Introduction

As a microscopic quantum many-body system, the atomic nucleus carries a wealth of symmetries and symmetry breakings. Spontaneous symmetry breaking plays a crucial role in understanding the structure of atomic nuclei. Reflection symmetry breaking of atomic nuclei occurs in nuclei with octupole deformations (such as pear-shaped nuclei) [1-3]. This is related to the charge parity (CP) symmetry violation beyond the standard model [4] and has been at the frontier of both nuclear physics and particle physics.

The study of reflection symmetry breaking in pear-shaped nuclei can be traced back to the 1950s and is characterized by the occurrence of interleaved positive- and negative-parity bands in even-even nuclei, parity doublet bands in odd-mass nuclei, and enhanced electric dipole (E1) and octupole (E3) moments [1-3]. The pear shapes of the nucleus can arise from the strong octupole correlations of the nucleons near the Fermi surface. They occupy states of opposite parity with the orbital and total angular momenta differing by $3\hslash$, i.e., $\Delta l=\Delta j=3\hslash$. Empirically, this condition occurs for a proton or neutron particle numbers: 34 ($g_{9/2}\leftrightarrow p_{3/2}$), 56 ($h_{11/2}\leftrightarrow d_{5/2}$), 88 ($i_{13/2}\leftrightarrow f_{7/2}$), and 134 ($j_{15/2}\leftrightarrow g_{9/2}$)[1]. To date, the octupole correlations and pear-shaped nuclei have been extensively studied in the A~80 mass region with $Z\approx 34$ [5, 6], in the A~150 mass region with $Z\approx 56$ and $N\approx 88$ [7-9], and in the A~220 mass region, with $Z\approx 88$ and $N\approx 134$, see the reviews [1-3].

Focusing on barium isotopes in A~150 mass region, the nucleus ¹⁴⁴Ba is of continuous interest due to its octupole double-magic character with proton and neutron numbers Z=56 and N=88, respectively. Experimentally, various signatures including the low-lying negative-parity states, the interleaved positive- and negative-parity bands with enhanced E1 connecting transitions [10-13], and especially the enhanced E3 transition strengths [14], represent an unambiguous static nuclear octupole deformation in ¹⁴⁴Ba. Theoretically, the octupole deformation and collectivity in ¹⁴⁴Ba have been studied by using various approaches, such as cranked Woods-Saxon-Bogoliubov theory [15], self-consistent cranked Hartree-Fock-Bogoliubov (HFB) calculation with parity projection [16], a one-dimensional collective model with phenomenological spin-dependent potentials [17-19], the quadrupole–octupole collective model [20, 21], cluster model [22, 23], potential energy surface calculations [24], the reflection-asymmetric relativistic mean-field (RAS-RMF) approach [25-28], the interacting boson model (IBM) [29, 30], the generator coordinate method (GCM) based on nonrelativistic [31], and relativistic density functional theories [32, 33].

The nuclear density functional theory (DFT) starts with a universal energy density functional and can achieve a self- consistent description for almost all nuclei [34, 35]. Over the past decades, relativistic or covariant version of DFT (CDFT) have been developed and widely applied to investigate a large variety of nuclear phenomena [36-46]. To describe the nuclear spectroscopic properties, cranking CDFT has been developed and widely applied to investigate both the ground states of nuclei and various rotational excitation phenomena [47-50]. Specifically, cranking CDFT in a three-dimensional (3D) lattice space was realized in Ref. [51] and has been widely applied to investigate the exotic shapes of nuclei and their excitation modes such as the nuclear linear chain [51, 52], toroidal structures [53], and nuclear chiral rotation [54]. It provides a useful way to understand the current focus of the observed rotational bands in octupole double-magic nucleus ¹⁴⁴Ba, since in 3D lattice calculations, the single-particle wave functions have no symmetry limitation in space and all deformation degrees of freedom of the nucleus are self-consistent included.

In this study, the cranking CDFT in 3D lattice space is used to investigate the octupole deformation and collectivity in the nucleus ¹⁴⁴Ba. The model is described briefly in Sect. 2. The numerical details and calculated results for the available data, including the I–ω relation and electromagnetic transition probabilities, are discussed in Sect. 3. A summary is provided in Sect. 4.

