# The Power of Symmetries in Nuclear Structure and Some of Its Problems

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## Abstract

**:**

## 1. Introduction

## 2. A Pedestrian Introduction to Symmetries

#### 2.1. Algebras, Casimir Operators, and Group Structure

#### 2.2. Dynamical Symmetries and Construction of Model Hamiltonians

**s**refers to a scalar auxiliary boson introduced in order to lead to operators that conserve the total number of bosons. The total number of bosons is given by

**s**∼$\sqrt{{N}_{s}}$ to the matrix element when N is large (${N}_{s}\approx N$).

#### 2.3. An Example of Mathematically Two Equal Models

## 3. Examples of Models Using Symmetries: Algebraic Models

#### 3.1. The Elliott Model

**Figure 1.**Experimental spectrum of ${}^{20}$Ne [21]. The ground state band is plotted in the first column, which is well described by the Elliott model; however, also by any algebraic model with a $L(L+1)$ term.

#### 3.2. The Interacting Boson Approximation

**s**and for the d-bosons they are ${\mathit{d}}_{m}^{\u2020}$ and ${\mathit{d}}^{m}$ ($m=-2,\dots ,+2$). Here, we deviate a bit from the standard notation and make a difference between lower and upper indices, due to the different properties under unitary transformation. The annihilation operator with a lower index is related to the one with an upper index via ${d}_{m}={(-1)}^{2-m}{d}^{-m}$.

**The $\mathit{U}\left(\mathbf{5}\right)$ limit:**

**The $\mathit{SU}\left(\mathbf{3}\right)$ limit:**

**The $\mathit{SO}\left(\mathbf{6}\right)$ limit:**

- Starting from a certain number of degrees of freedom;
- Constructing group chains;
- Setting up a model Hamiltonian.all of which are the result of a pure mathematical input. Some of the physics is yet to be included. This is achieved when we note that the bosons are nucleon pairs, information that we have still to exploit. This is especially important when there are other models, distinct from the IBA, with the same number of degrees of freedom.

#### 3.3. The Nuclear Vibron Model

- Identify the degrees of freedom. The relative vector is a spin-1 tensor and described by spin one p-bosons, i.e., there are three degrees of freedom. Add an auxiliary scalar boson s to it, such that the total number of bosons $N={n}_{p}+{n}_{s}$ is constant. Here, adding the scalar boson is just a trick to introduce a cut-off. In this way, the p-bosons vary from zero to N.
- Construct a group chain, which contains the angular momentum group $S{O}_{R}\left(3\right)$, where R refers to the relative motion degree of freedom. This leads to$$\begin{array}{c}\hfill {U}_{R}\left(3\right)\supset S{O}_{R}\left(3\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}.\end{array}$$
- Construct a Model Hamiltonian. The simplest one is$$\begin{array}{ccc}\hfill {\mathit{H}}_{\mathrm{vibron}}& =& a{\mathit{n}}_{p}-b{\mathit{C}}_{2}\left(SU\left(3\right)\right)+c{\mathit{L}}^{2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}.\hfill \end{array}$$
- Its eigenvalue is$$\begin{array}{ccc}\hfill E& =& a{n}_{p}-b\frac{2}{3}({\lambda}^{2}+\lambda \mu +{\mu}^{2}+3\lambda +3\mu )+cL(L+1)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\hfill \end{array}$$
**n**the p-particle number operator, one of the generators of $U\left(3\right)$, together with the generators of $SU\left(3\right)$.

#### 3.4. The Semimicroscopic Algebraic Cluster Model (SACM)

#### 3.5. A Particular Application of Nuclear Physics Methods to Particle Physics

#### 3.6. Final Remarks

## 4. Some Problems Which May Arise, Using Symmetries Without Critics

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Illustration of the Wildermuth condition with $\alpha {+}^{16}$O →${}^{20}$Ne. In their ground state, all four nucleons in the $\alpha $ particle are in the s state and for the O-nucleus there are four in the s-shell and 12 in the p-shell. In order to satisfy the PEP, there must be four nucleons in the sd-shell for the Ne-nucleus. The $\alpha $ particle contributes no oscillation quanta, but the O-nucleus has 12 quanta. The Ne-nucleus has a total of 20 quanta; thus, the difference of eight quanta must be in the relative motion. This computation results in a minimal number of relative oscillation quanta.

**Figure 4.**In the (

**left**) panel, the spectrum of ${}^{12}$C as a three $\alpha $-particle state, not considering the PEP, is plotted, and in the (

**right**) panel figure, the same spectrum is plotted by taking into account the PEP. (Figure taken from [40].)

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

0 | (0,4) |

1 | (3,3) |

2 | (2,4), (4,3), (6,2) |

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

4 | (4,4), (6,3), (8,2), (10,1), (12,0) |

5 | (5,4), (7,3), (13,0) |

6 | (6,4), (8,3), (10,2), (12,1) |

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Hess, P.O.
The Power of Symmetries in Nuclear Structure and Some of Its Problems. *Symmetry* **2023**, *15*, 1197.
https://doi.org/10.3390/sym15061197

**AMA Style**

Hess PO.
The Power of Symmetries in Nuclear Structure and Some of Its Problems. *Symmetry*. 2023; 15(6):1197.
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**Chicago/Turabian Style**

Hess, Peter O.
2023. "The Power of Symmetries in Nuclear Structure and Some of Its Problems" *Symmetry* 15, no. 6: 1197.
https://doi.org/10.3390/sym15061197