# The Role of the Pauli Exclusion Principle in Nuclear Physics Models

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## Abstract

**:**

## 1. Introduction

## 2. Nuclear Cluster Physics and Algebraic Models

#### 2.1. The Nuclear Vibron Model

#### 2.2. The Semimicroscopic Algebraic Cluster Model

#### 2.3. The Algebraic Cluster Model

#### 2.4. Applications to Multi-$Alpha$ Cluster Systems

#### 2.4.1. ${}^{12}$C

- Each $\alpha $-cluster is assumed to be in a (0,0) irrep of the $SU\left(3\right)$ shell model.
- Between the first two $\alpha $’s the Wildermuth condition requires a minimal number of four oscillation quanta. (${}^{8}$Be has 4 oscillation quanta within the shell model and each $\alpha $ contributes zero, thus, 4 oscillation quanta have to be in the relative motion). The same is the case for the relative motion of the third $\alpha $ particle with respect to the first two. ${}^{12}$C has 8 oscillation quanta and the 2-$\alpha $ subsystem already 4, thus, 4 more oscillation quanta have to be added in the relative motion of the second Jacobi-coordinate.
- The symmetric states of the three $\alpha $-system is obtained using the procedure as exposes in [30].
- The obtained list of possible $SU\left(3\right)$ irreps is compared to the shell model space (programs are available for that), resulting in the final model space.

- At low energy there is only one ${5}^{-}$ state in the SACM, while in the ACM there are 2, a consequence of ignoring the PEP.
- The spectrum in the ACM is denser at low energy than in the SACM. For example, several predicted parity doublets vanish when the PEP is taken into account.
- Within the ACM, the maximal number of bosons was used as a parameter, which is not allowed. The number N is a cut-off and convergence should be checked with increasing N. This is also a problem in the SACM: we used finite N, otherwise the space becomes too large. One conforms with a renormalization of the parameters when N increases.

#### 2.4.2. ${}^{16}$O

## 3. Algebraic Models in Nuclear Structure Physics

#### The Interacting Boson Model

**N**is the number operator of $U\left(n\right)$, ${\mathcal{C}}_{{G}_{k}}$ a Casimir operator of the group ${G}_{k}$ and ${\mathit{L}}^{2}$ the angular momentum operator, though in our example it is just the Casimir operator of a $SO\left(3\right)$ group.

**s**) and of angular momentum 2 (${\mathit{d}}_{m}^{\u2020}$, ${\mathit{d}}^{m}$ ($m=-2,-1,0,+1,+2$). In total there are six degrees of freedom and, thus, a $U\left(6\right)$ appears.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The experimental spectrum of ${}^{20}$Ne. Only those states are listed which appear in the reaction channel ${}^{16}$O$(\alpha ,\gamma )$. The left band corresponds to the (8,0) irrep in the $SU\left(3\right)$ model. The remaining states of positive and negative parity belong to other $SU\left(3\right)$ irreps. The experimental data were taken from [21].

**Figure 2.**The content of $SU\left(3\right)$ irreps of the states in the rotational bands in ${}^{20}$Ne. The vertical axis lists the percentage of contribution, while the horizontal axis lists the eigenvalue of the second order Casimir operator, given by $({\lambda}^{2}+\lambda \mu +{\mu}^{2}+2\lambda +3\mu )$. The relation of the symbols to the states with a given angular momentum is indicated on the upper right in each figure. The left panel is within the SACM and the right panel gives the result of the NVM. Clearly seen is that the states of the supposed ground state band within the NVM do not form a band.

**Figure 3.**A cartoon of the geometrical structure of the 3-$\alpha $ (${}^{12}$C) and 4-$\alpha $ (${}^{16}$O) particle system, according to the ACM. The $\overrightarrow{\lambda}$ and $\overrightarrow{\varrho}$ are Jacobi-coordinates in ${}^{12}$C, while ${\overrightarrow{\lambda}}_{k}$ ($k=1,2,3$) are Jacobi-coordinates in ${}^{16}$O.

