# Overview of Seniority Isomers

^{*}

## Abstract

**:**

## 1. Introduction

^{213}Pb using the seniority isomerism [43]. This paper is not an extensive review for all of such efforts but provides a general overview of the subject in and around semi-magic nuclei on the basis of seniority and generalized seniority. Many microscopic model explanations for these mass regions are also available, but in this overview, we mostly cover the discussion based on the simple concepts of seniority and generalized seniority. The details of quasi-spin algebra for both the single-j and multi-j shells in terms of seniority and generalized seniority, respectively, have briefly been reviewed in Section 2 along with the resulting electromagnetic (plus the seniority) selection rules. Section 3 compares the possibility of seniority isomerism in single-j to the generalized seniority isomerism in multi-j. Due to available advanced experimental and theoretical tools, the seniority isomers have been explored at the extremes of the binding energies; a few of them include [21,25,26,27,47], providing an opportunity to test the rigidity of spherical magic numbers. The examples of seniority isomers along with their structural details are given in Section 4. Section 5 reports a collection of predictions where seniority provides a useful guidance. Section 6 concludes the paper.

## 2. Quasi-Spin Algebra: Single-j and Multi-j

- A quasi-spin scalar if $l+{l}^{\prime}+L$ is odd.
- $\kappa =0$ component of the quasi-spin vector if $l+{l}^{\prime}+L$ is even.

## 3. Seniority (and Generalized Seniority) Isomers: Where and Why?

^{+}(fully coupled) state in the multi-j scenario for a two-body odd-tensor interaction ${V}_{ik}$ may be defined as ${V}_{0}=\langle {\tilde{j}}^{2}J=0|{V}_{ik}|{\tilde{j}}^{2}J=0\rangle $. In terms of a ${j}^{v}$ configuration, the matrix element for a ${j}^{n}$ configuration may be represented as [37]:

## 4. Seniority Isomers in Various Mass Regions

#### 4.1. Ca Isotopes

#### 4.2. Ni Isotopes

^{+}state in the case of ${}^{70}$Ni but not in ${}^{76}$Ni [17].

^{+}isomers for four ${g}_{9/2}$ particles/holes in ${}^{72}$Ni and ${}^{74}$Ni [20] has been provided using seniority mixing arguments [17]. As soon as four particles/holes start to occupy the ${g}_{9/2}$ orbital, the possible states would be ${0}^{+}$ (twice), ${2}^{+}$ (twice), ${3}^{+}$, ${4}^{+}$ (thrice), ${5}^{+}$, ${6}^{+}$ (thrice), and ${7}^{+}$, ${8}^{+}$, ${9}^{+}$, ${10}^{+}$, ${12}^{+}$ [62]. The ${8}^{+}$ states in both of these Ni isotopes are able to decay by the allowed and faster $\Delta v=2$ transition to the second ${6}^{+}$ state, which has $v=4$, since two levels for $J={4}^{+}$ and $J={6}^{+}$ exist and are fairly near in energy. According to Equation (14), the enhanced transition probability between the $v=2,{8}^{+}$ and $v=4,{6}^{+}$ states causes the 8

^{+}state’s half-life to be shorter, resulting in no isomer. For these 8

^{+}seniority isomers (states) in neutron-rich Ni isotopes, there are no moment measurements available.

