# Systematic shell-model study of 98-130Cd isotopes

Deepak Patel and Praveen C. Srivastava111Corresponding author:
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee 247667, India

###### Abstract

We have performed shell-model calculations for the systematic study of even-
even 98-130Cd isotopes with $\triangle$NNLO${}_{\text{GO}}$ interaction
derived from coupled cluster approach, where 84Se is taken as an inert core.
Results for these Cd-isotopes using a realistic effective shell-model
interaction, derived from the G-matrix approach with core 88Sr, have also been
shown. We have compared our calculated low-lying excited energy spectra and
electromagnetic properties with the experimental data. On the basis of
recently available experimental data, we have also predicted spins and
parities corresponding to unconfirmed states. We have also reported the
properties of $8^{+}$ isomeric states in 98-104,130Cd isotopes.

###### pacs:

21.60.Cs, 23.20.-g, 23.20.Lv, 27.60.+j

##  I Introduction

In recent years, the study of nuclear structure properties of Cd and Te
isotopes has increased both from the experimental and theoretical sides
because they are in the vicinity of the 100Sn, which is the heaviest $N=Z$
doubly magic nucleus Siciliano ; Vaman ; Togashi ; Nara Singh ; Faestermann .
The study of nuclear properties in Cd isotopes is also essential for the
understanding of stellar nucleosynthesis. Several similarities are pointed out
between Sn and Cd isotopic chains, like excitation energy spectra and electric
quadrupole transition probability Togashi ; Zuker ; Banu ; Doornenbal ;
Isacker ; Jain ; Rakesh ; Srivastava . The shell-model provides a better
platform to verify these properties.

In the past, the energy states in Cd-isotopes have been measured using
different advanced experimental techniques. Górska $et$ $al.$ Gorska1  studied
the $\gamma$-decay and $8^{+}$ isomer in 100Cd using a recoil catcher setup
inside the multidetector $\gamma$-array OSIRIS. For the first time, Alber $et$
$al.$ Alber  experimentally observed the low-lying spectra of neutron-
deficient 100-102Cd. Measurements of lifetimes of excited nuclear states are
important for computing $B(E2)$ transition rates. Lieb $et$ $al.$ Lieb
determined picosecond lifetimes of high-spin states in 102Cd isotope using the
reaction 58Ni(50Cr,$\alpha$ $2p$) and coincidence recoil distance Doppler-
shift technique (RDDS). Boelaert $et$ $al.$ Boelaert1  determined the
lifetimes of low-lying $2^{+}_{1}$ and $4^{+}_{1}$ states in 104Cd and
$2^{+}_{1}$ state in 102Cd. Müller $et$ $al.$ Muller  determined lifetimes of
high-spin states in 104Cd in the time range 0.3 $ps$$\leq\tau\leq$1.2 $ns$ for
the first time using RDDS and Doppler-shift attenuation methods (DSAM). They
have also discussed new experimental results in 104Cd in the frame of the
interacting boson plus broken pair model. Simons $et$ $al.$ Simons  measured
the lifetimes of excited states in the lowest-lying positive parity bands in
106,108Cd isotopes using the RDDS and also deduced the $B(E2)$ transition
rates. Benczer-Koller $et$ $al.$ Koller  remeasured and supplemented existing
lifetimes and magnetic moments of low-lying states in 106Cd. Fan $et$ $al.$
Fan  measured g-factors of rotational states of the yrast band from spin
$10^{+}$ to spin $16^{+}$ in 108Cd using transient-magnetic-field ion
implantation perturbed angular distribution method. Kumpulainen $et$ $al.$
Kumpulainen  investigated the low-lying low-spin collective states in even
106-112,116Cd isotopes using in-beam and off-beam $\gamma$-ray and conversion-
electron spectroscopy. Gade $et$ $al.$ Gade1  reported the first evidence for
the complete quadrupole-octupole coupled quintuplet of negative parity
$(2^{-}_{1}\otimes 3^{-}_{1})^{(J^{-})}$ with $J=1,2,3,4,5$ for 108Cd. Nomura
$et$ $al.$ Nomura  investigated the structure of even-even 108-116Cd isotopes
using a self-consistent mean-field approach. Corminboeuf $et$ $al.$
Corminboeuf  performed measurements consisting of $\gamma$-ray excitation
functions and angular distributions on 110Cd. They extracted the lifetimes for
16 excited states using the DSAM. Piiparinen $et$ $al.$ Piiparinen  measured
lifetimes of 20 yrast levels in 110Cd using the recoil-distance method and the
NORDBALL array of Compton-suppressed Ge-detectors. They also deduced the
reduced-transition probabilities and revealed the rotational and vibrational
behavior of 110Cd. After that, Regan $et$ $al.$ Regan  measured the
gyromagnetic ratio of yrast $10^{+}$ state in 110Cd using ion-implantation
perturbed-angular-distribution (IMPAD) technique. Garrett $et$ $al.$ Garrett
measured lifetimes of those levels which are below 4 MeV in 112Cd isotope
using DSAM following the inelastic scattering of monoenergetic neutrons.
Jamieson $et$ $al.$ Jamieson  studied the quadrupole-octupole coupled states
in 112Cd with 111Cd$(d,p)^{112}$Cd reaction using 22 MeV polarized deuterons.
Zamfir $et$ $al.$ Zamfir  studied the structure of 122Cd through the decay of
122Ag by $\gamma-\gamma$ coincidences, $\gamma$-ray angular correlations, and
$\beta-\gamma-\gamma$ fast electronic scintillation timing (FEST)
measurements. Scherillo $et$ $al.$ Scherillo  investigated microsecond isomers
in Cd isotopes in the mass range A=123 to 130 using the LOHENGRIN mass
spectrometer. Hoteling $et$ $al.$ Hoteling  identified the presence of
isomeric levels with half-lives in the microsecond range in 125-128Cd. In a
recent TITAN experiment at TRIUMF, the cadmium masses were measured for
125/mCd and 126Cd. Lascar $et$ $al.$ Lascar  presented the results of
precision mass measurements of 125-127Cd isotopes. They have also compared
their measurements with the $ab$ $initio$ shell-model calculations. Dunlop
$et$ $al.$ Dunlop  measured the $\beta$-decay half-lives of 128-130Cd isotopes
using GRIFFIN $\gamma$-ray spectrometer at the TRIUMF-ISAC facility. Dillmann
$et$ $al.$ Dillmann  performed the first $\beta$\- and $\gamma$-spectroscopic
decay of the $N=82$ r-process “waiting-point” nuclide 130Cd at CERN-ISOLDE.

In the previous theoretical works, Schmidt $et$ $al.$ Schmidt  performed
shell-model calculations for even-even 98-108Cd isotopes. Blazhev $et$ $al.$
Blazhev  identified a core excited $I^{\pi}$=(12+) spin-gap isomer in 98Cd.
They have also discussed the measured $E4$ and $E2$ strengths, 100Sn core
excitations, and the origin of empirical polarization charges in large-scale
shell-model calculations. Boelaert $et$ $al.$ Boelaert  performed shell-model
calculations for even-even 98-106Cd isotopes and calculated energy spectra and
transition probabilities. Clark $et$ $al.$ Clark  studied the gamma decay of
excited states in 100Cd with Gammasphere array following the
46Ti(58Ni,2$p$,2$n$) reaction at 215 MeV. They compared their results with
shell-model calculations. Maheshwari $et$ $al.$ Maheshwari  studied the energy
variation of $2^{+}_{1}$ states and $B(E2;0^{+}_{1}\rightarrow 2^{+}_{1})$
trends in the Cd, Sn and Te isotopic chains using generalized seniority
approach.

The aim of the present paper is to perform systematic shell-model calculations
for even 98-130Cd isotopes. We have investigated different properties of these
isotopes like excitation energy, reduced electromagnetic transition
probabilities, isomeric states, electric quadrupole, and magnetic moments from
neutron-deficient isotopes (near the proton drip line) to the neutron-rich
isotopes up to $N=82$. Recently, several new experimental data are available
in Cd isotopic chain for low-lying spin states. Therefore, we have focused
here on the low-lying spin states of these isotopes.

This paper is divided into the following sections. In Section II, the
formalism for coupled cluster method and the resulting similarity transformed
Hamiltonian is briefly discussed. In Section III, we introduce the monopole
shifts in the single-particle energies of valence proton orbitals for Cd-
isotopes. In Section IV, we present our calculated results of low-lying energy
spectra, electromagnetic properties, octupole collective $3^{-}$ states, and
$8^{+}$ isomeric states in the Cd chain. Finally, we summarize our results and
conclude the paper in section V.

##  II Formalism

The nuclear shell-model Hamiltonian can be expressed in terms of single-
particle energies and two nucleon interactions,

$H=\sum_{\alpha}\epsilon_{\alpha}c_{\alpha}^{\dagger}c_{\alpha}+\frac{1}{4}\sum_{\alpha\beta\gamma\delta
JT}\langle
j_{\alpha}j_{\beta}|V|j_{\gamma}j_{\delta}\rangle_{JT}c_{\alpha}^{\dagger}c_{\beta}^{\dagger}c_{\delta}c_{\gamma}$
(1)

where $\alpha=\\{n,l,j,t\\}$ stand for the single-particle orbitals and
$\epsilon_{\alpha}$ are corresponding single-particle energies. $c_{\alpha}$
and $c_{\alpha}^{\dagger}$ are the fermion annihilation and creation
operators. $\langle j_{\alpha}j_{\beta}|V|j_{\gamma}j_{\delta}\rangle_{JT}$
denote the antisymmetrized two-body matrix elements coupled to spin $J$ and
isospin $T$.

