Introduction

The majority of the nuclei on the nuclear chart are characterized by reflection-symmetric shapes, either spherical or quadrupolar, in their ground states. However, atomic nuclei with a proton number Z or neutron numbers $N\simeq \mathrm{34,}\mathrm{56,}88$, and 134 possess strong octupole correlations exhibiting dynamical or static octupole deformations [1, 2]. Signatures of the nuclei with the static octupole-deformed shape include the presence of a low-lying positive- and negative-parity doublets as well as strong electric the dipole (E1) and octupole (E3) transition strengths. According to this criterion, the observation of a low-lying 3^- state and enhanced E3 transitions in the ^144,146Ba [3-5], ²²²Ra [6], ²²⁴Ra [7] and ²²⁶Ra [8] suggest that these nuclei have a stable octupole deformation. By contrast, some atomic nuclei with slightly weaker E3 transitions but relatively larger excitation energy for the 3^- state are usually interpreted as octupole vibrators such as ²²⁸Ra [6], as well as ²⁰⁸Pb and ²²⁰Rn [7]. Additionally, low-lying negative-parity states or enhanced E1 transitions were measured, indicating that ^222-226Rn are vibrators [9] and ²²⁸Th is pear-shaped [10]. However, in suggested octupole deformed nuclei, a strict alternation between positive and negative-parity energy levels are not observed. To draw a solid conclusion as to whether these nuclei belong to octupole rotors or vibrators, additional measurements, and comprehensive theoretical studies on these nuclei are required.

Atomic nuclei with octupole correlations have been extensively studied with various nuclear models [11-17], including self-consistent mean-field (SCMF) methods based on different energy density functionals (EDFs)[18-31] and beyond, in combination with the interacting boson models[32-35] or collective Hamiltonians [36-39]. In particular, the generator coordinate method (GCM) implemented using quantum-number projection techniques, including parity, particle-number, and angular-momentum projections have been developed for the low-lying states of the atomic nuclei with octupole correlations based on different EDFs [40-44]. Within this multireference density functional theory (MR-DFT), it has been demonstrated that the low-lying states of ²⁰⁸Pb are multioctupole-phonon excitations [43]. By contrast, for ^144,146Ba, and ²²⁴Ra, the octupole shapes of positive-parity states rapidly stabilize with an increase in spin, gradually drifting toward those of negative-parity ones [41, 42]. Given this success, we extended the MR-DFT framework to study the low-lying states of ²²⁴Rn, including the energy spectrum and electric multipole transition strengths based on relativistic energy density functional (EDF) to shed light on whether the nucleus belong to octupole rotors or vibrators.

The remainder of this paper is organized as follows. In Sec. 2. introduction to MR-DFT based on the relativistic EDF is presented. In Sec. 3, we discuss the calculation results for the low-lying states of ²²⁴Rn compared with those of ²²⁴Ra. Finally, a summary is presented in Sec. 4.

Theoretical framework

In MR-DFT, the nuclear wave functions of the low-lying parity doublet states are constructed as linear combinations of sets of quantum numbers projected nonorthogonal mean-field states $|\text{q}\rangle$ around the equilibrium shape, $|{\text{Ψ}}_{\alpha }^{J\pi }\rangle ={\displaystyle \sum _{\text{q}}{f}_{\alpha }^{J\pi }}\left(\text{q}\right){\widehat{P}}_{MK}^J{\widehat{P}}^N{\widehat{P}}^Z{\widehat{P}}^{\pi }|\text{q}\left({\beta }_2\mathrm{,}{\beta }_3\right)\rangle \mathrm{,}$ (1) where α denotes the different quantum states corresponding to a given $J^{\pi }$. The symbol q represents the quadrupole, and octupole deformation parameters for each mean-field state. The operators ${\widehat{P}}_{MK}^J$, ${\widehat{P}}^{N\mathrm{,}Z}$, and ${\widehat{P}}^{\pi }$ are used to select the components of configurations with specific quantum numbers, namely the angular momentum J, neutron (proton) number N(Z) and parity π=± [45].