Theoretical Framework

The detailed formalism of the cranking CDFT in 3D lattice space has been outlined in Ref. [51]. The starting point of the CDFT is a standard effective nuclear Lagrangian density, where the nucleons are coupled with a meson exchange interaction [55-57] or zero-range point-coupling interaction [58-60] as follows, $\begin{array}{c}\mathcal{L}={\mathcal{L}}^{\text{free}}+{\mathcal{L}}^{\text{4f}}+{\mathcal{L}}^{\text{hot}}+{\mathcal{L}}^{\text{der}}+{\mathcal{L}}^{\text{em}}\\ =\overline{\psi }\left(i{\gamma }_{\mu }{\partial }^{\mu }-m\right)\psi -\frac{1}{2}{\alpha }_{\text{S}}\left(\overline{\psi }\psi \right)\left(\overline{\psi }\psi \right)-\frac{1}{2}{\alpha }_{\text{V}}\left(\overline{\psi }{\gamma }_{\mu }\psi \right)\left(\overline{\psi }{\gamma }^{\mu }\psi \right)\\ -\frac{1}{2}{\alpha }_{\text{TS}}\left(\overline{\psi }\overrightarrow{\tau }\psi \right)\left(\overline{\psi }\overrightarrow{\tau }\psi \right)-\frac{1}{2}{\alpha }_{\text{TV}}\left(\overline{\psi }\overrightarrow{\tau }{\gamma }_{\mu }\psi \right)\left(\overline{\psi }\overrightarrow{\tau }{\gamma }^{\mu }\psi \right)\\ -\frac{1}{3}{\beta }_{\text{S}}{\left(\overline{\psi }\psi \right)}^3-\frac{1}{4}{\gamma }_S{\left(\overline{\psi }\psi \right)}^4-\frac{1}{4}{\gamma }_{\text{V}}{\left[\left(\overline{\psi }{\gamma }_{\mu }\psi \right)\left(\overline{\psi }{\gamma }^{\mu }\psi \right)\right]}^2\\ -\frac{1}{2}{\delta }_{\text{S}}{\partial }_{\nu }\left(\overline{\psi }\psi \right){\partial }^{\nu }\left(\overline{\psi }\psi \right)-\frac{1}{2}{\delta }_{\text{V}}{\partial }_{\nu }\left(\overline{\psi }{\gamma }_{\mu }\psi \right){\partial }^{\nu }\left(\overline{\psi }{\gamma }^{\mu }\psi \right)\\ -\frac{1}{2}{\delta }_{\text{TS}}{\partial }_{\nu }\left(\overline{\psi }\overrightarrow{\tau }\psi \right){\partial }^{\nu }\left(\overline{\psi }\overrightarrow{\tau }\psi \right)-\frac{1}{2}{\delta }_{\text{TV}}{\partial }_{\nu }\left(\overline{\psi }\overrightarrow{\tau }{\gamma }_{\mu }\psi \right){\partial }^{\nu }\left(\overline{\psi }\overrightarrow{\tau }{\gamma }_{\mu }\psi \right)\\ -\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }-e\frac{1-{\tau }_3}{2}\overline{\psi }{\gamma }^{\mu }\psi A_{\mu }\mathrm{.}\end{array}$ (1) In Eq. (1), m denotes the nucleon mass, e denotes the charge unit of the protons, Aμ and $F_{\mu \nu }$ denote the four-vector potential and field strength tensor of the electromagnetic field, respectively. For the 11 coupling constants, αS, αV, αTS, αTV, βS, γS, γV, δS, δV, δTS, and δTV, α refers to the four-fermion term, β and γ refer to the third- and fourth-order terms, respectively, and δ refers to the derivative couplings. Subscripts S, V, and T indicate the symmetries of the couplings, i.e., S denotes a scalar, V denotes a vector, and T denotes an isovector.