**Figure 5.**Theoretical spectra of "${}^{12}$C for ACM (

**left panel**) and SACM (

**right panel**). Note, in the SACM there is only one ${5}^{-}$ state and several parity doublets are gone.

**Figure 6.**Spectrum of ${}^{16}$O according to a fit within the SACM, in the $SU\left(3\right)$ limit. States are listed up to 17 MeV. Below the bands their corresponding $SU\left(3\right)$ irrep notation is given by $\Delta {n}_{\pi}\left(\lambda ,\mu \right)$, where $\Delta {n}_{\pi}$ denotes the number of shell excitation and $\left(\lambda ,\mu \right)$ the $SU\left(3\right)$ irrep. Comparing with Table 3 one notes that none of the higher lying unphysical states occurs at low energy.

**Figure 7.**Spectrum of ${}^{16}$O up to 17MeV, according to the ACM. Note, the ordering of states is diferemt as done in the SACM (Figure 6). However, all states at low energy of the ACM also appear in the SACM, only associated to a different band. The parity doublets in the ACM are broken in the SACM.

$\mathit{n}\mathit{\hslash}\mathit{\omega}$ | $\left(\mathit{\lambda},\mathit{\mu}\right)$ |
---|---|

0 | (0,4) |

1 | (3,3) |

2 | (2,4), (6,2) |

3 | (3,4), (5,3), (9,1) |

4 | (0,6), (4,4), (6,3), (8,2), (12,0) |

5 | (3,5), (5,4), (7,3), (9,2), (11,1) |

6 | (2,6), (6,4)${}^{2}$, (8,3), (10,2), (12,1), (14,0) |

**Table 2.**Some calculated $B\left(E2\right)$ values of ${}^{12}$C, determined within the SACM and the ACM, compared to experimental available values. The numbers are in Weisskopf units (WU).

$\mathit{B}(\mathit{EL};{\mathit{J}}_{\mathit{i}}^{\mathit{\pi}}\to {\mathit{J}}_{\mathit{f}}^{\mathit{\pi}})$ | EXP.[WU] | $\mathit{SU}\left(3\right)$ [WU] | SACM [WU] | ACM |
---|---|---|---|---|

$B(E2;{2}_{1}^{+}\to {0}_{1}^{+})$ | $4.65\pm 0.26$ | 4.65 | 5.37 | 5.15 |

$B(E2;{0}_{2}^{+}\to {2}_{1}^{+})$ | $8.\pm 0.11$ | 0.0 | 7.73 | 0.8 |

$B(E3;{3}_{1}^{-}\to {0}_{1}^{+})$ | $12.\pm 2$ | 6.32 | 24.28 | 5.14 |

**Table 3.**The model space of the 4-$\alpha $ partcile system, according to [6] (left panel) and the SACM (right panel). The $SU\left(3\right)$ irrep in italic (left side) does not appear in the model space of the SACM, as the irreps in italic on the right side do not appear in the list given in [6]. In addition, some multiplicities of irrep in the SACM are larger than in [6]. We show only the irreps up to 4 excitation quanta, otherwise the table would get quite dense. Differences between the two lists appear from 3$\hslash \omega $ shell excitations on.

$\mathit{n}\mathit{\hslash}\mathit{\omega}$ | Kato | SACM |
---|---|---|

0 | $(0,0)$ | $(0,0)$ |

1 | $(2,1)$ | $(2,1)$ |

2 | $(2,0)\phantom{\rule{4pt}{0ex}}(3,1)\phantom{\rule{4pt}{0ex}}(0,4)\phantom{\rule{4pt}{0ex}}(4,2)$ | $(2,0)\phantom{\rule{4pt}{0ex}}(3,1)\phantom{\rule{4pt}{0ex}}(0,4)\phantom{\rule{4pt}{0ex}}(4,2)$ |

