# Systematic shell model study for $N=82$ and $N=126$ isotones and nuclear
isomers

Bharti<EMAIL_ADDRESS>and Praveen C.
<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Roorkee, Roorkee 247 667, India

###### Abstract

In the present work, we have done a systematic shell model study of $N=82$ and
$N=126$ isotones. For the $N=82$ isotones, we have performed calculations
using SN100PN interaction, while for $N=126$ isotones, we have used KHPE
interaction. Similarities between these two isotonic chains have been
reported, using the strong resemblance between the high-$j$ orbitals. Apart
from the nuclear spectroscopic properties, we have also explained different
isomeric states in these two regions. In the $N$ =82 region, we have mainly
discussed the properties of the $6^{+}$ and $17/2^{+}$ isomers, while in the
$N=126$ region for $8^{+}$, $11^{-}$, $21/2^{-}$ and $29/2^{+}$ isomers. We
have reported $B(E2)$, $B(E3)$, $g$-factor, and quadrupole moments of the
isomeric states for comparison in these two isotonic chains.

###### pacs:

21.60.Cs, 21.10.−k, 23.20.-g, 27.60.+j, 27.80.+w

##  I Introduction

The spectroscopy of 132Sn and 208Pb and its neighbors in many ways are similar
to each other. Several nuclear structure studies have been done theoretically
and experimentally by comparing the spectroscopy. Some similarities have been
pointed out in Coraggio1 ; Coraggio2 ; Coraggio3 ; Coraggio4 ; Leander ,
showing universal patterns of nuclear structure in both regions. Using the
correspondence between the high-$j$ orbitals located above the $Z$ = 50 and
$Z$ = 82 shell gaps, Astier et al. astier1  assigned the isomeric state of
217Pa as the fully aligned state of the $\pi h_{9/2}^{2}f_{7/2}$
configuration. They used this analogy to assign the isomeric state of 217Pa
with compared to the $J^{\pi}$= $17/2^{+}$ isomeric state measured in 139La.

The single-particle energy gaps in the two cases are comparably large, and the
orbitals above and below the shell gaps are similarly arranged blomqvist .
Every single particle state in the 132Sn region has its counterpart in the
vicinity of 208Pb. Several groups Coraggio1 ; Coraggio2 ; Coraggio3 ;
Coraggio4 ; Leander ; blomqvist ; erokhina  have worked to develop a general
theoretical description of nuclear structural properties near and far from the
$\beta$-stability line using the shell model. They have emphasized the
desirability of particular comparisons between experimental information around
the 132Sn and look for the corresponding nuclear properties in the 208Pb
region.

In semi-magic nuclei, the seniority quantum number distinguishes between the
different nuclear states with identical valence nucleons occupying one orbital
or a simple configuration according to the nucleon number Racah . This
property provides us with an appropriate framework to establish a similarity
between the 132Sn and 208Pb regions in terms of the orbital in each model-
space using the shell-model wave-function. Seniority $\nu$ is the number of
particles not in pairs coupled to angular momentum $J$ = 0. In the seniority
scheme, the properties of a nucleus can be described from the neighboring
isotopes or isotones with the addition or removal of valence nucleons Shalit ;
Talmi1 ; Talmi2 ; Shlomo . Studies of even-even semi-magic nuclei within the
generalized seniority scheme scholten ; sandul ; karta  have shown that the
seniority-zero state can define the ground state. And the seniority $\nu>0$
states by pairs coupled to two unpaired nucleons represent low-lying excited
states.

The $N$ = 82 region has long been the subject of shell-model studies due to
the remarkable regularities exhibited in the low-lying states of these nuclei
and also the strong doubly magic character of 132Sn. This region gives an
opportunity over a large number of nuclei ($A$ = 133 to 152) to study the
effect of adding protons to a doubly magic core. Extensive research of this
type was conducted over the past few years with the emergence of the shell
when particles and (or ) holes were added to the 132Sn Andreo1 ; Wildenthal1 .
In this context, experimental data of energy levels and electromagnetic
properties for light even-even $N$ = 82 isotones are rich, and have encouraged
many shell-model calculations sarkar ; Wildenthal2 ; Berant ; Brown ; Astier ;
Bianco . Shell-model calculations using effective interactions derived from
realistic nucleon-nucleon ($NN$) potentials have been performed in the $N$ =
82 region with some excellent results Andreo2 ; Covello1 ; holt ; Suhonen ;
Covello2 . The evolution of the low-energy properties of the $N$ = 82 isotones
as a function of $A$ has been well reproduced by the theory in some of these
studies Covello1 ; holt ; Suhonen .

The lead isotopes and the $N$=126 nuclei have long been the subject of both
experimental and theoretical studies. These stable or near stable nuclei have
been extensively investigated. A relatively large amount of experimental data
is available in this region. While the high-spin states in nuclei near 208Pb
have been studied extensively, data on electromagnetic transition rates
between the low-spin states are insufficient. The good doubly magic character
of 208Pb suggests successful shell model results and has motivated many shell-
model calculations. On the other hand, the heavy mass 208Pb region with
protons and neutrons filling different major shells offers favorable
conditions for the formation of isomeric states due to the presence of
high-$j$ orbitals. With the close resemblance between proton-neutron
multiplets in the 132Sn and 208Pb regions, an analogue of 208Pb Coraggio1 ;
Coraggio2 ; Coraggio3 ; Coraggio4 ; Cie , has led to a new focus on low-spin
states in the lead region. The energy levels of 210Bi, 212Bi, 212At Coraggio1
; Coraggio2 ; Coraggio3  were performed by employing an effective interaction
derived from Bonn-A $NN$ interaction. Using the highlights of previous works
by different groups, the structural information and the electromagnetic
properties can be predicted with the shell-model calculation for the Sn and Pb
region.

There are experimental indications for the similar properties of nuclei
belonging to $N$ = 82 and $N$ = 126 regions, but no systematic theoretical
results are available. Motivated with the recent experimental data for
spectroscopic properties, we have performed a systematic shell model study of
$N=82$ ($52\leq Z\leq 60$) and $N=126$ ($84\leq Z\leq 92$) isotones. The aim
of the present paper is to investigate the similarity between $N=82$ and
$N=126$ isotones. The present study will add more information to the earlier
works.

These two isotonic chains are very rich in terms of nuclear isomers astier1 ;
Jain ; Stuchbery ; Walker . Thus we have also reported shell model results of
nuclear isomers in these two regions. For the $N=82$ region we have discussed
$6^{+}$ and $17/2^{+}$ isomers, while for $N=126$ region the $8^{+}$,
$11^{-}$, $21/2^{-}$ and $29/2^{+}$ isomers. The role of $g_{7/2}^{2}$ and
$(g_{7/2}d_{5/2})^{n}$ configurations in the $N=82$ region, while
$h_{9/2}^{n}$, $(h_{9/2}f_{7/2})^{n}$ and $(h_{9/2}i_{13/2})^{n}$
configurations in $N=126$ are very crucial to decide nuclear isomers in these
two regions.

This paper is organized as follows. In the Sec. II, we present details about
theoretical formalism. Comprehensive discussions are reported in Sec. III. In
this section, we have discussed spectroscopy of $N=82$ and $N=126$ isotones in
terms of seniority. Finally, a summary and conclusions are drawn in Sec. IV.

##  II Formalism: model space and interactions

The shell-model calculations for the N = 82 isotones have been performed in
the 50-82 valence shell composed of the orbits $1g_{7/2}$, $2d_{5/2}$,
$2d_{3/2}$, $3s_{1/2}$, and $1h_{11/2}$ using the interaction SN100PN taken
from Brown et al.SN100PN1 ; Brown . This interaction has four parts,
corresponding to the neutron-neutron, neutron-proton, proton-proton component
of the nuclear forces and Coulomb repulsion between the protons. The residual
two-body interaction was obtained from the CD-Bonn G-matrix within the so-
called $\hat{Q}$ -box folded diagram theory and the further nn part was
multiplied by a factor of 0.90 to improve the agreement with the experiment
for 130Sn. The single-particle energies for the neutrons are -10.610, -10.289,
-8.717, -8.694, and -8.816 MeV for the $1g_{7/2}$, $2d_{5/2}$, $2d_{3/2}$,
$3s_{1/2}$, and $1h_{11/2}$ orbitals, respectively, and those for the protons
are 0.807, 1.562, 3.316, 3.224, and 3.605 MeV. We have performed calculations
with the SN100PN interaction in some previous studies in this region 2019 ;
2015 ; 2013 .

The shell-model calculations for the N = 126 isotones have been performed in
the valence shell 82-126 for proton and 126-184 for neutron using the
interaction KHPE Warburton1 . The model space here consists of
$1h_{9/2},2f_{7/2},2f_{5/2},3p_{3/2},3p_{1/2},1i_{13/2}$ proton orbitals and
$1i_{11/2},2g_{9/2},2g_{7/2},3d_{5/2},3d_{3/2},4s_{1/2},1j_{15/2}$ neutron
orbitals. The effective realistic residual interaction of Kuo and Herling Kuo1
; Kuo2  was derived from a free nucleon-nucleon potential of Hamada and
Johnston Hamada  with renormalization due to the finite extension of model
space by the reaction matrix techniques developed by Kuo and Brown Kuo3 . In
the present work, we have performed shell model calculations with KHPE
interaction without any truncation. Recently, we have performed an extensive
calculation for Rn isotopes with the KHPE interaction Rn . In the present
work, we have used the shell model code KShell kshell  for the diagonalization
of matrices.

##  III Results and discussion

###  III.1 $N$ = 82 even isotones

In Figures 1, 2 we present the calculated energy levels of $N$ = 82 isotones
for even and odd nuclei, respectively, obtained in the $50-82$ model space in
comparison with the experimental data. In the $N$ = 82 isotones, the states
correspond to proton excitations, for which the configurations are built
within the proton orbitals ($1g_{7/2}$, $2d_{5/2}$, $2d_{3/2}$, $3s_{1/2}$,
$1h_{11/2}$). The study of wave functions suggests that for all the low-lying
positive-parity states, the valence protons occupy the two orbitals, $\pi
g_{7/2}$ and $\pi d_{5/2}$. The main configurations of the low-lying positive-
parity states correspond to $(\pi g_{7/2})^{n}$, $\pi(g_{7/2})^{n-1}d_{5/2}$,
and $\pi(g_{7/2})^{n-2}d_{5/2}^{2}$ or configuration mixing of these
components, where $n$ is the number of valence protons. We can simply
represent these configurations as $\pi(g_{7/2}d_{5/2})^{n}$. The main
structure of negative-parity states in the $N$ = 82 isotones above 132Sn is
constructed by removing one proton from the $1g_{7/2}2d_{5/2}$ model space to
the $\pi h_{11/2}$ orbital. Then the lowest negative-parity state gives rise
to $(\pi g_{7/2}d_{5/2})^{n-1}h_{11/2}$ configuration. Our calculation shows
that different configurations identify these low-lying states with varied
natures. In our calculation, we take the low-lying yrast states, i.e., the
assembling of the lowest state for each spin. One can see that the shell-model
calculation very well reproduces the experimental energy levels.

