Energy density functional analysis of shape evolution in $N=28$ isotones

The structure of low-energy collective states in proton-deficient $N=28$ isotones is analyzed using structure models based on the relativistic energy density functional DD-PC1. The relativistic Hartree-Bogoliubov model for triaxial nuclei is used to calculate binding energy maps in the $\beta$-$\gamma$ plane. The evolution of neutron and proton single-particle levels with quadrupole deformation, and the occurrence of gaps around the Fermi surface, provide a simple microscopic interpretation of the onset of deformation and shape coexistence. Starting from self-consistent constrained energy surfaces calculated with the functional DD-PC1, a collective Hamiltonian for quadrupole vibrations and rotations is employed in the analysis of excitation spectra and transition rates of ${}^{46}$Ar, ${}^{44}$S, and ${}^{42}$Si. The results are compared to available data, and previous studies based either on the mean-field approach or large-scale shell-model calculations. The present study is particularly focused on ${}^{44}$S, for which data have recently been reported that indicate pronounced shape coexistence.

Figure 1

Self-consistent RHB triaxial quadrupole constrained energy surfaces of $N=28$ isotones in the $\beta -\gamma$ plane ($0⩽\gamma ⩽{60}^{\circ }$). For each nucleus energies are normalized with respect to the binding energy of the global minimum. The contours join points on the surface with the same energy (in MeV).

Figure 2

Single-neutron and single-proton energy levels of  ${}^{46}$Ar as functions of the deformation parameters along closed paths in the $\beta -\gamma$ plane. Solid (black) curves correspond to levels with positive parity, and (red) dashed curves denote levels with negative parity. The dot-dashed (blue) curves corresponds to the Fermi levels. The panels on the left and right display prolate ($\gamma =0^{\circ }$) and oblate ($\gamma ={60}^{\circ }$) axially symmetric single-particle levels, respectively. In the middle panel of each figure the neutron and proton levels are plotted as functions of $\gamma$ for a fixed value of the axial deformation $\left|\beta \right|$ at the approximate position of the mean-field minimum.

Figure 3

Same as described in the caption to Fig. 2 but for the nucleus ${}^{44}$S.

Figure 4

Same as described in the caption to Fig. 2 but for the nucleus ${}^{42}$Si.

Figure 5

Same as described in the caption to Fig. 2 but for the nucleus ${}^{40}$Mg.

Figure 6

The spectrum of ${}^{46}$Ar calculated with the DD-PC1 relativistic density functional (left), compared to data [44] (right) for the excitation energy of $2_1^{+}$, and the reduced electric quadrupole transition $B\left(E2\right)$ (in units of $e^2{\mathrm{fm}}^4$). The prediction for the electric monopole transition strength ${\rho }^2(E0;0_2^{+}\to 0_1^{+})$ is also included in the theoretical spectrum.

Figure 7

Same as described in the caption to Fig. 6 but for the nucleus ${}^{44}$S. The data are from Refs. [5, 13].

Figure 8

Same as described in the caption to Fig. 6 but for the nucleus ${}^{42}$Si. The data are from Ref. [11].

Figure 9

Evolution of the characteristic observables $E\left(2_1^{+}\right)$ and $B(E2;2_1^{+}\to 0_1^{+})$ (in $e^2{\mathrm{fm}}^4$) with proton number in $N=28$ isotones. The ratio between the excitation energies of the first $4^{+}$ and $2^{+}$ states is also displayed in the inset. The microscopic values calculated with the energy density functional DD-PC1 are shown in comparison with available data.

Figure 10

Probability distribution Eq. (20) in the $\beta -\gamma$ plane for the lowest collective states of ${}^{44}$S, predicted by DD-PC1 energy density functional.

Figure 11

Same as described in the caption to Fig. 10 but for the nucleus ${}^{42}$Si.