We have studied the structure of even-even Cd isotopes from the neutron
numbers 50 to 82 using two types of interactions and model spaces. The first
effective interaction is derived through the shell-model coupled cluster
(SMCC) approach based on similarity transformations. In the medium-mass
nuclei, the coupled cluster method is useful for $ab$ $initio$ calculations.
Here, we have taken a chiral interaction named $\Delta\text{NNLO}_{\text{GO}}$
Jiang  interaction. We have mainly focused on the application of this
interaction in the even-even 98-130Cd chain. The shell model code KSHELL
KShell  has been used for our calculations to diagonalize the matrices. The
highest dimension is 1.77 $\times$ 109 for 114Cd with
$\Delta\text{NNLO}_{\text{GO}}$ interaction. For the interaction
$\Delta\text{NNLO}_{\text{GO}}$, the model space consist of $1p_{3/2}$,
$1p_{1/2}$, $0g_{9/2}$ proton orbitals and $1d_{5/2}$, $2s_{1/2}$, $1d_{3/2}$,
$0g_{7/2}$, $0h_{11/2}$ neutron orbitals with core 84Se. We have performed our
calculations with this interaction without any truncation. The formalism to
find an effective Hamiltonian is given below.

The intrinsic Hamiltonian for coupled-cluster calculations Sun  can be written
as-

$H=\left(1-\frac{1}{A}\right)\sum_{i=1}^{A}\frac{p_{i}^{2}}{2m}+\sum_{i<j=1}^{A}\Big{\\{}V_{ij}-\frac{\vec{p}_{i}.\vec{p}_{j}}{mA}\Big{\\}}+\sum_{i<j<k}^{A}V_{ijk}$
(2)

Here, $A$ is the mass number, $m$ is the mass of a nucleon, $p_{i}$ and
$p_{j}$ are single-particle momentum, $V_{ij}$ is the nucleon-nucleon
interaction, and $V_{ijk}$ is the three-body interaction.

The normal-ordered Hamiltonian for shell-model coupled-cluster (SMCC)
calculations with respect to a Hartree-Fock reference state $|\phi_{0}\rangle$
can be expressed as-

$\mathcal{H}=E_{0}+\sum_{ab}g_{ab}\\{a^{\dagger}b\\}+\frac{1}{4}\sum_{abcd}V_{abcd}\\{a^{\dagger}b^{\dagger}dc\\}+\frac{1}{36}\sum_{abcdef}V_{abc,def}\\{a^{\dagger}b^{\dagger}c^{\dagger}def\\}.$
(3)

$a^{\dagger}$ is particle creation operator and $a$ is particle annihilation
operator on state $|\phi_{a}\rangle$. $g_{ab}$ denote the Fock-matrix
elements, while $V_{abcd}$ and $V_{abc,def}$ denote the two-body and three-
body matrix elements, respectively. For our calculations, the Hamiltonian
given by equation (3) is truncated at the two-body level. The energy
expectation value of the reference state is given by
$E_{0}=\langle\phi_{0}|H|\phi_{0}\rangle$ where $H$ is the intrinsic
Hamiltonian.

The wave function of the coupled-cluster theory can be written as an
exponential ansatz,

$|\phi\rangle=e^{T}|\phi_{0}\rangle$ (4)

where $T$ is the cluster operator.

The similarity-transformed Hamiltonian Sun  is given by,
$\mathcal{\overline{H}}\equiv e^{-T}\mathcal{H}e^{T}$, can be expanded as-

$\mathcal{\overline{H}}=E_{CC}+\sum_{ab}\mathcal{\overline{H}}_{ab}\\{a^{\dagger}b\\}+\sum_{abcd}\mathcal{\overline{H}}_{abcd}\\{a^{\dagger}b^{\dagger}dc\\}+\sum_{abcdef}\mathcal{\overline{H}}_{abcdef}\\{a^{\dagger}b^{\dagger}c^{\dagger}fed\\}+...,$
(5)

Here, $E_{CC}=\langle\phi_{0}|\mathcal{\overline{H}}|\phi_{0}\rangle$ is the
correlation energy.

To construct a shell-model effective interaction, we split infinite Hilbert
space into the model space $P$ and its complement $Q$. Which obey the
relation-

$P+Q=1$ (6)

To decouple the $P$ space from $Q$ space for the Hamiltonian, a secondary
similarity transformation is applied to the coupled-cluster Hamiltonian,

$\mathcal{\overline{\overline{H}}}=e^{-S}\mathcal{\overline{H}}e^{S}$ (7)

Where $S$ is an $A$-body operator.

The decoupling between $P$ and $Q$ space requires,

$Q\mathcal{\overline{\overline{H}}}P=0$ (8)

which results the effective Hamiltonian,

$H_{eff}=P\mathcal{\overline{\overline{H}}}P.$ (9)

More details about effective interaction is given in Ref. Sun .

We have also performed shell-model calculations with the G-matrix effective
interaction Boelaert , which is constructed by the effective microscopic
interaction derived from a charge-symmetry breaking nucleon-nucleon potential
Machleidt  and another interaction from Holt  (named v3rd), with further
modifications in the monopole part. This interaction consist of $1p_{1/2}$,
$0g_{9/2}$ proton orbitals and $1d_{5/2}$, $2s_{1/2}$, $1d_{3/2}$, $0g_{7/2}$,
$0h_{11/2}$ neutron orbitals with core 88Sr.

Figure 1:  Variation in the effective single particle energies in Cd isotopes
corresponding to $\Delta\text{NNLO}_{\text{GO}}$, and the G-matrix
interactions.

##  III Monopole shift

In the case of 98-130Cd, there are two valence proton holes and several
valence neutrons outside the doubly magic nucleus 100Sn. It can be expected
that the proton-neutron interaction will influence the effective single-
particle energies of the proton orbitals. The effective single-particle
energies depend on the proton-neutron effective interactions and the pairing
properties between the neutron Smirnova . The variation in the single-particle
energy of valence proton orbital as a function of neutron number is known as
the monopole shift Srivastava2 . The low-lying states carry a significant part
of the proton single-particle strength for these isotopes. In Cd isotopes,
valence protons in $j_{\pi}$ orbitals interact with a number of valence
protons and neutrons in proton orbitals $j_{\pi}$ and neutron orbitals
$j_{\nu}$, respectively. The Hamiltonian can be written as

$\hat{H}=\epsilon_{j_{\pi}}+\sum_{j_{\nu}}\epsilon_{j_{\nu}}\hat{n}_{j_{\nu}}+V_{\pi\pi}+V_{\pi\nu}+V_{\nu\nu}$
(10)

Here $\epsilon_{j_{\pi}}$ and $\epsilon_{j_{\nu}}$ are single-particle
energies of proton and neutron corresponding to $j_{\pi}$ and $j_{\nu}$
orbitals, respectively. $\hat{n}_{j_{\nu}}$ is the number operator for
neutrons. $V_{\pi\pi}$, $V_{\pi\nu}$ and $V_{\nu\nu}$ represent residual
proton-proton, proton-neutron, and neutron-neutron interactions, respectively.

For different multipole components of these interactions, $\hat{H}$ can be
written as

$\hat{H}=\hat{H}_{\lambda=0}+\hat{H}_{\lambda=1}+...+\text{higher
}\lambda\text{ multipoles of }\pi\pi\text{, }\pi\nu\text{ and }\nu\nu\text{
interaction}.$ (11)

