Bubble Structure in Isotopes of Lu to Hg

Abstract

Bubble nuclei, characterized by a depletion in nucleon density at the nuclear center, are investigated within the atomic number range $71\le Z\le 80$ using the Deformed Relativistic Hartree–Bogoliubov theory in continuum. This study extends previous investigations, which were limited to even–even isotopes, by incorporating even–odd, odd–even, and odd–odd nuclei within this range. The extension is achieved by introducing the blocking effect into the point-coupling approach to ensure self-consistency. Following previous studies, we define a nucleus as a bubble candidate if the bubble parameter exceeds ${\mathcal{B}}_p^{★}=20\%$, and identify five bubble nuclei in both even-Z and odd-Z nuclei groups, based on the highest ${\mathcal{B}}_p^{★}$ values. The formation of bubble structures is confirmed through an analysis of proton single-particle energy levels of the most centrally depleted nuclei across four categories: even–odd, even–even, odd–even, and odd–odd.

1. Introduction

The exploration of atomic nuclei, particularly those near the neutron drip line, has progressed significantly in recent years, driven by advancements in experimental facilities worldwide [1,2]. The discovery of over 3000 isotopes has broadened our understanding of the nuclear landscape, revealing previously uncharted regions of a nuclear structure. These findings play a crucial role in deepening our knowledge of the nuclear structure and the evolution of heavy nuclei. However, despite these advancements, the evolution of nuclear density distributions remains incompletely understood, particularly regarding the shape decoupling effect, which leads to distinct deformations between the core and the halo, and the existence of bound nuclei beyond the neutron drip line.

A key issue in nuclear structure research is understanding exotic nuclei, which exhibit unique phenomena such as bubble structures and shape coexistence in their ground states [3]. Bubble nuclei, in particular, arise when the central region of the nucleus exhibits a substantial depletion in nucleon density. These nuclei, characterized by reduced proton and neutron densities, have been extensively studied theoretically. It has been suggested that bubble formation originates from the reduced occupation of s states near the Fermi surface, which suppresses the central density [3,4,5,6,7,8,9,10,11,12]. Additionally, pairing correlations and nuclear deformation are known to further weaken the bubble structure [3,6,13,14,15,16,17].

In this study, we employ the deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) to investigate the nuclear density distributions and internal structure. This framework is well suited for studying exotic nuclear phenomena, such as neutron-rich nuclei and bubble structures, as it incorporates deformation and treats self-consistently continuum effects and pairing [18,19,20].

This paper is organized as follows. In Section 2, we introduce the DRHBc framework and discuss its application to bubble structures. Section 3 presents our results on bubble nuclei with atomic numbers ranging $71\le Z\le 80$, identifying those with the most significant central density depletion and analyzing their structural properties. Finally, a conclusion of our findings is provided in Section 4.

2. Theoretical Framework

2.1. DRHBc

In this study, we employ the relativistic mean-field (RMF) framework to investigate nuclear structure and density distributions while incorporating pairing correlations. The blocking effect is included to account for the presence of unpaired nucleons in odd-mass and odd–odd nuclei. This modifies the pairing potential and alters single-particle occupation probabilities, which in turn affect central density depletion in bubble structures. The Lagrangian density used in our calculations is given by [21]

$$\begin{array}{cc}\hfill \mathcal{L}& =\overline{\psi }\left(i{\gamma }^{\mu }{\partial }_{\mu }-m\right)\psi -{\displaystyle \frac{1}{2}}{\alpha }_S\left(\overline{\psi }\psi \right)\left(\overline{\psi }\psi \right)-{\displaystyle \frac{1}{2}}{\alpha }_V\left(\overline{\psi }{\gamma }_{\mu }\psi \right)\left(\overline{\psi }{\gamma }^{\mu }\psi \right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\alpha }_{TV}\left(\overline{\psi }\overrightarrow{\tau }{\gamma }_{\mu }\psi \right)\left(\overline{\psi }\overrightarrow{\tau }{\gamma }^{\mu }\psi \right)-{\displaystyle \frac{1}{3}}{\beta }_S{\left(\overline{\psi }\psi \right)}^3-{\displaystyle \frac{1}{4}}{\gamma }_S{\left(\overline{\psi }\psi \right)}^4\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}{\gamma }_V{\left[\left(\overline{\psi }{\gamma }_{\mu }\psi \right)\left(\overline{\psi }{\gamma }^{\mu }\psi \right)\right]}^2-{\displaystyle \frac{1}{2}}{\delta }_S{\partial }_{\nu }\left(\overline{\psi }\psi \right){\partial }^{\nu }\left(\overline{\psi }\psi \right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\delta }_V{\partial }_{\nu }\left(\overline{\psi }{\gamma }_{\mu }\psi \right){\partial }^{\nu }\left(\overline{\psi }{\gamma }^{\mu }\psi \right)-{\displaystyle \frac{1}{2}}{\delta }_{TV}{\partial }_{\nu }\left(\overline{\psi }\overrightarrow{\tau }{\gamma }_{\mu }\psi \right){\partial }^{\nu }\left(\overline{\psi }\overrightarrow{\tau }{\gamma }^{\mu }\psi \right)\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}{F}^{\mu \nu }{F}_{\mu \nu }-e{\displaystyle \frac{(1-{\tau }_3)}{2}}\overline{\psi }{\gamma }^{\mu }\psi A_{\mu },\hfill \end{array}$$