The mean-field states $|\text{q}\rangle$ are generated from the point-coupling relativistic mean field and BCS (PC-RMF+BCS) calculations with constraints on the average nucleon numbers and quadrupole-octupole moments using the variational principle $\delta \langle \text{q}|\widehat{H}-{\displaystyle \sum _{\tau =n\mathrm{,}p}{\lambda }_{\tau }}{\widehat{N}}_{\tau }-{\displaystyle \sum _{\lambda =\mathrm{1,2,3}}{C}_{\lambda }}{\left({\widehat{Q}}_{\lambda 0}-q_{\lambda }\right)}^2|\text{q}\rangle =0$ (2) with Lagrange multipliers ${\lambda }_{\tau }$ determined by the constraints $\langle \text{q}|{\widehat{N}}_{\tau }|\text{q}\rangle =N(Z)$. The position of the center-of-mass coordinate is fixed at the origin to decouple the spurious states using the constraint $\langle \text{q}|{\widehat{Q}}_{10}|\text{q}\rangle =0$. ${\widehat{N}}_{\tau }$ and ${\widehat{Q}}_{\lambda 0}\equiv r^{\lambda }{Y}_{\lambda 0}$ are the particle number and multipole moment operator, respectively. ${\text{q}}_{\lambda }$ is the constrained value of the multipole moment and $C_{\lambda }$ is the corresponding stiffness constant [45]. Deformation parameters ${\beta }_{\lambda }\left(\lambda =\mathrm{2,3}\right)$ are defined as follows: ${\beta }_{\lambda }\equiv \frac{4\pi }{3A{R}_0^{\lambda }}\langle \text{q}\left|{\widehat{Q}}_{\lambda 0}\right|\text{q}\rangle$ (3) with $R_0=r_0{A}^{1/3}$ and A represents the mass number of the nucleus and r₀=1.2 fm.

The weight function $f_{\alpha }^{J\pi }\left(\text{q}\right)$ is given by Eq. (1) is determined by solving the Hill–Wheeler–Griffin: (HWG) equation [46, 47]. ${\displaystyle \sum _{{\text{q}}_b}\left[{\mathcal{H}}^{J\pi }\left({\text{q}}_a\mathrm{,}{\text{q}}_b\right)-E_{\alpha }^{J\pi }{\mathcal{N}}^{J\pi }\left({\text{q}}_a\mathrm{,}{\text{q}}_b\right)\right]}{f}_{\alpha }^{J\pi }\left({\text{q}}_b\right)=\mathrm{0,}$ (4) where the kernels are expressed as ${\mathcal{O}}^{J\pi }\left({\text{q}}_a\mathrm{,}{\text{q}}_b\right)=\langle {\text{q}}_a|\widehat{O}{\widehat{P}}_{KM}^J{\widehat{P}}^{\pi }{\widehat{P}}^N{\widehat{P}}^Z\widehat{|}{\text{q}}_b\rangle$ (5) with operators $\widehat{O}$ representing $\widehat{H}$ and 1 for the the Hamiltonian kernel ${\mathcal{H}}^{J\pi }\left({\text{q}}_a\mathrm{,}{\text{q}}_b\right)$ and the norm kernel ${\mathcal{N}}^{J\pi }\left({\text{q}}_a\mathrm{,}{\text{q}}_b\right)$.