Table 1:  The calculated shell model wave functions corresponding to $2^{+}_{1}$ state in even-even 98-130Cd isotopes. Nuclei | State | Probability | $\Delta\text{NNLO}_{\text{GO}}$ | Probability | G-matrix
---|---|---|---|---|---
98Cd | $2^{+}_{1}$ | 98.8% | $\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})$ | 100% | $\pi(p_{1/2}^{2}g_{9/2}^{8})$
100Cd | $2^{+}_{1}$ | 52.2% |  $\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(g_{7/2}^{2})$ | 55.9% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{2})$
102Cd | $2^{+}_{1}$ | 36.3% |  $\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(g_{7/2}^{2}d_{5/2}^{2})$ | 27.8% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{4})$
104Cd | $2^{+}_{1}$ | 25.1% |  $\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(g_{7/2}^{4}d_{5/2}^{2})$ | 32.1% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{4}g_{7/2}^{2})$
106Cd | $2^{+}_{1}$ | 21.5% |  $\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(g_{7/2}^{4}d_{5/2}^{4})$ | 26.5% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{4}g_{7/2}^{4})$
108Cd | $2^{+}_{1}$ | 13.6% |  $\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(g_{7/2}^{6}d_{5/2}^{4})$ | 10.8% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{4}g_{7/2}^{6})$
110Cd | $2^{+}_{1}$ | 9.6% |  $\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(g_{7/2}^{6}d_{5/2}^{6})$ | 6.8% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{4}g_{7/2}^{6}h_{11/2}^{2})$
112Cd | $2^{+}_{1}$ | 4.0% |  $\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(g_{7/2}^{6}d_{5/2}^{6}h_{11/2}^{2})$ | 8.5% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}g_{7/2}^{4}d_{3/2}^{2}h_{11/2}^{2})$
114Cd | $2^{+}_{1}$ | 3.2% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{2})$ | 8.2% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{4}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{4})$
116Cd | $2^{+}_{1}$ | 3.0% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{4})$ | 11.3% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{4})$
118Cd | $2^{+}_{1}$ | 2.6% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{6})$ | 20.9% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{6})$
120Cd | $2^{+}_{1}$ | 2.9% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{2}h_{11/2}^{6})$ | 24.3% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{8})$
122Cd | $2^{+}_{1}$ | 3.7% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{2}h_{11/2}^{8})$ | 35.9% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{8})$
124Cd | $2^{+}_{1}$ | 4.8% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{2}h_{11/2}^{10})$ | 28.2% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{10})$
126Cd | $2^{+}_{1}$ | 6.4% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{2}h_{11/2}^{12})$ | 43.8% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{8}d_{3/2}^{2}h_{11/2}^{10})$
128Cd | $2^{+}_{1}$ | 12.9% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{4}h_{11/2}^{12})$ | 80.5% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{8}d_{3/2}^{4}h_{11/2}^{10})$
130Cd | $2^{+}_{1}$ | 91.1% |  $\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes$ $\nu(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{4}s_{1/2}^{2}h_{11/2}^{12})$ | 100% |  $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes$ $\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{8}d_{3/2}^{4}h_{11/2}^{12})$

We can write the monopole Hamiltonian $\hat{H}_{\lambda=0}$ as

$\hat{H}_{\lambda=0}=\sum_{j_{\pi}}\tilde{\epsilon}_{j_{\pi}}\hat{n}_{j_{\pi}}+\sum_{j_{\nu}}\tilde{\epsilon}_{j_{\nu}}\hat{n}_{j_{\nu}}.$
(12)

Here $\tilde{\epsilon}_{j_{\pi}}$ is the effective proton single-particle
energy, which is given as

$\tilde{\epsilon}_{j_{\pi}}=\epsilon_{j_{\pi}}+\sum_{j_{\pi}}\bar{E}(j_{\pi}j_{\pi})\hat{n}_{j_{\pi}}+\sum_{j_{\nu}}\bar{E}(j_{\pi}j_{\nu})\hat{n}_{j_{\nu}}.$
(13)

Where $\bar{E}(j_{\pi}j_{\pi})$ and $\bar{E}(j_{\pi}j_{\nu})$ are monopole
matrix elements Otsuka . They can be expressed as

$\bar{E}(j_{\pi}j_{\pi})=\frac{\sum_{J}(2J+1)\langle
j_{\pi}j_{\pi};J|V|j_{\pi}j_{\pi};J\rangle}{\sum_{J}(2J+1)}$ (14)

and

$\bar{E}(j_{\pi}j_{\nu})=\frac{\sum_{J}(2J+1)\langle
j_{\pi}j_{\nu};J|V|j_{\pi}j_{\nu};J\rangle}{\sum_{J}(2J+1)}$ (15)

In Fig. 1, we have shown the change in effective single-particle energy (ESPE)
of proton orbitals with respect to neutron number in Cd-isotopes for
$\Delta\text{NNLO}_{\text{GO}}$, and the G-matrix interactions. We can see
that the variation in effective single particle energy of the proton orbitals
is different from neutron-deficient to neutron-rich side.

##  IV Results

Figure 2:  Comparison between calculated and experimental energy NNDC  levels
for even-even nuclei of 98-104Cd.

###  IV.1 Low-lying energy spectra of 98-130Cd even isotopes

We have compared the experimental data of energy levels with two interactions
named $\Delta\text{NNLO}_{\text{GO}}$, and the G-matrix interaction for
98-130Cd isotopes as shown in Figs. 2-6. We have shown five states maximally
for a given spin-parity $J^{\pi}$. The wave function corresponding to
$2_{1}^{+}$ state for 98-130Cd isotopes are shown in Table 1.

Górska $et$ $al.$ Gorska  identified and studied excited states in 98Cd (two
proton holes from 100Sn) using in-beam spectroscopy for the first time. They
also populated states up to $8^{+}$ isomer and compared it to the shell-model
predictions. In the case of 98Cd, the G-matrix interaction shows good
agreement with the experimental data for the $2_{1}^{+}$ and $4_{1}^{+}$
states. For this interaction, the $6_{1}^{+}$ and $8_{1}^{+}$ states are also
comparable with the experimental data but compressed from the
$\Delta\text{NNLO}_{\text{GO}}$ interaction. The energy gap between
$8^{+}_{1}$ and $6^{+}_{1}$ is very close with both interactions as in the
experimental data. From both interactions, the ground state $0^{+}$ and other
excited states up to $8^{+}$ show $\pi(g_{9/2}^{-2})$ character, which
corresponds to the single-particle behavior of this isotope. We have also
performed shell-model calculations for $0^{+}_{2}$, $5^{-}_{1}$ and
$4^{-}_{1}$ states as reported in the Ref. Gorska . With G-matrix interaction,
the location of these states are comparable to the results obtained in Ref.
Gorska . From $\Delta\text{NNLO}_{\text{GO}}$ interaction, the $0_{2}^{+}$
state lies at very high-energy (3.762 MeV).

The ${}^{100}_{48}$Cd52 isotope is a good candidate for studying two valence
neutron-particle (proton-hole) configurations. For this isotope, the excited
$2^{+}_{1}$, $4^{+}_{2}$, $6^{+}_{2}$, and $8^{+}_{1}$ states from the
G-matrix interaction are comparable with the experimental data. The
$4^{+}_{1}$ and $6^{+}_{1}$ states lie slightly lower than the tentative
experimental data with this interaction. With $\Delta\text{NNLO}_{\text{GO}}$
interaction these states are compressed.  From the G-matrix interaction, the
configurations of $0^{+}_{1}$, $2^{+}_{1}$ and $4^{+}_{1}$ state are
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{2})_{0^{+},2^{+},4^{+}}$ with
50%, 55.9% and 67.9% probabilities, respectively. The $6^{+}_{1}$ state is
coming with configuration
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{1}g_{7/2}^{1})_{6^{+}}$. From
these configurations, it is clear that the yrast
$0^{+}_{1}-2^{+}_{1}-4^{+}_{1}-6^{+}_{1}$ states show neutron nature. Which
supports the results found in previous works Gorska1 ; Alber .  The excited
$10^{+}_{1}$ state lies 0.393 MeV lower than the experimental level from the
G-matrix interaction. From the G-matrix interaction, the configuration of the
$10^{+}_{1}$ state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{1}g_{7/2}^{1})$ with 63.1%
probability.  The little mixing is mainly due to the particle-hole character
of the proton-neutron interaction. With $\Delta\text{NNLO}_{\text{GO}}$
interaction, the $10^{+}_{1}$ state is compressed like other states. The
neutron occupancies for $\Delta\text{NNLO}_{\text{GO}}$ interaction
corresponding to $2^{+}_{1}$ state are 1.111, 0.665, 0.076, 0.025, and 0.123
in $g_{7/2}$, $d_{5/2}$, $d_{3/2}$, $s_{1/2}$ and $h_{11/2}$ orbitals,
respectively. For the G-matrix interaction, the neutron occupancies
corresponding to $2^{+}_{1}$ state are 1.259, 0.138, 0.446, 0.141, and 0.016
in $d_{5/2}$, $s_{1/2}$, $g_{7/2}$, $d_{3/2}$ and $h_{11/2}$ orbitals,
respectively. Here, neutron orbital $d_{5/2}$ is dominating for the G-matrix
interaction and $g_{7/2}$ for $\Delta\text{NNLO}_{\text{GO}}$ interaction.  We
have also calculated the branching ratios for $8^{+}_{1}\rightarrow 6^{+}_{1}$
and $8^{+}_{1}\rightarrow 6^{+}_{2}$ transitions with the G-matrix interaction
($e_{\pi}=1.7e$, $e_{\nu}=1.1e$) as 99.96% and 0.04%, respectively. From these
results, it seems that the $6^{+}_{2}$ state is fed by week $\gamma$-branches,
quite similar to the calculated results in previous work Clark . In Ref.
Gorska1 , the spacing between calculated $6_{1}^{+}$ and $8_{1}^{+}$ states is
reported as 71 keV, our calculated value with G-matrix and
$\Delta\text{NNLO}_{\text{GO}}$ interactions are 10 and 130 keV, respectively.

Figure 3:  Comparison between calculated and experimental energy NNDC  levels
for even-even nuclei of 106-112Cd.

