Beyond relativistic mean-field approach for nuclear octupole excitations

We report the first beyond-mean-field study of low-lying parity-doublet states in ${}^{224}\mathrm{Ra}$ by extending the multireference relativistic energy density functional method to include dynamical correlations related to symmetry restoration and quadrupole-octupole shape fluctuation with a generator coordinate method combined with parity, particle-number, and angular-momentum projections. We clarify microscopically that the origin of spin-dependent parity splitting in low-spin states is related to the octupole shape stabilization of positive-parity states, the shapes of which drift gradually toward those of negative-parity ones.

Figure 1

(a) Mean-field energy surface of ${}^{224}\mathrm{Ra}$ in ${\beta }_2-{\beta }_3$ deformation plane (normalized to energy minimum), where two neighboring lines are separated by 0.5 MeV. (b) Total nucleon density of the energy-minimum state with ${\beta }_2=0.179,{\beta }_3=0.125,{\beta }_4=0.146$ in the $y=0$ plane.

Figure 2

Low-lying energy spectra for ${}^{224}\mathrm{Ra}$. The available data from Ref. [3] and the results from full-configuration and the single energy-minimum configuration calculations are shown in panels (a), (b), and (c), respectively. The numbers on arrows are $E2$ (red [light gray] color) and $E3$ (blue [dark gray] color) transition strengths (Weisskopf units).

Figure 3

(a) Excitation energy ratio $R_{J/2}\equiv E_x\left(J\right)/E_x\left(2^{+}\right)$, (b) normalized staggering amplitude $S_{J/2}\equiv |E_x\left(J\right)-\frac{J+1}{2J+1}{E}_x(J-1)-\frac{J}{2J+1}{E}_x(J+1)|/E_x\left(2^{+}\right)$ [26], and (c) average deformation $\langle {\beta }_{\lambda }\rangle$ with $\lambda =2,3$ as functions of spin $J$ in ${}^{224}\mathrm{Ra}$.

Figure 4

(a) Excitation energy ratio $R_{J/2}$ and (inset) staggering amplitude $S_{J/2}$; (b) norm overlap $\langle {\beta }_2{\beta }_3\left|\widehat{P}\right|{\beta }_2{\beta }_3\rangle$ of parity operator in mean-field states; and (c) and (d) mean-field and projected energy curves onto particle numbers $N,Z$, angular momentum $J$, and parity $\pi ={(-1)}^J$ (solid lines for even $J$, dashed lines for odd $J$). The mean-field (or projected) energy curves are normalized to the $0^{+}$ state at ${\beta }_3=0$ (or minimum). The mean-field energy surfaces in panels (c) and (d) are shifted down by 3.43 and 4.15 MeV, respectively. Panels (e) and (f) show the absolute values of collective wave functions for parity-doublet states. Those for $\text{odd-}\pi (1^{-},3^{-},...,9^{-})$ states are close to each other.