Figure 1:  Comparison of calculated and experimental nndc  excitation energy
spectra for the $N$ = 82 even isotones using SN100PN interaction.

The 134Te contains two valence protons outside the 132Sn core. The present
calculations reproduce the characteristic of spherical even-even nuclei that
there is a large gap of about 1.2 MeV between ground state $0^{+}$ and the
first excited state $2^{+}$ as shown in Fig 1. The wave functions of the
calculated states are pure, indicating the dominance of single-particle
excitations in this nucleus. The lowest four states of 134Te arise from the
$\pi g_{7/2}^{2}$ configuration. The ground state of 134Te may be described by
two protons residing in the $\pi g_{7/2}$ orbital as one pair coupled to
angular momentum $J$ = 0, i.e., seniority $\nu$ = 0. For the yrast chain of
$\pi g_{7/2}^{2}$ configuration with seniority $\nu$ = 2, the $4^{+}_{1}$ and
$6^{+}_{1}$ levels are also reproduced well. The other positive-parity states
with energy between 2-3 MeV from proton excitation within the major shell are
reproduced well. These states have the main configuration of $\pi
g_{7/2}d_{5/2}$, with seniority $\nu$=2. The $2^{+}_{3}$ level at 3.055 MeV,
which is close to the experimental $2^{+}$ level at 2.934 MeV, has a
configuration of $\pi d_{5/2}^{2}$ (71%). The negative-parity states are
observed above 3.5 MeV. These states belong to the multiplets of $\pi
g_{7/2}h_{11/2}$, with seniority $\nu$=2. The levels of $9^{-}$, $7^{-}$ from
the $\pi g_{7/2}h_{11/2}$ multiplet have good accordance with experimental
data in both energy and order.

The low-lying states in 136Xe can be well understood in terms of seniority,
with a $N$ = 82 closed neutron shell and four protons beyond the $Z$ = 50
closed shell. The $\pi g_{7/2}^{4}$ configuration with seniorities $\nu$ = 0
and 2 gives rise to a state with $J^{\pi}$ = $0^{+}$ and a multiplet with
$J^{\pi}$ = $2^{+}_{1}$, $4^{+}_{1}$, $6^{+}_{1}$, $4^{+}_{2}$, $2^{+}_{3}$,
$8^{+}_{1}$ in order of increasing energy. These states are well reproduced,
with over 50-90% probability of this configuration in the wave functions of
the respective calculated states. For the $0^{+}$ ground state of 136Xe this
contribution amounts to 53%. The wave functions of the states are rather pure
with small admixtures, indicating the dominance of single-particle
excitations. Different multiplets arising from three protons in the $\pi
g_{7/2}$ orbital and a single proton in the $\pi d_{5/2}$ orbital ($\nu$ = 2
states) and two protons in each orbital ($\nu$ = 0 states) are also possible.
The $\pi g_{7/2}^{3}d_{5/2}$ $\nu$ = 2 configuration produces a $J^{\pi}$ =
$2^{+}_{2}$, $3^{+}_{1}$, $6^{+}_{2}$, $8^{+}_{2}$, $10^{+}_{1}$ multiplet,
where the even-spin states are reproduced well. The $6^{+}_{2}$ state at 2.262
MeV decays only to the $6^{+}_{1}$ state and not to the $4^{+}_{1}$ state, is
observed in nndc . The calculation shows that the states $9^{-}_{1}$,
$3^{-}_{1}$ are the member of the $\pi g_{7/2}^{3}h_{11/2}$ $\nu$ = 2
multiplet.

For 138Ba, the multiplets of $\pi g_{7/2}^{4}d_{5/2}^{2}$ are $J^{\pi}$ =
$0^{+}_{1}$, $2^{+}_{1}$, $2^{+}_{2}$, $4^{+}_{1}$, $6^{+}_{2}$. However, the
contribution of $\pi g_{7/2}^{6}$ is also not small in the $0^{+}_{1}$ ,
$2^{+}_{1}$, $4^{+}_{1}$ and $6^{+}_{2}$ states, with 20-30% probability of
this configuration. For the states $J^{\pi}$ = $6^{+}_{1}$, $4^{+}_{2}$,
$3^{+}_{1}$, $2^{+}_{3}$, $1^{+}_{1}$, $8^{+}_{1}$, $10^{+}_{1}$ the leading
configuration is $\pi g_{7/2}^{5}d_{5/2}$, contribution amounts to 40-60%. The
state $3^{-}_{1}$ is mainly due to the configuration $\pi
g_{7/2}^{4}d_{5/2}h_{11/2}$. The state $9^{-}_{1}$ is mainly due to one-proton
excitation from the $\pi g_{7/2}$ orbital to $\pi h_{11/2}$, with the
predominant configuration of $\pi g_{7/2}^{5}h_{11/2}$. One can see the
$0^{+}_{1}$ state has seniority $\nu$ = 0, and the even parity states as well
as the $3^{-}_{1}$, $9^{-}_{1}$ states have seniority $\nu$ = 2. There are a
few pair states which have two or three pairs, such as the $8^{+}_{1}$,
$10^{+}_{1}$ states with seniority $\nu$=4. This indicates that the
generalized seniority is a good quantum number here.

140Ce can be described similarly to 138Ba with two more protons in the valence
space. The $\pi g_{7/2}^{6}d_{5/2}^{2}$ $\nu$=0,2 configuration gives rise to
$J^{\pi}$ = $0^{+}_{1}$, $2^{+}_{1}$, $4^{+}_{1}$, $2^{+}_{2}$, $4^{+}_{2}$,
$6^{+}_{2}$ multiplets. The $\pi g_{7/2}^{5}d_{5/2}^{3}$ $\nu$=2 configuration
produces $J^{\pi}$ = $6^{+}_{1}$, $5^{+}_{1}$, $3^{+}_{1}$, $1^{+}_{1}$,
$2^{+}_{3}$, multiplet in order of increasing energy. However, these states
from the two configuration chain are arising from significant configuration
mixing with $\pi g_{7/2}^{4}d_{5/2}^{4}$ and $\pi g_{7/2}^{7}d_{5/2}$,
respectively for the two chains, with 12-20% probabilities of these
configurations mixing with over 35-45% of the leading configuration. Similar
to 138Ba, the states $9^{-}_{1}$ and $7^{-}_{1}$ are mainly due to one-proton
excitation from the $\pi g_{7/2}$ orbital to $\pi h_{11/2}$, with the
predominant configuration of $\pi g_{7/2}^{5}d_{5/2}^{2}h_{11/2}$. The
calculation shows that the state $3^{-}_{1}$ is a member of the $\pi
g_{7/2}^{6}d_{5/2}h_{11/2}$ multiplet.

For 142Nd, we compare selected calculated levels to their experimental
counterparts. The $\pi g_{7/2}^{6}d_{5/2}^{4}$ $\nu$=0,2 configuration gives
rise to a $J^{\pi}$ = $0^{+}_{1}$, $2^{+}_{1}$, $4^{+}_{1}$, $6^{+}_{2}$
multiplet. However, these states are arising from significant configuration
mixing with $\pi g_{7/2}^{6}d_{5/2}^{2}h_{11/2}^{2}$ with 10-15% probability
of this configuration mixing with over 20-35% of the leading configuration.
Contrast to 138Ba and 140Ce, the states $3^{-}_{1}$ and $7^{-}_{1}$ are mainly
due to one-proton excitation from the $\pi d_{5/2}$ orbital to $\pi h_{11/2}$
and the $9^{-}_{1}$ state is due to one-proton excitation from the $\pi
g_{7/2}$ orbital to $\pi h_{11/2}$ with the predominant configuration of $\pi
g_{7/2}^{6}d_{5/2}^{3}h_{11/2}$ and $\pi g_{7/2}^{5}d_{5/2}^{4}h_{11/2}$,
respectively. Thus these negative parity states are seniority $\nu$ = 2
states.

Interestingly, it can be seen that the $9^{-}_{1}$ isomeric state of 134Te and
136Xe is better represented by coupling the octupole phonon to the $6^{+}_{2}$
non-isomeric state rather than to the $6^{+}_{1}$ isomeric state, while the
$9^{-}_{1}$ isomeric state of 138Ba, 140Ce and 142Nd, is better represented by
coupling the octupole phonon to the $6^{+}_{1}$ isomeric state. Particularly,
it was suggested for 136Xe and 138Ba in Ref Hwang . From this calculation, a
broad understanding into the structure of the low-lying levels can be
obtained.

###  III.2 $N$ = 82 odd isotones

Figure 2:  Comparison of calculated and experimental nndc  excitation energy
spectra for the $N$ = 82 odd isotones using SN100PN interaction.