The study of 102Cd (with $N=54$) helps for the understanding of $\pi\nu$
interaction in $\pi(g_{9/2})$, $\nu(d_{5/2})$, and $\nu(g_{7/2})$ orbitals
Alber . In 102Cd, the location of first excited $2^{+}$ state is comparable
with $\Delta\text{NNLO}_{\text{GO}}$ and the G-matrix interactions. From the
G-matrix interaction, the $4^{+}_{1}$ state shows good agreement with the
experimental data. But, this state is compressed by 0.267 MeV than the
experimental level with $\Delta\text{NNLO}_{\text{GO}}$ interaction.  The
half-life of $4^{+}_{1}$ state should be less than 5.6 ps from the
experimental data NNDC . Our calculated half-lives are 1.44, 1.69, and 2.29 ps
from the G-matrix interaction with effective charges ($e_{\pi}$, $e_{\nu}$) =
(1.7, 1.1), (1.4, 1.1) and (1.4, 0.85), respectively. These calculated half-
lives may be useful to compare experimental half-life once it measures
accurately for $4^{+}_{1}$ state. In the Cd isotopic chain, the 102Cd isotope
lies in the transitional region between single-particle character (as in
nuclei 98Cd) and collective vibrational isotopes (with $N\geq 56$). The $E2$
isomers with spin-parity $8^{+}$ are observed in this transitional region. For
the isomeric state, $8^{+}_{1}$, showing good agreement with the G-matrix
interaction. With $\Delta\text{NNLO}_{\text{GO}}$ interaction, $8^{+}_{1}$
state is compressed. The order of
$0^{+}_{1}$-$2^{+}_{1}$-$4^{+}_{1}$-$6^{+}_{1}$ states are the same with both
interactions. Calculation from the G-matrix interaction favour $5^{+}_{1}$
state for the experimental $(5^{+},6^{+})$ state. With the
$\Delta\text{NNLO}_{\text{GO}}$ interaction, $6^{+}_{1}$ state is compressed
but $5^{+}_{1}$ state is comparable with the experimental $(5^{+},6^{+})$
state. In the formation of $2^{+}_{1}$ state, neutron orbital $d_{5/2}$
dominates with a maximum occupancy of 2.476 for the G-matrix interaction. But,
in the case of $\Delta\text{NNLO}_{\text{GO}}$ interaction neutron orbital
$g_{7/2}$ is dominating.

The 104Cd shows the typical vibrational behavior and nearly spherical shape at
low spins. For this isotope, the excited $2^{+}_{1}$, $4^{+}_{1}$ and
$4^{+}_{2}$ states reproduced well with $\Delta\text{NNLO}_{\text{GO}}$ and
the G-matrix interactions. From the experimental data, the half-life of
$4^{+}_{1}$ state should be less than 4.2 ps NNDC . With the three sets of
effective charges (1.7, 1.1), (1.4, 1.1), and (1.4, 0.85), the G-matrix
calculated half-lives are 2.37, 2.68, and 3.8 ps, respectively. From
$\Delta\text{NNLO}_{\text{GO}}$ interaction, the calculated half-life is 4.16
ps with effective charges (1.7, 1.1). We can notice that all calculated half-
lives are showing good agreement with the experimental data. The order of
$0^{+}_{1}$, $2^{+}_{1}$, $4^{+}_{1}$, $4^{+}_{2}$, $6^{+}_{1}$ and
$6^{+}_{2}$ states are reproduced well with the
$\Delta\text{NNLO}_{\text{GO}}$ interaction. The energy gap between
$4^{+}_{1}$ and $2^{+}_{1}$ is comparable with $\Delta\text{NNLO}_{\text{GO}}$
and the G-matrix interactions as in the experimental data. As reported in Ref.
Angelis , the $8^{+}_{1}$ state decays via $\gamma$-transition into
$6^{+}_{1,2}$ states, this $8^{+}_{1}$ state is slightly compressed with the
G-matrix interaction. With $\Delta\text{NNLO}_{\text{GO}}$ interaction, this
state lies lower by 0.714 MeV than the experimental level. In the formation of
$2^{+}_{1}$ state, the neutron occupancy is maximum in $d_{5/2}$ orbital with
the G-matrix interaction and $g_{7/2}$ orbital with
$\Delta\text{NNLO}_{\text{GO}}$ interaction like 100,102Cd. The excited
$8^{+}_{3}$ state is comparable with the experimental level from
$\Delta\text{NNLO}_{\text{GO}}$ interaction. From this interaction, the
configuration of $8^{+}_{3}$ state is
$\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(g_{7/2}^{2}d_{5/2}^{4})$
with 28.7% probability. Here, this state comes from one proton pair breaking
in $g_{9/2}$ orbital. Thus, seniority $\nu=2$. We notice that no low-lying
(approximately up to 3.5 MeV) states with negative parity are observed
experimentally from 98Cd to 104Cd. The reason behind this, there is no
significant role of $h_{11/2}$ orbital in the filling of valence neutrons.

In the cadmium chain, 106Cd is a good example of the onset of collectivity
Gray  with a simple structure.  This isotope keeps a small but active number
of valence nucleons. For this isotope, $2_{1}^{+}$, $4_{1}^{+}$, and
$2^{+}_{2}$ states are reproduced well with the G-matrix interaction.
Earlier, Benczer-Koller $et$ $al.$ Koller  performed shell-model calculation
for 106Cd considering 78Ni as a core. They have calculated the location of
$2^{+}_{1}$ and $4^{+}_{1}$ states taking two sets of neutron model spaces. In
our work with the G-matrix interaction, the calculated energy levels for
$2^{+}_{1}$ and $4^{+}_{1}$ states are close to the experimental data than
their results in both cases. With $\Delta\text{NNLO}_{\text{GO}}$ interaction,
these states lie higher than the experimental level. The $5^{+}_{1}$,
$6^{+}_{1}$ and $6^{+}_{2}$ states are comparable with
$\Delta\text{NNLO}_{\text{GO}}$ interaction. These states are compressed with
the G-matrix interaction. The wave function for
$\Delta\text{NNLO}_{\text{GO}}$ interaction, corresponding to $2^{+}_{1}$
state is
$\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(g_{7/2}^{4}d_{5/2}^{4})$
with maximum probability 21.46%. For the G-matrix interaction, the wave
function corresponding to $2^{+}_{1}$ state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{4}g_{7/2}^{4})$ with maximum
probability 26.53%. Here, neutron configuration is same in $g_{7/2}$ and
$d_{5/2}$ orbitals for $2^{+}_{1}$ state with both interactions. From the
G-matrix interaction, the configuration of octupole collective $3^{-}_{1}$
state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{3}g_{7/2}^{4}h_{11/2}^{1})$
with 10.0% probability. It is evident that there are collective structure in
106Cd with the occupation of atleast single valence neutron in $h_{11/2}$
orbital.

For 108Cd, the $2^{+}_{1}$ state is comparable with the G-matrix interaction.
With $\Delta\text{NNLO}_{\text{GO}}$ interaction, these states lie higher than
the experimental data. Gade $et$ $al.$ Gade  established the intruder band in
108Cd isotope using the Doppler shift attenuation method. The $0^{+}_{2}$
intruder state lies 0.229 and 0.334 MeV lower than the experimental data with
the G-matrix and $\Delta\text{NNLO}_{\text{GO}}$ interactions, respectively.
In 108Cd, the excited $4^{+}_{1}$ and $2^{+}_{2}$ states are two-phonon
states. The $4^{+}_{1}$ state is reproduced well, but the $2^{+}_{2}$ state
lies 0.154 MeV lower than the experimental level with the G-matrix
interaction. With $\Delta\text{NNLO}_{\text{GO}}$ interaction, the location of
$4^{+}_{1}$ and $2^{+}_{2}$ states are overpredicted. Location of the
$3^{-}_{1}$ state is showing good agreement with
$\Delta\text{NNLO}_{\text{GO}}$ and the G-matrix interactions. The $6^{+}_{1}$
state is comparable with $\Delta\text{NNLO}_{\text{GO}}$ interaction. With the
G-matrix interaction, this state lies slightly below.  It is important to
study the lowest quadrupole and octupole modes in 108Cd. The quadrupole-
octupole coupled $(2^{+}\otimes 3^{-})$ quintuplet states $5^{-}_{1}$ and
$5^{-}_{2}$ are underpredicted with the G-matrix interaction. In case of
$\Delta\text{NNLO}_{\text{GO}}$ interaction, the $5^{-}_{1}$ state is
compressed but the location of $5^{-}_{2}$ state is comparable with the
experimental data. Another member of this family $1^{-}$ state lies 0.249 MeV
higher than the experimental level with the G-matrix interaction. This state
is coming with configuration
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{4}g_{7/2}^{5}h_{11/2}^{1})$
and 14.6% probability. The involvement of two neutron orbitals $h_{11/2}$ and
$g_{7/2}$ with higher single-particle energy push the $1^{-}$ state to lie at
higher energy. With the G-matrix interaction,
$0^{+}_{1}-2^{+}_{1}-4^{+}_{1}-2^{+}_{2}-0^{+}_{2}-3^{+}_{1}$ states are in
the same order as in the experimental data. In the formation of $2^{+}_{1}$
state, $d_{5/2}$ orbital is dominating with maximum neutron occupancies of
4.274 for the G-matrix interaction. But, with $\Delta\text{NNLO}_{\text{GO}}$
interaction $g_{7/2}$ orbital is dominating with maximum neutron occupancy
4.452.

