Multidimensionally-constrained relativistic mean-field models and potential-energy surfaces of actinide nuclei

Background: Many different shape degrees of freedom play crucial roles in determining the nuclear ground state and saddle point properties and the fission path. For the study of nuclear potential energy surfaces, it is desirable to have microscopic and self-consistent models in which all known important shape degrees of freedom are included.

Purpose: By breaking both the axial and the spatial reflection symmetries simultaneously, we develop multidimensionally-constrained relativistic mean field (MDC-RMF) models.

Methods: The nuclear shape is assumed to be invariant under the reversion of $x$ and $y$ axes, i.e., the intrinsic symmetry group is $V_4$ and all shape degrees of freedom ${\beta }_{\lambda \mu }$ with even $\mu$, such as ${\beta }_{20}$, ${\beta }_{22}$, ${\beta }_{30}$, ${\beta }_{32}$, ${\beta }_{40}$, $...,$ are included self-consistently. The single-particle wave functions are expanded in an axially deformed harmonic oscillator (ADHO) basis. The RMF functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the nonlinear or density-dependent couplings. The pairing effects are taken into account with the BCS approach.

Results: The one-, two, and three-dimensional potential energy surfaces of ${}^{240}$Pu are illustrated for numerical checks and for the study of the effect of the triaxiality on the fission barriers. Potential energy curves of even-even actinide nuclei around the first and second fission barriers are studied systematically. Besides the first ones, the second fission barriers in these nuclei are also lowered considerably by the triaxial deformation. This lowering effect is independent of the effective interactions used in the RMF functionals. Further discussions are made about different predictions on the effect of the triaxiality between the macroscopic-microscopic and MDC-RMF models, possible discontinuities on PES's from self-consistent approaches, and the restoration of broken symmetries.

Conclusions: MDC-RMF models give a reasonably good description of fission barriers of even-even actinide nuclei. It is important to include both the nonaxial and the reflection asymmetric shapes simultaneously for the study of potential energy surfaces and fission barriers of actinide nuclei and of those in unknown mass regions such as, e.g., superheavy nuclei.

Figure 1

The potential energy curve of ${}^{240}$Pu from MDC-RMF calculations with different truncations of the ADHO basis. Both triaxial (TA) and reflection asymmetric (RA) deformations are allowed. The results calculated with $N_f=16$, 18, and 20 are depicted by dashed, dotted, and solid curves, respectively. The four subfigures show the detailed structure of the potential energy curve near the ground state (A), the inner barrier (B), the isomeric state (C), and the outer barrier (D). The results calculated with $N_f=22$ (dot-dashed curves) are also included in the subfigures. The width of each subfigure is 0.1 and the height is 1 MeV.

Figure 2

Mean field energy (i.e., $E_{\mathrm{c}}.\mathrm{m}.$ is not included) of ${}^{240}$Pu calculated by using basis with different basis deformation ${\beta }_{\mathrm{basis}}$ and different $N_f$. The results calculated with $N_f=16$, 18, and 20 are denoted by red diamonds, green squares, and blue dots, respectively. The deformation is constrained to ${\beta }_{20}=0.3$, 1.3, and 2.0, respectively.

Figure 3

Mean field energy (i.e., $E_{\mathrm{c}}.\mathrm{m}.$ is not included) of ${}^{240}$Pu calculated with different truncations on large (upper panel) and small component (lower panel). In the calculations for the upper panel, $N_g$ are fixed to 23, while the calculations for the lower panel are preformed with $N_f=10$. The deformation is constrained to ${\beta }_{20}=0.3$.

Figure 4

The potential energy curve of ${}^{240}$Pu calculated by using basis with different oscillator frequency $\hslash {\omega }_0$. The axial symmetry is imposed and $N_f=20$ is used. The results calculated with $R\equiv \hslash {\omega }_0/\left(41A^{-1/3}\mathrm{MeV}\right)=0.8$, 0.9, 1.0, 1.1, and 1.2 are depicted by dotted, dashed, solid, dot-dashed, and dash-dot-dotted curves, respectively. The four subfigures show the detailed structure of the potential energy curve near the ground state (A), the inner barrier (B), the isomeric state (C), and the outer barrier (D). The width of each subfigure is 0.1 and the height is 2 MeV.

Figure 5

Sections of the three-dimensional potential energy surface, $E=E({\beta }_{20},{\beta }_{22},{\beta }_{30})$, of ${}^{240}$Pu around the ground state, the inner barrier, and the fission isomer from MDC-RMF calculations. In each subfigure the energy is shown as a function of the deformation parameters ${\beta }_{22}$ and ${\beta }_{30}$ when ${\beta }_{20}$ is fixed at a certain value. The energies are normalized with respect to the binding energy of the ground state. The contour interval is 1 MeV.

Figure 6

Sections of the three-dimensional potential energy surface, $E=E({\beta }_{20},{\beta }_{22},{\beta }_{30})$, of ${}^{240}$Pu around the fission isomer and the outer barrier from MDC-RMF calculations. In each sub-figure the energy is shown as a function of the deformation parameters ${\beta }_{22}$ and ${\beta }_{30}$ when ${\beta }_{20}$ is fixed at a certain value. The energies are normalized with respect to the binding energy of the ground state. The contour interval is 1 MeV.

Figure 7

Potential energy curves of even-even actinide nuclei around the ground states and the first fission barriers from MDC-RMF calculations. The axially symmetric results are displayed by dotted curves, while those from the triaxial calculations are shown by solid curves. The binding energy is normalized with respect to that of the ground state of each nucleus. The empirical values of fission barriers are taken from Ref. [102] and shown by red dots.

Figure 8

Potential energy curves of even-even actinide nuclei around the fission isomers and the second fission barriers from MDC-RMF calculations. The reflection asymmetric shapes are allowed in the calculations. The axially symmetric results are displayed by dotted curves, while those from the triaxial calculations are shown by solid curves. The binding energy is normalized with respect to that of the ground state of each nucleus. The empirical values of fission barriers are taken from Ref. [102] and shown by red dots. Experimental values of energies of fission isomers are taken from Ref. [103] and shown by green horizontal lines. For ${}^{244}$Pu, only an energy range was given for the isomer.