The electric multipole transition probabilities B(Eλ) obtained from the initial state $\left(J_i{\pi }_i{\alpha }_i\right)$ into the final state $\left(J_f{\pi }_f{\alpha }_f\right)$ are calculated according to the Wigner–Eckart Theorem: $\begin{array}{c}B\left(E\lambda ;J_{\text{i}}{\pi }_{\text{i}}{\alpha }_{\text{i}}\to J_{\text{f}}{\pi }_{\text{f}}{\alpha }_{\text{f}}\right)\\ =\frac{e^2}{2J_i+1}|{{\displaystyle \sum _{{\text{q}}_{\text{i}}\mathrm{,}{\text{q}}_{\text{f}}}\left[f_{{\alpha }_{\text{i}}}^{J_{\text{i}}{\pi }_{\text{i}}}\left({\text{q}}_{\text{i}}\right)\right]}}^{\ast }\left[f_{{\alpha }_f}^{J_f{\pi }_f}\left({\text{q}}_{\text{f}}\right)\right]\\ {\times \langle {\text{Φ}}_{{\alpha }_{\text{f}}}^{J_{\text{f}}{\pi }_{\text{f}}}\left({\text{q}}_{\text{f}}\right)\Vert {\widehat{Q}}_{\lambda }\Vert {\text{Φ}}_{{\alpha }_{\text{i}}}^{J_{\text{i}}{\pi }_{\text{i}}}\left({\text{q}}_{\text{i}}\right)\rangle |}^2\end{array}$ (6) with a reduced transition matrix element, $\begin{array}{c}\langle {\text{Φ}}_{{\alpha }_{\text{f}}}^{J_{\text{f}}{\pi }_{\text{f}}}\left({\text{q}}_{\text{f}}\right)\Vert {\widehat{Q}}_{\lambda }\Vert {\text{Φ}}_{{\alpha }_{\text{i}}}^{J_{\text{i}}{\pi }_{\text{i}}}\left({\text{q}}_{\text{i}}\right)\rangle \\ ={\delta }_{{\pi }_{\text{i}}{\pi }_{\text{f}}\mathrm{,}{\left(-1\right)}^{\lambda }}\frac{\left(2J_{\text{f}}+1\right)\left(2J_{\text{i}}+1\right)}{2}{\displaystyle \sum _M\left(\begin{array}{ccc}{J}_{\text{f}}& \lambda & J_{\text{i}}\\ 0& M& -M\end{array}\right)}\\ \times {\displaystyle {\int }_0^{\pi }}d\theta \sin \left(\theta \right)d_{-M0}^{J_{\text{i}}\ast }\left(\theta \right)\langle \text{Φ}\left({\text{q}}_f\right)\left|{\widehat{Q}}_{\lambda M}{e}^{i\theta {\widehat{J}}_y}{\widehat{P}}^{\pi }{\widehat{P}}^N{\widehat{P}}^Z\right|\text{Φ}\left({\text{q}}_{\text{i}}\right)\rangle \mathrm{,}\end{array}$ (7) where ${\widehat{Q}}_{\lambda M}\equiv e{r}^2{Y}_{\lambda M}$ denotes the electric multipole moment operator of rank λ. More details on the MR-DFT for quadrupole-octupole nuclei can be found in Refs. [41-44, 48, 49].

Results and discussions

The Dirac spinors of the nucleons are expanded in a set of harmonic oscillator basis with 14 major shells. In the PC-RMF+BCS calculations, the relativistic EDF PC-PK1 [50] was employed. Only the degrees of freedom of the axial symmetry deformation were considered in the current study. Pairing correlations between nucleons are treated within the BCS approximation using a density-independent δforce with smooth cutoffs [51]. Strength parameters of the pairing force are set to V_n=-349.5 MeV fm³ for the neutrons and V_p=-330.0 MeV fm³ for protons. In the calculation of the projected kernels, $N_{\beta }=16$ mesh points are used for the rotation angle β, and $N_{\phi }=7$ for the gauge angle φ, both within the interval [0,π]. Since the low-lying parity-doublet bands of ²²⁴Rn are primarily characterized by prolate deformed configurations with β₂>0, the oblate deformed configurations with β₂<0 are excluded in the final configuration-mixing GCM calculations to reduce computational costs. The Pfaffian method [52, 53] is implemented to avoid the sign problem when calculating the norm kernel overlap.

Figure 1 presents the energies of the mean-field states for ²²⁴Rn normalized to the energy minimum in β₂–β₃ deformation plane. It is shown that, although the energy minimum is at β₃=0, the energy surface is soft along the β₃ direction around the minimum, which is similar to the findings in study using the relativistic Hartree-Bogoliubov (RHB) method [26]. The softness of the energy surface in ²²⁴Rn is attributed to the coupling of the proton orbitals $i_{13/2}-f_{7/2}$ and the neutron orbitals $j_{13/2}-g_{7/2}$ around the the Fermi surfaces [54]. The soft behavior indicates that the dynamic correlation effects, including symmetry restoration and quadrupole-octupole shape fluctuations can be significant in the low-lying states of ²²⁴Rn.