In the case of 110Cd, the location of the $2^{+}_{1}$ state is comparable with
the experimental data for the G-matrix interaction. With
$\Delta\text{NNLO}_{\text{GO}}$ interaction, this state lies higher than the
experimental level. This first excited state decays to the ground state
$0^{+}_{1}$ via $E2$ transition. Our calculated half-life of this state is
12.3 ps from the G-matrix interaction (with effective charges $e_{\pi}=1.7e$,
$e_{\nu}=1.1e$), which is comparable to the experimental half-life 5.42(16) ps
NNDC . For $\Delta\text{NNLO}_{\text{GO}}$ interaction, the $0^{+}_{2}$
intruder state lies 0.271 MeV higher than the experimental level with
configuration
$\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(g_{7/2}^{6}d_{5/2}^{6})$ and
10.9% probability. The description of the energy levels of this $0^{+}_{2}$
state is not good with the G-matrix interaction.  The 110Cd isotope is known
as collective nuclei showing multiphonon excitations. The order of 2-phonon
states $2^{+}_{2}-4^{+}_{1}-0^{+}_{3}$ is reproduced with the G-matrix
interaction, but these states are compressed. With this interaction, the
2-phonon state $0^{+}_{3}$ lies 0.174 MeV lower than the experimental level.
With $\Delta\text{NNLO}_{\text{GO}}$ interaction, this $0^{+}_{3}$ state
($\pi(p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(g_{7/2}^{6}d_{5/2}^{4}h_{11/2}^{2})$)
is formed by involvement of two neutrons in higher-lying $h_{11/2}$ orbital
unlike $0^{+}_{1}$ and $0^{+}_{2}$ state. Thus, this state is located at a
very high energy than the experimental data. The $3^{-}_{1}$ state is
reproduced well with $\Delta\text{NNLO}_{\text{GO}}$ and the G-matrix
interactions. With $\Delta\text{NNLO}_{\text{GO}}$ interaction, the
$6^{-}_{1}$ and $5^{+}_{1}$ states are comparable with the experimental data.
But, the order of these states is reversed. These energy levels are compressed
with the G-matrix interaction. The $8^{+}_{1}$ state is also well reproduced
with $\Delta\text{NNLO}_{\text{GO}}$ interaction.

Figure 4:  Comparison between calculated and experimental NNDC  energy levels
for even-even nuclei of 114-120Cd.

For 112Cd, the order of excited $0^{+}_{1}-2^{+}_{1}-4^{+}_{1}$ states are
reproduced well with the G-matrix interaction as in the experimental data. The
first excited $2^{+}_{1}$ state lies 0.203 MeV lower than the experimental
data with this interaction. We have calculated the half-life of this state,
which equals 37.17 ps from the G-matrix interaction (with effective charges
$e_{\pi}=1.7e$, $e_{\nu}=1.1e$). Our calculated half-life is slightly higher
than the experimental half-life (6.46 ps NNDC ). Other excited states are
compressed with the G-matrix interaction. The octupole state $3^{-}_{1}$ lies
0.182 MeV higher than the experimental data with the
$\Delta\text{NNLO}_{\text{GO}}$ interaction.  The energy level of the
quadrupole-octupole coupled state $1^{-}$ should be near the sum of the
energies of the quadrupole and octupole phonon states ($2^{+}_{1}$ and
$3^{-}_{1}$, respectively) Garrett1 . The G-matrix calculated $1^{-}_{1}$
state is underpredicted. But, the sum of the $2^{+}_{1}$ and $3^{-}_{1}$
energies (2.170 MeV) is close to the energy level of the $1^{-}_{1}$ state
(2.108 MeV). The $4^{-}_{1}$ state is also a member of quadrupole-octupole
coupled quintuplet Jamieson . The location of the $4^{-}_{1}$ state is 0.180
MeV compressed with the $\Delta\text{NNLO}_{\text{GO}}$ interaction. From this
interaction, the excited $4^{-}_{1}$ state is coming with configuration
$\pi(p_{3/2}^{4}p_{1/2}^{1}g_{9/2}^{9})\otimes\nu(g_{7/2}^{6}d_{5/2}^{6}h_{11/2}^{2})$
and 8.6% probability. Here, the $4^{-}_{1}$ state is formed by the coupling of
one unpaired proton in $p_{1/2}$ orbital and one unpaired proton in $g_{9/2}$
orbital, having seniority $\nu=2$. It is expected that the negative parity
states $5^{-}_{1}$ and $6^{-}_{1}$ are formed by contribution of $h_{11/2}$
neutron orbital. From the G-matrix interaction, the configuration of
$5^{-}_{1}$ and $6^{-}_{1}$ states is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{4}g_{7/2}^{6}d_{3/2}^{1}h_{11/2}^{3})$
with 10% and 7% probabilities, respectively. In the configuration of both
states, we can see that these two negative-parity states ($5^{-}_{1}$ and
$6^{-}_{1}$) are coming from the coupling of valence neutrons in $h_{11/2}$
and $d_{3/2}$ orbitals. For both states, the seniority $\nu=2$.

In the Cd chain, the 114Cd is situated in the middle of the neutron number
$N=50$ and $N=82$. In 114Cd, the excited energy levels are compressed with the
G-matrix interaction, but the order of
$0^{+}_{1}-2^{+}_{1}-4^{+}_{1}-3^{+}_{1}$ states are reproduced well. The
excited $4^{-}_{1}$ state is showing good agreement with the experimental data
calculated from the $\Delta\text{NNLO}_{\text{GO}}$ interaction. With this
interaction, the configuration of $4^{-}_{1}$ state is
$\pi(p_{3/2}^{4}p_{1/2}^{1}g_{9/2}^{9})\otimes\nu(g_{7/2}^{6}d_{5/2}^{6}h_{11/2}^{4})$
with 6.4% probability. Here, this $4^{-}_{1}$ state is coming from the
coupling of one unpaired proton in $g_{9/2}$ orbital and one unpaired proton
in $p_{1/2}$ orbital. Thus, the seniority $\nu=2$. The excited $5^{-}_{1}$
state lies 0.219 MeV lower than the experimental data with
$\Delta\text{NNLO}_{\text{GO}}$ interaction.  The $7^{-}_{1}$ state is coming
with configuration
$\pi(p_{3/2}^{4}g_{9/2}^{10})\otimes(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{1}h_{11/2}^{1})$
from $\Delta\text{NNLO}_{\text{GO}}$ interaction. We can see that no valence
protons are occupied in $p_{1/2}$ orbital, and they jump to the $g_{9/2}$
orbital. This excitation of protons takes more energy to form the $7^{-}$
state. So, this state lies at higher energy in comparison to other negative
parity states from $\Delta\text{NNLO}_{\text{GO}}$ interaction. It is expected
that the $h_{11/2}$ neutron orbital plays a dominating role in the mid-shell
isotopes of the Cd chain, even at low spin Jungclaus . In the configuration of
low-spin states (like $2^{+}_{1}$ state) of 112-114Cd, we can see that the
$h_{11/2}$ orbital starts to dominate.

Figure 5:  Comparison between calculated and experimental NNDC  energy levels
for even-even nuclei of 122-126Cd.

For 116Cd, the order of $0^{+}_{1}-2^{+}_{1}-4^{+}_{1}-4^{+}_{2}$ states are
reproduced with the G-matrix interaction. But, these states are compressed.
The $3^{-}_{1}$ state is also very compressed with the G-matrix interaction
but with $\Delta\text{NNLO}_{\text{GO}}$ interaction, lies 0.327 MeV higher
than the experimental data. The $5^{-}_{1}$ state is comparable with
$\Delta\text{NNLO}_{\text{GO}}$ interaction. From
$\Delta\text{NNLO}_{\text{GO}}$ interaction, the configuration of $5^{-}_{1}$
and $4^{-}_{1}$ states is
$\pi(p_{3/2}^{4}p_{1/2}^{1}g_{9/2}^{9})\otimes\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{4})$
with 4.0% and 5.1% probabilities, respectively. From this interaction, both
states are formed by the coupling of one unpaired proton in $p_{1/2}$ orbital
and one unpaired proton in $g_{9/2}$ orbital like 112,114Cd with seniority
$\nu=2$. With the G-matrix interaction, these states are very compressed.  We
can notice that the energy spectra are compressed with the G-matrix
interaction, but the location of $1^{+}_{1}$ state
($\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{6}s_{1/2}^{1}g_{7/2}^{6}d_{3/2}^{1}h_{11/2}^{4})$)
is higher in comparison to other states. One can also see that the $1^{+}_{1}$
state
($\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{4})$)
is underpredicted with $\Delta\text{NNLO}_{\text{GO}}$ interaction unlike the
most of the states are located at higher energy. The reason may be that this
$1^{+}_{1}$ state is formed by the coupling of valence protons in low-lying
$p_{3/2}$ and $p_{1/2}$ orbitals, which compress the location of this state
with $\Delta\text{NNLO}_{\text{GO}}$ interaction.