135I is a single-closed nucleus with three valence protons. The present
results well explain those measured experimental energies of the low-lying
positive-parity states. Most of the experimental states in 135I are tentative,
and our calculation confirms all the proposed states with spin-parity in
correct order. In 135I, the low-lying energies are obtained mainly in terms of
the three configurations, $\pi g_{7/2}^{3}$, $\pi g_{7/2}^{2}d_{5/2}$, and
$\pi g_{7/2}^{2}h_{11/2}$, which are mostly pure states with dominant
configuration probability over 75-100$\%$. The lowest-energy configuration
$\pi g_{7/2}^{3}$ can create $J^{\pi}$ = $5/2^{+}$, $7/2^{+}$, $9/2^{+}$,
$11/2^{+}$ and $15/2^{+}$. Similar to $J^{\pi}$ = $0^{+}$ with seniority $\nu$
= 0 in 134Te, the ground state $J^{\pi}$ = $7/2^{+}$ with seniority single-
proton, $\nu$ = 1 in 135I. Our calculated wave functions indicate that the
$5/2^{+}_{2}$, $7/2^{+}_{1}$, $9/2^{+}_{1}$, $11/2^{+}_{1}$ and $15/2^{+}_{1}$
states have the predominant configuration $\pi g_{7/2}^{3}$. The $\nu$ = 3
multiplet contains the $5/2^{+}_{2}$ , $9/2^{+}_{1}$, $11/2^{+}_{1}$ and
$15/2^{+}_{1}$ states where the corresponding compositions are over 90$\%$.
These states, therefore correspond to the single-particle nature. The dominant
configuration of the lower $5/2^{+}_{1}$ state is $\pi g_{7/2}^{2}d_{5/2}$
($\nu$ = 1). The $17/2^{+}_{1}$ state is found to be the member of the same
multiplet as $5/2^{+}_{1}$ state, but $\nu$ = 3. The negative-parity states
$19/2^{-}_{1}$, $23/2^{-}_{1}$, and $21/2^{-}_{1}$ between 3 and 4 MeV are due
to one proton excitation from the orbital $\pi g_{7/2}$ into the orbital $\pi
h_{11/2}$. The predominant configuration is $\pi g_{7/2}^{2}h_{11/2}$, with
seniority $\nu$ = 3.

For 137Cs, the calculated low-lying yrast states are due to configuration
mixing. The $J^{\pi}$ = $7/2^{+}_{1}$, $5/2^{+}_{2}$, $3/2^{+}_{1}$,
$11/2^{+}_{1}$, and $15/2^{+}_{1}$ states have a leading configuration of $\pi
g_{7/2}^{5}$, but with some differences. The $7/2^{+}_{1}$ and $11/2^{+}_{1}$
states are mixed with a considerable amount of $\pi g_{7/2}^{3}d_{5/2}^{2}$
component, while this component is small in $15/2^{+}_{1}$. The $7/2^{+}_{1}$
ground state, in which two $0^{+}$ proton pairs are coupled to the odd
$g_{7/2}$ proton is seniority $\nu$ = 1 state. As in 135I, the $\nu$ = 3
states are $J^{\pi}$ = $5/2^{+}_{2}$, $3/2^{+}_{1}$, $11/2^{+}_{1}$, and
$15/2^{+}_{1}$, excitations with a $\pi g_{7/2}$ broken pair coupled to a
proton in the $\pi g_{7/2}$. In 137Cs an extra pair of valence protons is
available to generate high-spin yrast states. The $5/2^{+}_{1}$,
$1/2^{+}_{1}$, $17/2^{+}_{1}$, and $19/2^{+}_{1}$ states belong to the $\pi
g_{7/2}^{4}d_{5/2}$ multiplet with seniority $\nu$ = 3, with a $\pi g_{7/2}$
broken pair coupled to a proton in the $\pi d_{5/2}$, except for
$21/2^{+}_{1}$ which is seniority $\nu$ = 5 state. The $23/2^{+}_{1}$ is a
member of the $\pi g_{7/2}^{3}d_{5/2}^{2}$, $\nu$ = 5. The negative parity
excitations involve one proton from $\pi g_{7/2}$ to $\pi h_{11/2}$ orbital,
with $\nu$ = 3. The negative parity states $J^{\pi}$ = $9/2^{-}_{1}$,
$19/2^{-}_{1}$, $21/2^{-}_{1}$ and $23/2^{-}_{1}$ are multiplets of $\pi
g_{7/2}^{4}h_{11/2}$ configuration.

For 139La, the calculated low-lying yrast states for $J$= $1/2$ to $23/2$ are
due to the mixing of several configurations, except for $19/2^{+}_{1}$,
$21/2^{+}_{1}$ and $23/2^{+}_{1}$, where leading configuration dominates. The
$J^{\pi}$ = $7/2^{+}_{1}$, $5/2^{+}_{2}$, $11/2^{+}_{1}$, $15/2^{+}_{1}$,
$19/2^{+}_{1}$, and $23/2^{+}_{1}$ states have a leading configuration of $\pi
g_{7/2}^{5}d_{5/2}^{2}$, configuration probability ranges from 39-82$\%$. The
$1/2^{+}_{1}$ state is arising from $\pi g_{7/2}^{4}d_{5/2}^{2}s_{1/2}$
configuration with 24% probability. The $7/2^{+}_{1}$ ground state and
$1/2^{+}_{1}$ are seniority $\nu$ = 1 state, in which $0^{+}$ proton pairs are
coupled to the odd $g_{7/2}$ and $s_{1/2}$ proton, respectively. The $\nu$ = 3
states with $J^{\pi}$ = $5/2^{+}_{2}$, $11/2^{+}_{1}$, $15/2^{+}_{1}$ and
$\nu$ = 5 states $19/2^{+}_{1}$, $23/2^{+}_{1}$, are arising due to coupling
of $\pi g_{7/2}$ or $\pi d_{5/2}$ broken pair (or pairs) with $\pi g_{7/2}$.
The $\pi g_{7/2}^{4}d_{5/2}^{3}$ seniority $\nu$ = 3 and $\nu$ = 5, gives rise
to the multiplet $3/2^{+}_{1}$, and $21/2^{+}_{1}$, respectively. The
$J^{\pi}$ = $5/2^{+}_{1}$, $9/2^{+}_{1}$, $13/2^{+}_{1}$ and $17/2^{+}_{1}$
states have a leading configuration of $\pi g_{7/2}^{6}d_{5/2}$, excitations
with a $\pi g_{7/2}$ broken pair coupled to a proton in the $\pi d_{5/2}$. The
negative-parity states are due to one proton excitation from the orbital $\pi
g_{7/2}$ or $\pi d_{5/2}$ into the orbital $\pi h_{11/2}$. The dominant
configuration $\pi g_{7/2}^{4}d_{5/2}^{2}h_{11/2}$, with seniority $\nu$ = 3
gives rise to the multiplet $9/2^{-}_{1}$, $11/2^{-}_{1}$ and $23/2^{-}_{1}$.
The $11/2^{-}_{2}$ and $21/2^{-}_{1}$ are $\nu$ = 3 states, with dominant
configuration $\pi g_{7/2}^{5}d_{5/2}h_{11/2}$. The $19/2^{-}_{1}$ state is a
$\nu$ = 3 state with dominant configuration $\pi g_{7/2}^{6}h_{11/2}$
(41$\%$), and has a significant contribution from $\pi
g_{7/2}^{4}d_{5/2}^{2}h_{11/2}$ (34$\%$) configuration.

For 141Pr, the excitation energies and ordering of the positive-parity yrast
states are well reproduced in the present calculations. The negative parity
states are with some discrepancies, as they are compressed in energy with
respect to the experimental data. The 141Pr has similar multiplet structure as
139La, as well the calculated wave functions have a significant admixture of
configurations involving excitations of the $\pi g_{7/2}$ or $\pi d_{5/2}$
orbital. Compared to low-lying states, relatively less configuration mixing is
observed at high-lying states. Whereas, in 141Pr, the number of multiplets and
maximum spin achievable is large with an extra pair of valence protons in $\pi
g_{7/2}$ or $\pi d_{5/2}$ orbital. In 141Pr, $5/2^{+}$ state becomes the
ground state as the number of protons increases, in which $0^{+}$ proton pairs
are coupled to the odd $d_{5/2}$ is a seniority $\nu$ = 1 state, with
configuration $\pi g_{7/2}^{6}d_{5/2}^{3}$. Similar to 139La, the $\pi
g_{7/2}^{6}d_{5/2}^{3}$ multiplets are seniority $\nu$ = 3 or 5, with a $\pi
g_{7/2}$ broken pair(s) coupled to a proton in the $\pi d_{5/2}$, with
$J^{\pi}$ = $3/2^{+}_{1}$, $9/2^{+}_{1}$, $17/2^{+}_{1}$, $19/2^{+}_{1}$ and
$21/2^{+}_{1}$, where $19/2^{+}_{1}$ and $21/2^{+}_{1}$ are $\nu$ = 5 states.
The $\pi g_{7/2}^{5}d_{5/2}^{4}$ multiplets are $J^{\pi}$ = $5/2^{+}_{2}$,
$7/2^{+}_{1}$, $11/2^{+}_{1}$, $15/2^{+}_{1}$ with $\nu$ = 3 and
$23/2^{+}_{1}$ is $\nu$ = 5 state. The $1/2^{+}_{1}$ is a $\nu$ =1 state of
$\pi g_{7/2}^{6}d_{5/2}^{2}s_{1/2}$ configuration, and $13/2^{+}_{1}$ is a
$\nu$ =3 state of $\pi g_{7/2}^{7}d_{5/2}^{2}$ configuration. As in 139La with
two added protons, the negative-parity states are mainly due to $\pi
g_{7/2}^{5}d_{5/2}^{3}h_{11/2}$ and $\pi g_{7/2}^{6}d_{5/2}^{2}h_{11/2}$
configurations with seniority $\nu$ = 3. The first group has members $J^{\pi}$
= $17/2^{-}_{1}$, $19/2^{-}_{1}$, $21/2^{-}_{1}$ and $23/2^{-}_{1}$ states,
the states with $J^{\pi}$ = $11/2^{-}_{1}$, $13/2^{-}_{1}$ and $15/2^{-}_{1}$
are member of second group.

###  III.3 $N$ = 126 even isotones

Figure 3:  Comparison of calculated and experimental nndc  excitation energy
spectra for the $N$ = 126 even isotones using KHPE interaction.

Isotones with $N$ = 126 represent an excellent example of the applicability of
the seniority scheme with $\nu\geq 2$, as the low-lying spectrum follows the
seniority scheme for the semi-magic nuclei above the 208Pb core. Calculations
in the entire range of proton configurations using KHPE interaction showed a
good agreement with the experimental data. This calculation confirmed the
seniority conservation for the isotones up to $Z$ = 92.

