Description of induced nuclear fission with Skyrme energy functionals: Static potential energy surfaces and fission fragment properties

Eighty years after its experimental discovery, a description of induced nuclear fission based solely on the interactions between neutrons and protons and quantum many-body methods still poses formidable challenges. The goal of this paper is to contribute to the development of a predictive microscopic framework for the accurate calculation of static properties of fission fragments for hot fission and thermal or slow neutrons. To this end, we focus on the ${}^{239}\mathrm{Pu}\left(n,f\right)$ reaction and employ nuclear density functional theory with Skyrme energy densities. Potential energy surfaces are computed at the Hartree-Fock-Bogoliubov approximation with up to five collective variables. We find that the triaxial degree of freedom plays an important role, both near the fission barrier and at scission. The impact of the parametrization of the Skyrme energy density and the role of pairing correlations on deformation properties from the ground state up to scission are also quantified. We introduce a general template for the quantitative description of fission fragment properties. It is based on the careful analysis of scission configurations, using both advanced topological methods and recently proposed quantum many-body techniques. We conclude that an accurate prediction of fission fragment properties at low incident neutron energies, although technologically demanding, should be within the reach of current nuclear density functional theory.

Figure 1

Two-dimensional potential energy surface of ${}^{240}\mathrm{Pu}$ in the $(q_{20},q_{22})$ plane for the SkM* EDF. The energy is relative to the ground-state value. The dashed line represents the symmetric, triaxial least-energy pathway.

Figure 2

Two-dimensional potential energy surface of ${}^{240}\mathrm{Pu}$ in the $(q_{20},q_{40})$ plane for the SkM* functional. The energy is relative to $-1830$ MeV. The dashed line represents the least-energy pathway.

Figure 3

Two-dimensional potential energy surface of ${}^{240}\mathrm{Pu}$ in the $(q_{20},q_{30})$ plane for the SkM* functional. The energy is given relative to $-1840$ MeV. The dashed line represents the least-energy pathway.

Figure 4

Energy along the least-energy fission pathway in ${}^{240}\mathrm{Pu}$ for the SkM* EDF: axial symmetric path (black squares), triaxial symmetric path (red circles), and triaxial asymmetric path (blue triangles). Energy curves are given relative to the ground state. The value of the octupole and hexadecapole moments are also shown along the symmetric and asymmetric paths.

Figure 5

Variation of the total HFB energy as a function of the hexadecapole moment $q_{40}$ along the least-energy fission pathway in ${}^{240}\mathrm{Pu}$.

Figure 6

Energy along the least-energy fission pathway in ${}^{240}\mathrm{Pu}$ for three parametrizations of the Skyrme functional: SkM* [54], UNEDF0 [47] and UNEDF1 [48]. All curves are given relative to their ground-state value.

Figure 7

Energy along the least-energy fission pathway in ${}^{240}\mathrm{Pu}$ for five parametrizations of the pairing force and the SkM* parametrization of the Skyrme functional. All curves are given relative to their ground-state value.

Figure 8

Two-dimensional potential energy surface of ${}^{240}\mathrm{Pu}$ in the $(q_{20},q_{22})$ plane for the SkM* functional around the least-energy fission pathway. The energy is normalized arbitrarily at $-1820$ MeV.

Figure 9

One-dimensional potential energy surface of ${}^{240}\mathrm{Pu}$ along the $q_{22}$ direction for the SkM* functional around the least-energy fission pathway.

Figure 10

Potential energy surface in the $(q_{30},q_N)$ plane just before scission for the SkM* functional. The axial quadrupole moment ${\widehat{Q}}_{20}$ is fixed at $q_{20}=345$ b, the triaxial quadrupole ${\widehat{Q}}_{22}$ and hexadecapole moments ${\widehat{Q}}_{40}$ are unconstrained. The energy is normalized arbitrarily at $-1827$ MeV.

Figure 11

Total energy as a function of the density of particles in the neck $q_N$ along the least-energy fission pathway for the SkM*, UNEDF0, and UNEDF1 functionals. All curves are normalized relative to their respective values at $q_N=4.5$. Inset contour plots show the density profile at $q_N=4.0$ and $q_N=0.3$.

Figure 12

Two small functions $f(x,y)$ and $g(x,y)$ on a triangular mesh (dashed lines) shown as contour plots and their respective contour trees. Circled (a) and squared (b) points represent the values of the function on the mesh.

Figure 13

Joint contour slabs found by intersecting the slabs of functions $f$ and $g$ of Fig. 12 (a). Joint contour net obtained by analyzing the connectivity of the slabs; see text for details (b).

Figure 14

JCN graphs near the scission for ${}^{240}\mathrm{Pu}$ at $q_N=4.0$ (a) and at $q_N=0.1$ (b). The principal feature visible is that the single branch for high isovalues of the densities (upper right side of top figure) at $q_N=4.0$ has forked into two distinct high isovalues branches (upper right side of bottom figure) at $q_N=0.1$, each branch featuring starbursts.

Figure 15

Total, left, and right fragment densities before (plain lines) and after (dashed lines) the localization of q.p.'s at the $q_N=0.4$ point of ${}^{240}\mathrm{Pu}$. Calculations for the SkM* functional.

Figure 16

Skyrme interaction energy between the fission fragments in ${}^{240}\mathrm{Pu}$ as a function of the number of particles in the neck for the SkM*, UNEDF0, and UNEDF1 functionals. Plain curves correspond to the calculation before the localization is applied, dashed curves to after it has been applied.

Figure 17

Direct Coulomb interaction energy between the fission fragments in ${}^{240}\mathrm{Pu}$ as a function of the number of particles in the neck for the SkM*, UNEDF0, and UNEDF1 functionals. Plain curves correspond to the calculation before the localization is applied, dashed curves after it has been applied.

Figure 18

Skyrme interaction energy between fission fragments in ${}^{240}\mathrm{Pu}$ as a function of the number of particles in the neck for the SkM* functional. The parameters of reference for the localization procedure were chosen as $\Delta E=2$ MeV, ${\ell }_{\text{max}}=0.75,N_{\text{max}}=0.005$, and five iterations. Top left: dependence on $\Delta E$, all other parameters being fixed to their reference value; top right: dependence on ${\ell }_{\text{max}}$; bottom left: dependence on the number of iterations; bottom right: dependence on $N_{\text{max}}$. See text for further details.