In 118Cd, the excited $2^{+}_{1}$ state calculated with the G-matrix
interaction lies 0.279 MeV lower than the experimental data. The location of
2-phonon states $4^{+}_{1}-2^{+}_{2}-0^{+}_{2}$ are also underpredicted with
the G-matrix interaction. But, these states are overpredicted with
$\Delta\text{NNLO}_{\text{GO}}$ interaction.  The excited $2^{+}_{1,2}$ states
($\pi(p_{3/2}^{3}p_{1/2}^{1}g_{9/2}^{10})\otimes\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{6})$)
are showing $\pi(p^{-1}_{3/2}p^{-1}_{1/2})$ character in 34-50 proton model
space with $\Delta\text{NNLO}_{\text{GO}}$ interaction. The G-matrix
calculated $7^{-}_{1}$ state is comparable with the tentative experimental
level at 1.269 MeV. The quadrupole interaction between protons and neutrons is
responsible for the excitation energies of intruder states. Earlier, Wang $et$
$al.$ Wang  studied the intruder states in 118Cd. The G-matrix calculated
intruder state $0^{+}_{3}$ is compressed by 0.396 MeV than the experimental
data. As reported in Ref. Wang , the intruder state $2^{+}_{3}$ (at 1916 keV)
is compressed with the G-matrix interaction. But, this state lies at very high
energy with $\Delta\text{NNLO}_{\text{GO}}$ interaction. In the configuration
of the $2^{+}_{1}$ state, the neutron orbitals $g_{7/2}$ and $h_{11/2}$ are
dominant for $\Delta\text{NNLO}_{\text{GO}}$ interaction. In the neutron-rich
region of Cd-isotopes, we can see that the excited energy levels are lying at
very high energy with $\Delta\text{NNLO}_{\text{GO}}$ interaction. One reason
could be that this chiral interaction is truncated at the two-body level, and
the residual three-body term is not included.

For 120Cd, the order of $0^{+}_{1}-2^{+}_{1}-4^{+}_{1}$ states are reproduced
well with the G-matrix interaction as in the experimental data. But, the
excited $2^{+}_{1}$ and $4^{+}_{1}$ states are compressed with this
interaction. The 2-phonon state $2^{+}_{2}$ lies 0.235 MeV lower than the
experimental data with the G-matrix interaction. From the G-matrix
interaction, the configuration of this state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{6})$
with 28.7% probability, which shows the $2_{2}^{+}$ state comes from one
proton pair breaking in $g_{9/2}$ orbital with seniority $\nu=2$. This state
lies at very high energy with $\Delta\text{NNLO}_{\text{GO}}$ interaction.
The intruder state $0^{+}_{3}$ is overpredicted by 0.243 MeV than the
experimental level. With $\Delta\text{NNLO}_{\text{GO}}$ interaction, excited
$8^{+}_{1}$ state lies 0.434 MeV above the experimental level (2.886 MeV). The
location of $10^{+}_{1}$ state is 0.247 MeV higher with
$\Delta\text{NNLO}_{\text{GO}}$ interaction from the experimental level.  But,
this state is underpredicted with the G-matrix interaction. From
$\Delta\text{NNLO}_{\text{GO}}$ and the G-matrix interactions, this state
comes with configuration
$\pi(p_{3/2}^{4}g_{9/2}^{10})\otimes(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{2}h_{11/2}^{6})$
and
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes(d_{5/2}^{6}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{8})$,
respectively. In the configuration of $10^{+}_{1}$ state there is no proton
occupy in $p_{1/2}$ orbital with $\Delta\text{NNLO}_{\text{GO}}$ interaction.
So, there is an excitation of two valence protons from $p_{1/2}$ to $g_{9/2}$
orbital. But, no such type of excitation occurs with the G-matrix interaction,
and this $10^{+}_{1}$ state is compressed like other states. The order of
$6^{+}_{1}-8^{+}_{1}-10^{+}_{1}$ is same with $\Delta\text{NNLO}_{\text{GO}}$
interaction as in the experimental data.

The low-lying states in 122Cd have been studied up to $J^{\pi}=8^{+}$ in Ref.
Zamfir . For this isotope, the excited energy states lie higher than the
experimental level with $\Delta\text{NNLO}_{\text{GO}}$ interaction. The
$3^{-}_{1}$ state lies 0.365 MeV higher with $\Delta\text{NNLO}_{\text{GO}}$
interaction.  If we include the three-body forces in this chiral interaction
and make some appropriate changes in the adopted model space for our
calculations, then we may be able to predict the location of excited states in
the neutron-rich isotopes more accurately. With the G-matrix interaction,
$3^{-}_{1}$ state is compressed by 0.268 MeV.  But, in comparison to other
states, the $3^{-}_{1}$ state
($\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes(d_{5/2}^{6}g_{7/2}^{6}d_{3/2}^{3}h_{11/2}^{9})$)
lies at high energy with this interaction because it comes from the coupling
of higher-lying neutron orbitals $\nu h_{11/2}$ and $\nu d_{3/2}$. On the
other side, the $3^{-}_{1}$ state calculated with
$\Delta\text{NNLO}_{\text{GO}}$ interaction is coming with
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{2}h_{11/2}^{8})$
configuration. This state is formed by the coupling of valence protons in
$p_{3/2}$ and $g_{9/2}$ orbitals, which have lower single-particle energies.
The $2^{+}_{1}$, $4_{1}^{+}$, $6^{+}_{1}$ and $8^{+}_{1}$ states are also
compressed with the G-matrix interaction. In the formation of $2^{+}_{1}$
state, $h_{11/2}$ orbital is dominant with maximum neutron occupancies for
both interactions.

The level scheme of 124Cd has been reported in previous works Stoyer ;
Vancraeyenest . In 124Cd, the location of $2^{+}_{1}$, $4^{+}_{1}$,
$6^{+}_{1}$ and $8^{+}_{1}$ states lie higher than the experimental data with
$\Delta\text{NNLO}_{\text{GO}}$ interaction. With
$\Delta\text{NNLO}_{\text{GO}}$ interaction, the $5^{-}_{1}$ state
($\pi(p_{3/2}^{4}p_{1/2}^{1}g_{9/2}^{9})\otimes(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{2}h_{11/2}^{10})$)
is comparable with the experimental data. This state is formed by the coupling
of one valence proton in $g_{9/2}$ orbital and one valence proton in $p_{1/2}$
orbital with seniority $\nu=2$. The $5^{-}_{1}$ state is compressed by 0.259
MeV with the G-matrix interaction.  But, the $5^{-}_{2}$ state
($\pi(p_{3/2}^{2}p_{1/2}^{2}g_{9/2}^{10})\otimes(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{1}h_{11/2}^{11})$)
is overpredicted with $\Delta\text{NNLO}_{\text{GO}}$ interaction in
comparison to experimentally observed ($5^{-}$, $6^{+}$) state at 2.560 MeV.
This may be due to proton excitation from $p_{3/2}$ to $g_{9/2}$ orbital.
Another aspect is also important in the formation of the $5^{-}_{2}$ state
that this state is showing neutron character and coming from the coupling of
higher lying $\nu h_{11/2}$ and $\nu d_{3/2}$ orbital, unlike $5^{-}_{1}$
state. The order of excited $0^{+}_{1}$-$2^{+}_{1}$-$4^{+}_{1}$-$2^{+}_{2}$
states are well reproduced by the G-matrix interaction. Like 120,122Cd, the
$h_{11/2}$ orbital is dominant in the formation of $2^{+}_{1}$ state for both
the interactions.

Figure 6:  Comparison between calculated and experimental NNDC  energy levels
for even-even nuclei of 128,130Cd.

Kautzsch $et$ $al.$ Kautzsch  calculated the excited energy levels for 126Cd
using shell-model. In our study, the $2^{+}_{1}$ state is compressed by 0.270
MeV with the G-matrix interaction for this isotope. With
$\Delta\text{NNLO}_{\text{GO}}$ interaction, this state lies higher than the
experimental data.  But, the calculated half-life (6.73 ps) of $2^{+}_{1}$
state with $\Delta\text{NNLO}_{\text{GO}}$ interaction is close to the
experimental half-life 8.9(+27-17) ps. The $2^{+}_{2}$ state is comparable
with the $\Delta\text{NNLO}_{\text{GO}}$ interaction. The $5^{-}_{1}$ state is
comparable with the G-matrix interaction. From this interaction, the
configuration of $5^{-}_{1}$ state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{6}s_{1/2}^{1}g_{7/2}^{8}d_{3/2}^{2}h_{11/2}^{11})$
with 28.6% probability. Here, the $5^{-}_{1}$ state arises from the coupling
of unpaired neutrons in $s_{1/2}$ and $h_{11/2}$ orbitals. With
$\Delta\text{NNLO}_{\text{GO}}$ interaction, this state lies 0.243 MeV higher
than the experimental level. The energy spectra of all states are in the same
order with the G-matrix interaction.  Hoteling $et$ $al.$ Hoteling  suggested
a new energy state $7^{-}$ at 1950 keV. With the G-matrix interaction, this
state is very close to the tentative experimental level with an energy
difference of 15 keV. This state comes from
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{8}d_{3/2}^{3}h_{11/2}^{9})$
configuration with 59.3% probability. From $\Delta\text{NNLO}_{\text{GO}}$
interaction, this state is overpredicted.