In Figures 3, 4 we present the calculated energy levels of $N$ = 126 isotones
for even and odd nuclei, respectively, obtained in the shell-model
configuration space in comparison with the experimental data. In the $N$ = 126
isotones, the low-lying states correspond to proton excitations, for which the
configurations are built within the proton orbitals
($1h_{9/2},2f_{7/2},2f_{5/2},3p_{3/2},3p_{1/2},1i_{13/2}$). The study of wave
functions suggests that for all the low-lying positive-parity states, the
valence protons occupy the three orbitals, $\pi h_{9/2}$, $\pi f_{7/2}$ and
$\pi i_{13/2}$. The main configurations of the low-lying positive-parity
states correspond to $(\pi h_{9/2})^{n}$, $\pi(h_{9/2})^{n-1}f_{7/2}$,
$\pi(h_{9/2})^{n-2}f_{7/2}^{2}$, and
$\pi(h_{9/2})^{n-4}f_{7/2}^{2}i_{13/2}^{2}$ or combinations of them, where $n$
is the number of valence protons. We can simply represent these configurations
as $(\pi h_{9/2}f_{7/2}i_{13/2})^{n}$. The main structure of negative-parity
states in the $N$ = 126 isotones above 208Pb is simply constructed by exciting
one proton from the $\pi h_{9/2}f_{7/2}$ model space to the $\pi i_{13/2}$
orbital, and also from $\pi(h_{9/2}f_{7/2})^{n}$ for odd isotones. Our
calculation shows that these low-lying states are identified by different
configurations with a combination of mixed and pure states. In our
calculation, we take the low-lying yrast states. One can see the experimental
energy levels are very well reproduced for the $N$ = 126 isotones by the
shell-model calculation.

210Po contains two valence protons outside the 208Pb core. The low-lying yrast
states, $0^{+}_{1}$, $2^{+}_{1}$, $4^{+}_{1}$, $6^{+}_{1}$ and $8^{+}_{1}$ are
constructed by proton-dominated two-particle excitations of the $\pi
h_{9/2}^{2}$ configuration, in a similar fashion to the situation in 134Te.
The wave functions of the states are pure, indicating the dominant single-
particle excitations in 210Po. In contrast, two protons need to occupy the
$\pi i_{13/2}$ orbital to make states above $10^{+}$. The $0^{+}_{1}$ ground
state is a seniority $\nu$ = 0 state. The states $J^{\pi}$ = $2^{+}_{1}$,
$4^{+}_{1}$, $6^{+}_{1}$ and $8^{+}_{1}$ are seniority $\nu$ = 2 states, with
one broken proton pair. The $8^{+}_{2}$ state is a member of the $\pi
h_{9/2}f_{7/2}$ configuration and seniority $\nu$ = 2. The $11^{-}_{1}$ state
consists of the $\pi h_{9/2}i_{13/2}$ configuration.

The low-lying states in 212Rn are made up of four valence protons beyond the
$Z$ = 82 closed shell, with a $N$ = 126 closed neutron shell should be well
understood in terms of seniority structure, same as 136Xe. In the first four
excited states $J^{\pi}$ = $2^{+}_{1}$, $4^{+}_{1}$, $6^{+}_{1}$ and
$8^{+}_{1}$ the percentage of the dominant $\pi h_{9/2}^{4}$ configurations,
ranges from 60% to 65% while it becomes 42% in the ground state. All the
levels arising from the $\pi h_{9/2}^{4}$ configuration have a configuration
mixing with the $\pi h_{9/2}^{2}i_{13/2}^{2}$ and $\pi h_{9/2}^{2}f_{7/2}^{2}$
configurations. The ground state of 212Rn may be described by four protons
residing in the $\pi h_{9/2}$ orbital as two pairs coupled to angular momentum
$J$ = 0, i.e., seniority $\nu$ = 0. The excited states are seniority $\nu$ = 2
states of one broken pair with the dominant $\pi h_{9/2}^{4}$ configuration.
The $8^{+}_{2}$ excited state dominated by the $\pi h_{9/2}^{3}f_{7/2}$
configuration is a seniority $\nu$ = 2 state with one odd $\pi h_{9/2}$ proton
coupled to $\pi f_{7/2}$, with 71% contribution. The $8^{+}_{2}$ state has a
significant mixing with $\pi h_{9/2}f_{7/2}i_{13/2}^{2}$ and $\pi
h_{9/2}f_{7/2}^{3}$ configurations. The $10^{+}_{1}$ state is dominated by
$\pi h_{9/2}^{4}$ configuration, with over 95% contribution. The $10^{+}_{1}$
is seniority $\nu$ =4 state with the breaking of two proton pairs. The low-
lying $11^{-}$ dominated by $\pi h_{9/2}^{3}i_{13/2}$ configuration is a
seniority $\nu$ = 2 state.

For 214Ra, 216Th and 218U, the low-lying states are due to significant
configuration mixing. In 214Ra, the components of $\pi h_{9/2}^{6}$ are
$J^{\pi}$ = $2^{+}_{1}$, $4^{+}_{1}$, $6^{+}_{1}$, $8^{+}_{1}$. These states
are arising from significant configuration mixing with $\pi
h_{9/2}^{4}f_{7/2}^{2}$ and $\pi h_{9/2}^{4}i_{13/2}^{2}$, with 24%
probabilities of these configurations mixing with over 28-32% of the leading
configuration. The ground state of 214Ra can be described by $\pi
h_{9/2}^{4}i_{13/2}^{2}$ with 24% probability as two pairs coupled to angular
momentum $J$ = 0, i.e., seniority $\nu$ = 0. However, the $\pi h_{9/2}^{6}$
(21%) and $\pi h_{9/2}^{4}f_{7/2}^{2}$ (21%) components are also not small for
the ground state. The $8^{+}_{2}$ excited state dominated by the $\pi
h_{9/2}^{5}f_{7/2}$ configuration is a seniority $\nu$ = 2 state with two
proton pairs and one odd $\pi h_{9/2}$ proton coupled to $\pi f_{7/2}$, with
over 43% contribution. The state $11^{-}_{1}$ is mainly due to one-proton
excitation from the $\pi h_{9/2}$ orbital to $\pi i_{13/2}$, with the
predominant configuration of $\pi h_{9/2}^{5}i_{13/2}$. One can see the
$0^{+}_{1}$ state is seniority $\nu$ = 0 state, and the even parity states as
well as the $11^{-}_{1}$ are seniority $\nu$ = 2 states.

216Th can be described similarly to 214Ra with two more protons in the valence
space. The $\pi h_{9/2}^{6}f_{7/2}^{2}$, $\nu$=2 configuration gives rise to a
$J^{\pi}$ = $2^{+}_{1}$, $4^{+}_{1}$, $6^{+}_{1}$, $8^{+}_{1}$ multiplets.
However, these states are arising from significant configuration mixing with
$\pi h_{9/2}^{6}i_{13/2}^{2}$ and $\pi h_{9/2}^{4}f_{7/2}^{2}i_{13/2}^{2}$,
with over 16-20% probabilities of these configurations mixing with over 20-22%
of the leading configuration. Similar to 214Ra with two extra proton in $\pi
f_{7/2}$ and $\pi i_{13/2}$ orbital, the ground state can be described by
seniority $\nu$ = 0 with $\pi h_{9/2}^{4}f_{7/2}^{2}i_{13/2}^{2}$
configuration and the $8^{+}_{2}$ is a seniority $\nu$ = 2 state dominated by
the $\pi h_{9/2}^{5}f_{7/2}i_{13/2}^{2}$ configuration. The $0^{+}_{1}$ is due
to admixture of $\pi h_{9/2}^{6}f_{7/2}^{2}$ and $\pi h_{9/2}^{6}i_{13/2}^{2}$
configurations with almost the same contribution 16-17% as the dominant one
(18%). The $11^{-}_{1}$ is a seniority $\nu$ = 2 state of $\pi
h_{9/2}^{5}i_{13/2}^{3}$ configuration with 23%, including the admixture of
$\pi h_{9/2}^{5}f_{7/2}^{2}i_{13/2}$ and $\pi h_{9/2}^{7}i_{13/2}$
configurations with over 20-22% contribution.

Showing similar structure as 216Th the multiplets from the configuration $\pi
h_{9/2}^{6}f_{7/2}^{2}$ including the ground state now arises from $\pi
h_{9/2}^{6}f_{7/2}^{2}i_{13/2}^{2}$ in 218U with two added valence protons.
The multiplets $J^{\pi}$ = $0^{+}_{1}$, $2^{+}_{1}$, $4^{+}_{1}$, $6^{+}_{1}$,
$8^{+}_{2}$ are due to significant mixing with several configurations, with
8-9% probabilities of these configurations mixing with 19-22% of the leading
configuration. In 218U the contribution of configurations apart from the
dominant one has decreased compared to 214Ra and 216Th. The seniority scheme
here is the same as the 214Ra and 216Th. In contrast to other even $N$ = 126
isotones, the $8^{+}_{1}$ state is a multiplet of $\pi
h_{9/2}^{5}f_{7/2}^{3}i_{13/2}^{2}$, highly mixed with $\pi
h_{9/2}^{7}f_{7/2}^{3}$ and $\pi h_{9/2}^{7}f_{7/2}i_{13/2}^{2}$
configurations with almost the same contribution 13-15% as the dominant one
(over 15%). The $11^{-}_{1}$ is a multiplet of $\pi
h_{9/2}^{5}f_{7/2}^{2}i_{13/2}^{3}$ with 19%, including the admixture of $\pi
h_{9/2}^{7}f_{7/2}^{2}i_{13/2}$ and $\pi h_{9/2}^{7}i_{13/2}^{3}$
configurations with over 16% contributions.

###  III.4 $N$ = 126 odd isotones

Figure 4:  Comparison of calculated and experimental nndc  excitation energy
spectra for the $N$ = 126 odd isotones using KHPE interaction.

The spectra of the odd-even isotones calculated assuming the pure
$h_{9/2}^{n}$ configuration with seniority $\nu$ = 1 and 3 are compared with
the experimental data. The experimental spectra themselves no longer contain
all the levels predicted by the seniority scheme. These isotones belong to the
region of neutron-rich short-lived nuclei and, the precise determination of
their spectra of excited states is a complex experimental task.

