Axial shape asymmetry and high-spin states in nuclei with $Z=100$ suggested by the projected total energy surface approach

The axial-shape asymmetry of yrast states in ${}^{246–256}\mathrm{Fm}$ is studied by performing the projected total-energy surface (PTES) calculations, which consider the beyond-mean-field effects associated with the restoration of rotational symmetry and shape variation at the same time. The results show a large elongation deformation but also a considerable large triaxiality for their ground and high spin states, the triaxial deformation $\gamma \approx {11}^{\circ }$ in average. In comparison, the TRS calculations have also been performed for these nuclei, and the results show a well-established axial quadrupole shape in their ground states. The presence of the significant triaxial deformation can be attributed to the beyond-mean-field effects generated by the angular-momentum projection. The axial asymmetric shape for the yrast states of nuclei with $Z=100$, suggested by the present variation after projection (VAP) calculations, indicates that the triaxial degree of freedom may also play a significant role in other transfermium and even superheavy nuclei. The present PTES calculations have well reproduced the available experimental energies of the ground-band states and predict the rest yrast states up to spin 30 in each nucleus. The calculated yrast bands of ${}^{246–256}\mathrm{Fm}$ present the back bending phenomenon at about the state ${18}^{+}$, caused by the alignment excitations of the two quasiparticle neutrons of $\nu j_{15/2}\left[743\right]7/2$ or of $\nu h_{11/2}\left[761\right]1/2$. It is worth confirming the predicted band structures by the future spectroscopic experiments in the transfermium nuclei for the study of the single-particle structure in the superheavy mass region.

Figure 1

Contour plot of total energy in units of MeV for the ground state of ${}^{248}\mathrm{Fm}$, the local minimum is marked by “$+$.” The $\gamma$ axis is given in units of degrees.

Figure 2

Neutron Nilsson diagram showing single-particle orbitals as deformation ${\varepsilon }_2$ varies. Orbitals with positive (negative) parity are shown by solid (dashed) curves.

Figure 3

Calculated yrast bands, the energy levels, for ${}^{246–256}\mathrm{Fm}$ isotopes compared with the experimental data.

Figure 4

Moment of inertia as function of rotational frequency for ${}^{246–256}\mathrm{Fm}$. The upper panel shows the results of the present calculations, each symbol point corresponds to an initial state of the $E2$ transition, whose $I^{\pi }$ value is marked above symbols for few points only as guide. As a comparison, the bottom panel shows the results of the CRHB calculations, and the data are measured by reading the part of Fm isotopes of Fig. 12 in Ref. [32].

Figure 5

Contour plot of total Routhian in units of MeV for ${}^{248}\mathrm{Fm}$, calculated at the rotational frequency $\hslash \omega =0.02\hslash {\omega }_0$, the minimum is marked by “$+$.”

Figure 6

Energy surfaces for states with angular momentum $I^{\pi }=0^{+},2^{+},...,{10}^{+}$. The energies are calculated as function of (a) quadrupole deformation ${\varepsilon }_2$ with $\gamma =0^{\circ }$, and (b) $\gamma$ deformation with ${\varepsilon }_2=0.255$. Full (dashed) curves correspond to projected (unprojected) calculations. The $\gamma$ axis is given in units of degrees.

Figure 7

Calculated g.s. bands (open squares) and their $\gamma$ bands (open triangles), the energy versus spin for ${}^{254}\mathrm{Fm}$ and ${}^{256}\mathrm{Fm}$ and a comparison with the experimental data (solid symbols).

Figure 8

Component energy surfaces of the total energy for the ground state of ${\mathrm{I}}^{\pi }=0^{+}$ in ${}^{248}\mathrm{Fm}$: (a) $E_{LD}+E_{\mathrm{shell}}$ and (b) $E_{\mathrm{rot}}$. The energy is in units of MeV. The $\gamma$ axis is given in units of degrees.