Fig. 1Full-screen View

(Color online) The mean-field energy surface of ²²⁴Rn in β₂-β₃ deformation plane normalized to the energy minimum. Two neighboring contour lines are separated by 0.4 MeV

Figure 2 shows the energy surfaces of ²²⁴Rn with projections to good nucleon numbers and spin parity $J^{\pi }{=0}^{+}{\mathrm{,1}}^{-}{\mathrm{,2}}^{+}$, and 3^-. Since the mean-field configurations with very small values of β₃ are predominated by components with positive parity, the energies of the negative parity states projected from these mean-field configurations are not shown. The energy minimum of the 0⁺ state shifts to an octupole-deformed shape with ${\beta }_2=\mathrm{0.1,}{\beta }_3=\pm 0.05$. The energy gained from the restoration of broken symmetries for the energy-minim state is ~4.23 MeV. The energy surface of the 2⁺ state is similar to that of the of 0⁺ state, except that it is softer along the quadrupole deformation ${\beta }_2\simeq [\mathrm{0.10,0.20}]$ with octupole deformation ${\beta }_3\simeq [-\mathrm{0.10,}+0.10]$. A similar result was observed in the potential energy surfaces (PESs) with J=1 (cf. Fig. 2(c)), and J=3 (cf. Fig. 2(d)), where the absolute minima are well separated along β₃-direction. The projected PESs with J=1 and J=3 show the soft structures in octupole ${\beta }_3\simeq \pm 0.15$ with a quadrupole deformation ranging from ${\beta }_2\simeq 0.10$ to 0.25.

Fig. 2Full-screen View

(Color online) The energies of states in ²²⁴Rn with projections onto good nucleon numbers, different spin parities with (a) $J^{\pi }{=0}^{+}$, (b) $J^{\pi }{=2}^{+}$, (c) $J^{\pi }{=1}^{-}$, and (d) $J^{\pi }{=3}^{-}$ in β₂-β₃ deformation plane normalized to the energy minimum of each J state

The quadrupole-octupole deformed configurations with good quantum numbers serve as the basis for expanding the wave functions of the low-lying state within the GCM. Figure 3 shows the excitation energies of the positive- and negative-parity bands calculated by solving the HWG equations (4) for three different configuration-mixing schemes. Calculation Results are compared with data from Ref. [9]. The calculation by mixing configurations with different β₂ and fixed β₃=0.05 provides very spread energy spectrum. In particular, the negative-parity states are very high in energy. By contrast, by mixing configurations with different β₃ values but fixed β₂=0.15, the energy spectrum were significantly compressed. In the full quadrupole-octupole configuration mixing calculation, the negative-parity states shift and approached the data.

Fig. 3Full-screen View

(Color online) The energy spectra of low-lying states in ²²⁴Rn obtained from the GCM calculations with (b) (β₂, β₃), (c) (β₂=0.15, β₃), and (d) (β₂, β₃=0.05) as generating coordinates, respectively. The data from Ref. [9] are shown in (a)

Figure 4 shows a detailed comparison of the low-lying parity-doublet states including electric multipole transition strength B(Eλ). It can be observed that the results calculated using the configuration-mixing GCM (Fig. 4(b)) and a single energy-minimum configuration (Fig. 4(c)) show similar parity doublet bands with rotational characteristics. In contrast to the results of GCM calculations, the positive-parity band is more compressed in single-configuration calculations, where the negative-parity band becomes slightly lower than that obtained from GCM calculation. Quantitatively, the excitation energy E(1^-) of the negative-parity band head is 0.47 MeV and 0.39 MeV from the GCM and single-configuration calculations, respectively. For the 3^- state, the calculated excitation energy E(3^-) from the GCM is 0.63 MeV, which is in good agreement with the data of E(3^-)=0.65 MeV. The electric octupole transition strength of ²²⁴Rn is $B\left(E{3;3}^{-}\to 0^{+}\right)=43$ W.u., which is comparable to that for ²²⁴Ra $B\left(E{3;3}^{-}\to 0^{+}\right)=42(3)$ W.u.. This provides evidence for the increased strength of octupole correlations in the area surrounding A=224 mass nuclei, even though the negative-parity 3^- state of ²²⁴Rn has a much higher excitation energy. These results are obtained for ²²⁴Ra, however, the transition strengths are reproduced perfectly [41]. In future experiments, it is important to further verify the calculated intraband E2 transitions in the same parity band, as well as the interband E1 and E3 transitions connected to the ground-state band of ²²⁴Rn. This investigation indicates that the calculated energies for the excited states are slightly higher than expected. The discrepancy is likely due to the omission of triaxial and time reversal symmetry-breaking components in the model calculations, as discussed in previous studies on other nuclei with different GCM approaches [55, 56], however, when considering these symmetry breaks require consideration of the GCM calculations with cranking or particle-hole excitation configurations and the inclusion of three-dimensional angular-momentum projection (3DAMP), which is beyond the framework of our proposed model.