The low-lying levels of 211At can be described by the coupling of three
valence protons residing outside the $Z$ = 82, $N$ = 126 closed shell. The
level scheme of 211At, 213Fr and 215Ac are very similar. In 211At, all the
spins considered are pure states with 96-100% contribution and for ground
state over 67% contribution. The 210Po and 211At both are confined to a single
orbital $\pi h_{9/2}$ for the low-lying states. The results are explained
relating in a $\pi h_{9/2}^{2}$ configuration (210Po) to those in a $\pi
h_{9/2}^{3}$ configuration (211At). The $J^{\pi}$ = $9/2^{-}_{1}$,
$13/2^{-}_{1}$, $17/2^{-}_{1}$ and $21/2^{-}_{1}$ states are identically
defined by three protons in $\pi h_{9/2}$ with seniority $\nu$ = 3, except for
$9/2^{-}_{1}$. For $9/2^{-}_{1}$, there are two independent states with
seniority $\nu$ = 1 and 3, respectively. In 211At, the $9/2^{-}_{1}$ is the
ground state and we can assume it has seniority $\nu$ = 1. The $19/2^{-}_{1}$
and $23/2^{-}_{1}$ states are formed by $\pi h_{9/2}^{2}f_{7/2}$ configuration
with seniority $\nu$ = 3. The $29/2^{+}_{1}$ and $25/2^{+}_{1}$ are seniority
$\nu$ = 3 states with one proton excitation from $\pi h_{9/2}$ to $\pi
i_{13/2}$ orbital, which is $\pi h_{9/2}^{2}i_{13/2}$ configuration. The
energy of the $19/2^{-}_{1}$ state is higher compared to other first negative
parity states. Because the single-particle energy of the $\pi f_{7/2}$ orbital
is higher than that of the $\pi h_{9/2}$ orbital, the energy of the
$19/2^{-}_{1}$ state is higher compared to the $17/2^{-}_{1}$ and
$21/2^{-}_{1}$ states. The $17/2^{-}_{1}$ and $21/2^{-}_{1}$ states consist of
the $\pi h_{9/2}^{3}$ configuration. In contrast the $19/2^{-}_{1}$ state
cannot be constructed by the $\pi h_{9/2}^{3}$ configuration.

In 213Fr, the calculated low-lying yrast states are due to configuration
mixing. The $J^{\pi}$ = $9/2^{-}_{1}$, $13/2^{-}_{1}$, $17/2^{-}_{1}$ and
$21/2^{-}_{1}$ states have a leading configuration of $\pi h_{9/2}^{5}$, with
38-60% probability which is the lowest for the ground state. The $9/2^{-}_{1}$
ground state, in which two $0^{+}$ proton pairs are coupled to the odd
$h_{9/2}$ proton, is seniority $\nu$ = 1 state. As in 211At, the seniority
$\nu$ = 3 states are $J^{\pi}$ = $13/2^{-}_{1}$, $17/2^{-}_{1}$ and
$21/2^{-}_{1}$, excitations with a broken pair coupled to a proton in the $\pi
h_{9/2}$ orbital. The $19/2^{-}_{1}$ and $23/2^{-}_{1}$ states belong to the
$\pi h_{9/2}^{4}f_{7/2}$ multiplet is seniority $\nu$ = 3 state, with a broken
pair coupled to a proton in the $\pi f_{7/2}$ orbital. The $29/2^{+}_{1}$ is a
seniority $\nu$ = 3 state having a leading configuration of $\pi
h_{9/2}^{4}i_{13/2}$, where two protons in the $\pi h_{9/2}$ orbital with spin
eight coupled to the last proton in the $\pi i_{13/2}$ orbital. The maximum
spin of this configuration is 29/2.

For 215Ac, the calculated yrast states are due to large configuration mixing.
The ground state $9/2^{-}_{1}$ has the leading configuration $\pi
h_{9/2}^{5}i_{13/2}^{2}$, although another configuration $\pi
h_{9/2}^{5}f_{7/2}^{2}$ has a significant probability almost the same as the
leading term of 22%. This shows the importance of $\pi i_{13/2}$ orbital for
the formation of low-lying states around $Z$ = 90 and above it. The $J^{\pi}$
= $13/2^{-}_{1}$, $17/2^{-}_{1}$ and $21/2^{-}_{1}$ states have a leading
configuration of $\pi h_{9/2}^{5}f_{7/2}^{2}$, with probability over 27$\%$.
However, these states are coming from the mixing of two more configurations,
$\pi h_{9/2}^{5}i_{13/2}^{2}$ and $\pi h_{9/2}^{7}$, with over 19-24%
contribution. The $23/2^{-}_{1}$ and $19/2^{-}_{1}$ states belong to the $\pi
h_{9/2}^{6}f_{7/2}$ multiplet, and the $29/2^{+}_{1}$ state belongs to the
$\pi h_{9/2}^{6}i_{13/2}$ multiplet, with over 38% contribution. The seniority
scheme for the states followed here is the same as those in 211At and 213Fr.

For 217Pa, the calculated yrast states are also due to large configuration
mixing. The $J^{\pi}$ = $9/2^{-}_{1}$, $13/2^{-}_{1}$ and $17/2^{-}_{1}$
states have a leading configuration of $\pi
h_{9/2}^{5}f_{7/2}^{2}i_{13/2}^{2}$, with probability from 20-23$\%$, which is
the lowest for the ground state. Similar to the 215Ac, these states are coming
from the mixing of two more configurations $\pi h_{9/2}^{7}i_{13/2}^{2}$ and
$\pi h_{9/2}^{7}f_{7/2}^{2}$, with over 12-18% contribution with the leading
term. The $23/2^{-}_{1}$, $19/2^{-}_{1}$ and $21/2^{-}_{1}$ states belong to
the $\pi h_{9/2}^{6}f_{7/2}i_{13/2}^{2}$ multiplet with seniority $\nu$ = 3,
with a $\pi h_{9/2}$ broken pair coupled to a proton in the $\pi f_{7/2}$. The
$29/2^{+}_{1}$ state belongs to the $\pi h_{9/2}^{6}f_{7/2}^{2}i_{13/2}$
multiplet with seniority $\nu$ = 3, where a broken pair in the $\pi h_{9/2}$
orbital coupled to the last proton in the $\pi i_{13/2}$ orbital. In contrast
to the other odd $N$ = 126 isotones, the excitation energy of the isomeric
state $23/2^{-}_{1}$ is lower than the $21/2^{-}_{1}$ ($\pi h_{9/2}^{3}$) at
$Z$ = 91. Previously, this was also expected in the experimental work 217pa .

###  III.5 Similarity between Sn and Pb regions

The low-lying states are expected to rise only from the three high-$j$
orbitals $\pi g_{7/2}$, $\pi d_{5/2}$ and $\pi h_{11/2}$ for the $N$ = 82
isotones with $Z\geq$ 50\. For the $N$ = 82 isotones ($52\leq Z\leq 60$), the
breaking of a first proton pair generates different multiplets occupying the
$\pi g_{7/2}$ and $\pi d_{5/2}$ orbitals depending on the number of valence
protons. In doubly even nuclei, the low-lying excited states $J^{\pi}\leq
6^{+}$ are generated from breaking of one proton pair with seniority $\nu\leq
2$ from the $\pi(g_{7/2}d_{5/2})^{n}$ configuration. The positive parity
states with $J^{\pi}\geq 6^{+}$ are possible in the same configuration space
coexisting with core-couple states. Hence, for the states, $J^{\pi}\geq 6^{+}$
seniority must be $\nu\geq 4$ with breaking a second or more proton pair (up
to $J\leq 12$). Although the admixture of configuration from the $\pi d_{3/2}$
and $\pi s_{1/2}$ orbitals are small but very crucial to generate the correct
level scheme 140Ce ; 141Pr . In doubly even nuclei, odd parity states
$J^{\pi}\leq 9^{-}$ are generated by the ($\pi h_{11/2},\pi d_{5/2}$) and
($\pi h_{11/2},\pi g_{7/2}$) multiplets.

Figure 5: Calculated excitation energies of the fully-aligned states with one
broken proton pair in the $N$ = 82 isotones with $52\leq Z\leq 60$ (a), and in
the $N$ = 126 isotones with $84\leq Z\leq 92$ (b) in comparison with
experimental excitation energies for $N$ = 82 isotones (c), and $N$ = 126
isotones (d). The isomeric states are highlighted with filled symbols.

Figure 6:  Variation of proton effective single-particle energies for
different orbitals of $N$ = 82 and $N$ = 126 isotones.

Similarly, in the Pb-region, particular orbitals are consistently responsible
for forming the low-lying states and isomers. The low-lying states are
expected to rise only from the three high-$j$ orbitals $\pi h_{9/2}$, $\pi
f_{7/2}$ and $\pi i_{13/2}$ for the $N$ = 126 isotones with $Z\geq$ = 82. For
the $N$ = 126 isotones ($84\leq Z\leq 92$), the breaking of a first proton
pair generates different multiplets occupying the $\pi h_{9/2}$ and $\pi
f_{7/2}$ orbitals depending on the number of valence protons. In doubly even
nuclei, the low-lying excited states $J^{\pi}\leq 8^{+}$ are generated from
breaking of one proton pair with seniority $\nu\leq 2$ from the
$\pi(h_{9/2}f_{7/2})^{n}$ configuration. The positive parity states with
$J^{\pi}\geq 8^{+}$ are possible in the same configuration space coexisting
with core-couple states. Hence, for the states $J^{\pi}\geq 8^{+}$ seniority
must be $\nu\geq 4$ with breaking of a second or more proton pair (up to
$J\leq 14$). In doubly even nuclei, odd parity states $J^{\pi}\leq 11^{-}$ are
generated by the ($\pi i_{13/2},\pi h_{9/2}$) and ($\pi i_{13/2},\pi f_{7/2}$)
multiplets, with seniority $1\leq\nu\leq 4$ for states up to $J^{\pi}\leq
17^{-}$.

Fig. 5 shows the similarity between the shell structures in the 132Sn and
208Pb regions for the fully-aligned states with one broken proton pair in the
$N$ = 82 isotones with $52\leq Z\leq 60$, involving the three orbitals just
above the $Z$ = 50 gap, $\pi g_{7/2},\pi d_{5/2}$, and $\pi h_{11/2}$, and in
the $N$ = 126 isotones with $84\leq Z\leq 92$, involving the three orbitals
just above the $Z$ = 82 gap, $\pi h_{9/2},\pi f_{7/2}$, and $\pi i_{13/2}$. In
these two regions the orbital angular momentum differ by one unit, with the
same orientation of the intrinsic spin, $\pi g_{7/2}\rightarrow\pi h_{9/2},\pi
d_{5/2}\rightarrow\pi f_{7/2}$, and $\pi h_{11/2}\rightarrow\pi i_{13/2}$.
Similar evolution can be seen for the same seniority $\nu$ = 2 states in the
two $6^{+}$ states of the $N$ = 82 isotones and the two $8^{+}$ states of the
$N$ = 126 isotones. Similarly for the odd isotones with seniority $\nu$ =3 the
behavior between the $15/2^{+}$ and $17/2^{+}$ states of the $N$ = 82 isotones
and the $21/2^{-}$ and $23/2^{-}$ states of the $N$ = 126 isotones.

