Multidimensionally constrained relativistic mean-field study of triple-humped barriers in actinides

Background: Potential energy surfaces (PES's) of actinide nuclei are characterized by a two-humped barrier structure. At large deformations beyond the second barrier, the occurrence of a third barrier was predicted by macroscopic-microscopic model calculations in the 1970s, but contradictory results were later reported by a number of studies that used different methods.

Purpose: Triple-humped barriers in actinide nuclei are investigated in the framework of covariant density functional theory (CDFT).

Methods: Calculations are performed using the multidimensionally constrained relativistic mean field (MDC-RMF) model, with the nonlinear point-coupling functional PC-PK1 and the density-dependent meson exchange functional DD-ME2 in the particle-hole channel. Pairing correlations are treated in the BCS approximation with a separable pairing force of finite range.

Results: Two-dimensional PES's of ${}^{226,228,230,232}\mathrm{Th}$ and ${}^{232,234,236,238}\mathrm{U}$ are mapped and the third minima on these surfaces are located. Then one-dimensional potential energy curves along the fission path are analyzed in detail and the energies of the second barrier, the third minimum, and the third barrier are determined. The functional DD-ME2 predicts the occurrence of a third barrier in all Th nuclei and ${}^{238}\mathrm{U}$. The third minima in ${}^{230,232}\mathrm{Th}$ are very shallow, whereas those in ${}^{226,228}\mathrm{Th}$ and ${}^{238}\mathrm{U}$ are quite prominent. With the functional PC-PK1 a third barrier is found only in ${}^{226,228,230}\mathrm{Th}$. Single-nucleon levels around the Fermi surface are analyzed in ${}^{226}\mathrm{Th}$, and it is found that the formation of the third minimum is mainly due to the $Z=90$ proton energy gap at ${\beta }_{20}\approx 1.5$ and ${\beta }_{30}\approx 0.7$.

Conclusions: The possible occurrence of a third barrier on the PES's of actinide nuclei depends on the effective interaction used in multidimensional CDFT calculations. More pronounced minima are predicted by the DD-ME2 functional, as compared to the functional PC-PK1. The depth of the third well in Th isotopes decreases with increasing neutron number. The origin of the third minimum is due to the proton $Z=90$ shell gap at relevant deformations.

Figure 1

Energy curve of ${}^{226}\mathrm{Th}$ computed with the MDC-RMF model using the functional PC-PK1. When axial symmetry is imposed, the energy denoted by the dash-dotted (black) curve is obtained [axial symmetric (AS) and reflection asymmetric (RA)], whereas triaxial calculations yield the solid (blue [gray]) curve [nonaxial (NA) and reflection asymmetric (RA)]. The ADHO basis contains $N_f=16$ oscillator shells.

Figure 2

The axially symmetric energy curve of ${}^{226}\mathrm{Th}$ computed with the MDC-RMF model, using the functional PC-PK1. Results obtained with different truncations of the ADHO basis, i.e., with $N_f=16$, 18, 20, and 22 shells are plotted by the dash-dotted, dotted, dashed, and solid curves, respectively.

Figure 3

Self-consistent MDC-RMF energy surfaces of U and Th isotopes in the $({\beta }_{20},{\beta }_{30})$ plane, calculated with the relativistic functionals PC-PK1 (a) and DD-ME2 (b). For each nucleus the energy is normalized with respect to the binding energy of the absolute minimum. The contours join points on the surface with the same energy and the energy difference between neighboring contours is 0.5 MeV. The calculation has been performed in the ADHO basis with $N_f=16$ shells.

Figure 4

The odd-even differences of binding energies of Th isotopes computed with the MDC-RMF model, using the functional PC-PK1. Results obtained using different pairing strength parameters $g/g_0=1.0$, 1.1, and 1.2, where $g_0=728$ MeV ${\text{fm}}^3$, are plotted as dots, triangles, and stars respectively. The squares denote the experimental values obtained from Ref. [71]. The calculation has been performed in the ADHO basis with $N_f=20$ shells.

Figure 5

The odd-even differences of binding energies of Th isotopes computed with the MDC-RMF model, using the functional DD-ME2. Results obtained using different pairing strength parameters $g/g_0=1.0$, 1.1, and 1.2, where $g_0=728$ MeV ${\text{fm}}^3$, are plotted as dots, triangles, and stars respectively. The squares denote the experimental values obtained from Ref. [71]. The calculation has been performed in the ADHO basis with $N_f=20$ shells.

Figure 6

MDC-RMF energy curves of ${}^{226,228,230,232}\mathrm{Th}$ and ${}^{232,234,238}\mathrm{U}$. The calculation has been performed in the ADHO basis with $N_f=20$ shells and the pairing strength parameter $g/g_0=1.1$ for PC-PK1 and $g/g_0=1.2$ for DD-ME2. For each nucleus the energy is normalized with respect to the binding energy at the absolute minimum.

Figure 7

The neutron (upper panel) and proton (lower panel) single-particle levels of ${}^{226}\mathrm{Th}$ near the Fermi surface along the static fission path. For ${\beta }_{20}\le 0.6$, only reflection-symmetric deformations are considered and the solid (dotted) curves represent positive (negative) parity states. When ${\beta }_{20}>0.6$, the octupole deformation ${\beta }_{30}$ has a nonvanishing value and parity is not a good quantum number. The dash-dotted curves denote the Fermi energy, and the symbols in the lower panel are used to guide the eye. The MDC-RMF calculation has been performed with the functional DD-ME2.