Fission dynamics within time-dependent Hartree-Fock. II. Boost-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 induced fission processes, using quadrupole boosts in the nuclide ${}^{240}\mathrm{Pu}$ as an example.

Methods: Following upon the work presented in Goddard et al. [Phys. Rev. C 92, 054610 (2015)], quadrupole-constrained Hartree-Fock calculations are used to create a potential energy surface. An isomeric state and a state beyond the second barrier peak are excited by means of instantaneous as well as temporally extended gauge boosts with quadrupole shapes. The subsequent deexcitation is studied in a time-dependent Hartree-Fock simulation, with emphasis on fissioned final states. The corresponding fission fragment mass numbers are studied.

Results: In general, the energy deposited by the quadrupole boost is quickly absorbed by the nucleus. In instantaneous boosts, this leads to fast shape rearrangements and violent dynamics that can ultimately lead to fission. This is a qualitatively different process than the deformation-induced fission. Boosts induced within a finite time window excite the system in a relatively gentler way and do induce fission but with a smaller energy deposition.

Conclusions: The fission products obtained using boost-induced fission in time-dependent Hartree-Fock are more asymmetric than the fragments obtained in deformation-induced fission or the corresponding adiabatic approaches.

Figure 1

(a) Slice of the 3D particle density. The isolines are separated by 0.05 $\text{particles}/{\mathrm{fm}}^3$. (b) Current vectors, $\mathit{j}\left(\mathit{r}\right)$, for a quadrupole velocity field applied instantaneously. Both pictures are taken for the isomeric state, ${\beta }_{20}=0.682$, at time $t=0$. The current vectors have been normalized to a visually instructive length.

Figure 2

Time evolution of (a) quadrupole, (b) octupole, and (c) hexadecapole deformation parameters following instantaneous quadrupole excitations upon the isomeric state. The threshold for fission is between 175 and 200 MeV. Scission occurs between 950 and 1000 $\text{fm}/c$ for the $E=200$ MeV boost.

Figure 3

Slices of the total particle density for various times, following an instantaneous $200\text{-MeV}$ quadrupole excitation upon the isomeric state. The isolines are separated by 0.05 $\text{particles}/{\mathrm{fm}}^3$.

Figure 4

(a)–(j) Evolution of the integrated contributions to the EDF following instantaneous quadrupole excitations upon the isomeric state. Vertical lines in panels (g) and (h) correspond to the point of scission. Units in all panels are MeV. See text for more details.

Figure 5

One-dimensional slices of the particle density along the principal axis for various times, following the application of an instantaneous $200\text{-MeV}$ quadrupole excitation upon the isomeric state. Densities at a specified time (solid lines) are compared to the density in the previous time (dotted line) to highlight the oscillatory nature of the prefission configuration. The final panel is very close to the scission point.

Figure 6

Current vectors corresponding to the particle density slices presented in Fig. 3. The normalization factors for the vector arrows from $t=250$ to $t=900$ are the same (=$\mathcal{N}$), and for other times the normalization factor has been rescaled so that the panels are visually instructive. The normalisation factors are $\mathcal{N}/10,3\mathcal{N}/10$, $3\mathcal{N}/10$, $4\mathcal{N}/10$, and $3\mathcal{N}/10$ for $t=0,50,100,1000$, and 1100 $\text{fm}/c$, respectively.

Figure 7

Time evolution of (a) quadrupole, (b) octupole, and (c) hexadecapole deformation parameters following a large-amplitude quadrupole excitation upon the initial state with ${\beta }_{20}=0.890$. The threshold for fission is between 200 and 225 MeV. Scission occurs around 1700 $\text{fm}/c$, and the measurements of the multipole deformation parameters are cut off at this point.

Figure 8

Slices of the total particle density for various times, following an instantaneous $225\text{-MeV}$ quadrupole excitation upon the state with initial deformation ${\beta }_{20}=0.890$. The isolines are separated by 0.05 $\text{particles}/{\mathrm{fm}}^3$.

Figure 9

(a)–(j) Evolution of the integrated contributions to the EDF following instantaneous quadrupole excitations of the state with initial deformation ${\beta }_{20}=0.890$. The dashed line corresponds to a nonfissioning state, whereas the dash-dotted line represents a fissioning configuration. Vertical lines in panels (g) and (h) correspond to the point of scission. Units in all panels are MeV. See text for more details.