3 | $(3,0)\phantom{\rule{4pt}{0ex}}(0,3)\phantom{\rule{4pt}{0ex}}(2,2)\phantom{\rule{4pt}{0ex}}(4,1)\phantom{\rule{4pt}{0ex}}(3,3)\phantom{\rule{4pt}{0ex}}(6,0)\phantom{\rule{4pt}{0ex}}\left(5.2\right)\phantom{\rule{4pt}{0ex}}(2,5)\phantom{\rule{4pt}{0ex}}(6,3)\phantom{\rule{4pt}{0ex}}$ | $\phantom{\rule{4pt}{0ex}}(3,0)\phantom{\rule{4pt}{0ex}}(0,3)\phantom{\rule{4pt}{0ex}}(2,2)\phantom{\rule{4pt}{0ex}}(4,1)\phantom{\rule{4pt}{0ex}}(1,4)\phantom{\rule{4pt}{0ex}}(3,3)\phantom{\rule{4pt}{0ex}}(6,0)\phantom{\rule{4pt}{0ex}}(5,2)\phantom{\rule{4pt}{0ex}}(2,5)\phantom{\rule{4pt}{0ex}}(6,3)$ |

4 | $(0,2)\phantom{\rule{4pt}{0ex}}(1,3)\phantom{\rule{4pt}{0ex}}{(4,0)}^{2}\phantom{\rule{4pt}{0ex}}(3,2)\phantom{\rule{4pt}{0ex}}{(2,4)}^{2}\phantom{\rule{4pt}{0ex}}{(5,1)}^{2}\phantom{\rule{4pt}{0ex}}(4,3)$ | $(0,2)\phantom{\rule{4pt}{0ex}}(2,1)\phantom{\rule{4pt}{0ex}}{(1,3)}^{2}\phantom{\rule{4pt}{0ex}}{(4,0)}^{2}\phantom{\rule{4pt}{0ex}}{(3,2)}^{2}\phantom{\rule{4pt}{0ex}}{(2,4)}^{4}\phantom{\rule{4pt}{0ex}}{(5,1)}^{3}\phantom{\rule{4pt}{0ex}}{(4,3)}^{2}\phantom{\rule{4pt}{0ex}}(1,6)$ |

$(3,5)\phantom{\rule{4pt}{0ex}}{(6,2)}^{2}\phantom{\rule{4pt}{0ex}}(5,4)\phantom{\rule{4pt}{0ex}}(0,8)\phantom{\rule{4pt}{0ex}}(8,1)\phantom{\rule{4pt}{0ex}}(4,6)\phantom{\rule{4pt}{0ex}}(7,3)\phantom{\rule{4pt}{0ex}}(8,4)\phantom{\rule{4pt}{0ex}}$ | ${(3,5)}^{2}\phantom{\rule{4pt}{0ex}}{(6,2)}^{3}\phantom{\rule{4pt}{0ex}}(5,4)\phantom{\rule{4pt}{0ex}}(0,8)\phantom{\rule{4pt}{0ex}}(4,6)\phantom{\rule{4pt}{0ex}}(7,3)\phantom{\rule{4pt}{0ex}}(8,4)$ |

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Berriel-Aguayo, J.R.M.; Hess, P.O.
The Role of the Pauli Exclusion Principle in Nuclear Physics Models. *Symmetry* **2020**, *12*, 738.
https://doi.org/10.3390/sym12050738

**AMA Style**

Berriel-Aguayo JRM, Hess PO.
The Role of the Pauli Exclusion Principle in Nuclear Physics Models. *Symmetry*. 2020; 12(5):738.
https://doi.org/10.3390/sym12050738

**Chicago/Turabian Style**

Berriel-Aguayo, Josué R. M., and Peter O. Hess.
2020. "The Role of the Pauli Exclusion Principle in Nuclear Physics Models" *Symmetry* 12, no. 5: 738.
https://doi.org/10.3390/sym12050738