(1)

where m is the nucleon mass, and ${\alpha }_S$, ${\alpha }_V$, and ${\alpha }_{TV}$ are the coupling constants for four-fermion contact interactions. The terms with ${\beta }_S$, ${\gamma }_S$, and ${\gamma }_V$ account for density-dependent effects, while those with ${\delta }_S$, ${\delta }_V$, and ${\delta }_{TV}$ describe the finite-range effects. $A_{\mu }$ and $F^{\mu \nu }$ represent the four-vector potential and the field strength tensor of the electromagnetic field, respectively. The subscripts S, V, and $TV$ denote scalar, vector, and isovector components. Further details on the DRHBc theory can be found in Refs. [19,20], and numerical implementations are described in Ref. [21].

Both automatic and orbital-fixed blocking methods are employed in the DRHBc framework. The automatic method iteratively excludes the lowest-energy quasiparticle state, ensuring self-consistency during convergence. The orbital-fixed method, on the other hand, enforces the occupation of a specific single-particle state. These approaches enable reliable calculations for odd-mass and odd–odd nuclei. Applying the mean-field approximation to the Lagrangian density and performing a Legendre transformation, we derive the mean-field Hamiltonian density. By employing the variational method and the Bogoliubov transformation, we obtain the relativistic Hartree–Bogoliubov (RHB) equation [22], given by

$$\begin{array}{c}\left(\begin{array}{cc}{h}_D-\lambda & \Delta \\ -{\Delta }^{*}& -h_D^{*}+\lambda \end{array}\right)\left(\begin{array}{c}{U}_k\\ V_k\end{array}\right)=E_k\left(\begin{array}{c}{U}_k\\ V_k\end{array}\right),\end{array}$$

(2)

where $h_D$ is the Dirac Hamiltonian, $\lambda$ is the chemical potential, $\Delta$ represents the pairing potential, and $U_k,V_k$ are the quasiparticle wavefunctions.

To account for nuclear deformation, the DRHBc framework expands the density distribution in terms of Legendre polynomials as

$$f\left(\mathbf{r}\right)=\sum _{\lambda }{f}_{\lambda }\left(r\right)P_{\lambda }\left(\cos \theta \right),\lambda =0,2,4,\dots .$$

(3)

where $f_{\lambda }\left(r\right)$ are the expansion coefficients, and $P_{\lambda }\left(\cos \theta \right)$ are the Legendre polynomials.

In odd-mass and odd–odd nuclei, the blocking effect changes the pairing field by excluding specific quasiparticle states. This leads to modifications in single-particle occupation probabilities and the energy spectrum. The automatic blocking method is particularly effective in cases where a well-defined lowest quasiparticle state exists, facilitating rapid convergence. However, when several low-energy quasiparticle states are closely spaced, the orbital-fixed blocking method is used to investigate the impact of blocking different states on nuclear properties. This ensures a robust and reliable description of pairing suppression and structural evolution in exotic nuclei [20].

2.2. Bubble Parameter

Proton bubble nuclei are identified by a significant reduction in central proton density relative to the maximum proton density within the nucleus. The central density depletion is quantified using the proton depletion fraction, which serves as a key parameter for characterizing bubble structures. In previous studies [6,8,9,14], the bubble parameter was defined as the ratio of the maximum proton density, which is uniquely defined and does not depend on the direction. However, for deformed nuclei described within the DRHBc framework, the proton density distribution varies with the polar angle $\theta$, leading to anisotropic density profiles. Consequently, the maximum proton density also depends on $\theta$, making the original definition unsuitable for deformed nuclei.

