#
Divergence in the Relativistic Mean Field Formalism: A Case Study of the Ground State Properties of the Decay Chain of ^{214,216,218}U Isotopes

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

^{214}U has been recently observed experimentally. This opens the window to theoretically investigate the ground-state properties of the lightest known even–even neutron deficient

^{214,216,218}U isotopes and to examine $\alpha $-particle clustering around the shell closure. The decay half-lives are calculated within the preformed cluster-decay model (PCM). To obtain the $\alpha $-daughter interaction potential, the RMF densities are folded with the newly developed R3Y and the well-known M3Y NN potentials for comparison. The alpha preformation probability $\left({P}_{\alpha}\right)$ is calculated from the analytic formula of Deng and Zhang. The WKB approximation is employed for the calculation of the transmission probability. The individual binding energies (BE) for the participating nuclei are estimated from the relativistic mean-field (RMF) formalism and those from the finite range droplet model (FRDM) as well as WS3 mass tables. In addition to $Z=84$, the so-called abnormal enhancement region, i.e., $84\le Z\le 90$ and $N<126$, is normalised by an appropriately fitted neck-parameter $\Delta R$. On the other hand, the discrepancy sets in due to the shell effect at (and around) the proton magic number $Z=82$ and 84, and thus a higher scaling factor ranging from ${10}^{-8}$–${10}^{-5}$ is required. Additionally, in contrast with the experimental binding energy data, large deviations of about 5–10 MeV are evident in the RMF formalism despite the use of different parameter sets. An accurate prediction of $\alpha $-decay half-lives requires a Q-value that is in proximity with the experimental data. In addition, other microscopic frameworks besides RMF could be more reliable for the mass region under study. $\alpha $-particle clustering is largely influenced by the shell effect.

## 1. Introduction

## 2. Theoretical Framework

#### Preformed Cluster-Decay Model (PCM)

**Figure 1.**The total nucleus-nucleus interaction potential V (MeV) and its components, namely, nuclear and Coulomb potentials as a function of radial separation R (fm) for R3Y (NL${3}^{*}$) and M3Y NN potentials as a representative case of ${}^{214}$U →${}^{210}$Th + $\alpha $. The inset shows a magnified view of the barrier height and position.

## 3. Calculations and Discussions

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CSF | Common Scaling Factor |

PCM | Preformed Cluster-decay Model |

WKB | Wentzel–Kramers–Brillouin |

FRDM | Finite-Range-Droplet Model |

RMF | Relativistic Mean-Field |

M3Y | Michigan-3-Yukawa |

R3Y | Relativistic (Mean-Field)-3-Yukawa |

OPEP | One-Pion Exchange Potential |

NN | Nucleon–Nucleon |

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**Figure 5.**The logarithmic half-lives (upper panel) and the ${Q}_{\alpha}$ for the $\alpha $-decay chain of ${}^{214}$U, ${}^{216}$U and ${}^{218}$U, obtained of from the RMF formalism (NL${3}^{*}$) in contrast with those from the FRDM [52], WS3 [53] and the recent experimental measurement of Zhang et al. [15].

**Figure 8.**Upper panel: The driving potential ($V\left({R}_{a}\right)-{Q}_{FRDM}$) for the decay chains of ${}^{214,216,218}\mathrm{U}$ isotopes for R3Y (in black squares) and M3Y (in red circles) NN potentials as a function of the parent nuclei A. The lower panel illustrates the rms radii for matter distribution ${r}_{m}$ and charge distribution ${r}_{c}$ for the ${}^{214,216,218}\mathrm{U}$ isotopes using the relativistic mean-field formalism RMF (NL${3}^{*}$).

**Table 1.**The RMF (NL${3}^{*}$) predictions for the binding energy (BE), pairing energy ${E}_{pair}$, deformation ${\beta}_{2}$, root-mean-square radii (rms), charge radii ${r}_{c}$, proton radii ${r}_{p}$ and neutron radii ${r}_{n}$ in comparison with their results from FRDM [52] and WS3 [53]. The energy is in MeV and radii in fm.