To describe nuclear rotations in the cranking approximation, the effective Lagrangian density of Eq. (1) is transformed into a rotating frame with a constant rotational frequency ω around the rotational axis. The equation of the single-particle motion can be derived from the Lagrangian in the rotating frame: ${\widehat{h}}^{\prime }{\psi }_k=\left({\widehat{h}}_0-\omega \cdot \widehat{\text{J}}\right){\psi }_k={{\epsilon }^{\prime }}_k{\psi }_k\mathrm{,}$ (2) with $\widehat{h}\text{'}$ denoting the cranking single-particle Hamiltonian, $-\text{ω}\cdot \widehat{\text{J}}$ denoting the Coriolis or cranking term, ${{\epsilon }^{\prime }}_k$ denoting the single-particle Routhians, and $\widehat{\text{J}}=\widehat{\text{l}}+\frac{1}{2}\widehat{\sum }$ denoting the total angular momentum of the nucleon spinors. The single-particle Hamiltonian ${\widehat{h}}_0$ is ${\widehat{h}}_0=\text{α}\cdot \left[-\text{i}\nabla -\text{V}\left(\text{r}\right)\right]+\beta \left[m_N+S\left(\text{r}\right)\right]+V_0\left(\text{r}\right)\mathrm{.}$ (3) The relativistic scalar s(r) and vector field V_μ(r) are connected in a self-consistent manner to the nucleon density and current distribution. By solving the cranking Dirac equation (2) self-consistently at a given rotational frequency, the single-particle Routhians, expectation values of the angular momentum, and quadrupole and octupole moments can be obtained, see Refs. [34, 61-63] for the detailed formalism. Specifically, single-particle wave functions have no symmetry limitation in space and all nuclear deformation degrees of freedom including octupole deformation are self-consistently obtained in the present 3D lattice cranking CDFT calculation. It provides a powerful way to investigate the evolution of octupole shapes with spin in the current octupole double-magic nucleus ¹⁴⁴Ba. The quadrupole moments (Q₂₀, Q_2±1, Q_2±2) and octupole moments (Q₃₀, Q_3±1, Q_3±2, Q_3±3) can be calculated as: $Q_{20}=\sqrt{\frac{5}{16\pi }}\langle 3z^2-r^2\rangle \mathrm{,}$ (4) $Q_{21}=-Q_{2-1}^{*}=-\sqrt{\frac{15}{32\pi }}\langle \left(x+iy\right)z\rangle \mathrm{,}$ (5) $Q_{22}=Q_{2-2}=\sqrt{\frac{15}{32\pi }}\langle x^2-y^2\rangle \mathrm{,}$ (6) $Q_{30}=\sqrt{\frac{7}{16\pi }}\langle \left(2z^2-3x^2-3y^2\right)z\rangle \mathrm{,}$ (7) $Q_{31}=-Q_{3-1}^{*}=-\sqrt{\frac{21}{64\pi }}\left[\langle \left(4z^2-x^2-y^2\right)x\rangle +\text{i}\langle \left(4z^2-x^2-y^2\right)y\rangle \right]\mathrm{,}$ (8) $Q_{32}=Q_{3-2}^{*}=\sqrt{\frac{105}{32\pi }}\left[\langle \left(x^2-y^2\right)z\rangle +\text{i}\langle xyz\rangle \right]\mathrm{,}$ (9) $Q_{33}=-Q_{3-3}^{*}=-\sqrt{\frac{35}{64\pi }}\left[\langle \left(x^2-3y^2\right)x\rangle +\text{i}\langle \left(3x^2-y^2\right)y\rangle \right]\mathrm{.}$ (10) Since the nuclei are placed in a 3D lattice space, we additionally constrain the center of mass of the entire nucleus at the origin and align the principal axes with the coordinate axes to remove redundant degrees of freedom. The values of Q₂₁ and Q_2-1 always disappear. The deformation parameters ${\beta }_{\lambda \mu }$ can be obtained from the corresponding multipole moments [64] ${\beta }_{\lambda \mu }=\frac{4\pi }{3N{R}^{\lambda }}{Q}_{\lambda \mu }$ (11) with R=1.2A^1/3 fm and N denoting the number of protons neutrons, or nucleons.