In Fig. 6, we have shown the variation of proton effective single-particle
energies for $N$ = 82 and $N$ = 126 isotones corresponding to SN100PN and KHPE
interactions, respectively. For $N=82$ isotones, as we increase proton number,
the gap between proton effective single-particle energies for $\pi g_{7/2}$
and $\pi h_{11/2}$ orbital decreases. This effect can be seen as a decrease in
energy (see Fig. 5 ) for $9^{-}$ and $23/2^{-}$ states having configurations
composed of $\pi g_{7/2}$ and $\pi h_{11/2}$. For $N$ = 126 isotones with KHPE
interaction, the gap between proton effective single-particle energies for
$\pi h_{9/2}$ and $\pi i_{13/2}$ orbital decreases by 436 keV as we move from
$Z=84$ to $Z=92$, this is also reflected from the decrease in the energies of
$11^{-}$ and $29/2^{+}$ isomers.

Table 1: The calculated (SM) half-life with configuration for $6^{+}$ state in Sn-region and $8^{+}$ state Pb- region with respect to the experimental data (Expt.) nndc . $N$ = 82 | $\pi g_{7/2}^{2}$ | $\pi g_{7/2}d_{5/2}$ | Transition | T1/2 | T1/2 |  $N$ = 126 | $\pi h_{9/2}^{2}$ | $\pi h_{9/2}f_{7/2}$ | Transition | T1/2 | T1/2
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (SM) | (Expt.) |  |  |  |  | (SM) | (Expt.)
134Te | $6^{+}_{1}$ | - | $E2$ | 126.90 ns | 164.1(9) ns |  210Po | $8^{+}_{1}$ | - | $E2$ | 106.16 ns | 98.9(25) ns
| - | $6^{+}_{2}$ | $M1+E2$ | 1.17 ps |  $<$ 16 ps |  | - | $8^{+}_{2}$ | $M1+E2$ | 33.70 ps | -
136Xe | $6^{+}_{1}$ | - | $E2$ | 1.47 $\mu$s | 2.95(17) $\mu$s |  212Rn | $8^{+}_{1}$ | - | $E2$ | 1.83 $\mu$s | 0.91(3) $\mu$s
| - | $6^{+}_{2}$ | $M1+E2$ | 10.72 ps |  $\leq$ 50 ps |  | - | $8^{+}_{2}$ | $M1$ | 37.56 ps | -
138Ba | - | $6^{+}_{1}$ | $E2$ | 3.60 $\mu$s | 0.85(16) $\mu$s |  214Ra | $8^{+}_{1}$ | - | $E2$ | 130.88 $\mu$s | 67.8(15) $\mu$s
| $6^{+}_{2}$ | - | $M1+E2$ | 11.48 ps | 55(17) ps |  | - | $8^{+}_{2}$ | $M1$ | 58.23 ps | -
140Ce | - | $6^{+}_{1}$ | $E2$ | 13.95 $\mu$s | 7.31(15) $\mu$s |  216Th | $8^{+}_{1}$ | - | $E2$ | 41.90 $\mu$s | 134 (4) $\mu$s
| $6^{+}_{2}$ | - | $M1+E2$ | 6.26 ps |  |  | - | $8^{+}_{2}$ | $M1$ | 25.16 ps | -
142Nd | - | $6^{+}_{1}$ | $E2$ | 5279 $\mu$s | 16.5 $\mu$s |  218U | - | $8^{+}_{1}$ | $E2$ | 26.39 $\mu$s | 0.56${}^{+26}_{-14}$ ms
| $6^{+}_{2}$ | - | $M1+E2$ | 3.94 ps |  |  | $8^{+}_{2}$ | - | $M1$ | 17.90 ps | -

It should be noted that the configuration $(\pi g_{7/2}d_{5/2})^{n}$ is more
appropriate than taking the two orbitals individually as the two orbitals are
close in energy. Similarly, for the configuration $(\pi h_{9/2}f_{7/2})^{n}$
in the Pb-region. Since we have two different orbitals in the configurations
of the even isotones, two different combinations of orbitals can give the same
angular momentum. Such as, $J^{\pi}=6^{+}$ from $\pi g_{7/2}^{2}$ and $\pi
g_{7/2}d_{5/2}$ in Sn-region, and $J^{\pi}=8^{+}$ from $\pi h_{9/2}^{2}$ and
$\pi h_{9/2}f_{7/2}$ in Pb-region. It is found that the low-lying $6^{+}_{1}$
is an isomeric state, regardless of the configuration.

The different natures and the evolution of the two $6^{+}$ and $8^{+}$ states
can be confirmed by the different values of the calculated half-lives in Table
1. We can see the exchange of configuration in both the Sn-region (between two
$6^{+}$) and Pb-region (between two $8^{+}$) at almost in the middle of the
range of isotones $52\leq Z\leq 60$ for $N$ = 82 and $84\leq Z\leq 92$ for $N$
= 126. This is because of the small gap in the single-particle energy between
the orbitals $\pi g_{9/2}$ and $\pi d_{5/2}$ in the Sn-region and $\pi
h_{9/2}$ and $\pi f_{7/2}$ in the Pb-region. Although, from the shell-model
calculation, the configuration exchange for $8^{+}$ is taking place at $Z$ =
92. The half-lives of $6^{+}_{1}$ and $8^{+}_{1}$ states are always larger
than $6^{+}_{2}$ and $8^{+}_{2}$ states, confirming the isomeric nature for
the first yarst states of these spins, despite the change of configuration.
The $6^{+}_{2}$ state decays with the $M1+E2$ transition, and the $M1$
transition shows a large impact by reducing the half-life from ns to ps.
Similarly, the $8^{+}_{2}$ state is solely dominated by the $M1$ transition,
thus giving a smaller half-life than the $8^{+}_{1}$ state. Our calculation
supports the experimental counterpart of the $B(E2)$ and half-lives except for
142Nd and 218U, whose half-lives show large discrepancies with the
experimental values. For 142Nd, the energy levels are reproduced well, but the
$B(E2)$ transition value is very small (0.0003 W.u.). This could be the reason
for the large magnitude of the calculated half-life. In the case of 218U, as
there is no experimental information available for the electromagnetic
transition of $8^{+}$, we can not say due to which part discrepancy is
arising.

We have calculated $M1$ transition for the $8^{+}_{2}$ state of 218U, which is
not measured experimentally. In comparison with the isomeric nature of
$8^{+}_{1}$ state in 216Th, $\pi h_{9/2}f_{7/2}$ configuration has been
proposed in 218U  for the 218U. Following the behavior of $6^{+}$ states in
the Sn-region and our shell-model results, we can propose the configuration of
$8^{+}_{2}$ state for 218U to be based on $\pi h_{9/2}^{2}$ (or $\pi
h_{9/2}^{6}f_{7/2}^{2}i_{13/2}^{2}$ ).

In Fig. 7, the occupancy of different orbitals for the $6^{+}$ isomers in
$N$=82 and $8^{+}$ isomers in $N$=126 regions are shown. The behavior of the
dominant part of the wave function is similar in both the regions, i.e., the
$\pi g_{7/2}$ orbital for $6^{+}$ isomers, while the $\pi h_{9/2}$ orbital for
$8^{+}$ isomers. The contribution of $\pi h_{11/2}$ orbital in the Sn-region
and $\pi i_{13/2}$ orbital in the Pb-region is continuously increasing with
the increase in $Z$ number. This shows the importance of these two high-$j$
orbitals even in the formation of low-lying isomeric states, while the
dominant orbitals $\pi g_{7/2}$ and $\pi h_{9/2}$ are expected to be
sufficient in the formation of low-lying isomeric states, giving a pure
configuration. As we focus precisely on the contribution of different orbitals
in the configuration, it appears that fragmentation of the dominant wave
function into different orbitals increases significantly with the increase in
the $Z$ number. This is reflected in the $B(E2)$ values of these two isomeric
states discussed ahead. In Fig. 8, we can see a sharp decrease in the $B(E2)$
transition values for the $6^{+}$ and $8^{+}$ isomeric state, after $Z$ = 52
and 84, respectively. The $B(E2)$ values are higher at $Z$ = 52 and 84, as the
configuration is pure, without any fragmentation into the other orbitals.
There might be other factors affecting the behavior of the $B(E2)$ transition
value.

Figure 7: The occupancy for different orbitals corresponding to $6^{+}$ and
$8^{+}$ isomers.  Figure 8:  Comparison of calculated and experimental nndc
$B(E2)$ values of different transitions for $N$ = 82 and $N$ = 126 isotones.

Figure 9:  Comparison of calculated and experimental nndc
$B(E2:2^{+}\rightarrow 0^{+})$ and $B(E2:4^{+}\rightarrow 2^{+})$ transition
probabilities and first $2^{+}$ excitation energy for $N$ = 82 and $N$ = 126
isotones.

###  III.6 $B(E2)$ and $B(E3)$

The reduced electric transition probabilities provide much information. Such
as, a small $B(E2)$ value indicates a single-particle structure, while a
larger value provides an understanding of collective motion. It also tells
about the accuracy of the shell-model wave functions. Here, the
$B(E2;J\rightarrow J-2)$ transitions are calculated, interconnecting different
even and odd parity states. The results for different transition rates,
displayed in Fig. 8 and the result of $B(E2:2^{+}\rightarrow 0^{+})$ in Fig.
9. These values are calculated using the standard effective charges
$e_{\pi}=1.5e$ and $e_{\nu}=0.5e$.

