Shape evolution and coexistence in neutron-deficient Nd and Sm nuclei

The evolution of shapes and low-energy shape coexistence is analyzed in neutron-deficient Nd and Sm nuclei, using a five-dimensional quadrupole collective Hamiltonian (5DCH). Deformation energy surfaces, calculated with the relativistic energy density functional PC-PK1 and a separable finite-range pairing interaction, exhibit a transition from spherical shapes near $N=80$, to $\gamma$-soft shapes, and to prolate deformed minima in lighter isotopes. The corresponding 5DCH model calculation, based on the self-consistent mean-field potentials, reproduces the empirical isotopic trend of characteristic collective observables, and predicts significantly different deformations for the first two $0^{+}$ states in the $N=74$ isotones ${}^{134}\mathrm{Nd}$ and ${}^{136}\mathrm{Sm}$. In addition to bands based on the triaxial $\gamma$-soft ground state, in excellent agreement with data, the occurrence of a low-energy rotational band is predicted, built on the prolate deformed ($\beta \sim 0.4$) excited state $0_2^{+}$.

Figure 1

Collective potential energy $V_{\text{coll}}$ of the even-even isotopes ${}^{126–140}\mathrm{Nd}$ in the $(\beta ,\gamma )$ plane, obtained by constrained relativistic mean-field plus BCS calculations. For each nucleus energies are normalized to the absolute minimum. The energy difference between neighboring contours is $0.5\mathrm{MeV}$.

Figure 2

Same as in the caption to Fig. 1 but for the neutron-deficient isotopes of Sm.

Figure 3

The energy ratio $R_{42}=\frac{E\left(4^{+}\right)-E\left(0^{+}\right)}{E\left(2^{+}\right)-E\left(0^{+}\right)}$ (a) and $B(E2;2^{+}\to 0^{+})$ values (b) for the ground-state band and the band based on the $0_2^{+}$ state, the energy ratio $E\left(2_{\gamma }^{+}\right)/E\left(4_1^{+}\right)$ (c), the excitation energy of the $0_2^{+}$ state $E\left(0_2^{+}\right)$ (d), the values of the quadrupole deformation parameters $\beta$ (e), and $\gamma$ (f) for the $0_1^{+}$ and $0_2^{+}$ state, as functions of the neutron number in Nd isotopes. The data are from Ref.  [41].

Figure 4

Same as in the caption to Fig. 3 but for the chain of Sm isotopes.

Figure 5

5DCH energy spectra of ${}^{134}\mathrm{Nd}$ and ${}^{136}\mathrm{Sm}$ (PC-PK1), compared to available data  [41, 42] (Exp.), and to results obtained with the 5DCH based on the Gogny force (D1S) [43].

Figure 6

Distribution of the probability density ${\rho }_{J\alpha }(\beta ,\gamma )$ Eq. (5) for $0_1^{+}, 0_2^{+}, 2_1^{+}$, and $2_3^{+}$ collective states of ${}^{134}\mathrm{Nd}$ and ${}^{136}\mathrm{Sm}$.

Figure 7

Neutron (upper panel) and proton (lower panel) single-nucleon energy levels of ${}^{134}\mathrm{Nd}$, as functions of the deformation parameters along a closed path in the $\beta -\gamma$ plane. Solid (black) curves correspond to levels with positive parity, and dashed (red) curves denote negative-parity levels. The dotted (blue) curves correspond 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 the proton and neutron levels are plotted as functions of $\gamma$ for a fixed value $\left|\beta \right|=0.25$, corresponding to the approximate position of the mean-field minimum.

Figure 8

Single-neutron and single-proton levels in ${}^{134}\mathrm{Nd}$ as functions of the axial deformation parameter $\beta$. Solid (black) and dashed (red) curves denote the positive- and negative-parity levels, respectively. The dotted (blue) curves are the corresponding Fermi levels.

Figure 9

Single-neutron levels in ${}^{132,134,136}\mathrm{Nd}$ isotopes as functions of the axial deformation parameter $\beta$. Solid (black) and dashed (red) curves correspond to the positive- and negative-parity levels, respectively. The dotted (blue) curves denote the Fermi levels.