#
Origin of Low- and High-Energy Monopole Collectivity in ^{132}Sn

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The FRSA Model

## 3. Results and Discussion

**B**in Table 3. One can see that the wave functions of the excited ${0}_{1,2,3}^{+}$ states of ${}^{132}$Sn are dominated by two-phonon configurations (>$90\%$). The ${[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ configuration is the crucial component of the wave function of the LTP ${0}^{+}$ state since the ${2}_{1}^{+}$ RPA state is the lowest collective excitation, which leads to the minimal two-phonon energy and the maximal matrix elements coupling one- and two-phonon configurations. The energy for the LTP ${0}^{+}$ state is slightly less than the double energy of the ${2}_{1}^{+}$ RPA state (see Table 1). As a result, the inclusion of the 2p-2h neutron configuration ${\left\{2{f}_{\frac{7}{2}}2{f}_{\frac{7}{2}}1{h}_{\frac{11}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ plays a considerable role in the calculations of the first excited ${0}^{+}$ state of ${}^{132}$Sn. We found that the two-phonon configurations ${[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ and ${[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ are split between the excited ${0}_{2}^{+}$ and ${0}_{3}^{+}$ states. On the other hand, the wave function of the ${0}_{2,3}^{+}$ states particularly mix 2.1% of the ${\left[{0}_{6}^{+}\right]}_{RPA}$ and 2.9% of the ${\left[{0}_{8}^{+}\right]}_{RPA}$ configurations into the ${0}_{2}^{+}$ state; and 1.0% of the ${\left[{0}_{6}^{+}\right]}_{RPA}$ and 1.2% of the ${\left[{0}_{8}^{+}\right]}_{RPA}$ configurations into the ${0}_{3}^{+}$ state. This small change in the structure has a large effect on the ${\left|\langle {0}_{2,3}^{+}|{\widehat{M}}_{\lambda =0}|{0}_{g.s.}^{+}\rangle \right|}^{2}$ values. To check the structure for the low-lying ${0}^{+}$ states, the PPC calculations were performed in the space of all the one- and two-phonon configurations with energies up to 25 MeV (column

**A**in Table 3) and 30 MeV (column

**B**in Table 3), respectively. The obtained results demonstrate the expected unimportance of the high-energy configurations. On the other hand, the structure of LTP ${0}^{+}$ states of ${}^{132}$Sn is dependent on the choice of Skyrme parametrizations (see column

**B**in Table 3). It is well known that the single-particle spectra around the Fermi level are key for microscopic analyses. In the case of the force SkM${}^{*}$, we found that the main components of the ${0}_{1}^{+}$ wave function are the ${[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ and ${[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ configurations. In addition, the structure of the first ${0}^{+}$ state comes from the 2.5% mix of the ${\left[{0}_{6}^{+}\right]}_{RPA}$ and 2.7% of the ${\left[{0}_{8}^{+}\right]}_{RPA}$ configurations. These configurations give the crucial contribution in the ${\left|\langle {0}_{1}^{+}|{\widehat{M}}_{\lambda =0}|{0}_{g.s.}^{+}\rangle \right|}^{2}$ value, as shown in Figure 5a. Clearly, further theoretical and experimental studies are required to provide additional insight into the low-energy structure of excited ${0}^{+}$ states.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The calculated ISGMR strength distributions for ${}^{118}$Sn (panel (

**a**)), ${}^{120}$Sn (panel (

**b**)), and ${}^{122}$Sn (panel (

**c**)) are compared with the experimental data from [10]. The QRPA results (dashed curves) are scaled down by a factor of 2 for a better display. The solid curves correspond to the results obtained by means of the coupling between the one- and two-phonon configuration. The distributions were calculated with the Skyrme force SLy4. The smoothing parameter 1 MeV was used for the strength distribution described by the Lorentzian function.

**Figure 2.**The calculated ISGMR strength distributions in ${}^{124}$Sn are compared with the experimental data from [10] using the Skyrme forces SLy4 (panel (

**a**)), SGII (panel (

**b**)), and SkM${}^{*}$ (panel (

**c**)). The QRPA results (dashed curves) are scaled down by a factor of 2 for a better display. The solid curves correspond to the results obtained by means of the coupling between the one- and two-phonon configuration. The smoothing parameter 1 MeV was used for the strength distribution described by the Lorentzian function.

**Figure 3.**As in Figure 1, for ${}^{126}$Sn (panel (

**a**)), ${}^{128}$Sn (panel (

**b**)), and ${}^{130}$Sn (panel (

**c**)), the distributions were calculated with the Skyrme force SLy4.

**Figure 4.**The phonon–phonon coupling effect on the ISGMR of ${}^{132}$Sn. The RPA results (top panel (

**a**)) are scaled down by a factor of 2 for a better display. The results obtained with inclusion of the PPC effects are given in the bottom panel (

**b**). The distributions were calculated with the Skyrme force SLy4. The unit of transition probabilities (the strength function) refers to the vertical axis on the left-hand (right-hand) side. The smoothing parameter 1 MeV was used for the strength distribution described by the Lorentzian function.