To address this issue, a modified bubble parameter ${\mathcal{B}}_p^{★}$ was introduced in the previous study [3]. This incorporates an angle-averaged maximum proton density, providing a more consistent measure of central density depletion in deformed nuclei:

$${\mathcal{B}}_p^{★}\equiv \left(1-{\displaystyle \frac{{\rho }_{p,\mathrm{c}}}{{\overline{\rho }}_{p,\max }}}\right)\times 100[\%],$$

(4)

where the angle averaged maximum proton density ${\overline{\rho }}_{p,\max }$ is defined as

$${\overline{\rho }}_{p,\max }={\displaystyle \frac{\int {\rho }_p(r,\theta )\delta (r-r_{\max }\left(\theta \right))dV}{\int \delta (r-r_{\max }\left(\theta \right))dV}}.$$

(5)

Here, $r_{\max }\left(\theta \right)$ denotes the radial coordinate where the proton density reaches its local maximum for a given $\theta$. This formulation ensures that ${\overline{\rho }}_{p,\max }$ represents an effective maximum density, accounting for directional variations in deformed nuclei. For spherical nuclei, where the proton density distribution is isotropic, the bubble parameter ${\mathcal{B}}_p^{★}$ naturally reduces to its conventional definition, i.e., ${\mathcal{B}}_p^{★}={\mathcal{B}}_p$, ensuring consistency with the previous studies.

3. Results

Using Equation (4), we systematically calculate the bubble parameter for nuclei with $71\le Z\le 80$ described in DRHBc and analyze the result.

Table 1. List of the nuclei which show the highest 5 bubble parameters ${\mathcal{B}}_p^{★}$ for even-Z and odd-Z nuclei, respectively.

Even-Z Nuclei			Odd-Z Nuclei
Nuclei	${\mathit{\beta }}_{\mathbf{2}}$	${\mathit{B}}_{\mathbf{p}}^{*}\mathbf{[}\mathbf{\%}\mathbf{]}$	Nuclei	${\mathit{\beta }}_{\mathbf{2}}$	${\mathit{B}}_{\mathbf{p}}^{\mathbf{*}}\mathbf{[}\mathbf{\%}\mathbf{]}$
263Hg	0.000	34.6	255Lu	0.000	30.1
261Hg	0.000	32.1	257Ta	0.002	29.6
256Hf	0.000	28.9	254Re	0.087	29.2
258W	0.000	28.3	256Ta	0.031	28.6
253W	0.086	28.2	259Re	0.003	28.5

represents the list of nuclei, which show the highest 5 bubble parameter ${\mathcal{B}}_p^{★}$ for even-Z and odd-Z nuclei, respectively. We adopt the point coupling density functional (PC-PK1) [23] to extend and compare the previous study. The neutron numbers of the nuclei in Table 1 are close to the neutron magic number 184, with the bubble parameter exceeding 28%. Compared to previous study [3], which identified Hf as the most depleted nucleus, these results reveal that the presence of unpaired nucleons, particularly in odd-Z or odd-N nuclei, significantly contributes to the bubble parameter. In other words, this finding aligns with earlier studies, which indicates that paired nucleons tend to limit the bubble structure. For even-Z nuclei, the nucleus with the most central depletion is Hg, while for odd-Z nuclei, Lu exhibits the most central depletion.

Additionally, we observe the effect of quadrupole deformation (${\beta }_2$) on the bubble parameter. For even-Z nuclei, a comparison between ²⁵⁸W and ²⁵³W indicates that ²⁵³W exhibits a smaller bubble parameter. Note that ²⁵⁸W has a magic neutron number, but ²⁵³W has a non-magic neutron number. Similarly, for odd-Z nuclei, a comparison between ²⁵⁷Ta and ²⁵⁶Ta shows that ²⁵⁶Ta, with an odd neutron number, has a lower value. These effects can be attributed to the quadrupole deformation (${\beta }_2$), which reduces the bubble parameter. Additionally, the presence of an unpaired nucleon can alter pairing correlations and subsequently influence the bubble parameter, consistent with previous studies [3,6,13,14,15,16,17].

The density distributions of ²⁶³Hg, ²⁵⁶Hf, ²⁵⁵Lu, and ²⁵⁴Re are presented in Figure 1, representing even–odd, even–even, odd–even, and odd–odd nuclei, respectively. The scaled density distributions of protons ${\rho }_{\mathrm{p}}A/Z$ and neutrons ${\rho }_{\mathrm{n}}A/N$ are provided to facilitate comparison with the total baryon density. Since nuclear interactions depend on the nucleon composition, direct comparisons between different nuclides are challenging due to variations in the mass numbers, which correspond to different nuclear systems. Nevertheless, a qualitative analysis provides useful insights. First, both ²⁵⁵Lu and ²⁵⁶Hf have a neutron magic number of 184. Compared to the other two nuclides, their neutron density distributions appear relatively smooth. This suggests that deviation from a neutron magic number significantly affects the overall nuclear density distribution. Furthermore, an analysis of proton density distributions indicates that both odd and even proton numbers consistently result in lower central densities. However, inherent differences between these nuclear systems make direct quantitative comparisons challenging.