Based on the quadrupole moments, B(E2) transition probabilities can be derived in semiclassical approximation [50, 65] $\begin{array}{c}B\left(E2\right)=\frac{3}{8}{\left[Q_{20}^{\text{p}}{{\displaystyle \sin }}^2\theta +\sqrt{\frac{2}{3}}{Q}_{22}^{\text{p}}\cos 2\phi \right]}^2\\ +{\left(Q_{22}^{\text{p}}\cos \theta \sin 2\phi \right)}^2\mathrm{,}\end{array}$ (12) where $Q_{20}^{\text{p}}$ and $Q_{22}^{\text{p}}$ denote the quadrupole moments of the protons, and θ and φ denote the polar and azimuthal angles of the total angular momentum in the intrinsic frame, respectively. Similarly, B(E3) transition probabilities can be derived in semiclassical approximation: $\begin{array}{c}B\left(E3\right)=[\frac{\sqrt{5}}{4}{{\displaystyle \sin }}^3\theta Q_{30}^{\text{p}}\\ +\frac{\sqrt{15}}{4}{{\displaystyle \sin }}^2\theta \cos \theta \left(\cos \phi \text{Re}\left[Q_{31}^{\text{p}}\right]+\sin \phi \text{Im}\left[Q_{31}^{\text{p}}\right]\right)\\ +\frac{\sqrt{6}}{4}\sin \theta \left(1+{{\displaystyle \cos }}^2\theta \right)\left(\cos 2\phi \text{Re}\left[Q_{32}^{\text{p}}\right]+\sin 2\phi \text{Im}\left[Q_{32}^{\text{p}}\right]\right)\\ {+\frac{1}{4}\left(3+{{\displaystyle \cos }}^2\theta \right)\cos \theta \left(\cos 3\phi \text{Re}\left[Q_{33}^{\text{p}}\right]+\sin 3\phi \text{Im}\left[Q_{33}^{\text{p}}\right]\right)]}^2\\ +[\frac{\sqrt{15}}{4}{{\displaystyle \sin }}^2\theta \left(\cos \phi \text{Im}\left[Q_{31}^{\text{p}}\right]-\sin \phi \text{Re}\left[Q_{31}^{\text{p}}\right]\right)\\ +\frac{\sqrt{6}}{2}\sin \theta \cos \theta \left(\cos 2\phi \text{Im}\left[Q_{32}^{\text{p}}\right]-\sin 2\phi \text{Re}\left[Q_{32}^{\text{p}}\right]\right)\\ {+\frac{1}{4}\left(1+3{{\displaystyle \cos }}^2\theta \right)\left(\cos 3\phi \text{Im}\left[Q_{33}^{\text{p}}\right]-\sin 3\phi \text{Re}\left[Q_{33}^{\text{p}}\right]\right)]}^2\mathrm{,}\end{array}$ (13) where $Q_{30}^{\text{p}}$, $Q_{3\pm 1}^{\text{p}}$, $Q_{3\pm 2}^{\text{p}}$, and $Q_{3\pm 3}^{\text{p}}$ are the octupole moments of protons.

Results and Discussion

A successful density functional PC-PK1 [60] is employed in the CDFT calculation. The Dirac spinors of the nucleons and potentials in the single-particle Hamiltonian (2) are represented in 3D lattice space. The step size and grid number along x, y, and z axes are chosen as 1 fm and 30, respectively. The convergence of the iteration is realized by ensuring that the energy uncertainty for every occupied single-particle state is smaller than 10^-9 MeV² and the maximum absolute difference between the mean potentials at two adjacent iterations is smaller than 10^-3 MeV. To provide the potential energy surface in the (β₂₀, β₃₀) plane, a deformation-constrained CDFT calculation in 3D lattice space is performed in the region ${\beta }_{20}\in \left[-\mathrm{0.3,}0.4\right]$ and ${\beta }_{30}\in \left[\mathrm{0.0,}0.3\right]$ with a step size of 0.05. For cranking CDFT calculations, the rotational frequency is varied from ω=0.0 to 0.45 MeV.

