Fission barriers in covariant density functional theory: Extrapolation to superheavy nuclei

Systematic calculations of fission barriers allowing for triaxial deformation are performed for even-even superheavy nuclei with charge number $Z=112–120$ using three classes of covariant density functional models. The softness of nuclei in the triaxial plane leads to an emergence of several competing fission paths in the region of the inner fission barrier in some of these nuclei. The outer fission barriers are considerably affected by triaxiality and octupole deformation. General trends of the evolution of the inner and the outer fission barrier heights are discussed as a function of the particle numbers.

Figure 1

Potential energy surface of the ${}^{240}$Pu nucleus with the NL3* parametrization of the RMF Lagrangian. The energy difference between two neighboring equipotential lines is equal to 0.5 MeV. The pink dashed line with solid circles shows the lowest-energy solution as a function of ${\beta }_2$. Small red stars show the positions of deformation points at which the numerical results were obtained. The figure is based on the results of Ref. [1]. Further details are given in the text.

Figure 2

The difference between experimental and calculated heights of inner fission barriers as a function of neutron number $N$. The results of the calculations are compared to estimated fission barrier heights given in the RIPL-3 database [49], which is used for this purpose in the absolute majority of theoretical studies on fission barriers in actinides. The results of the calculations within the microscopic + macroscopic method (MM(Dobrowolski) [6] and MM(Möller) [7]), covariant density functional theory (CDFT [1]), and density functional theory based on the finite-range Gogny force (Gogny DFT [14]) are shown. Thick dashed lines are used to show the average trend of the deviations between theory and experiment as a function of neutron number. The average deviation per barrier $\delta$ (in MeV) is defined as $\delta ={\sum }_{i=1}^N|B_f^i\left(\text{th}\right)-B_f^i\left(\text{exp}\right)|/N$, where $N$ is the number of the barriers with known experimental heights, and $B_f^i\left(\text{th}\right)$ [$B_f^i\left(\text{exp}\right)$] are calculated (experimental) heights of the barriers. Long-dashed lines represent the trend of the deviations between theory and experiment as a function of neutron number. They are obtained via linear regression based on a least-squares fit.

Figure 3

The same as in Fig. 2 but as a function of proton number $Z$.

Figure 4

Potential energy surfaces of selected nuclei. The energy difference between two neighboring equipotential lines is equal to 0.5 MeV. The saddles along the Ax, Tr-A, and Tr-B fission paths are shown by solid circles, triangles, and squares, respectively. The solid diamonds show the outer fission barrier saddles. The saddles are defined via the immersion method [7], while the fission paths are defined as minimum energy paths [63, 64] which represent the most probable pathway connecting two minima via a given saddle. Note that in contrast to other nuclei there are two possible fission paths for the outer fission barrier in the ${}^{300}$116 nucleus.

Figure 5

Deformation energy curves for the $Z=120,N=172$ nucleus obtained with the NL3* parametrization. The black solid, red dashed, and blue dot-dashed lines display the deformation energy curves for the axially symmetric, triaxial, and axial octupole deformed solutions. We show the deformation energy curves for the last two solutions only in the range of ${\beta }_2$ values where it is lower in energy than the deformation energy curve of the axially symmetric solution. Note that the Tr-B fission path via the saddle at ${\beta }_2=0.47,\gamma ={26}^{\circ }$ (see text for details) is not shown since this saddle is lower than the one of the Ax fission path only by 0.2 MeV (see Table ); the results of Ref. [67] suggest that spontaneous fission exploiting this path is less likely than the one along the Ax fission path.

Figure 6

Deformation energy curves for the $Z=112,114,$ and 116 nuclei obtained with the NL3* and DD-PC1 parametrizations of the RMF Lagrangian. Solid lines correspond to axial solutions with reflection symmetry (A), dashed lines to triaxial solutions with reflection symmetry (T), and dotted lines to octupole deformed solutions with axial symmetry (O). Note that the T and O solutions are shown only in the deformation range in which they are lower in energy than the axial solution.

Figure 7

The same as Fig. 6 but for the NL3* and DD-ME2 parametrizations.

Figure 8

The same as in Fig. 4 but for the nucleus ${}^{294}$116. The results are shown for the parametrizations NL3*, DD-PC1, and DD-ME2. The energy difference between two neighboring equipotential lines is equal to 0.2 MeV.

Figure 9

(a) The deformations of the ground states of even-even $Z=112–120$ nuclei as a function of neutron number. (b) The energies of axial (large solid symbols, thick lines) and triaxial (either Ax-Tr or Tr-A; shown by small open symbols and thin lines) saddles with respect to the spherical or weakly (normal) deformed minima in these nuclei as a function of neutron number. Note that the same type of symbols is used for axial and triaxial saddles in a given isotope chain.

Figure 10

The heights of outer fission barriers of even-even $Z=112–120$ nuclei relative to spherical or weakly (normal) deformed minima as a function of neutron number. The results of calculations of type A (axial), T (triaxial), and O (octupole) obtained with the NL3* parametrization are presented.

Figure 11

The same as in Fig. 5 but for the nuclei ${}^{298}$120 and ${}^{306}$120. Note that for simplicity only the axial solution (A) is shown at ${\beta }_2<0.5$.

Figure 12

The same as in Fig. 4 but for the nucleus ${}^{302}$120. In order to show the potential energy surface between the axial $\beta \sim 0.2$ and triaxial (${\beta }_2\sim 0.40,\gamma \sim {25}^{\circ }$) hills in greater detail the energy difference between two neighboring equipotential lines is set to 0.1 MeV and only one fourth of the deformation plane of Fig. 4 is shown.