Nuclei | RMF | FRDM | WS3 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{E}}_{\mathit{p}\mathit{a}\mathit{i}\mathit{r}}$ | ${\mathit{E}}_{\mathbf{c}.\mathbf{m}.}$ | B.E | B.E/A | ${\mathit{r}}_{\mathit{c}}$ | ${\mathit{r}}_{\mathit{n}}$ | ${\mathit{r}}_{\mathit{p}}$ | rms | ${\mathit{\beta}}_{2}$ | B.E | B.E/A | ${\mathit{\beta}}_{2}$ | B.E | B.E/A | ${\mathit{\beta}}_{2}$ | |

${}^{214}\mathrm{U}$ | 16.49 | −5.141 | 1640.51 | 7.67 | 5.66 | 5.73 | 5.60 | 5.68 | 0.001 | 1630.22 | 7.62 | −0.115 | 1629.70 | 7.62 | −0.106 |

${}^{210}\mathrm{Th}$ | 16.59 | −5.173 | 1618.15 | 7.71 | 5.62 | 5.70 | 5.56 | 5.64 | 0.045 | 1610.37 | 7.67 | −0.135 | 1610.53 | 7.67 | −0.124 |

${}^{206}\mathrm{Ra}$ | 16.36 | −5.207 | 1596.11 | 7.75 | 5.58 | 5.67 | 5.52 | 5.61 | 0.095 | 1589.89 | 7.72 | −0.125 | 1590.06 | 7.72 | −0.123 |

${}^{202}\mathrm{Rn}$ | 16.35 | −5.241 | 1573.09 | 7.79 | 5.54 | 5.64 | 5.49 | 5.58 | 0.109 | 1568.87 | 7.77 | −0.115 | 1569.16 | 7.77 | 0.096 |

${}^{198}\mathrm{Po}$ | 16.40 | −5.276 | 1549.31 | 7.83 | 5.50 | 5.61 | 5.44 | 5.54 | 0.111 | 1547.43 | 7.82 | 0.075 | 1547.97 | 7.82 | 0.074 |

${}^{194}\mathrm{Pb}$ | 16.31 | −5.312 | 1524.98 | 7.86 | 5.47 | 5.58 | 5.41 | 5.51 | 0.124 | 1526.18 | 7.87 | 0.000 | 1526.38 | 7.87 | 0.083 |

${}^{216}\mathrm{U}$ | 15.81 | −5.125 | 1659.59 | 7.68 | 5.66 | 5.76 | 5.61 | 5.69 | 0.000 | 1648.69 | 7.63 | −0.073 | 1648.16 | 7.63 | −0.084 |

${}^{212}\mathrm{Th}$ | 16.00 | −5.157 | 1637.16 | 7.72 | 5.62 | 5.72 | 5.57 | 5.66 | 0.003 | 1628.74 | 7.68 | −0.094 | 1628.38 | 7.68 | −0.101 |

${}^{208}\mathrm{Ra}$ | 15.89 | −5.190 | 1614.15 | 7.76 | 5.59 | 5.69 | 5.53 | 5.62 | 0.059 | 1608.11 | 7.73 | −0.125 | 1608.51 | 7.73 | −0.116 |

${}^{204}\mathrm{Rn}$ | 15.84 | −5.224 | 1590.80 | 7.80 | 5.55 | 5.66 | 5.489 | 5.59 | 0.083 | 1586.94 | 7.78 | −0.115 | 1587.23 | 7.78 | −0.101 |

${}^{200}\mathrm{Po}$ | 15.73 | −5.258 | 1566.81 | 7.83 | 5.51 | 5.63 | 5.45 | 5.55 | 0.089 | 1565.52 | 7.83 | −0.063 | 1565.66 | 7.83 | −0.065 |

${}^{196}\mathrm{Pb}$ | 15.43 | −5.294 | 1542.13 | 7.87 | 5.47 | 5.60 | 5.41 | 5.52 | 0.096 | 1543.48 | 7.88 | 0.000 | 1543.96 | 7.88 | 0.008 |

${}^{218}\mathrm{U}$ | 15.10 | −5.109 | 1677.21 | 7.69 | 5.67 | 5.78 | 5.62 | 5.71 | 0.001 | 1666.17 | 7.64 | 0.000 | 1665.89 | 7.64 | −0.004 |

${}^{214}\mathrm{Th}$ | 15.29 | −5.141 | 1655.61 | 7.74 | 5.63 | 5.75 | 5.58 | 5.68 | 0.000 | 1646.39 | 7.69 | −0.063 | 1645.97 | 7.69 | −0.072 |

