# Spherical, Axial, and Triaxial Symmetries in the Study of Halo Nuclei with Covariant Density Functional Theory

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## Abstract

**:**

## 1. Introduction

## 2. Symmetries in the CDFT for Halo Nuclei

#### 2.1. Basic Framework of the CDFT

#### 2.2. Solving the RHB Equation

#### 2.3. Spherical Symmetry

#### 2.4. Axial Symmetry

#### 2.5. Triaxial Symmetry

## 3. Applications to Halo Nuclei

#### 3.1. The RCHB Theory

#### 3.2. The DRHBc Theory

#### 3.3. The TRHBc Theory

## 4. Summary and Prospect

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CDFT | Covariant density functional theory |

RCHB theory | Relativistic continuum Hartree–Bogoliubov theory |

DRHBc theory | Deformed relativistic Hartree–Bogoliubov theory in continuum |

TRHBc theory | Triaxial relativistic Hartree–Bogoliubov theory in continuum |

HO basis | Harmonic oscillator basis |

SWS basis | Schrödinger Woods–Saxon basis |

DWS basis | Dirac Woods–Saxon basis |

AMP | Angular momentum projection |

FAM | Finite amplitude method |

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**Figure 1.**Experimentally known nuclear landscape from helium to phosphorus, where stable nuclei and experimentally confirmed/suggested neutron as well as proton halo nuclei/candidates are indicated in gray, olive/green, and orange/yellow colors, respectively. Taken from Ref. [4].

**Figure 2.**Calculated and experimental density distributions in ${}^{11}$Li and ${}^{9}$Li. The solid line shows the result of ${}^{11}$Li, while the dashed line corresponds to the calculation of ${}^{9}$Li. The shaded area gives the experimental results with error bars. Taken from Ref. [14].

**Figure 3.**Density distributions of ${}^{44}$Mg with the z axis as the symmetry axis. (

**a**) The proton density (for $x<0$) and the neutron density (for $x>0$); (

**b**) the density of the neutron core; and (

**c**) the density of the neutron halo. In each plot, a dotted circle is drawn to guide the eye. Taken from Ref. [39].

**Figure 4.**Neutron density distributions in $xy$, $xz$, and $yz$ planes contributed by the core (

**a**–

**c**) and the halo (

**d**–

**f**) of ${}^{42}$Al. In each plot, a circle in a dotted line is drawn to guide the eye. With the rms radius and deformation parameters, $\beta $ and $\gamma $ from the densities, the corresponding schematic shapes for the core and the halo are given in the left, in which s, i, and l, respectively, represent the short, intermediate, and long axes. Taken from Ref. [55].

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**MDPI and ACS Style**

Xiang, Y.; Luo, Q.; Yang, S.; Zhang, K.
Spherical, Axial, and Triaxial Symmetries in the Study of Halo Nuclei with Covariant Density Functional Theory. *Symmetry* **2023**, *15*, 1420.
https://doi.org/10.3390/sym15071420

**AMA Style**

Xiang Y, Luo Q, Yang S, Zhang K.
Spherical, Axial, and Triaxial Symmetries in the Study of Halo Nuclei with Covariant Density Functional Theory. *Symmetry*. 2023; 15(7):1420.
https://doi.org/10.3390/sym15071420

**Chicago/Turabian Style**

Xiang, Yifeng, Qingjin Luo, Siqi Yang, and Kaiyuan Zhang.
2023. "Spherical, Axial, and Triaxial Symmetries in the Study of Halo Nuclei with Covariant Density Functional Theory" *Symmetry* 15, no. 7: 1420.
https://doi.org/10.3390/sym15071420