Fission dynamics within time-dependent Hartree-Fock: Deformation-induced fission

Background: Nuclear fission is a complex large-amplitude collective decay mode in heavy nuclei. Microscopic density functional studies of fission have previously concentrated on adiabatic approaches based on constrained static calculations ignoring dynamical excitations of the fissioning nucleus and the daughter products.

Purpose: We explore the ability of dynamic mean-field methods to describe fast fission processes beyond the fission barrier, using the nuclide ${}^{240}\text{Pu}$ as an example.

Methods: Time-dependent Hartree-Fock calculations based on the Skyrme interaction are used to calculate nonadiabatic fission paths, beginning from static constrained Hartree-Fock calculations. The properties of the dynamic states are interpreted in terms of the nature of their collective motion. Fission product properties are compared to data.

Results: Parent nuclei constrained to begin dynamic evolution with a deformation less than the fission barrier exhibit giant-resonance-type behavior. Those beginning just beyond the barrier explore large-amplitude motion but do not fission, whereas those beginning beyond the two-fragment pathway crossing fission to final states which differ according to the exact initial deformation.

Conclusions: Time-dependent Hartree-Fock is able to give a good qualitative and quantitative description of fast fission, provided one begins from a sufficiently deformed state.

Figure 1

Schematic of typical potential energy surface obtained when increasing the elongation (which corresponds to quadrupole deformation) of an actinide nucleus. The energies $E_A$, $E_B$, and $E_{II}$ relate to properties of the fission barriers, and correspond to the definitions in Ref. [10].

Figure 2

Resulting PES for ${}^{240}\text{Pu}$ following a constraint of the quadrupole deformation parameter ${\beta }_{20}$. The top panels display the dependence of the constrained energy on the (a) quadrupole, (b) octupole, and (c) hexadecapole moments. Once the second barrier is overcome, a significant octupole deformation (corresponding to mass asymmetry) develops. Bottom panels (d)–(k) display two-dimensional (2D) slices of the 3D density for increasing quadrupole deformation. The isolines correspond to $0.05\mathrm{particles}/{\mathrm{fm}}^3$. It is interesting to note that, for states far beyond the second barrier, scission has not yet occurred.

Figure 3

(a) Octupole deformation as a function of quadrupole deformation and (b) hexadecapole deformation as a function of quadrupole deformation for the calculated PES of ${}^{240}\text{Pu}$. The octupole deformation, corresponding to a mass asymmetry, rapidly onsets after the second fission barrier is passed. Panel (c) displays the corresponding $\gamma$ deformation parameter.

Figure 4

Panels (a)–(c) One-fragment (solid circles) and two-fragment (solid squares) fission pathways for ${}^{240}\text{Pu}$. The arrows in panel (a) show the direction in which the PES is explored. Beyond ${\beta }_{20}=1.25$ the one-fragment pathway jumps into the two-fragment pathway, and this state is used as the initial configuration for investigating the latter pathway. Sample density slices on the competing pathways with the same ${\beta }_{20}$ are shown in the panels (d), (e) and (f), (g). The isolines are separated by $0.05\mathrm{particles}/{\mathrm{fm}}^3$.

Figure 5

Time evolution of quadrupole (left column), octupole (center column), and hexadecapole (right column) multipole parameters for initial states which are solutions to CHF calculations, with initial quadrupole deformation labeled on the right-hand side. All of the initial states are deformed beyond the peak of the static fission barrier, ${\beta }_{20}>0.86$.

Figure 6

Static one- and two-fragment fission pathways (solid circles and squares, respectively). For TDHF calculations, three regions may be defined when following the one-fragment pathway. In a first region, fission is forbidden within TDHF time scales. Beyond the maximum in the PES at ${\beta }_{20}=0.86$, the time scale for fission is inhibiting. Finally, beyond the crossing of two pathways at ${\beta }_{20}=1.085$, fission is allowed. See text for more details.

Figure 7

Slices of the total particle density for various times, starting from the static case with ${\beta }_{20}=1.19$. It takes between 775 and $800\mathrm{fm}/c$ for scission (as defined in the text) to occur. The isolines are separated by $0.05\mathrm{particles}/{\mathrm{fm}}^3$.

Figure 8

Current vectors corresponding to the slices of the particle density presented in Fig. 7. The vectors have been normalized to the same scale in each panel to a visually instructive length. Note that the dimensions of the grid exceed what is presented in these panels.

Figure 9

Time evolution of (a) quadrupole, (b) octupole, and (c) hexadecapole deformation parameters for various static configurations observed to fission upon time evolution. The evolution is stopped at the scission point, as defined in the text.

Figure 10

Evolution of the decomposed energy density functional for the fissioning systems. The calculations are terminated when the fragments are separated by 100 fm. For reference, the vertical lines in the panels corresponding to kinetic energies show the corresponding scission points. The total energy is conserved within fluctuations no greater than 4 MeV.

Figure 11

Fits to the obtained collective kinetic energy for the initial state with deformation ${\beta }_{20}=1.10$. Fits are performed over three different ranges: from the point where the separation of the fragments exceeds 30, 50, and 60 fm, respectively.

Figure 12

(a) Experimental independent fission yields for neutron-induced fission at $E=0.0253$ eV. The data are from Ref. [67]. (b) Theoretical mass fragment distributions. The red bars correspond to the binned TDHF results and the blue bar corresponds to the static two-fragment mass split. See text for more details.

Figure 13

Summed energy density functionals for the region of space corresponding to the (a) heavy and (b) light fission fragments. The quadrupole deformation of the initial state is labeled. The drift in the energy can be attributed mainly to the Coulomb interaction between the two fragments. See text for more details.

Figure 14

Comparison of the summed $\mathrm{\Delta }E$ from Table 7 (solid circles) to the mean interpolated collective kinetic energy from Table 3) (solid triangles) for each of the fissioning cases. Indicative error bars are displayed. See text for details.

Figure 15

Experimental kinetic energy distribution taken from Ref. [3]. The shaded box corresponds to the extremes of the range of kinetic energy values displayed in Fig. 14.

Figure 16

Decomposed contributions to the energy of the system for the case with initial deformation ${\beta }_{20}=1.25$. A Galilean transformation has been applied to remove the linear momentum of the individual fragments once the separation between the fragment center of mass reaches 100 fm. The calculation is terminated once the separation exceeds 105 fm.

Figure 17

Evolution of (a) quadrupole, (b) octupole, and (c) hexadecapole deformation parameters for the heavy fission fragment. The initial deformation was ${\beta }_{20}=1.25$. The measurement time is significantly extended by applying the Galilean transformation to remove the linear momentum of the fragments. See text for more details.

Figure 18

Evolution of (a) quadrupole, (b) octupole, and (c) hexadecapole deformation parameters for the light fission fragment. The initial deformation is ${\beta }_{20}=1.25$.

Figure 19

Power spectra corresponding to Fig. 17 (heavy fission fragment). The resolution is significantly improved owing to the longer measurement time available with the use of Galilean transformations to remove the linear momentum of the fragments.

Figure 20

Power spectra corresponding to Fig. 18 (light fission fragment).