Fission dynamics of ${}^{240}\mathrm{Pu}$ from saddle to scission and beyond

Calculations are presented for the time evolution of ${}^{240}\mathrm{Pu}$ from the proximity of the outer saddle point until the fission fragments are well separated, using the time-dependent density functional theory extended to superfluid systems. We have tested three families of nuclear energy density functionals and found that all functionals exhibit a similar dynamics: The collective motion is highly dissipative and with little trace of inertial dynamics, due to the one-body dissipation mechanism alone. This finding justifies the validity of using the overdamped collective motion approach and to some extent the main assumptions in statistical models of fission. This conclusion is robust with respect to the nuclear energy density functional used. The configurations and interactions left out of the present theory framework only increase the role of the dissipative couplings. An unexpected finding is varying the pairing strength within a quite large range has only minor effects on the dynamics. We find notable differences in the excitation energy sharing between the fission fragments in the cases of spontaneous and induced fission. With increasing initial excitation energy of the fissioning nucleus, more excitation energy is deposited in the heavy fragment, in agreement with experimental data on average neutron multiplicities.

Figure 1

The cumulative number of quasiparticle states for neutrons and protons obtained in a full diagonalization of the (initial) stationary quasiparticle Hamiltonian with constraints [see Eq. (A6)] for neutrons and protons in a discrete variable representation used in (TD)SLDA (solid line) and using the configuration space approach [61] (dashed line) in case of SkM* NEDF. The insets show the cumulative number of quasiparticle energies for SeaLL1 (red), SLy4 (blue), and SkM* (black) NEDFs respectively. The maximum quasiparticle energy is $\approx 400\mathrm{MeV}$ in (TD)SLDA while in HFBTHO is $\approx 100\mathrm{MeV}$. The total number of quasiparticle states, for either neutrons or protons, is $2N_x{N}_y{N}_z=2\times {24}^2\times 48=55296$ in the present numerical implementation. In SLDA, the single-particle level density increases sharply above the nucleon separation energy, as physically expected. As a result, above the nucleon separation energy the configuration space approach severely underestimates the single-particle level density.

Figure 2

Fission pathway for ${}^{240}\mathrm{Pu}$ along the mass quadrupole moment $Q_{20}$ calculated with SeaLL1, SkM*, and UNEDF1. With black horizontal lines labeled by $E_A$, $E_B$, and $E_{II}$, we show the values of the inner and outer fission barriers and the energy of the fission isomer with respect to the ground-state energy.

Figure 3

Schematic evolution of sp levels of nucleons (upper panel) and the total nuclear energy (lower panel) as a function of deformation parameter $q$ [77, 78]. The thick line represents the Fermi level and the up and down arrows depict the Cooper pairs of nucleons on the Fermi level only, in time-reversed orbits $(m,-m)$.

Figure 4

The left three columns shows the induced fission of ${}^{240}\mathrm{Pu}$ with normal pairing strength, which lasts up to 14 000 fm/c ($\approx 47\times {10}^{-21}\mathrm{s}$) from saddle to scission. The columns show sequential frames of the density (first column), the magnitude of the pairing field (second column), and the phase of the corresponding pairing field (third column). The upper and lower parts of each frame show the neutron and proton densities, the magnitudes of neutron and proton pairing fields, and the phase of the pairing field, respectively [62]. The right three columns show the corresponding snapshots of the induced fission of ${}^{240}\mathrm{Pu}$ with enhanced pairing strength, which lasts about 1 400 fm/c.

Figure 5

Fission trajectories for SeaLL1 (upper panel) and SkM* (lower pannel). The red (SeaLL1-1) and cyan lines (SeaLL1-2) correspond to initial configurations slightly above and below the outer fission barrier. The SeaLL1-1sy trajectory had a small initial left-right asymmetry. The green lines in the lower panel correspond to ${\mathrm{SKM}}^{*}$-1asy and and the red lines to the trajectories obtained with ${\mathrm{SkM}}^{*}$-2asy. The white dashed lines show the path linking the ground-state minimum, the fission isomer minimum, and the inner and the outer saddle points. In both panels, we also show an example of a symmetric fission trajectory and in the low panel also of a trajectory which wondered inside the fission isomer well.

Figure 6

We compare here the average neutron multiplicity $\overline{\nu }\left(A\right)$ emitted by FFs in the case of a default CGMF simulation, which assumes no $E_n$ dependence for the energy sharing, with the one extracted using the the excitation energy sharing between the FFs in our calculation with NEDF SeaLL1, as a function of the equivalent incident neutron energy in the ${}^{239}\mathrm{Pu}(n,f$) reaction along with available experimental data for the reaction ${}^{239}\mathrm{Pu}(n_{\text{th}},f$) from Refs. [99, 100, 101, 102]. The fragment mass $A$ is before neutron emission.