Fig. 4Full-screen View

(Color online) Low-lying energy spectra for ²²⁴Rn. The available data are collected from Ref. [9] and the results calculated from configuration-mixing GCM and single energy-minimum configuration are shown in (a), (b), and (c) columns, respectively. The numbers on the arrows are intraband E2 (blue color for positive-parity and red color for negative-parity bands) and interband E1 (green color) or E3 (violet color) transition strengths connecting to the ground state band. All transition strengths are in Weisskopf units

Figure 5 shows the collective nuclear wave functions ${\left|g_{\alpha }^{J\pi }\right|}^2$ for low-lying parity doublet states of the angular momentum and parity in ²²⁴Rn, where the orthonormal collective wave function $g_{\alpha }^{J\pi }$ is constructed as $g_{\alpha }^{J\pi }\left({\text{q}}_a\right)={\displaystyle \sum _{{\text{q}}_b}{\left[{\mathcal{N}}^{J\pi }\right]}_{{\text{q}}_a\mathrm{,}{\text{q}}_b}^{1/2}}{f}_{\alpha }^{J\pi }\left({\text{q}}_b\right)\mathrm{.}$ (8) The distribution of the collective wave functions in the β₂–β₃ plane is usually adopted to analyze staggering behaviors in low-lying parity doublet states. The wave functions of the states become increasingly concentrated in a quadrupole-octupole deformed configuration with an increase in angular momentum, demonstrating a picture of rotation-induced shape stabilization. Similar to ²²⁴Ra [41], from the perspective of collective nuclear wave functions, the radioactive nucleus ²²⁴Rn exhibits a transition from a gentle octupole deformation to a stable pear shape. We examined the nuclear wave functions of ²²⁴Rn and found that the calculations involving configurations with different β₃ and a fixed β₂ yield results similar to those of ²²⁴Ra (cf. Figs. 4(e) and 4(f) in Ref. [41]). As the spin increases, the dominant configuration for the positive-parity states gradually shifted from weak octupole configurations to those with large octupole shapes. Conversely, for the negative-parity states, the collective wave functions are zero at β₃=0 and become concentrated around large octupole-deformed configurations. This corresponds with the evolution trend of the collective wave functions with the spins from the full GCM calculations, as shown in Fig. 5.

Fig. 5Full-screen View

(Color online) Collective wave functions of the parity-doublet states (a) with J^π=0⁺, 2⁺, …, 8⁺ and (b) with J^π=1^-, 3^-, …, 9^- in the β₂-β₃ deformation plane. See text for details

