Microscopic study of induced fission dynamics of ${}^{226}\mathrm{Th}$ with covariant energy density functionals

Static and dynamic aspects of the fission process of ${}^{226}\mathrm{Th}$ are analyzed in a self-consistent framework based on relativistic energy density functionals. Constrained relativistic mean-field calculations in the collective space of axially symmetric quadrupole and octupole deformations, based on the energy density functional PC-PK1 and a $\delta$-force pairing, are performed to determine the potential energy surface of the fissioning nucleus, the scission line, the single-nucleon wave functions, energies, and occupation probabilities, as functions of deformation parameters. Induced fission dynamics is described using the time-dependent generator coordinate method in the Gaussian overlap approximation. A collective Schrödinger equation, determined entirely by the microscopic single-nucleon degrees of freedom, propagates adiabatically in time the initial wave packet built by boosting the ground-state solution of the collective Hamiltonian for ${}^{226}\mathrm{Th}$. The position of the scission line and the microscopic input for the collective Hamiltonian are analyzed as functions of the strength of the pairing interaction. The effect of static pairing correlations on the preneutron emission charge yields and total kinetic energy of fission fragments is examined in comparison with available data, and the distribution of fission fragments is analyzed for different values of the initial excitation energy.

Figure 1

Self-consistent RMF+BCS quadrupole and octupole constrained deformation energy surface (in MeV) of ${}^{226}\mathrm{Th}$ in the ${\beta }_2-{\beta }_3$ plane.

Figure 2

Same as in the caption to Fig. 1, but plotted as a contour map. The red curve is the static fission path and the density distributions for selected deformations along the fission path are also shown.

Figure 3

The scission contour of ${}^{226}\mathrm{Th}$ in the ${\beta }_2-{\beta }_3$ plane.

Figure 4

The calculated total kinetic energy of the nascent fission fragments for ${}^{226}\mathrm{Th}$ as a function of fragment mass, in comparison to the data [53].

Figure 5

Potential energy surfaces of ${}^{226}\mathrm{Th}$ in the ${\beta }_2-{\beta }_3$ plane, calculated with the functional PC-PK1 and for three parametrizations of the pairing force: $(V_n,V_p)=(324,340.2)$ (top), (360, 378) (middle), and (396, 415.8) (bottom), in units of $\mathrm{MeV}{\mathrm{fm}}^3$.

Figure 6

Collective masses $B_{22}^{-1}$ and $B_{33}^{-1}$ related to vibrations in ${\beta }_2$ and ${\beta }_3$, respectively, along the static fission path for three values of the pairing strength.

Figure 7

Pairing gaps for neutrons (top) and protons (bottom) along the static fission path.

Figure 8

The scission lines for ${}^{226}\mathrm{Th}$ in the ${\beta }_2-{\beta }_3$ plane, obtained in calculations with three different values of the pairing strength.

Figure 9

Comparison between experimental and calculated total kinetic energy of nascent fission fragments for ${}^{226}\mathrm{Th}$, as a function of fragment mass and pairing strength.

Figure 10

Preneutron emission charge yields for photoinduced fission of ${}^{226}\mathrm{Th}$. The results of calculations for three different values of the pairing strength are compared to the data [53].

Figure 11

Total flux as a function of time for three different pairing strengths.

Figure 12

Charge distribution of fission fragments for different excitation energies. The original pairing strength is used.

Figure 13

Same as in the caption to Fig. 12 but for the mass yield.

Figure 14

(a) Self-consistent RMF+BCS binding energy of ${}^{226}\mathrm{Th}$ for two extremely deformed configurations $({\beta }_2,{\beta }_3)=(5.01,0.01)$ and $(3.01,2.01)$, as a function of the number of major shells of the axially deformed harmonic oscillator basis. (b) Self-consistent RMF+BCS binding energies of ${}^{226}\mathrm{Th}$ as functions of ${\beta }_2$ for different values of $a$ in Eq. (A13), with $N_f=20$ and ${\beta }_3=0.01$. (c) Same as in the caption to (b) but for ${\beta }_3=1.73$.