**Table 1.**Energies, transition probabilities, and structures of the RPA low-lying states: ${2}_{1}^{+}$, ${3}_{1}^{-}$, ${4}_{1}^{+}$, and ${5}_{1}^{-}$ of ${}^{132}$Sn using the Skyrme forces SLy4, SGII, and SkM${}^{*}$. The $B(E\lambda ;{\lambda}_{1}^{\pi}\to {0}_{gs}^{+})$ values are given in Weisskopf units. The two-quasiparticle configuration contributions greater than 10% are given.

${\mathit{\lambda}}_{1}^{\mathit{\pi}}$ | Energy (MeV) | $\mathit{B}(\mathit{E}\mathit{\lambda};{\mathit{\lambda}}_{1}^{\mathit{\pi}}\to {0}_{\mathbf{gs}}^{+})$ (W.u.) | Structure | |
---|---|---|---|---|

SLy4 | ${\left[{2}_{1}^{+}\right]}_{RPA}$ | 4.5 | 6.9 | 61%${\left\{2{f}_{\frac{7}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ |

33%${\left\{2{d}_{\frac{5}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

${\left[{3}_{1}^{-}\right]}_{RPA}$ | 5.5 | 28.0 | 12%${\left\{1{i}_{\frac{13}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ | |

12%${\left\{2{f}_{\frac{7}{2}}3{s}_{\frac{1}{2}}\right\}}_{\nu}$ | ||||

16%${\left\{1{h}_{\frac{11}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

11%${\left\{1{g}_{\frac{7}{2}}2{p}_{\frac{1}{2}}\right\}}_{\pi}$ | ||||

${\left[{4}_{1}^{+}\right]}_{RPA}$ | 5.1 | 11.3 | 66%${\left\{2{f}_{\frac{7}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ | |

19%${\left\{2{d}_{\frac{5}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

${\left[{5}_{1}^{-}\right]}_{RPA}$ | 6.9 | 8.5 | 85%${\left\{2{f}_{\frac{7}{2}}2{d}_{\frac{3}{2}}\right\}}_{\nu}$ | |

SkM${}^{*}$ | ${\left[{2}_{1}^{+}\right]}_{RPA}$ | 4.3 | 6.0 | 70%${\left\{2{f}_{\frac{7}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ |

25%${\left\{2{d}_{\frac{5}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

${\left[{3}_{1}^{-}\right]}_{RPA}$ | 4.6 | 22.0 | 22%${\left\{2{f}_{\frac{7}{2}}3{s}_{\frac{1}{2}}\right\}}_{\nu}$ | |

14%${\left\{2{f}_{\frac{7}{2}}2{d}_{\frac{3}{2}}\right\}}_{\nu}$ | ||||

11%${\left\{1{i}_{\frac{13}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ | ||||

10%${\left\{1{h}_{\frac{11}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

${\left[{4}_{1}^{+}\right]}_{RPA}$ | 4.8 | 9.5 | 75%${\left\{2{f}_{\frac{7}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ | |

13%${\left\{2{d}_{\frac{5}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

${\left[{5}_{1}^{-}\right]}_{RPA}$ | 5.5 | 4.9 | 92%${\left\{2{f}_{\frac{7}{2}}2{d}_{\frac{3}{2}}\right\}}_{\nu}$ | |

SGII | ${\left[{2}_{1}^{+}\right]}_{RPA}$ | 4.3 | 7.0 | 63%${\left\{2{f}_{\frac{7}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ |

27%${\left\{2{d}_{\frac{5}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

${\left[{3}_{1}^{-}\right]}_{RPA}$ | 4.5 | 27.5 | 16%${\left\{2{f}_{\frac{7}{2}}3{s}_{\frac{1}{2}}\right\}}_{\nu}$ | |

12%${\left\{1{i}_{\frac{13}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ | ||||

11%${\left\{1{h}_{\frac{11}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

11%${\left\{1{g}_{\frac{7}{2}}2{p}_{\frac{1}{2}}\right\}}_{\pi}$ | ||||

${\left[{4}_{1}^{+}\right]}_{RPA}$ | 4.8 | 13.3 | 49%${\left\{2{f}_{\frac{7}{2}}1{h}_{\frac{11}{2}}\right\}}_{\nu}$ | |

24%${\left\{1{g}_{\frac{7}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

13%${\left\{2{d}_{\frac{5}{2}}1{g}_{\frac{9}{2}}\right\}}_{\pi}$ | ||||

${\left[{5}_{1}^{-}\right]}_{RPA}$ | 6.2 | 10.6 | 75%${\left\{2{f}_{\frac{7}{2}}2{d}_{\frac{3}{2}}\right\}}_{\nu}$ |

**Table 2.**Energies and transition probabilities for the low-lying states ${2}_{1}^{+}$, ${3}_{1}^{-}$, ${4}_{1}^{+}$, and ${5}_{1}^{-}$ of ${}^{132}$Sn using the Skyrme forces SLy4, SGII, and SkM${}^{*}$. The PPC effects are taken into account. The experimental data are taken from [44,57,58]. The $B(E\lambda ;{\lambda}_{1}^{\pi}\to {0}_{gs}^{+})$ values are given in Weisskopf units.