#### 4.3. Sn Isotopes

#### 4.4. Pb Isotopes

#### 4.5. N = 28 Isotones

#### 4.6. N = 50 Isotones

#### 4.7. N = 82 Isotones

#### 4.8. N = 126 Isotones

#### 4.9. Cd and Te Isotopes

#### 4.10. Hg and Po Isotopes

#### 4.11. N = 48 and N = 52 Isotones

#### 4.12. N = 80 and N = 84 Isotones

#### 4.13. N = 124 and N = 128 Isotones

## 5. Predictions and Open Issues

- The measurement gaps in the systematics of seniority isomers require special attention from the experimentalists, since this missing piece of information would be crucial for the theoretical developments. The presence of $v=2,{8}^{+}$ isomers in neutron-rich ${}^{128}$Pd and ${}^{130}$Cd, for example, strongly suggests the same $v=2$ isomers in more neutron-rich ${}^{126}$Ru and ${}^{124}$Mo, $N=82$ isotonic nuclei. Similarly, future experimental data on the $v=2,{10}^{+}$ isomer in neutron-rich ${}^{124}$Ru $(Z=44,N=80)$ (due to two neutron holes in ${h}_{11/2}$) will be immensely helpful in understanding the function of pairing in such a limiting and exceedingly neutron-rich nucleus. The existence of ${10}^{+},v=2$ isomers in ${}^{128}$Cd also suggests the dominance of the neutron ${h}_{11/2}^{-2}$ configuration for the lower lying ${8}^{+}$ state. Any information on the ${8}^{+}$ state due to two proton ${g}_{9/2}$ holes in ${}^{128}$Cd $(Z=48,N=80)$ would be equally important for assessing the competitiveness between proton and neutron two-body configurations. Similar investigation for the ${8}^{+}$ states in ${}^{126}$Pd would also be encouraging to understand the structural evolution.
- The particle-number independent variation of the magnetic moments for the good (generalized) seniority states can be used to predict the g-factors for the gaps in measurements. For example, the g-factor for the ${6}^{+}$ isomer in ${}^{46}$Ca should be in the similar order to the g-factor of the ${6}^{+}$ isomer in ${}^{42}$Ca. Despite the fact that the two ${6}^{+}$ states in ${}^{44}$Ca have different seniorities, the g-factor should be equal to neutron ${f}_{7/2}$ owing to the pure-j configuration. Similarly, the g-factor in ${}^{70,76}$Ni isotopes for the ${8}^{+}$ isomers should be comparable and, if measured, would represent the nature of the implicated neutron ${g}_{9/2}$ orbital. The same can be said for the ${8}^{+}$ states in ${}^{72,74}$Ni isotopes, which are not isomeric due to the additional and permitted decay branch. Similar would be true for the seniority isomers in medium to heavy mass nuclei such as the ${10}^{+}$ isomers in ${}^{120,122,124,126,130}$Sn isotopes, the ${12}^{+}$ isomers in ${}^{190}$Pb, and lighter ${}^{186,184,182,\dots}$Pb isotopes. The same is true for the g-factors of seniority isomers in various isotonic chains.
- If the isomers have the same origin and only differ in terms of an extra odd-particle, the g-factor of odd-A isomers will be in the same order as that of even-A isomers for a given isotopic or isotonic chain. The g-factor for the ${27/2}^{-}$ isomers in odd-A ${}^{117,119,121,123,125,127,129}$Sn isotopes, for example, would be of the same order as the ${10}^{+}$ isomers in even-A Sn isotopes. The same will hold true for the odd-A ${33/2}^{+}$ isomers in lighter odd-A ${}^{183,185,187,189,191}$Pb isotopes due to their similarity to the neighboring even-A ${12}^{+}$ isomers. Similarly, the g-factor for the ${10}^{+}$ and ${27/2}^{-}$ isomers in respective even-A $Z=66,68,70,72$, and odd-A $Z=67,69,71,73$, $N=82$ isotones should be almost equal to each other. The Schmidt value for proton ${h}_{11/2}$ is $+1.42$ n.m., although the GSSM estimate for the mixed proton ${h}_{11/2}\otimes {d}_{3/2}\otimes {s}_{1/2}$ configuration is +1.27 n.m. Such future moment measurements would provide the complete understanding of a nuclear structure for these isotonic isomers.
- To address the similarities and differences in the behavior of ${11/2}^{-}$ states in Cd, Sn and Te isotopes, the Q-moment measurements in heavier Te isotopes are of current experimental interest, particularly when similar measurements for Cd and Sn isotopes are now known with great precision at the ISOLDE facility [81,115]. Since the $v=1,{11/2}^{-}$ states are found to occur quite regularly in Cd, Sn and Te isotopes for the range of $N=65-81$, one can expect the higher seniority isomers such as $v=2,{10}^{+}$, $v=3,{27/2}^{-}$, $v=4,{15}^{-}$ in the Cd and Te isotopes, similar to the Sn isotopes.
- The lack of experimental data on E2 decay properties of the first ${2}^{+},{4}^{+}$ states below the seniority isomers in $N=50$ isotones prevents a conclusion on the seniority conservation in $j=9/2$ from being established. Similarly, in other heavier mass regions, firm E2 assignments below the most-aligned seniority isomer are not yet available. The E2 properties for the states below the neutron-rich ${6}^{+}$ seniority isomers in Sn isotopes beyond ${}^{132}$Sn and the states below the ${8}^{+}$ seniority isomers in Pb isotopes beyond ${}^{208}$Pb, for example, will undoubtedly contribute to realistic and effective nuclear shell model interactions. To fully comprehend the neutron–neutron/proton–proton as well as neutron–proton two-body matrix elements, comprehensive spectroscopic information for isomers and states below isomers in two-particles/holes nuclei with respect to semi-magic nuclei such as Cd and Te isotopes, Hg and Po isotopes, $N=48$, 52 isotones, $N=80$, 84 isotones, $N=124$, 128 isotones is necessary.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(Color online) Schematic level scheme for (9/2)

