Performance of the Fourier shape parametrization for the fission process

The availability of realistic potential energy landscapes in restricted deformation space is the prerequisite starting point for modeling several nuclear properties and reactions, namely large-amplitude phenomena. The achievement of a macroscopic-microscopic approach, employing an innovative four-dimensional (4D) nuclear shape parametrization based on a Fourier expansion, and a realistic potential-energy prescription, is presented. A systematic analysis of the 4D deformation energy landscapes over an extended region of the nuclear chart from Pt to Pu is performed, searching for fission valleys, as well as exotic ground and metastable states. The significance of the approach for predicting mass partitioning in low-energy fission is demonstrated. The ability of the model to address shape-driven effects, like stable octupole and very elongated isomeric configurations, is discussed, too. The proposed approach constitutes an efficient framework for an extended model of fission dynamics over a wide range of fissioning mass, excitation energy, and angular momentum.

Figure 1

Schematic visualization in cylindrical coordinates of the parameters entering the definition of the profile function defined with Eqs. (1, 2, 3, 4). The quantities $z_l$ and $z_r$ localize the mass centers of left and right nascent fragment entering the definition of $R_{12}=z_r-z_l$.

Figure 2

Relative importance of the contributions to the shape function ${\rho }_s^2\left(z\right)$, Eq. (1), from different orders $a_n$ of the Fourier series, for the left-right asymmetric shape given in the top part of the picture. Profiles are plotted as a function of the reduced variable $u=(z-z_{sh})/z_0$. The shape of the nucleus obtained within the Strutinsky optimal shapes theory [13] is shown as well with the thick dashed line.

Figure 3

Comparison of liquid drop three-quadratic saddle-point shapes (black full contours) for different values of the nuclear fissility $x$, and shapes generated through the Fourier series, Eq. (1), limited to $a_4$ (blue dotted) or $a_6$ (red dashed).

Figure 4

LSD deformation energy relative to the spherical ground state as a function of $R_{12}/R_0$ for ${}^{232}\mathrm{Th}$ with the Fourier series limited up to $a_4$ (blue dotted), $a_6$ (red dashed), and $a_8$ (black full line).

Figure 5

LSD potential energy relative to the spherical ground state, in units of the surface energy in the ($a_2,a_4$) deformation plane for a nucleus with fissility $x$=0.8. The LSD fission path runs from the top right to the bottom left end. Corresponding values of the Funny-Hills elongation variable $c$ [9] are indicated along the path.

Figure 6

LSD potential energy in the ($q_2,q_4$) (top left), ($q_2,q_3$) (top right), ($q_2,q_6$) (bottom left), and ($q_2,q_5$) (bottom right) deformation subspaces for a nucleus with fissility $x$=0.8. In each ($q_i,q_j$) plot, the collective coordinates $q_k$ with $k\ne i,j$ are set to zero.

Figure 7

Macroscopic-microscopic potential energy as a function of $q_2$ for the nucleus ${}^{240}\mathrm{Pu}$ after minimization with respect to the other deformation parameters. The lowest-energy left-right asymmetric path (solid line) as well as the path imposing left-right symmetry (dashed line) are shown.

Figure 8

(Top) Macroscopic-microscopic potential energy in the ($q_2,q_3$) plane for ${}^{228}\mathrm{Ra}$. The two identified fission paths are paved by dashed and solid curves. (Bottom) Sequence of ($q_3,q_4$) potential energy maps for selected values of $q_2$ along the fission paths identified in the top panel: $q_2=0.2$ (b), $q_2=0.7$ (c), $q_2=1.0$ (d), $q_2=1.3$ (e) and (h), $q_2=1.7$ (f) and (i), $q_2=2.0$ (g) and (j). Representative shapes along the asymmetric and symmetric paths are displayed in the top and bottom of the upper figure, respectively, and labeled with small letters along the paths (a)–(j).

Figure 9

Macroscopic-microscopic potential energy in the ($q_2, q_3$) plane for several isotopes of Hg (left column), Po (middle column), and Th (right column). The identified fission paths are indicated by a solid black line. Insets show, where available, experimental data [34, 40, 46, 50] on fission-fragment mass or charge distribution at low excitation energy.

Figure 10

Macroscopic-microscopic potential energy in the ($q_2, q_3$) plane for the isotopes ${}^{238,242,246}\mathrm{Pu}$. The shapes close to scission ($q_2$=2.0) in the two asymmetric channels are shown for ${}^{246}\mathrm{Pu}$ in the bottom right panel.

Figure 11

Predicted octupole deformation coordinate $q_3$ for the ground states of Ra (red full), Th (blue dashed), and U (green dotted line) isotopes as a function of neutron number.

Figure 12

Macroscopic-microscopic potential energy in the ($q_2, q_4$) plane for four isotopes of Th for $q_3=0$ and $\eta =0$.