# Isotopic Shift in Hg-Isotopes within Brückner versus Relativistic Energy Density Functional

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## Abstract

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## 1. Introduction

## 2. Relativistic Mean Field Approach

#### 2.1. Spherical Equivalent Density Using Wood–Saxon Fitting

#### 2.2. Brückner’s Prescription and Symmetry Energy

#### 2.3. The Coherent Density Fluctuations Model

## 3. Results and Discussions

#### 3.1. Shift in Binding Energy and Charge Radius

**Figure 2.**(Color online) The binding energy deviation ($B{E}_{Expt.}$-$B{E}_{Cal.}$), the isotopic shifts of binding energy (BE) and charge distribution radius (${R}_{ch}$) in fm are shown in the upper, middle and lower panels respectively for the ground state of Hg-isotopes. The available experimental data are given for comparison [51,52]. The predictions by Goodacre et al. [9] for the isotopic shift at N = 126 are also given for comparison. The energies are in MeV and radius in fm.

**Figure 3.**(Color online) The binding energy (BE) and charge distribution radius (${R}_{ch}$) are shown along with their respective shifts ($\Delta BE$, $\Delta {R}_{ch}^{2}$) for the isomeric state of Hg-isotopes. The energies in MeV and radius in fm.

#### 3.2. Symmetry Energy and Its Shift

**Figure 4.**(Color online) The symmetry energy (${S}^{A}$) and its volume ${S}_{V}$ and surface ${S}_{S}$ components for IOPB-I, DD-ME2 and DD-PC1 parameter sets are shown along with the shift corresponding to the total symmetry energy ($\Delta {S}^{A}$) for the ground state of Hg-isotopes.

**Figure 5.**(Color online) Same as Figure 4, but for the existing isomeric states of Hg-isotopes.

**Figure 6.**(Color online) The total symmetry energy (${S}^{A}$) and its shift within the Brückner functional and the Relativistic energy density functional used to obtain the volume ${S}_{V}$ and surface ${S}_{S}$ symmetry energies with Danielwiz’s prescription are shown for the ground state of Hg-isotopes using RMF (NL3) parameter set.

**Figure 7.**(Color online) Same as Figure 6, but for isomeric states of Hg-isotopes.

#### 3.3. Single Particle Energy and Its Occupancy

**Figure 8.**(Color online) The ground state neutron single particle energies ${\u03f5}_{n}$ of ${}^{206}$Hg near the Fermi level with IOPB-I, DD-ME2, DD-PC1 and NL3 parameter sets.

**Figure 9.**(Color online) The ground state neutron occupation probabilities of 2${g}_{9/2}$ & 1${i}_{11/2}$ orbitals as a function of their respective single particle energies ${\u03f5}_{n}$ for the considered IOPB-I, DD-ME2, DD-PC1 and NL3 parameter sets.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

E-RMF | Effective Relativistic Mean Field |

NM | Nuclear Matter |

CDFM | Coherent Density Fluctuation Model |

GS | Ground State |

IS | Isomeric State |

BE | Binding Energy |

EDF | Energy density functional |

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**MDPI and ACS Style**

Pattnaik, J.A.; Majekodunmi, J.T.; Bhuyan, M.; Patra, S.K.
Isotopic Shift in Hg-Isotopes within Brückner versus Relativistic Energy Density Functional. *Foundations* **2022**, *2*, 898-911.
https://doi.org/10.3390/foundations2040061

**AMA Style**

Pattnaik JA, Majekodunmi JT, Bhuyan M, Patra SK.
Isotopic Shift in Hg-Isotopes within Brückner versus Relativistic Energy Density Functional. *Foundations*. 2022; 2(4):898-911.
https://doi.org/10.3390/foundations2040061

**Chicago/Turabian Style**

Pattnaik, Jeet Amrit, Joshua T. Majekodunmi, Mrutunjaya Bhuyan, and Suresh Kumar Patra.
2022. "Isotopic Shift in Hg-Isotopes within Brückner versus Relativistic Energy Density Functional" *Foundations* 2, no. 4: 898-911.
https://doi.org/10.3390/foundations2040061