${}^{210}\mathrm{Ra}$ | 15.46 | −5.173 | 1632.02 | 7.77 | 5.59 | 5.71 | 5.56 | 5.64 | 0.019 | 1625.80 | 7.74 | −0.084 | 1625.57 | 7.74 | −0.096 |

${}^{206}\mathrm{Rn}$ | 15.38 | −5.207 | 1608.23 | 7.81 | 5.55 | 5.68 | 5.49 | 5.60 | 0.047 | 1604.44 | 7.79 | −0.094 | 1604.86 | 7.79 | −0.097 |

${}^{202}\mathrm{Po}$ | 15.22 | −5.241 | 1583.85 | 7.84 | 5.51 | 5.65 | 5.45 | 5.57 | 0.056 | 1582.71 | 7.84 | −0.063 | 1582.92 | 7.84 | −0.070 |

${}^{198}\mathrm{Pb}$ | 14.66 | −5.276 | 1558.98 | 7.87 | 5.47 | 5.62 | 5.42 | 5.54 | 0.075 | 1560.26 | 7.88 | 0.000 | 1560.75 | 7.88 | 0.022 |

**Table 2.**The R3Y and M3Y predictions of the $\alpha $-decay half-lives ${T}_{1/2}$ within the PCM ($T=0$) and the calculation details for the decay chains of the neutron-deficient even–even ${}^{214,216,218}\mathrm{U}$ isotopes from RMF (NL${3}^{*}$) in comparison with the experimental data [15,77,78]. The Q-values also are calculated using the binding energies from FRDM [52] and WS3 [53] for comparison.

R3Y | M3Y | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$-Transition | Experiment | Q Values (MeV) | $\Delta \mathit{R}$ | Scaling Factor | ${log}_{\mathbf{10}}{\mathbf{T}}_{\mathbf{1}/\mathbf{2}}$ | $\Delta \mathit{R}$ | Scaling Factor | ${log}_{\mathbf{10}}{\mathit{T}}_{\mathbf{1}/\mathbf{2}}$ | ||||||||

Parent | Daughter | ${\mathbf{Q}}_{\mathbf{\alpha}}^{\mathit{expt}}$ | ${log}_{\mathbf{10}}{\mathit{T}}_{\mathbf{1}/\mathbf{2}}$ | RMF | FRDM | WS3 | RMF | FRDM | WS3 | RMF | FRDM | WS3 | ||||

${}^{214}\mathrm{U}$ | ${}^{210}\mathrm{Th}$ | 8.533 | −3.284 | 5.942 | 8.446 | 9.126 | 1.270 | ${10}^{-4}$ | 5.672 | −3.285 | −4.950 | 0.540 | ${10}^{-4}$ | 5.291 | −3.197 | −4.721 |

${}^{210}\mathrm{Th}$ | ${}^{206}\mathrm{Ra}$ | 8.069 | −1.790 | 6.247 | 7.816 | 7.831 | 1.537 | ${10}^{-4}$ | 3.913 | −1.789 | −1.835 | 0.590 | ${10}^{-4}$ | 3.654 | −1.846 | −1.889 |

${}^{206}\mathrm{Ra}$ | ${}^{202}\mathrm{Rn}$ | 7.415 | −0.620 | 5.280 | 7.276 | 7.394 | 1.737 | ${10}^{-4}$ | 8.001 | −0.633 | −1.020 | 0.836 | ${10}^{-4}$ | 7.622 | −0.833 | −1.209 |

${}^{202}\mathrm{Rn}$ | ${}^{198}\mathrm{Po}$ | 6.774 | 1.093 | 4.518 | 6.856 | 7.099 | 0.728 | ${10}^{-4}$ | 9.098 | 1.094 | 0.717 | 1.487 | ${10}^{-4}$ | 12.858 | 1.088 | 0.230 |

${}^{198}\mathrm{Po}$ | ${}^{194}\mathrm{Pb}$ | 6.309 | 2.270 | 3.966 | 7.046 | 6.710 | 1.469 | ${10}^{-8}$${}^{a}$ | 18.299 | 2.269 | 3.404 | 1.302 | ${10}^{-7}$${}^{a}$ | 18.525 | 2.271 | 3.433 |

