy for the configuration $A$ at $Q_2=149\mathrm{b}$ Dashed red lines: $N_{\mathrm{neck}}=2.00$; solid black lines: $N_{\mathrm{neck}}=2.25$.