Figure 6 shows the energy ratio R_J/2 of the excitation energy of each state with an angular momentum J relative to that of positive-parity 2⁺ states for ²²⁴Rn. For comparison, the experimental data for ²²⁴Ra are also provided. The ratio is defined as $R_{J/2}=E_x\left(J^{\pi }\right)/E_x\left(2^{+}\right)$, where π=+ and - indicate positive and negative parity, respectively. The phenomenon of interleaving the positive and negative parity bands is a common method used to study the nuclei with octupole correlation. In idealized interleaving parity doublet bands, the ratio R_J/2 shows a quadratic-like function as angular momentum J increases. First, the energy levels of positive- and negative-parity bands remain decoupled, resulting in odd-even staggering and the staggering amplitude globally decreases with increasing J in ²²⁴Rn. This feature is similar to that obtained from the GCM calculations for ²²⁴Ra [41] and Ba isotopes [42]. The tendency of the staggering amplitude of R_J/2 as a function of the angular momentum is reproduced qualitatively for ²²⁴Rn. However, the staggering amplitude deviates from the experimental data. The single energy-minimum configuration overestimates the staggering amplitude of ratio R_J/2 resulting from the lower E(2⁺) value of positive-parity 2⁺ state obtained (cf. Fig. 4(c)). However, after considering the configuration-mixing effects, as shown in Fig. 4(b), an overestimation of the excitation energy $2_1^{+}$ state results in an underestimation of the R_J/2-staggering amplitude in the GCM calculations. The inset panel of the Fig. 6 shows the normalized staggering S_J/2 between the positive- and negative-parity bands. It is defined as [57] $S_{J/2}=\left|E_x\left(J^{\pm }\right)-\frac{\left(J+1\right)E_x{\left(J-1\right)}^{\mp }-J{E}_x{\left(J+1\right)}^{\mp }}{2J+1}\right|/E_x\left(2^{+}\right)\mathrm{.}$ (9) Superscripts denote the parity of the two bands. This quantity reflects the octupole deformation stability changing with angular momentum $J(\hslash )$. It is clear that the normalized staggering S_J/2 decreases as the angular momen $J(\hslash )$ increases. In brief, variations in the staggering R_J/2 and S_J/2 with increasing angular momentum $J(\hslash )$ show a behavior characteristic from octupole vibration at a lower $J(\hslash )$ to octupole rotation at a higher $J(\hslash )$ in ²²⁴Rn.

Fig. 6Full-screen View

(Color online) Ratio R_J/2 of the excitation energy of each $J^{\pm }$ state to that of positive-parity 2⁺ state as a function of the angular momentum $J(\hslash )$ for ²²⁴Rn. The inset panel is the normalized staggering amplitude S_J/2 as a function of the angular momentum $J(\hslash )$. The results are calculated from configuration-mixing GCM and single energy-minimum configuration. Experimental data of ²²⁴Ra [7] are also given for comparison

Figure 7 shows the correlation between the excitation energies of ²²⁴Rn and ²²⁴Ra. Both the calculated and the experimental data deviated slightly from the diagonal line. As the spin increases, the excitation energies of the positive- and negative-parity states in ²²⁴Rn and ²²⁴Ra increased at a similar rate. In Fig. 7(a), it is clear that our calculations overestimate the excitation energies of positive parity states. However, a linear relationship of the excitation energies between ²²⁴Rn and ²²⁴Ra is consistent with that of experimental data. This phenomenon has also been observed in negative-parity bands, as shown in Fig. 7(b). Furthermore, we plot Fig. 8 to demonstrate the relationship of the electric transition strengths between ²²⁴Ra and ²²⁴Rn. A linear increasing relationship is also found in the intraband transitions $B\left(E2;L^{\pm }\to {\left(L-2\right)}^{\pm }\right)$ or the interband transition $B\left(E3;L^{-}\to {\left(L-3\right)}^{+}\right)$ and $B\left(E1;L^{-}\to {\left(L-1\right)}^{+}\right)$ in the parity doublet bands. Moreover, for the interband B(E1) and B(E3) transitions between the positive- and negative-parity doublet bands, the linearity gradually deviates from the diagonal line as the spin increases, as shown in Fig. 8(b). However, the intraband transitions B(E2) of positive- and negative-parity bands tend to exhibit a diagonal distribution with increasing spin in Fig. 8(a). It appears that the fundamental structure associated with the quadrupole-octupole correlations in ²²⁴Rn exhibits similar behavior to that of ²²⁴Ra.

Fig. 7Full-screen View

(Color online) (a) Theoretical and experimental excitation energies of positive-parity states in ²²⁴Ra against the values in ²²⁴Rn. (b) Same as (a) but for negative-parity states. The results of calculations for ²²⁴Ra are taken from Ref. [41] and the available data are taken from Refs. [7, 9]

Fig. 8Full-screen View

(Color online) (a) Same as Fig. 7 but for electric multipole transition strengths. (a) Intraband transitions $B\left(E2;L^{\pm }\to {\left(L-2\right)}^{\pm }\right)$ with even L=2, 4, 6, 8 and odd L=3, 5, 7 stand for positive- and negative-parity bands, respectively. (b) Interband transition $B\left(E3;L^{-}\to {\left(L-3\right)}^{+}\right)$ or $B\left(E1;L^{-}\to {\left(L-1\right)}^{+}\right)$ with L=3, 5, 7 connect to parity-doublet bands