Figure 2 presents the proton single-particle levels for the ground states of ²⁶³Hg, ²⁵⁶Hf, ²⁵⁵Lu, and ²⁵⁴Re. The states above the Fermi surface appear due to the inclusion of the pairing potential in the DRHBc framework. For the spherical nuclei ²⁶³Hg, ²⁵⁶Hf, and ²⁵⁵Lu, only the $l=0$ orbital contributes to the central density because angular momentum l is a conserved quantum number. In contrast, ²⁵⁴Re exhibits slight quadrupole deformation (${\beta }_2=0.087$), leading to the splitting of single-particle states. Despite this deformation, a low occupation probability of $3s$ orbital is observed in all nuclei, resulting in a depleted central proton density. For even-Z nuclei, the $1s$ and $2s$ states are fully occupied because their energy levels lie far below the Fermi surface, consistent with previous findings [3]. In odd-Z nuclei, however, one proton occupies either the $11/2^{-}$ or $9/2^{-}$ state near the Fermi surface with an occupation probability of 0.5, rather than the $1/2^{+}$ state. These high-angular-momentum states ($l>9/2$) contribute minimally to the central density due to their spatial distribution, effectively acting as blocked protons. This suggests that a lower central density reduces destabilizing interactions between the surface and center, leading to a more stable nuclear structure. In other words, structures with lower central proton densities tend to be more stable than those with denser centers.

In the case of the deformed nucleus ²⁵⁴Re, the angular momentum l is not a good quantum number, but its projection onto the symmetry axis remains well defined. Consequently, the $1/2^{+}$ states in deformed nuclei can still contribute to the central density. The occupancy of the s state in deformed nuclei is estimated using the formalism from [24]:

$$N_{nlj}^{\mathrm{DRHBc}}=\langle \Psi |{\widehat{N}}_{nlj}|\Psi \rangle =\langle \Psi |\sum _m{c}_{njlm}^{†}{c}_{njlm}|\Psi \rangle ,$$

(6)

where m stands for the total angular momentum projection on the symmetry axis. For ²⁵⁴Re, as shown in Figure 2, the occupation of $3s$ states remains low, leading to a depleted central region. Both ²⁵⁵Lu, and ²⁵⁴Re are odd-Z nuclides, exhibiting the odd-proton blocking effect. As previously mentioned, ²⁵⁴Re features more single-particle levels in its ground state than ²⁶³Hg, ²⁵⁶Hf, and ²⁵⁵Lu. This is attributed to quadrupole deformation, which alters the nuclear structure and enables the inclusion of the $1/2^{+}s$ state. These results suggest that in odd-Z nuclides, nuclear deformation influences the bubble structure but does not necessarily suppress it, instead modifying the spatial distribution of nucleons within the nucleus.

4. Conclusions

In this study, we systematically calculated the bubble parameter for nuclei with $71\le Z\le 80$, extending the scope of the previous research [3]. We first identified the five nuclei with the highest bubble parameter in both even-Z and odd-Z nuclei groups. These ten nuclei were found to be either spherical or only slightly deformed, consistent with previous findings that nuclear deformation weakens the bubble structure. Next, we selected ²⁶³Hg, ²⁵⁶Hf, ²⁵⁵Lu, and ²⁵⁴Re as representative examples of even–odd, even–even, odd–even, and odd–odd bubble nuclei. These nuclei were analyzed in detail through their density distributions and proton single-particle levels. As anticipated from previous studies, the depletion in central density is primarily driven by lower occupations of s states near the Fermi surface. Furthermore, our results emphasized that that quadrupole deformation (${\beta }_2$) plays a critical role in moderating the bubble structure, with more deformed nuclei exhibiting weaker central density depletion. Additionally, for odd-Z nuclei such as ²⁵⁵Lu and ²⁵⁴Re, the proton blocking effect was observed, influencing the distribution of single-particle levels and contributing to the distinct characteristics of central density depletion in these nuclei. These findings provide further insight into the interplay between deformation, single-particle structure, and central density depletion, advancing our understanding of bubble structures in exotic nuclei.