Microscopic analysis of nuclear quantum phase transitions in the $\mathit{N}\approx 90$ region

The analysis of shape transitions in Nd isotopes, based on the framework of relativistic energy-density functionals and restricted to axially symmetric shapes in T. Nikšić, D. Vretenar, G. A. Lalazissis, and P. Ring [Phys. Rev. Lett. 99, 092502 (2007)], is extended to the region $Z=60,62,64$ with $N\approx 90$ and includes both $\beta$ and $\gamma$ deformations. Collective excitation spectra and transition probabilities are calculated starting from a five-dimensional Hamiltonian for quadrupole vibrational and rotational degrees of freedom, with parameters determined by constrained self-consistent relativistic mean-field calculations for triaxial shapes. The results reproduce available data and show that there is an abrupt change of structure at $N=90$ that can be approximately characterized by the X(5) analytic solution at the critical point of the first-order quantum phase transition between spherical and axially deformed shapes.

Figure 1

Self-consistent RMF+BCS triaxial quadrupole binding energy maps of the even-even ${}^{144-154}\mathrm{Nd}$ isotopes in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$). All energies are normalized with respect to the binding energy of the absolute minimum (red filled circle). The contours join points on the surface with the same energy (in MeV).

Figure 2

Self-consistent RMF+BCS binding energy curves of the even-even ${}^{144-154}\mathrm{Nd}$ isotopes as functions of the axial deformation parameter $\beta$. Energies are normalized with respect to the binding energy of the absolute minimum for a given isotope. Negative values of $\beta$ correspond to the oblate $(\beta >0,\hspace{0.5em}\gamma ={60}^{°})$ axis on the $\beta \text{−}\gamma$ plane.

Figure 3

Self-consistent RMF+BCS binding energy curves of the ${}^{150}\mathrm{Nd}$ nucleus, as functions of the deformation parameter $\gamma$ for two values of the axial deformation $\beta =0.2$ and 0.25.

Figure 4

Evolution of the characteristic collective observables $R_{4/2}$ and $B(E2;2_1^{+}\to 0_1^{+})$ (in W.u.) with neutron number in Nd isotopes. The microscopic values calculated with the PC-F1 energy-density functional are shown in comparison with data [34, 41, 42].

Figure 5

Evolution of the first excited $0^{+}$ state and the ratio $E(6_1^{+})/E(0_2^{+})$ with neutron number in Nd isotopes. The microscopic values calculated with the PC-F1 energy-density functional are compared to data [41, 42].

Figure 6

The spectrum of ${}^{150}\mathrm{Nd}$ calculated with the PC-F1 relativistic density functional (left) compared with the data [34] (middle) and the X(5)-symmetry predictions (right) for the excitation energies and intraband and interband $B(E2)$ values (in W.u.) of the ground-state ($s=1$) and ${\beta }_1$ ($s=2$) bands. The theoretical spectra are normalized to the experimental energy of the state $2_1^{+}$, and the X(5) transition strengths are normalized to the experimental $B(E2;2_1^{+}\to 0_1^{+})$.

Figure 7

$B(E2;L\to L-2)$ values (upper panel) and excitation energies (lower panel) for the yrast states in ${}^{148}\mathrm{Nd}$, ${}^{150}\mathrm{Nd}$, and ${}^{152}\mathrm{Nd}$, calculated with PC-F1 and compared with those predicted by the U(5), X(5), and SU(3) symmetries (open symbols).

Figure 8

Neutron single-particle levels in Nd isotopes as functions of the axial deformation parameter $\beta$. Thick dot-dashed curves denote the position of the Fermi level.

Figure 9

Neutron (left) and proton (right) single-particle levels in ${}^{150}\mathrm{Nd}$, as functions of the axial deformation parameter $\beta$. Thick dot-dashed curves denote the position of the corresponding Fermi levels.

Figure 10

Self-consistent RMF+BCS triaxial quadrupole binding energy maps of ${}^{150,152,154}\mathrm{Sm}$ in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$). All energies are normalized with respect to the binding energy of the absolute minimum (red filled circle). The contours join points on the surface with the same energy (in MeV).

Figure 11

Same as described in the caption to Fig. 10 but for the isotopes ${}^{152,154,156}\mathrm{Gd}$.

Figure 12

The spectrum of ${}^{152}\mathrm{Sm}$ calculated with the PC-F1 relativistic density functional (left) compared with data (middle) and the X(5)-symmetry predictions (right) for the excitation energies and intraband and interband $B(E2)$ values (in W.u.) of the ground-state ($s=1$) and ${\beta }_1$ ($s=2$) bands. The theoretical spectra are normalized to the experimental energy of the state $2_1^{+}$, and the X(5) transition strengths are normalized to the experimental $B(E2;2_1^{+}\to 0_1^{+})$.

Figure 13

Same as described in the caption to Fig. 12 but for the nucleus ${}^{154}\mathrm{Gd}$.

Figure 14

Same as described in the caption to Fig. 7 but for the isotopes ${}^{150,152,154}\mathrm{Sm}$.

Figure 15

Same as described in the caption to Fig. 7 but for the isotopes ${}^{152,154,156}\mathrm{Gd}$.