${}^{216}\mathrm{U}$ | ${}^{212}\mathrm{Th}$ | 8.384 | −2.326 | 5.863 | 8.346 | 8.513 | 1.810 | ${10}^{-4}$ | 7.004 | −2.319 | −2.779 | 0.914 | ${10}^{-4}$ | 6.796 | −2.387 | −2.837 |

${}^{212}\mathrm{Th}$ | ${}^{208}\mathrm{Ra}$ | 7.958 | −1.499 | 5.287 | 7.666 | 8.424 | 1.393 | ${10}^{-4}$ | 8.513 | −1.496 | −3.643 | 0.610 | ${10}^{-4}$ | 8.219 | −1.505 | −3.553 |

${}^{208}\mathrm{Ra}$ | ${}^{204}\mathrm{Rn}$ | 7.273 | 0.137 | 4.951 | 7.126 | 7.019 | 1.913 | ${10}^{-4}$ | 10.236 | 0.119 | 0.492 | 1.110 | ${10}^{-4}$ | 10.291 | 0.214 | 0.584 |

${}^{204}\mathrm{Rn}$ | ${}^{200}\mathrm{Po}$ | 6.546 | 2.013 | 4.303 | 6.876 | 6.727 | 0.700 | ${10}^{-4}$ | 10.568 | 2.014 | 2.205 | 0.956 | ${10}^{-6}$${}^{a}$ | 15.236 | 2.014 | 2.543 |

${}^{200}\mathrm{Po}$ | ${}^{196}\mathrm{Pb}$ | 5.981 | 3.794 | 3.614 | 6.256 | 6.589 | 0.797 | ${10}^{-6}$${}^{a}$ | 16.290 | 3.795 | 3.123 | 1.103 | ${10}^{-6}$${}^{a}$ | 20.254 | 3.794 | 2.513 |

${}^{218}\mathrm{U}$ | ${}^{214}\mathrm{Th}$ | 8.775 | −3.292 | 6.692 | 8.516 | 8.366 | 1.318 | ${10}^{-4}$ | 2.609 | −3.293 | −2.896 | 0.592 | ${10}^{-4}$ | 2.316 | −3.293 | −2.920 |

${}^{214}\mathrm{Th}$ | ${}^{210}\mathrm{Ra}$ | 7.827 | −1.060 | 4.708 | 7.706 | 7.896 | 1.812 | ${10}^{-4}$ | 12.896 | −1.060 | −1.641 | 0.872 | ${10}^{-4}$ | −9.977 | −1.060 | −1.631 |

${}^{210}\mathrm{Ra}$ | ${}^{206}\mathrm{Rn}$ | 7.151 | 0.585 | 4.500 | 6.936 | 7.591 | 1.622 | ${10}^{-4}$ | 13.039 | 0.585 | −1.599 | 0.828 | ${10}^{-4}$ | 12.905 | 0.585 | −1.558 |

${}^{206}\mathrm{Rn}$ | ${}^{202}\mathrm{Po}$ | 6.384 | 2.740 | 3.924 | 6.566 | 6.353 | 0.964 | ${10}^{-7}$${}^{a}$ | 16.978 | 2.741 | 3.486 | 1.188 | ${10}^{-5}$${}^{a}$ | 18.083 | 2.740 | 2.735 |

${}^{202}\mathrm{Po}$ | ${}^{198}\mathrm{Pb}$ | 5.701 | 5.143 | 3.422 | 5.846 | 6.130 | 1.655 | ${10}^{-6}$${}^{a}$ | 21.800 | 5.142 | 3.913 | 0.840 | ${10}^{-6}$${}^{a}$ | 21.656 | 5.144 | 3.934 |

^{a}A higher scaling factor between the range 10

^{−8}–10

^{−5}is required to calculate the a-decay half-lives with daughter nuclei near (and at) proton shell closure Z = 82.

**Table 3.**Details of the random fitting of the neck-length $\Delta $R and scaling factor due to the deviations found in RMF (only).