In Fig. 1, the potential energy surface (PES) in the (β₂₀, β₃₀) plane for ¹⁴⁴Ba calculated by the constrained CDFT calculation in a 3D lattice with the successful density functional PC-PK1 is shown [60]. It shows that the global minimum of the PES locates at β₂₀=0.22 and β₃₀=0.13. It should be noted that the single-particle wave functions calculated in the present CDFT calculation have no symmetry limitation, and all deformation degrees of freedom of the nucleus are self-consistent included. Apart from the constrained axially symmetric quardupole and octupole deformations β₂₀ and β₃₀, the axially symmetric hexadecapole deformation β₄₀=0.13 is also obtained self-consistently for the ground state of ¹⁴⁴Ba. All axial asymmetric deformations, i.e., ${\beta }_{2\mu }\mathrm{,}{\beta }_{3\mu }\mathrm{,}{\beta }_{4\mu }\mathrm{,}\left(\mu \ne 0\right)$, are found to be zero, which implies that the ground state of ¹⁴⁴Ba corresponds to an axially symmetric and reflection-asymmetric shape. Around the minimum, the PES exhibits a relatively soft character. The energy difference between the ground state (β₂₀=0.22, β₃₀=0.13) and the lowest reflection symmetric energy position (β₂₀=0.2, β₃₀=0.0) is less than 1.0 MeV. Moreover, a secondary minimum (∼3.0 MeV) is observed at (β₂₀=-0.15, β₃₀=0.05). Experimentally, a value of ${\beta }_{30}=0.17\left({}_{-6}^{+4}\right)$ has been derived [14] by using the relationship of the octupole moment (with the standard assumption of axial symmetry) and the commonly used β₃₀ parameters [66]. The obtained ground-state octupole deformation β₃₀=0.13 is consistent with the β₃₀ value within the experimental error range. As analyzed in Ref. [25], this octupole deformation minimum is a consequence of strong octupole–octupole interactions between pairs of single-particle orbitals near the Fermi surface with $\Delta l=\Delta j=3\hslash$ around the Fermi level, i.e., proton ($h_{11/2}\leftrightarrow d_{5/2}$) and neutron ($i_{13/2}\leftrightarrow f_{7/2}$).

Fig. 1Full-screen View

Potential energy surface for ¹⁴⁴Ba calculated by the CDFT in 3D lattice [51] with density functional PC-PK1 [60]. The contour separation is 0.25 MeV and the pentagram corresponds to the location of the ground state

To investigate the observed rotational spectroscopic properties and the evolution of the octupole deformation and collectivity with respect to nuclear rotation, cranking CDFT calculations in 3D lattice are performed. Figure 2 shows the calculated total angular momentum as a function of the rotational frequency for the ground-state band in ¹⁴⁴Ba in comparison with the available data [13]. It can be seen that the experimental data are slightly overestimated by the present calculated results, which give an excessive moment of inertia. A better agreement with the experimental data is expected by taking into account the pairing correlation [67] for which the moments of inertia will be depressed.

Fig. 2Full-screen View

Total angular momentum as a function of the rotational frequency for the ground state band of ¹⁴⁴Ba calculated by the cranking CDFT in 3D lattice (line) in comparison with the experimental interleaved positive-parity (solid symbols) and negative-parity bands (open symbols)

With the quadrupole and octupole moments obtained self-consistently, the reduced transition probabilities B(E2) and B(E3) can be calculated in semiclassical approximation according to Eqs. (12) and (13). In Fig. 3 (a), the calculated B(E2) values are compared with available data [14]. It is found that the resulting B(E2) values are in good agreement with the data on the order of magnitude. As shown in Fig. 3 (a), the calculated B(E2) values remain nearly constant with increasing rotational frequency, which can be further understood by the changes in the quadrupole deformation shown in Fig. 4 (a). With the increasing rotational frequency, the nucleus undergoes a nearly unchanged β₂₀ deformation from 0.22 to 0.20.