${\mathit{\lambda}}_{1}^{\mathit{\pi}}$ | Energy (MeV) | $\mathit{B}(\mathit{E}\mathit{\lambda};{\mathit{\lambda}}_{1}^{\mathit{\pi}}\to {0}_{\mathbf{gs}}^{+})$ (W.u.) | ||||||
---|---|---|---|---|---|---|---|---|

Expt. | Theory | Expt. | Theory | |||||

SLy4 | SkM${}^{*}$ | SGII | SLy4 | SkM${}^{*}$ | SGII | |||

${2}_{1}^{+}$ | 4.04 | 4.4 | 4.0 | 3.8 | $5.5\pm 1.5$ | 6.8 | 5.6 | 6.3 |

$4.4\pm 1.0$ | ||||||||

${3}_{1}^{-}$ | 4.35 | 5.2 | 4.2 | 3.8 | $>7.1$ | 26.2 | 20.1 | 24.1 |

${4}_{1}^{+}$ | 4.42 | 5.0 | 4.6 | 4.4 | $7.7\pm 0.4$ | 11.2 | 9.0 | 12.3 |

${5}_{1}^{-}$ | 4.94 | 6.9 | 5.5 | 6.0 | – | 8.5 | 4.8 | 9.7 |

**Table 3.**Energies, dominant RPA components of phonon structures, and NEWSR fraction of the LTP ${0}^{+}$ states of ${}^{132}$Sn using the Skyrme forces SLy4, SGII, and SkM${}^{*}$. Columns (

**A**) and (

**B**) give the values calculated in the space of all the one- and two-phonon configurations with energies up to 25 MeV and 30 MeV, respectively.

${\mathit{\lambda}}_{\mathit{i}}^{\mathit{\pi}}$ | Energy (MeV) | Structure | NEWSR Fraction | ||||
---|---|---|---|---|---|---|---|

A | B | A | B | A | B | ||

SLy4 | ${0}_{1}^{+}$ | 8.8 | 8.7 | 96$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | 93$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | 1.1% | 1.6% |

${0}_{2}^{+}$ | 10.0 | 9.9 | 61$\%{[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ | 49$\%{[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ | 5.0% | 6.0% | |

+27$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | +34$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | ||||||

${0}_{3}^{+}$ | 10.4 | 10.4 | 55$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | 47$\%{[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ | 3.2% | 1.9% | |

+37$\%{[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ | +43$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | ||||||

SkM${}^{*}$ | ${0}_{1}^{+}$ | 8.1 | 8.0 | 46$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | 42$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | 6.5% | 7.8% |

+39$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | +38$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | ||||||

${0}_{2}^{+}$ | 8.6 | 8.6 | 52$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | 59$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | 1.9% | 1.6% | |

+43$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | +36$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | ||||||

${0}_{3}^{+}$ | 9.6 | 9.6 | 94$\%{[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ | 92$\%{[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ | 0.2% | 0.2% | |

SGII | ${0}_{1}^{+}$ | 7.7 | 7.6 | 62$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | 60$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | 10.2% | 11.4% |

+17$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | +16$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | ||||||

${0}_{2}^{+}$ | 8.5 | 8.5 | 80$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | 81$\%{[{2}_{1}^{+}\otimes {2}_{1}^{+}]}_{RPA}$ | 0.3% | 0.3% | |

+19$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | +18$\%{[{3}_{1}^{-}\otimes {3}_{1}^{-}]}_{RPA}$ | ||||||

${0}_{3}^{+}$ | 9.5 | 9.5 | 93$\%{[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ | 92$\%{[{4}_{1}^{+}\otimes {4}_{1}^{+}]}_{RPA}$ | 0.2% | 0.2% |

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**MDPI and ACS Style**

Arsenyev, N.N.; Severyukhin, A.P.
Origin of Low- and High-Energy Monopole Collectivity in ^{132}Sn. *Universe* **2021**, *7*, 145.
https://doi.org/10.3390/universe7050145

**AMA Style**

Arsenyev NN, Severyukhin AP.
Origin of Low- and High-Energy Monopole Collectivity in ^{132}Sn. *Universe*. 2021; 7(5):145.
https://doi.org/10.3390/universe7050145

**Chicago/Turabian Style**

Arsenyev, Nikolay N., and Alexey P. Severyukhin.
2021. "Origin of Low- and High-Energy Monopole Collectivity in ^{132}Sn" *Universe* 7, no. 5: 145.
https://doi.org/10.3390/universe7050145