Parent Nuclei | Expt. | R3Y | M3Y | ||||
---|---|---|---|---|---|---|---|

${log}_{10}{\mathit{T}}_{1/2}$ | $\mathbf{\Delta}$R (fm) | Scaling Factor | ${log}_{10}{\mathit{T}}_{1/2}$ | $\mathbf{\Delta}$R (fm) | Scaling Factor | ${log}_{10}{\mathit{T}}_{1/2}$ | |

${}^{214}\mathrm{U}$ | −3.284 | 1.249 | ${10}^{5}$ | −3.285 | 0.544 | ${10}^{5}$ | −3.200 |

${}^{210}\mathrm{Th}$ | −1.790 | 1.208 | ${10}^{1}$ | −1.791 | 1.390 | ${10}^{3}$ | −1.850 |

${}^{206}\mathrm{Ra}$ | −0.620 | 1.070 | ${10}^{3}$ | −0.636 | 0.560 | ${10}^{4}$ | −0.833 |

${}^{202}\mathrm{Rn}$ | 1.093 | 0.728 | ${10}^{4}$ | 1.098 | 1.010 | ${10}^{7}$ | 1.104 |

${}^{198}\mathrm{Po}$ | 2.270 | 1.480 | ${10}^{8}$${}^{b}$ | 2.276 | 1.050 | ${10}^{9}$${}^{b}$ | 2.267 |

${}^{216}\mathrm{U}$ | −2.326 | 1.156 | ${10}^{4}$ | −2.319 | 0.492 | ${10}^{4}$ | −2.326 |

${}^{212}\mathrm{Th}$ | −1.499 | 1.399 | ${10}^{6}$ | −1.499 | 0.730 | ${10}^{6}$ | −1.501 |

${}^{208}\mathrm{Ra}$ | 0.137 | 1.911 | ${10}^{6}$ | 0.189 | 0.980 | ${10}^{6}$ | 0.127 |

${}^{204}\mathrm{Rn}$ | 2.013 | 1.030 | ${10}^{5}$ | 2.008 | 0.961 | ${10}^{7}$ | 2.225 |

${}^{200}\mathrm{Po}$ | 3.794 | 1.710 | ${10}^{10}$${}^{b}$ | 3.816 | 1.400 | ${10}^{11}$${}^{b}$ | 3.796 |

${}^{218}\mathrm{U}$ | −3.292 | 0.793 | ${10}^{1}$ | −3.280 | 0.625 | ${10}^{2}$ | −3.296 |

${}^{214}\mathrm{Th}$ | −1.060 | 0.591 | ${10}^{9}$ | −1.061 | 1.020 | ${10}^{5}$ | −1.129 |

${}^{210}\mathrm{Ra}$ | 0.585 | 1.090 | ${10}^{7}$ | 0.549 | 0.730 | ${10}^{8}$ | 0.581 |

${}^{206}\mathrm{Rn}$ | 2.740 | 1.364 | ${10}^{9}$ | 2.734 | 1.200 | ${10}^{8}$ | 2.742 |

${}^{202}\mathrm{Po}$ | 5.143 | 0.690 | ${10}^{7}$ | 5.144 | 0.780 | ${10}^{10}$${}^{b}$ | 5.194 |

^{b}The need for a higher ‘random’ scaling factor is evident to estimate the a-decay half-lives with daughter nuclei near (and at) proton shell closure Z = 82.

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**MDPI and ACS Style**

Joshua, T.M.; Jain, N.; Kumar, R.; Anwar, K.; Abdullah, N.; Bhuyan, M.
Divergence in the Relativistic Mean Field Formalism: A Case Study of the Ground State Properties of the Decay Chain of ^{214,216,218}U Isotopes. *Foundations* **2022**, *2*, 85-104.
https://doi.org/10.3390/foundations2010004

**AMA Style**

Joshua TM, Jain N, Kumar R, Anwar K, Abdullah N, Bhuyan M.
Divergence in the Relativistic Mean Field Formalism: A Case Study of the Ground State Properties of the Decay Chain of ^{214,216,218}U Isotopes. *Foundations*. 2022; 2(1):85-104.
https://doi.org/10.3390/foundations2010004

**Chicago/Turabian Style**

Joshua, Tolulope Majekodunmi, Nishu Jain, Raj Kumar, Khairul Anwar, Nooraihan Abdullah, and Mrutunjaya Bhuyan.
2022. "Divergence in the Relativistic Mean Field Formalism: A Case Study of the Ground State Properties of the Decay Chain of ^{214,216,218}U Isotopes" *Foundations* 2, no. 1: 85-104.
https://doi.org/10.3390/foundations2010004