The transition octupole moment $Q_3\left(3^{-}\to 0^{+}\right)$ can be derived from the transition matrix elements: $\langle 3^{-}\Vert \widehat{M}\left(E3\right)\Vert 0^{+}\rangle$, which corresponds to the $3^{-}\to 0^{+}$ transitions. We plot Fig. 9 to show the relationship between the transition octupole moment $Q_3\left(3^{\to }{0}^{+}\right)$ and the energy E(3^-) of the negative-parity state in ²²⁴Rn, which is denoted by an open square. The solid squares show the measured Q₃ values with error bars for the nuclei with mass A > 200 in the $Z\simeq 88$ and $N\simeq 134$ regions, as reported in [7, 6]. For comparison, we have also included the calculated Q₃ value for ²²⁴Ra, as given in Ref. [41]. ²⁰⁸Pb has a typical dynamic pear-shaped octupole vibrator is characterized by the highest E(3^-) energy with a minor Q₃ value. For the nuclei such as ²²⁰Rn, ^230,232Th, and ²³⁴U with similar Q₃ values to that of ²⁰⁸Pb are considered octupole vibrators. Larger Q₃ and smaller E(3^-) values for ²²²Ra, ²²⁴Ra, and ²²⁶Ra indicate an enhancement in octupole collectivity that is consistent with the onset of octupole deformation in this mass region. Although a more stretched negative-parity band is obtained, our results indicate that the E(3^-) energy of the experimental data is reproduced very well for ²²⁴Rn [cf. Fig. 4]. The predicted transition octupole moment Q₃ of ²²⁴Rn shows as larger as that of ²²⁴Ra, indicating that ²²⁴Rn is likely to exhibit strong octupole correlations. Therefore, ²²⁴Rn has a high probability of being a rotor in our theoretical calculations. Further measurements of Q_λ(λ=1 or 3) are required to confirm the possibility of enhanced octupole collectivity for ²²⁴Rn.

Fig. 9Full-screen View

(Color online) Relationship between the transition octupole moment $Q_3\left(3^{-}\to 0^{+}\right)$ and the energy E(3^-) of the negative-parity state in ²²⁴Rn, denoted by an open square. The solid squares indicate the measured Q₃ values with error bars for nuclei with mass A > 200 reported in [7, 6]. The calculated Q₃ value for ²²⁴Ra [41] (open square) have been included for comparison. The horizontal dashed line represents the Q₃ value of octupole vibrator ²⁰⁸Pb

Summary

In this study, we present a beyond-mean-field study of the low-lying parity doublet bands in ²²⁴Rn with a multireference covariant density functional theory, in which the dynamic correlations related to symmetry restoration and quadrupole-octupole shape fluctuations were treated using the generator coordinate method, combined with the parity, particle number, and angular momentum projections. The low-lying energy spectrum is reasonably reproduced when the shape fluctuations in both the quadrupole and octupole shapes are considered. Collective nuclear wave functions and the low-lying spectrum-related energy ratio R_J/2 and the normalized staggering S_J/2 suggest a transition from gentle octupole deformation to a stable pear-shaped structure. The results of ²²⁴Rn were compared to those of ²²⁴Ra. We have found these two nuclei have similar electric octupole (E3) transition strength. Specifically, $B\left(E{3;3}^{-}\to 0^{+}\right)=43$ W.u. for ²²⁴Rn, comparable to the experimental value of ²²⁴Ra (42(3) W.u.). This result indicates that ²²⁴Rn may possess a similar strong octupole correlation to that in ²²⁴Ra, even though the excitation energy of 3^- in ²²⁴Rn is approximately twice that of ²²⁴Ra. However, a more solid conclusion can only be drawn based on related octupole criteria such as the electric dipole E1 and octupole E3 transition probabilities, which will be measured in the future. This study suggests that ²²⁵Rn atom, similar to ²²⁵Ra atoms [58, 59], can serve as another candidate for measuring permanent atomic electric dipole moment.