For 128Cd, the prediction of the first excited $2^{+}$ state with the G-matrix
interaction is comparable. The $2^{+}_{1}$ state lies above than the
experimental data with $\Delta\text{NNLO}_{\text{GO}}$ interaction. From the
configurations as shown in Table 1, it is clear that the $2^{+}_{1}$ state is
formed by the coupling of one valence proton in $p_{3/2}$ and one valence
proton in $p_{1/2}$ orbitals with $\Delta\text{NNLO}_{\text{GO}}$ interaction.
In the case of the G-matrix interaction, this state comes with one proton pair
breaking in $g_{9/2}$ orbital. But, the seniority is conserved ($\nu=2$) in
all cases. The $4^{+}_{1}$ state lies 0.278 MeV lower than the experimental
data with the G-matrix interaction. With $\Delta\text{NNLO}_{\text{GO}}$
interaction, this state lies at very high energy. The order of
$8^{+}_{1}-10^{+}_{1}$ states is reversed with both interactions. But, the
energy gap between $8^{+}_{1}$ and $10^{+}_{1}$ is very close with
$\Delta\text{NNLO}_{\text{GO}}$ and the G-matrix interactions as in the
experimental data.  As expected, the positive-parity states
$0^{+}_{1}-8^{+}_{1}$ are showing predominent $\pi(g_{9/2}^{-2})$ character
from the G-matrix interaction. The $10^{+}_{1}$ state shows
$\nu(h_{11/2}^{-2})$ character with this interaction. The two negative parity
states $5^{-}_{1}$ and $7^{-}_{1}$ are coming from
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{8}d_{3/2}^{3}h_{11/2}^{11})$
configuration with 39.3% and 84.9% probabilities, respectively. Both states
are showing $\nu(d_{3/2}^{-1}h_{11/2}^{-1})$ character.

In 130Cd, the order of energy spectra with both interactions is the same as in
the experimental data.  The ground state $0^{+}_{1}$ comes from two different
configurations with both interactions. The G-matrix calculated $0^{+}_{1}$
state shows $\pi(g_{9/2}^{-2})$ nature but showing mixed character with
$\pi(p_{1/2}^{-2})$ configuration from $\Delta\text{NNLO}_{\text{GO}}$
interaction, also similar result is obtained in Ref. Jungclaus1 .  The
location of $2^{+}_{1}$ and $4^{+}_{1}$ states reproduced very well with the
G-matrix interaction. With $\Delta\text{NNLO}_{\text{GO}}$ interaction,
$2^{+}_{1}$ state is slightly compressed. The $6^{+}_{1}$ and $8^{+}_{1}$
states lie slightly higher with the G-matrix interaction. With
$\Delta\text{NNLO}_{\text{GO}}$ interaction, these states lie at high energy
than the experimental data. The energy gap between $4^{+}_{1}-6^{+}_{1}$, and
$6^{+}_{1}-8^{+}_{1}$ states are very close as in the experimental data with
$\Delta\text{NNLO}_{\text{GO}}$ interaction.  From both the interactions,
these three states ($4^{+}_{1}$, $6^{+}_{1}$ and $8^{+}_{1}$) are showing
$\pi(g_{9/2}^{-2})$ character. We have also calculated excited $5^{-}_{1}$ and
$4^{-}_{1}$ states, which are not observed experimentally yet. For the
G-matrix and $\Delta\text{NNLO}_{\text{GO}}$ interaction, the location of
these two states are comparable to each other. Thus, our results may help to
compare new experimental data in the future.

We can notice that our energy spectra are deviating from the experimental data
from 110Cd and start to compress up to 120Cd from the G-matrix interaction.
The reason behind this is that when we perform shell-model calculations using
realistic interactions and increase the number of particles (neutrons), our
results deteriorate rapidly Caurier . In the Cd isotopes after the 120Cd, the
increment in neutron particles is treated like a decrease in neutron holes
from the magic nuclei 132Sn. So, our excited states are rising upward with
respect to the ground state.

In Fig. 7, we have shown the occupancy of different neutron orbitals
corresponding to $2^{+}_{1}$ state from $N=50$ to $N=82$ in the Cd isotopic
chain. The $d_{5/2}$ and $g_{7/2}$ orbitals are dominant in the neutron-
deficient region for both interactions. With the increase of neutron numbers,
occupancy increases in all orbitals. In the mid of the Cd chain, the dominance
of the $h_{11/2}$ orbital can be observed. As we further increase the neutron
number, the occupancy of $h_{11/2}$ orbital becomes more than the other
orbitals, which shows the importance of $h_{11/2}$ orbital in the formation of
$2^{+}_{1}$ state in neutron-rich Cd isotopes.

Figure 7:  The occupancy for different neutron orbitals corresponding to
excited $2^{+}_{1}$ state in the Cd isotopic chain with
$\Delta\text{NNLO}_{\text{GO}}$, and the G-matrix interactions. Figure 8:
Comparison of the theoretical and experimental excitation energy ratios of
$4^{+}_{1}$ and $2^{+}_{1}$ states in Cd chain. The experimental data are
taken from NNDC .

In Fig. 8, we have compared the experimental and theoretical excitation energy
ratios of $4^{+}_{1}$ and $2^{+}_{1}$ states in the Cd isotopic chain. We can
notice that the $4^{+}_{1}/2^{+}_{1}$ energy ratio ($R_{4/2}$) follows the
same trend from the $N=50$ closed shell toward more collective structures with
the G-matrix interaction as in the experimental data. The value of $R_{4/2}$
ratio is less than $2.0$ for 98,100Cd isotopes with both interactions, which
corresponds to a single-particle structure near closed shells. For 102-108Cd
isotopes, the $R_{4/2}$ ratio lies between 2.0 and 2.6 with the G-matrix
interaction, which shows that these isotopes are quasi-deformed. For 106Cd,
the experimental value of $R_{4/2}$ ratio is 2.360. From our calculation using
the G-matrix interaction, the $R_{4/2}$ ratio is 2.393. The predicted
$R_{4/2}$ ratio from the pure vibrational and rotational model are 2.00 and
3.33, respectively Koller . Here, our calculated result shows good agreement
with the experimental data. For 108Cd, this ratio is 2.516. But, there is a
decrement in the $R_{4/2}$ ratio at $N=62$, which indicates 108Cd has large
deformation. In the neutron-rich isotopes of Cd, the behavior of $R_{4/2}$
ratio is almost constant with $\Delta\text{NNLO}_{\text{GO}}$ interaction.
But, the G-matrix interaction results are staggering from $N=70$ to $N=78$. In
120,124Cd, the configuration of $4^{+}_{1}$ states are
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{6}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{8})$
and
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{6}d_{3/2}^{2}h_{11/2}^{10})$
with the G-matrix interaction, respectively. There are no unpaired nucleons in
valence orbitals. So, it needs more energy to form the $4^{+}_{1}$ state,
which suddenly increases the $R_{4/2}$ ratio for 120,124Cd isotopes. We can
also notice that there is a divergence shown at $N=82$ with
$\text{NNLO}_{\text{GO}}$ interaction. We need to include more valence neutron
orbitals beyond $N=82$ to get converged results as in the experimental data.

It is important to add here the chiral effective shell-model interactions
provide advances in nuclear structure theory to study medium mass region and
help to investigate the unexplored fraction of the nuclear chart. Despite
this, these chiral interactions struggle to accurately reproduce the nuclear
properties in the neutron-rich region observed from the experiments. We found
that the $\Delta\text{NNLO}_{\text{GO}}$ interaction fails to reproduce the
location of energy levels in neutron-rich Cd isotopes. We have tested this
chiral interaction for a full isotopic chain for the first time without any
truncation. Our study reveals that this interaction needs further improvements
and the inclusion of $3N$ forces for better results.