Figure 7

The collective flow energy evaluated for NEDFs with realistic pairing SLy4 [62] (dash-dotted line), enhanced pairing SLy4* (dashed line), SkM*(dotted and dash-doted lines with error bars), and SeaLL1 (solid and dashed lines with errors bars) sets. The error bars illustrate the size of the variations due to different initial conditions in the cases of SeaLL1-1,2 and SkM*-1,2 NEDFs. In the case of enhanced pairing NEDF Sly4*, the time has been scaled by a factor of 1/10.

Figure 8

The intrinsic energy $E_{\text{int}}\left(t\right)$ along the fission path for SeaLL1-1 (dotted line with error bars) and SeaLL1-2 in the upper panel and SkM*-2 (dash line with error bars) and SkM*-1 (dotted line with error bars) in the lower panel. The error bars illustrate the size of the variations due to different initial conditions in the cases of SeaLL1-1,2 and SkM*-1,2 NEDFs. The collective potential energy determined in a constrained calculation, see Eq. (5), is represented in both panels with either dotted or dashed lines for the corresponding SeaLL1 and SkM* NEDFs. In the case of an adiabatic evolution along the fission path, $E_{\text{int}}\left(t\right)$ would trace rather closely the collective potential energy determined in the constrained calculation. Scission configuration corresponds to a quadrupole momentum of the entire nuclear system $Q_{20}\approx 400$ b.

Figure 9

Upper panel: At times indicated with red dots we have applied collective quadrupole momentum kicks to both neutrons and protons, see Eq. (23), with random values of $\eta$. Lower panel: The time evolution of the total energy of the nucleus, in the rest frame of the nucleus, after we have applied collective kicks to both neutrons and protons with random values of $\eta$.

Figure 10

The evolution of the quadrupole $Q_{20}$ and octupole $Q_{30}$ moments of the light (solid lines) and heavy (dashed lines) FFs before and after scission. The time $t_0$ stands for the moment when the distance between the two fragments is about 15 fm and the neck of mother nucleus is formed. Scission occurs at $t-t_0\approx 300\mathrm{fm}/\mathrm{c}$. The solid lines represent the multipoles moments of the light fragment and the dashed lines show the multipole moments of the heavy fragment in the case of the SeaLL1-1,2 and SkM*-1-2 respectively. The error bars illustrate the size of the variations due to different initial conditions in case of SeaLL1-1,2 and SkM*-1,2 NEDFs.

Figure 11

The evolution of the quadrupole $Q_{20}$ and octupole $Q_{30}$ moments of the light (solid lines) and heavy (dashed lines) FFs before and after scission, here as a function of the separation between the FFs. At scission, the separation between the FFs is $\approx 17$ fm. The meaning of various lines and error bars are explained in the caption to Fig. 10.

Figure 12

Upper panel: The number of neutrons emitted predominantly after scission. The error bars quantify the size of the fluctuations between trajectories corresponding to different initial conditions. Lower panel: the number of neutrons as a function of time emitted in each trajectories in the inset of upper one.

Figure 13

Occupation probabilities of the nucleon levels within a window of approximately 5 MeV were prescribed random values as in Eqs. (D3), (D4), and (D6). The red bars show the size of the sp energy window, in the case of each deformation, where stochastic fluctuations are allowed. This is a copy of the Fig. 1 from the Supplemental Material of Ref. [49].

Figure 14

Average neutron occupation probabilities $n_k$ (blue) for a system with $N=150$, a Fermi energy ${\varepsilon }_F=35\mathrm{MeV}$, and an almost constant average sp level density at a temperature 1 MeV, the ${\rho }_{kk}=n_k\pm {\sigma }_{kk}$ (black) and ${\sigma }_{kk}$ (red), and a typical random realization of the stochastic occupation probabilities $n_k+{\xi }_{kk}$ (black dots) chosen in an energy window ($-5$, 5) MeV around the Fermi level.

Figure 15

The “occupation probabilities” ${\nu }_m$ (D14) after the diagonalization of one stochastic realization of Eq. (D4) for $n_{k=1...30}=1$ and $n_{l=31...60}=0$. Two horizontal lines are at 0 and 1 levels. The shape of this spectrum of the “occupation probabilities” changes little from one stochastic realization to another. The range of ${\nu }_m$ values increases with the size of the energy interval over which fluctuations are allowed, covering the entire real axis, if the fluctuations are allowed over the entire sp spectrum.