Fig. 3Full-screen View

Calculated (a) B(E2) and (b) B(E3) values as functions of the total angular momentum in the cranking CDFT calculations compared with the data for ¹⁴⁴Ba [14]

Fig. 4Full-screen View

Evolution of (a) quadrupole deformation β₂₀ and (b) octupole deformation β₃₀ for the calculated yrast states in ¹⁴⁴Ba as functions of the rotational frequency

The octupole collectivity is also demonstrated by B(E3) values shown in Fig. 3(b) and the evolution of octupole deformation β₃₀ in Fig. 4(b). As shown in Fig. 3(b), the corresponding B(E3) data are effectively reproduced by the cranking CDFT calculation. As shown in Fig. 4(b), the calculated octupole deformation β₃₀ increases slightly and then decreases with increasing spin. With an increase in the rotational frequency to $\hslash \omega =0.45$ MeV (spin $I\approx 24\hslash$), the average octupole deformation is approximately β₃₀=0.128, indicating the robustness of the octupole shape with respect to nuclear rotation.

To understand the evolution of the octupole deformation of the yrast states in ¹⁴⁴Ba, the single-proton and single-neutron Routhians are shown as functions of rotational frequency in Figs. 5 (a) and 5 (b), respectively. The levels near the Fermi surface are labeled by Nilsson-like notation Ω[Nnzml] of the largest component. It is noted that these levels can mix different components with opposite parity due to the existence of octupole deformations. As shown in Fig. 5 (a) for single-proton Routhians, an energy gap Z=56 fenced with several levels near the Fermi surface, is found at the ground state and persistently presented with an increase in the rotational frequency. Furthermore, an energy gap N=88 for the single-neutron Routhians shown in Fig. 5(b) is found not only in the ground state, but also in the high-spin region. Therefore, it is concluded that these two energy gaps, Z=56 and N=88, near the Fermi surfaces are responsible for the octupole minimum and robustness of the octupole shape against nuclear rotations in ¹⁴⁴Ba.

Fig. 5Full-screen View

Single-particle Routhians for (a) proton and (b) neutron of ¹⁴⁴Ba as functions of the rotational frequency. The levels near the Fermi surface are labeled by Nilsson-like notations Ω[Nnzml] of the largest component

Summary

In summary, the cranking covariant density functional theory in a three-dimensional lattice is first applied to investigate the octupole deformation and collectivity in octupole double-magic nucleus ¹⁴⁴Ba. With the electric octupole moments obtained by self-consistently solving the cranking Dirac equation, the reduced transition probabilities B(E3) are derived in semiclassical approximation for the first time. The available data, including the I-ω relation, the B(E2) and B(E3) values, are well reproduced by the cranking CDFT calculations. The potential energy surface (PES) in (β₂₀, β₃₀) plane, calculated by the constrained CDFT calculation in 3D lattice, provide a static axial octupole and quadruple deformed ground state for ¹⁴⁴Ba. With the increase of the rotational frequency (up to $\hslash \omega =0.45$ MeV), the calculated octupole deformation is nearly unchanged and give an average value β₃₀=0.128, indicating the robustness of the octupole shape against nuclear rotation. By analyzing the single-proton and single-neutron Routhians with the increasing rotational frequency, two energy gaps, Z=56 and N=88, near the Fermi surfaces are found to be responsible for the octupole minimum and the robustness of the octupole shape against nuclear rotation in ¹⁴⁴Ba.

It should be noted that the pairing correlations were neglected in the present calculations. It would be interesting to introduce, for example, the shell-model-like approach [65, 67] to the present cranking CDFT in 3D lattice to investigate the effects of pairing correlations. The current cranking CDFT describes the physics only on average in a rotating mean field. The total angular moment is treated in semiclassical approximation and the total parity is not a good quantum number. For the interleaved positive- and negative-parity bands in octupole deformed nuclei, the total parity and parity splitting cannot be calculated using the present 3D lattice cranking CDFT calculation. Additionally, it will be interesting to introduce the parity projection beyond 3D lattice cranking CDFT in the future.