Table 2: The calculated (SM) $B(E3)$ transition strengths in W.u. for Sn and Pb region isotones compared to the experimental data (Expt.) nndc  using the effective charge $e_{\pi}=1.5e$. $N$ = 82 | Transition | $B(E3)$ | $B(E3)$ |  $N$ = 126 | Transition | $B(E3)$ | $B(E3)$ |  $N$ = 126 | Transition | $B(E3)$ | $B(E3)$
---|---|---|---|---|---|---|---|---|---|---|---
|  | (SM) | (Expt.) | Even |  | (SM) | (Expt.) | Odd |  | (SM) | (Expt.)
134Te | $9^{-}_{1}\rightarrow 6^{+}_{1}$ | 1.43 | 3.80(14) |  210Po | $11^{-}_{1}\rightarrow 8^{+}_{1}$ | 0.34 | 3.71(10) |  211At | ${29/2}^{+}_{1}\rightarrow{23/2}^{-}_{1}$ | 7.71 | 21.5(19)
| $9^{-}_{1}\rightarrow 6^{+}_{2}$ | 6.70 | 8.2(3) |  | $11^{-}_{1}\rightarrow 8^{+}_{2}$ | 7.62 | 19.7(11) |  |  |  |
136Xe | $9^{-}_{1}\rightarrow 6^{+}_{1}$ | 1.52 |  |  212Rn | $11^{-}_{1}\rightarrow 8^{+}_{1}$ | 0.22 | $1.8^{+6}_{-4}$ |  213Fr | ${29/2}^{+}_{1}\rightarrow{23/2}^{-}_{1}$ | 7.18 | 25.8(23)
| $9^{-}_{1}\rightarrow 6^{+}_{2}$ | 5.83 |  |  | $11^{-}_{1}\rightarrow 8^{+}_{2}$ | 7.11 | $27^{+10}_{-7}$ |  |  |  |
138Ba | $9^{-}_{1}\rightarrow 6^{+}_{1}$ | 5.72 |  |  214Ra | $11^{-}_{1}\rightarrow 8^{+}_{1}$ | 0.19 | 3.09(9) |  215Ac | ${29/2}^{+}_{1}\rightarrow{23/2}^{-}_{1}$ | 6.52 | 27.4(9)
| $9^{-}_{1}\rightarrow 6^{+}_{2}$ | 0.76 |  |  | $11^{-}_{1}\rightarrow 8^{+}_{2}$ | 6.47 | 21.8(6) |  |  |  |
140Ce | $9^{-}_{1}\rightarrow 6^{+}_{1}$ | 4.96 |  |  216Th | $11^{-}_{1}\rightarrow 8^{+}_{1}$ | 0.32 | 21(2) |  217Pa | ${29/2}^{+}_{1}\rightarrow{23/2}^{-}_{1}$ | 5.70 |
| $9^{-}_{1}\rightarrow 6^{+}_{2}$ | 0.12 |  |  | $11^{-}_{1}\rightarrow 8^{+}_{2}$ | 5.57 | 5.0(15) |  |  |  |
142Nd | $9^{-}_{1}\rightarrow 6^{+}_{1}$ | 3.42 |  |  218U | $11^{-}_{1}\rightarrow 8^{+}_{1}$ | 4.45 |  |  |  |  |
| $9^{-}_{1}\rightarrow 6^{+}_{2}$ | 0.04 |  |  | $11^{-}_{1}\rightarrow 8^{+}_{2}$ | 0.55 |  |  |  |  |

A similar general trend of decreasing $B(E2)$ value is observed in Fig. 8 as
the proton number increases for both the $N$ = 82 and 126 isotones. For the
spins, $J\geq 2$ the decrement of $B(E2)$ values away from the shell-closure
is an outcome of the seniority scheme. The $B(E2)$ transition that is even-
tensor, one body transition between the states of the same seniority, will
follow the trend determined by the matrix elements. Relating to this, the
orbitals involved in the wave-functions of the initial and final states differ
by one unit of angular momentum and the formation of the involved states in
transition rates for both regions follows the same seniority scheme.
Surprisingly, the $B(E2)$ transition rates for the same seniority states in
the two different closed-shell regions $N$ = 82 and 126 follow the same trend.
The trend of $B(E2)$ transition rates suggests that seniority is a good
quantum number for $N$ = 82 and 126 shell closure. This argument is only valid
near the shell closure where valence nucleons are small because collectivity
increases for a large number of valence particles and away from the shell
closure.

In Fig. 9, we can observe increment of the $B(E2:2^{+}\rightarrow 0^{+})$
transition probabilities as the proton number increases. The large transition
rates in 138Ba, 140Ce and 142Nd in the Sn-region and 216Th and 218U in the Pb-
region indicate the collective nature of these nuclei. A weak
$B(E2:2^{+}\rightarrow 0^{+})$ transition rate is observed for the 134Te in
the Sn-region similar to 210Po in the Pb-region. Overall, larger $B(E2)$
strengths are reached for larger $Z$ in the $N$ = 126 isotones. Their
strengths are similar as for the $N$ = 82 isotones in the 132Sn mass region.

From the results shown in Fig. 9, it is worth noticing that analogous increase
in the first $2^{+}$ excitation energy is observed for $N$ = 82 and 126
regions as the proton number increases. For instance, 1.211 MeV in 134Te and
1.200 MeV in 210Po to 1.522 MeV in 142Nd and 1.764 MeV in 218U. The increase
of $2^{+}$ and decrease of $4^{+}$ energy level and the larger
$B(E2:2^{+}\rightarrow 0^{+})$ value than the $B(E2:4^{+}\rightarrow 2^{+})$
shows the spherical character of these nuclei in both the region. The increase
in $B(E2:2^{+}\rightarrow 0^{+})$ values is very rapid in both regions. In
contrast, $2^{+}$ excitation energy shows a gradual increment. This behavior
indicates the emergence of collectivity.

In Table 2 we have reported shell model results for $B(E3)$ transition
strengths corresponding to the experimental data. The octupole transitions in
$N$ = 82 isotones resembles that for $N$ = 126 JP134Te ; Bergstrom . The
$B(E3)$ transitions are retarded due to the spin-flip single particle
transformation $\pi h_{11/2}\rightarrow\pi g_{7/2}$ and $\pi
i_{13/2}\rightarrow\pi h_{9/2}$, in comparison with the transformation $\pi
h_{11/2}\rightarrow\pi d_{5/2}$ and $\pi i_{13/2}\rightarrow\pi f_{7/2}$ which
gives rise to fast $E3$ transition, in $N$ = 82 and $N$ = 126 isotones,
respectively. Thus, the $B(E3:9^{-}_{1}\rightarrow 6^{+}_{1})$ value increases
with proton number (despite it decreases for 140Ce and 142Nd, because of the
exchange of configuration), while the $B(E3:9^{-}_{1}\rightarrow 6^{+}_{2})$
value decreases. Similar behaviour can be seen for $N$ = 126 isotones with
weak $B(E3)$ transition. The $B(E3:11^{-}_{1}\rightarrow 8^{+}_{1})$ value
slows down with proton number and dominated by single proton spin-flip
transitions (despite it increases for 216Th and 218U), while a strong
$B(E3:11^{-}_{1}\rightarrow 8^{+}_{2})$ value is reproduced.

The $11_{1}^{-}$ and $8^{+}_{2}$ are supposed to arise due to octupole
coupling with $3^{-}$ core excitation. The $E3$ transition is expected to be
very fast for the state generated from the coupling between $f_{7/2}$ orbital
and the $3^{-}$ collective state. Thus, strong $11_{1}^{-}\rightarrow
8^{+}_{2}$ transitions are observed for the $11^{-}$ isomeric state, which are
due to single particle transformation $\pi i_{13/2}\rightarrow\pi f_{7/2}$.
For 216Th, a fast $E3$ decay from $11^{-}$ state to $8^{+}_{1}$ state and a
weak decay to the $8^{+}_{2}$ state has been observed experimentally. This is
because the wave functions of the $8^{+}_{1}$ and $11^{-}$ should come from
the contribution of $\pi f_{7/2}$, and mixing of $\pi h_{9/2}^{7}f_{7/2}$ and
$\pi h_{9/2}^{8}$ for the $8^{+}_{2}$ state. In contrast, in our shell-model
calculation, the wave-function of the two $8^{+}$ states is reversed for
216Th, giving a weak decay to the $8^{+}_{1}$ state and fast decay for the
$8^{+}_{2}$ state. In the case of 218U, our calculation replicates the
observed behavior for the 216Th.

In our shell-model calculation, the deficiency in the $B(E3)$ transitions is
due to the need for larger model space, especially the particle-hole core-
excitations across the shell gaps. This could be examined in a different work
by including a larger model space in the shell model calculation PRL_Brown .

Figure 10:  Trend of the $g$-factor for the two $6^{+}$ (in Sn region) and $8^{+}$ (in Pb region) in comparison with the experimental data nndc . Table 3:  The calculated $g-$factor (in $n.m.$) and quadrupole moment (in $eb$) for isomeric states in $N=82$ and $N=126$ isotones compared to the experimental data nndc . Shell-model calculations are performed with $g_{l}^{\pi}$=1 and $g_{s}^{\pi}$ = 5.585; $e_{\pi}=1.5e$. A | $J^{\pi}$ | $g-factor$ | SM | $Q(eb)$ | SM
---|---|---|---|---|---
134Te | $6^{+}_{1}$ |  | 0.525 |  | -0.373
| $9^{-}_{1}$ |  | 1.057 |  | -0.673
136Xe | $6^{+}_{1}$ |  | 0.560 |  | -0.125
| $9^{-}_{1}$ |  | 1.055 |  | -0.548
138Ba | $6^{+}_{1}$ | 0.98(2) | 0.964 |  | -0.097
| $9^{-}_{1}$ |  | 1.056 |  | -0.414
140Ce | $6^{+}_{1}$ |  | 1.029 |  | 0.120
| $9^{-}_{1}$ |  | 1.056 |  | -0.315
142Nd | $6^{+}_{1}$ |  | 1.028 |  | 0.397
| $9^{-}_{1}$ | +1.05 (13) | 1.056 |  | -0.233
135I | $17/2^{+}$ |  | 0.911 |  | -0.569
137Cs | $17/2^{+}$ |  | 0.908 |  | -0.267
139La | $17/2^{+}$ |  | 0.904 |  | 0.062
141Pr | $17/2^{+}$ |  | 0.896 |  | 0.411
210Po | $8^{+}_{1}$ | +0.919(6) | 0.737 | -0.552(20) | -0.577
| $11^{-}_{1}$ |  | 1.024 | -0.86(11) | -0.967
212Rn | $8^{+}_{1}$ | +0.894(2) | 0.586 | -0.18(2) | -0.308
| $11^{-}_{1}$ |  | 1.038 |  | -0.779
214Ra | $8^{+}_{1}$ | 0.885(4) | 0.589 |  | -0.060
| $11^{-}_{1}$ | 1.088(6) | 1.038 |  | -0.599
216Th | $8^{+}_{1}$ |  | 0.599 |  | 0.146
| $11^{-}_{1}$ |  | 1.037 |  | -0.421
218U | $8^{+}_{1}$ |  | 0.973 |  | 0.035
| $11^{-}_{1}$ |  | 1.037 |  | -0.248
211At | ${9/2}^{-}$ | +0.92 BDA  | 0.584 | -0.33(12)Cubiss  | -0.264
| ${21/2}^{-}$ | +0.910(8) | 0.588 | 0.53(5) | -0.524
213Fr | ${9/2}^{-}$ | +0.887(2)BDA  | 0.585 | -0.14(2) | -0.121
| ${21/2}^{-}$ | 0.888(3) | 0.5908 |  | -0.178
215Ac | ${9/2}^{-}$ | +0.920(13) | 0.587 | +0.04(10) | 0.093
| ${21/2}^{-}$ | 0.910(10) | 0.605 |  | 0.101
217Pa | ${9/2}^{-}$ |  | 0.590 |  | 0.126
| ${21/2}^{-}$ |  | 0.795 |  | 0.194