Figure 10

One-dimensional slices of the particle density along the principal axis for various times, following the application of an instantaneous $225\text{-MeV}$ quadrupole excitation upon the state with initial deformation ${\beta }_{20}=0.890$. Densities at a specified time (solid lines) are compared to the density in the previous time (dotted line).

Figure 11

The same as Fig. 10 for an excitation of $E=400$ MeV.

Figure 12

Time evolution of (a) quadrupole, (b) octupole, and (c) hexadecapole deformation parameters following time-dependent (${\tau }_0=500$ $\text{fm}/c$ and $\mathrm{\Delta }\tau =150$ $\text{fm}/c$) quadrupole excitations upon the isomeric state. The field with scaling parameter $\eta =0.0095$ is seen to induce fission, and the evolution of the multipole moments are sharply cut off at the point of scission at 1050 $\text{fm}/c$.

Figure 13

Slices of the total particle density for various times, following a time-dependent boost with scaling constant $\eta =0.0095$ upon the isomeric state. The isolines are separated by 0.05 $\text{particles}/{\mathrm{fm}}^3$.

Figure 14

Current vectors corresponding to the particle density slices presented in Fig. 13. The external field peaks at ${\tau }_0=500$ $\text{fm}/c$ with width $\mathrm{\Delta }\tau =150$ $\text{fm}/c$, so the excitation field has reduced to a negligible magnitude by $t=800$ $\text{fm}/c$. The vectors have been normalized to be visually instructive. The normalization factor $\left(\mathcal{N}\right)$ is the same in each panel, other than $t=1100$ $\text{fm}/c$, where it is $3\mathcal{N}/5$.

Figure 15

(a)–(j) Time evolution of the integrated contributions to the EDF when applying a time-dependent external field to the isomeric state. The time profile of the external field is displayed in panel (k). The case with scaling constant $\eta =0.0095$, which corresponds to an excitation of $\approx 52$ MeV, is seen to fission. For this case, the calculation is terminated once the fragment separation exceeds 100 fm. Vertical lines in panels (g) and (h) correspond to the point of scission. Units in all panels are MeV. See text for more details.

Figure 16

(a)–(j) Time evolution of the integrated contributions to the EDF when applying different time-dependent external field to the isomeric state. The (normalized) time profile of the external fields is displayed in panel (k). The boost's strength factor, $\eta$, is adjusted in each case and corresponds to the minimum value needed to induce fission. Vertical lines in panels (g) and (h) correspond to the point of scission. Units in all panels are MeV. See text and Table 3 for further details.

Figure 17

Time evolution of (a) quadrupole, (b) octupole, and (c) hexadecapole deformation parameters following time-dependent (${\tau }_0=500$ $\text{fm}/c$ and $\mathrm{\Delta }\tau =150$ $\text{fm}/c$) quadrupole excitations upon the initial state with ${\beta }_{20}=0.890$. The field with scaling parameter $\eta =0.007$ is seen to induce fission, and the evolution of the multipole moments is sharply cut off at the point of scission at 2050 $\text{fm}/c$.

Figure 18

Slices of the total particle density for various times, following a time-dependent boost with scaling constant $\eta =0.007$ upon the state with initial deformation ${\beta }_{20}=0.890$. The isolines are separated by 0.05 $\text{particles}/{\mathrm{fm}}^3$.

Figure 19

(a)–(j) Evolution of the integrated contributions to the EDF following a time-dependent external field boost upon the state with initial deformation ${\beta }_{20}=0.890$. The time profile of the external fields is displayed in panel (k). Vertical lines in panels (g) and (h) correspond to the point of scission. Units in all panels are MeV. See text for more details.

Figure 20

(a) Experimental independent fission yields for neutron-induced fission at $E=0.0253\text{-eV}$ energies. The data are from Ref. [39]. (b) Theoretical mass fragments obtained from the DIF process in Ref. [14]. The red bars correspond to the binned TDHF results and the blue bar corresponds to the static two-fragment mass split. (c) Theoretical mass fragments obtained from different BIF processes upon the state with initial deformation ${\beta }_{20}=0.890$.