Fission barriers of two odd-neutron actinide nuclei taking into account the time-reversal symmetry breaking at the mean-field level

Background: For a long time, fission barriers of actinide nuclei have been mostly microscopically calculated for even-even fissioning systems. Calculations in the case of odd nuclei have been performed merely within a so-called equal-filling approximation (EFA) as opposed to an approach taking explicitly into account the time-reversal-breaking properties at the mean-field level—and for only one single-particle configuration.

Purpose: We study the dependence of the fission barriers on various relevant configurations (e.g., to evaluate the so-called specialization energy). In addition, we want to assess the relevance of the EFA approach as a function of the deformation, which has been already found for the ground-state deformation.

Methods: Calculations within the Hartree–Fock plus BCS approach with self-consistent particle blocking have been performed by using the ${\mathrm{SkM}}^{*}$ Skyrme effective interaction in the particle-hole channel and a seniority force in the particle-particle channel. Axial symmetry has been imposed throughout the whole fission path while the intrinsic parity symmetry has been allowed to be broken in the outer fission barrier region.

Results: Potential-energy curves have been determined for six different configurations in ${}^{235}\mathrm{U}$ and four in ${}^{239}\mathrm{Pu}$. Inner and outer fission barriers have been calculated along with some spectroscopic properties in the fission isomeric well. These results have been compared with available data. The influence of time-reversal-breaking mean fields on the solutions has been investigated.

Conclusions: A sizable configuration dependence of the fission barrier (width and height) has been demonstrated. A reasonable agreement with available systematic evaluations of fission-barrier heights has been found. The EFA approach has been validated at the large elongations occurring at the outer-barrier region.

Figure 1

Deformation-energy curves of ${}^{239}\mathrm{Pu}$ as functions of $Q_{20}$ calculated with the ${\mathrm{SkM}}^{*}$ parametrization and without taking the rotational energy correction into account. The $K^{\pi }$ labels refer to the quantum numbers in the parity-symmetrical region (unfilled circles). The plotted solutions when this symmetry is broken (filled circles) are obtained by continuity as functions of $Q_{20}$.

Figure 2

Same as Fig. 1 but for ${}^{235}\mathrm{U}$.

Figure 3

Cuts for given values of $Q_{20}$ in the potential-energy surface around the top of the outer barrier as a function of the octupole moment $Q_{30}$ (given in ${\mathrm{barns}}^{3/2}$) of the 5/2 blocked configuration of ${}^{239}\mathrm{Pu}$. The ${\mathrm{SkM}}^{*}$ parametrization has been used.

Figure 4

A portion of the deformation-energy curves of the blocked $K=7/2$ configurations in ${}^{235}\mathrm{U}$ from the fission-isomeric well up to beyond the top of the outer barrier. The filled symbols refer to the local minima as a function of $Q_{30}$ for fixed elongation $Q_{20}$ while the unfilled symbols refer to the solutions obtained by imposing a left-right symmetry. The solid line connects the lowest-energy solutions when the left-right symmetry is broken.

Figure 5

(top) Evolution of the single-particle energies for two $\mathrm{\Omega }=7/2$ states near the BCS chemical potential (marked with crosses) as functions of $Q_{20}$ obtained in the parity-asymmetric calculations of ${}^{235}\mathrm{U}$. The solid line connects the blocked single-particle states as a function of deformation. (bottom) Average parity of the above considered blocked single-particle states as a function of $Q_{20}$.

Figure 6

Deformation-energy curves of ${}^{238,240}\mathrm{Pu}$ (with $K^{\pi }=0^{+}$) and ${}^{239}\mathrm{Pu}$ (with $K^{\pi }=1/2^{+}$, $5/2^{+}$, $7/2^{-}$, and $7/2^{+}$) as functions of $Q_{20}$ in barns. The Belyaev moments of inertia have been increased by a factor of two. (left panel) Absolute energy scale. (right panel) Relative scale, taking the normal-deformed minimum as the origin of energy for all curves.

Figure 7

Energy differences between various contributions [see Eqs. (21) to (23) for definitions] to the inner-barrier height and isomeric energy obtained in the default minimal time-odd scheme and the full time-odd scheme for several blocked configurations in ${}^{239}\mathrm{Pu}$ with the ${\mathrm{SkM}}^{*}$ parametrization. The difference in the inner-barrier heights $\mathrm{\Delta }{E}_A^{{}^{\prime }}$ and fission-isomeric energy $\mathrm{\Delta }{E}_{\mathrm{IS}}^{{}^{\prime }}$ between the two schemes are also given for each blocked configuration.

Figure 8

Same as Fig. 7 for the SIII parametrization.

Figure 9

Bandhead energy spectra of ${}^{239}\mathrm{Pu}$ calculated with the SLy5*, ${\mathrm{SkM}}^{*}$, and SIII parametrizations in the isomeric well with the inclusion of the rotational correction. The standard Thouless–Valatin correction of Ref. [34] beyond the Belyaev result has been taken into account for the moments of inertia of each band. The rotational spectra built upon the lowest-energy $5/2^{+}$ state (rot band) are also shown in the second column of each Skyrme force. The experimental data are taken from Refs. [62, 63]. The fission-isomeric energy defined as the energy difference between the lowest-energy solution in the ground state and the fission-isomeric well is denoted by $E_{\mathrm{II}}$.

Figure 10

Same as Fig. 9 but for ${}^{235}\mathrm{U}$.

Figure 11

Example of calculated transition states at the top of the inner barrier of ${}^{239}\mathrm{Pu}$.