^{2}configuration in single-j shell. The levels are identified by the seniority quantum number on the extreme right. $\Delta v=0$ and $\Delta v=2$ refer to the seniority-preserving and seniority-changing transitions, respectively.

**Figure 2.**(Color online) Schematic variation of electromagnetic reduced transition probabilities in both single-j shell and multi-j shell by using the electromagnetic and (generalized) seniority selection rules. The additional possibility of odd electric tensor decaying seniority isomers has been pointed out in the upper panel.

**Figure 3.**(Color online) Empirical energy systematics for the 0

^{+}to 10

^{+}states in Sn isotopes from N = 52 to 80, where v is the seniority quantum number. The major contributing orbitals are also pointed out before and after the middle of the neutron 50–82 valence space.

**Figure 4.**(Color online) B(E2) variation of v = 2 isomers in various semi-magic chains; (

**a**) for $Z=50$ isotopes, (

**b**) for $Z=82$ isotopes, (

**c**) for $N=82$ isotones, and (

**d**) for $N=126$ isotones. GS and S refer to the generalized seniority and seniority results, respectively. The uncertainties in the experimental data [80] are shown but mostly lie within the size of symbol.

**Figure 5.**(Color online) g-factor variation of v = 2 isomers in different semi-magic chains. GSSM refers to the estimates from the Generalized Seniority Schmidt Model. Exp. data have been taken from [82].

**Figure 6.**(Color online) A comparison of the experimental [80] and seniority calculated B(E2) trends from the g

_{9/2}orbital for the seniority v = 2, 8

^{+}isomers in both the N = 50 isotones (

**upper**panel) and Pb isotopes (

**lower**panel).

**Figure 7.**(Color online) Empirical energy systematics for the yrast and yrare 8

^{+}states in N = 48 isotones. The isotonic nuclei, where magnetic moment measurements for these ${8}_{1}^{+}$ isomers are known, are shown with green shade. The negative sign for these moments strongly supports the neutron ${g}_{9/2}^{-2}$ configuration.

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Maheshwari, B.; Nomura, K.
Overview of Seniority Isomers. *Symmetry* **2022**, *14*, 2680.
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Maheshwari B, Nomura K.
Overview of Seniority Isomers. *Symmetry*. 2022; 14(12):2680.
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Maheshwari, Bhoomika, and Kosuke Nomura.
2022. "Overview of Seniority Isomers" *Symmetry* 14, no. 12: 2680.
https://doi.org/10.3390/sym14122680