###  III.7 g-factor and quadrupole moment

In the Table 3, we have reported shell model results for $g$\- factor and
quadrupole moments for the $6^{+}_{1}$, $9^{-}_{1}$ and $17/2^{+}$ isomers in
$N$ = 82 isotones and $8^{+}_{1}$, $11^{-}_{1}$, ${21/2}^{-}$ isomers with
${9/2}^{-}$ ground state in $N$ = 126 isotones corresponding to the
experimental data.

The $g$-factors are very sensitive to which orbital unpaired particles (or
holes) are occupying. Thus, the $g-$factor measurements explain the purity of
a particular configuration and the contribution of the component in the wave
function. The $g-$factors are particularly sensitive to the core polarization
effects that couple the spin-orbit partners. The influence of $1p-1h$
excitation between spin-orbit (spin-flip) partners across a magic shell gap on
the magnetic moment is called the first-order core polarization effect. The
influence of $2p-2h$ excitations is called second-order core polarization. And
the $g-$factors are not very sensitive to second-order core polarization
effects (quadrupole particle-core coupling interactions). However, quadrupole
moments are extremely sensitive to this. Therefore, the $g$-factor is found to
be constant in a chain of isotopes or isotones.

According to the additivity rule, $g$-factors of the isomeric states with a
rather pure $\pi(g_{7/2})^{n}$ or $\pi(h_{9/2})^{n}$ configuration should all
be the same. Examples of the additivity of nuclear magnetic moments are given
in Ref. Heyde . The $9^{-}$ and $11^{-}$ isomers are from proton excitations
out of the $Z$=50 and 82 core, respectively, so their $g-$factors are usually
large and positive. These two isomers show almost the same $g-$factor
throughout the isotonic chain. The $g-$factors of $9^{-}$ and $11^{-}$ isomers
could be affected by particle-octupole vibration coupling admixtures. It needs
to be investigated more from the theoretical perspective. In some isotones,
due to first-order core polarization, Pauli-blocking can cause a breakdown of
the additivity rule for $g$-factors. This can result in a changing $g$-factor
as a function of $Z$. From our calculation we have seen that the
configurations of the isomers are getting more mixed with increasing $Z$, in
particular mixing with the $\pi(g_{7/2})^{n-1}d_{5/2}$ configuration in the
Sn-region or $\pi(h_{9/2})^{n-1}f_{7/2}$ configuration in the Pb-region.

This mixing alone is not able to explain the breakdown of additivity. Other
types of mixing are needed to be considered in the wave function to describe
the deviation from the additivity rule. This configuration mixing could be
where the particle-core couple to octupole-vibrations and (or) first-order
core polarization effects couple to valence particles. A detailed discussion
can be found in Stuchbery . The main impact of the octupole mixing into the
wave function is that fewer protons will occupy the pure orbital. This results
in a reduced Pauli blocking and thus better agreement with the experimental
data. The main configuration of the isomers is expected to be pure. The
$g$-factor of these isomers should be the same for all of them (additivity
theorem).

The $g-$factors of $\pi g_{7/2}$ and $\pi h_{9/2}$ increase with the
increasing proton number. In the Sn-region, $g-$factor of one proton in the
$g_{7/2}$ orbital is smaller than that of one proton in the $d_{5/2}$ orbital.
Thus, when we take coupled $g$-factor with increasing $Z$ number, the
$g$-factor of $6^{+}$ from the ($g_{7/2}$,$d_{5/2}$) configuration space will
be larger than the ($g_{7/2}$,$g_{7/2}$). Therefore, the $g-$factor for the
$6^{+}_{1}$ increases with the proton number while it decreases for the
$6^{+}_{2}$. In the Pb-region, the $g$-factor of $8^{+}$ from the
($h_{9/2}$,$f_{7/2}$) configuration space gives a larger $g$-factor than the
($h_{9/2}$,$h_{9/2}$). Similar to the Sn-region, the $g-$factor for the
$8^{+}_{1}$ increases with the proton number while it decreases for the
$8^{+}_{2}$. This shows the sensitivity of $g-$factors to the coupling of
spin-flip partners (first-order core polarization) into the wave function. In
Fig. 10 the evolution of the g-factor for two $6^{+}$ and $8^{+}$ states has
been shown with respect to the $Z$ number. From the figure, we can see that
for the $8^{+}_{1}$ isomeric state, a deviation towards lower neutron numbers
also appears. The more nucleons occupy the orbital, the higher the first-order
core-polarization effect; thus deviation of $g$ \- factor appears towards a
higher nucleon number. This parabolic curve might indicate that the correction
in the $g$ \- factor is related to the maximum number of particles (or holes).

Our calculated results of quadrupole moments, as shown in Table 3 are in a
reasonable agreement with the available experimental data. Out of seven
experimentally measured quadrupole moments, six quadrupole moments from the
shell model are within the error bars and measured signs. In 211At, for the
$21/2^{-}$ state, the sign is not reproduced in our calculation. In the Sn-
region, no experimental data is available for the quadrupole moments. Unlike
the $g$-factor, quadrupole moments change in the isotonic chain with the
increase in the $Z$ number. The deformation increases with the increase in the
Z number above the shell-closure, and the quadrupole moment indicates the
deformation. If we increase proton number beyond shell-closure, it is
important to include the effect of second-order core polarization by including
lower orbitals in shell-model space. The effect of second-order core
polarization on the quadrupole moments can be seen in Table 3. The deviation
of the calculated quadrupole moment increases with the increase in the $Z$
number.

##  IV Conclusions

In the present work, we have performed a new type of investigation for the
similarities between the $N=82$ ($52\leq Z\leq 60$) and $N=126$ ($84\leq Z\leq
92$) isotones in the framework of the nuclear shell model with well known
SN100PN and KHPE interactions, respectively. The isotones with $N$ = 82 and
$N$ = 126 are semi-magic nuclei. From comparing the spectroscopy of Sn and Pb
regions, similar patterns of nuclear structure are found in the framework of
the shell-model. The low-lying part of the spectrum should obey the seniority
scheme, which is reflected in our present study. For these two regions, we
have discussed spectroscopic and electromagnetic properties. Our study
confirms several isomeric states in $N=82$ and $N=126$ isotones are due to the
breaking of high-$j$ nucleon pairs and well described in terms of seniority
quantum number. A similar evolution of excitation energies can be seen between
the same seniority states in the Sn and Pb regions. We propose the wave
function of $8^{+}_{2}$ state for 218U to be $\pi
h_{9/2}^{6}f_{7/2}^{2}i_{13/2}^{2}$.

Using the correspondence between the Sn and Pb regions, the high-$j$ orbitals
above the shell gaps show similar behavior, eventually deciding other
spectroscopic properties. The decrease in $B(E2;J\rightarrow J-2)$ transition
value with proton number for the seniority $\nu$=2 and 3 multiplet is an
outcome of the seniority scheme and also due to the gradual decrease in the
occupation of $\pi g_{7/2}$ and $\pi h_{9/2}$ orbitals in the Sn and Pb
region, respectively. The rest of the occupation is scattered to the
$d_{5/2},h_{11/2}$ and $f_{7/2},i_{13/2}$ orbitals in Sn and Pb region,
respectively.

A strong resemblance can be seen in the growth of $B(E2:2^{+}\rightarrow
0^{+})$ transition probabilities and in the variation of the first $2^{+}$
excitation energy. The $6^{+}_{1}$, $6^{+}_{2}$ and $9^{-}$ states in the Sn
region and their counterpart $8^{+}_{1}$, $8^{+}_{2}$ and $11^{-}$ states in
the Pb region show several similarities in the structural formation. Their
evolution leads to a similar trend of $g-$factor and $B(E3)$ in both regions.
Also, the structure of $9^{-}$ in the Sn region and $11^{-}$ states in the Pb
region are related to the octupole coupling with two $6^{+}$ and $8^{+}$,
respectively. Depending on the two different configurations, one among the two
$6^{+}$ and $8^{+}$ states will be the octupole coupled core-excited state,
which couples to $9^{-}$ and $11^{-}$ states in the Sn and Pb region,
respectively. The $B(E3)$ value is crucial for a better understanding and
explaining the similarities between these two regions. The retardation of the
$B(E3)$ value in both regions is due to the spin-flip matrix element. The
breakdown in additivity for the $g-$factor could be explained with the first-
order core polarization with mixing to the octupole coupling effects. These
correlations can not be accounted in the present shell-model calculation.
Overall, shell model results agree with the available experimental data. We
have also predicted electromagnetic properties where experimental data are not
available. Our study will be beneficial for future experimental study.

## Acknowledgments

B. Bhoy acknowledges financial support from MHRD, Government of India. P. C.
Srivastava acknowledges a research grant from SERB (India), CRG/2019/000556.
We would like to thank Prof. Larry Zamick for several illuminating
discussions.

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