###  IV.2 Electromagnetic properties

Figure 9:  Behavior of $B(E2;0^{+}_{1}\rightarrow 2^{+}_{1})$ values along Cd isotopic chain. The experimental data are taken from Pritychenko ; Maheshwari . Figure 10:  Electric quadrupole moments for $2^{+}_{1}$ state in Cd isotopic chain. The experimental data for 102,104Cd and 106-116Cd are taken from Ekstrom  and NDS , respectively. Figure 11:  Magnetic moments for $2^{+}_{1}$ state in Cd isotopic chain. The experimental data are taken from NDS . Table 2: Comparison of theoretical and experimental $B(E2)$ transitions in cadmium isotopes. In our calculations we have taken ($e_{p}$, $e_{n}$)$=$(1.7, 1.1) Boelaert . For the G-matrix interaction, another two sets of effective charges (1.4, 1.1) and (1.4, 0.85) are used Zuker . The units are in $e^{2}b^{2}$. Isotope | $J_{i}^{\pi}\rightarrow J_{f}^{\pi}$ | $\Delta\text{NNLO}_{\text{GO}}$ | G-matrix | Exp.
---|---|---|---|---
|  |  | (1.7, 1.1) | (1.4, 1.1) | (1.4, 0.85) |
98Cd |  $0^{+}_{1}$$\rightarrow$$2^{+}_{1}$ | 0.069 | 0.069 | 0.047 | 0.047 | \-
|  $2^{+}_{1}$$\rightarrow$$4^{+}_{1}$ | 0.032 | 0.033 | 0.022 | 0.022 | \-
|  $4^{+}_{1}$$\rightarrow$$6^{+}_{1}$ | 0.018 | 0.018 | 0.012 | 0.012 | 0.018(3)
100Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.097 | 0.167 | 0.135 | 0.107 | 0.33(2)
|  $2_{1}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.048 | 0.047 | 0.040 | 0.029 | \-
|  $4_{1}^{+}$$\rightarrow$$6_{1}^{+}$ | 0.029 | 0.021 | 0.018 | 0.013 | \-
102Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.115 | 0.274 | 0.232 | 0.173 | 0.257(23)
|  $2_{1}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.059 | 0.143 | 0.122 | 0.090 |  $>$0.038
|  $4_{1}^{+}$$\rightarrow$$6_{1}^{+}$ | 0.035 | 0.001 | 0.001 | 0.001 | 0.057(45)
104Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.108 | 0.383 | 0.334 | 0.240 | 0.341(40)
|  $2_{1}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.063 | 0.198 | 0.175 | 0.123 | 0.058 $>$
|  $4_{1}^{+}$$\rightarrow$$6_{1}^{+}$ | 0.035 | 0.154 | 0.137 | 0.096 | 0.037 $>$
106Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.092 | 0.444 | 0.391 | 0.278 | 0.407(12)
|  $2_{1}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.068 | 0.230 | 0.205 | 0.143 | 0.247(32)
|  $4_{1}^{+}$$\rightarrow$$6_{1}^{+}$ | 0.017 | 0.021 | 0.019 | 0.013 | \-
108Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.080 | 0.487 | 0.431 | 0.304 | 0.419(14)
|  $2_{1}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.064 | 0.255 | 0.228 | 0.158 | 0.225(33)
110Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.055 | 0.547 | 0.491 | 0.340 | 0.426(21)
|  $2_{1}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.048 | 0.228 | 0.209 | 0.141 | 0.237(51)
112Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.024 | 0.619 | 0.566 | 0.382 | 0.501(22)
|  $2_{1}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.020 | 0.303 | 0.278 | 0.186 | 0.364(46)
114Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.030 | 0.704 | 0.646 | 0.434 | 0.536(25)
116Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.043 | 0.753 | 0.690 | 0.464 | 0.580(26)
118Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.051 | 0.754 | 0.690 | 0.465 | 0.578(44)
120Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.050 | 0.751 | 0.686 | 0.464 | 0.473(55)
122Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.041 | 0.603 | 0.544 | 0.374 | 0.44(20)
124Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.025 | 0.499 | 0.443 | 0.311 | 0.35(19)
126Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.016 | 0.359 | 0.312 | 0.225 | 0.263(60)
128Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.010 | 0.190 | 0.158 | 0.121 | \-
130Cd |  $0_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.048 | 0.080 | 0.054 | 0.054 | \-
Table 3: Comparison of theoretical and experimental electric quadrapole and magnetic moments in cadmium isotopes. In our calculations we have taken ($e_{p}$, $e_{n}$)$=$(1.7, 1.1); $g_{s}^{p}=3.91$, $g_{s}^{n}=-2.678$. For the G-matrix interaction, another two sets of effective charges (1.4, 1.1) and (1.4, 0.85) are used. |  |  |  Q (eb) |  $\mu$ ($\mu_{N}$)  |  |  |
---|---|---|---|---|---|---|---
Nuclei | $A$ | $J^{\pi}$ | $\Delta\text{NNLO}_{\text{GO}}$ | G-matrix | Exp. | $\Delta\text{NNLO}_{\text{GO}}$ | G-matrix | Exp. |
|  |  |  | (1.7, 1.1) | (1.4, 1.1) | (1.4, 0.85) |  |  |  |  |
Cd | 98 | $2^{+}_{1}$ | -0.218 | -0.219 | -0.180 | -0.180 | - | 2.671 | 2.647 | - |
|  | $4^{+}_{1}$ | -0.119 | -0.139 | -0.115 | -0.115 | - | 5.293 | 5.293 | - |
|  | $6^{+}_{1}$ | 0.088 | 0.088 | 0.072 | 0.072 | - | 7.940 | 7.940 | - |
|  | $8^{+}_{1}$ | 0.438 | 0.438 | 0.361 | 0.361 | - | 10.587 | 10.587 | - |
| 100 | $2^{+}_{1}$ | -0.231 | -0.291 | -0.259 | -0.234 | - | 2.418 | 1.405 | - |
|  | $4^{+}_{1}$ | -0.165 | -0.361 | -0.330 | -0.287 | - | 4.937 | -0.232 | - |
|  | $6^{+}_{1}$ | 0.104 | -0.572 | -0.534 | -0.453 | - | 7.504 | 0.220 | - |
|  | $8^{+}_{1}$ | 0.397 | 0.659 | 0.590 | 0.530 | - | 10.226 | 10.154 | 9.9(5) |
| 102 | $2^{+}_{1}$ | -0.266 | -0.441 | -0.406 | -0.351 | -0.20(1) | 2.085 | 1.288 | - |
|  | $4^{+}_{1}$ | -0.223 | -0.527 | -0.489 | -0.418 | - | 4.221 | 2.436 | - |
|  | $6^{+}_{1}$ | 0.038 | -0.473 | -0.437 | -0.376 | - | 6.566 | 0.572 | - |
|  | $8^{+}_{1}$ | 0.575 | 0.738 | 0.683 | 0.586 | 0.76(9) | 9.882 | 9.117 | 10.3(2) |
| 104 | $2^{+}_{1}$ | -0.280 | -0.538 | -0.503 | -0.426 | -0.23(1) | 1.905 | 1.145 | - |
|  | $4^{+}_{1}$ | -0.245 | -0.675 | -0.634 | -0.533 | - | 3.992 | 1.844 | - |
| 106 | $2^{+}_{1}$ | -0.281 | -0.565 | -0.529 | -0.447 | -0.28(8) | 1.782 | 1.108 | 0.79(6) |
|  | $4^{+}_{1}$ | -0.255 | -0.691 | -0.650 | -0.545 | - | 3.740 | 1.935 | 0.9(2) |
| 108 | $2^{+}_{1}$ | -0.267 | -0.570 | -0.534 | -0.450 | -0.45(8) | 1.702 | 1.019 | 0.78(6) |
|  | $4^{+}_{1}$ | -0.250 | -0.604 | -0.564 | -0.478 | - | 3.350 | 1.781 | - |
| 110 | $2^{+}_{1}$ | -0.209 | -0.407 | -0.378 | -0.323 | -0.40(4) | 1.988 | 0.738 | 0.81(6) |
|  | $4^{+}_{1}$ | -0.202 | -0.217 | -0.195 | -0.181 | - | 3.648 | 0.295 | - |
| 112 | $2^{+}_{1}$ | -0.074 | -0.087 | -0.071 | -0.072 | -0.37(4) | 2.432 | 0.307 | 0.72(5) |
|  | $4^{+}_{1}$ | -0.161 | -0.157 | -0.132 | -0.129 | - | 3.239 | 0.179 | - |
| 114 | $2^{+}_{1}$ | 0.009 | -0.269 | -0.248 | -0.214 | -0.348(12) | 2.146 | 0.217 | 0.65(4) |
| 116 | $2^{+}_{1}$ | 0.044 | -0.505 | -0.478 | -0.399 | -0.42(4) | 1.832 | 0.139 | 0.59(5) |
| 118 | $2^{+}_{1}$ | 0.050 | -0.682 | -0.650 | -0.537 | - | 1.640 | 0.067 | - |
| 120 | $2^{+}_{1}$ | 0.039 | -0.753 | -0.719 | -0.592 | - | 1.552 | 0.077 | - |
| 122 | $2^{+}_{1}$ | 0.025 | -0.601 | -0.567 | -0.474 | - | 1.547 | 0.217 | - |
| 124 | $2^{+}_{1}$ | 0.027 | -0.512 | -0.479 | -0.406 | - | 1.611 | 0.478 | - |
| 126 | $2^{+}_{1}$ | -0.015 | -0.386 | -0.353 | -0.307 | - | 1.469 | 0.514 | - |
| 128 | $2^{+}_{1}$ | 0.094 | -0.325 | -0.292 | -0.261 | - | 1.704 | 0.620 | - |
| 130 | $2^{+}_{1}$ | 0.024 | -0.238 | -0.196 | -0.196 | - | 3.758 | 2.647 | - |
Table 4: Comparison between theoretical and experimental $B(M1)$ (in $\mu_{N}^{2}$) values in cadmium isotopes. In our calculations we have taken $g_{s}^{p}=3.91$, $g_{s}^{n}=-2.678$. Isotope | $J_{i}^{\pi}\rightarrow J_{f}^{\pi}$ | $\Delta\text{NNLO}_{\text{GO}}$ | G-matrix | Exp.
---|---|---|---|---
98Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.134 | 0.818 | \-
|  $4_{2}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.178 | 0.477 | \-
100Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.246 | 0.033 | \-
|  $4_{2}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.267 | 0.134 | \-
102Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.162 | 0.043 | \-
|  $5_{1}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.001 | 0.002 | \-
104Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.162 | 0.043 | \-
|  $4_{2}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.041 | 0.069 | \-
106Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.192 | 0.015 | 0.013(3)
|  $4_{2}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.001 | 0.001 |  $>$0.047
108Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.008 | 0.025 | 0.014(9)
|  $6_{1}^{-}$$\rightarrow$$5_{1}^{-}$ | 0.392 | 0.021 | 0.0027(18)
110Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.002 | 0.095 | 0.024(4)
|  $4_{3}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.040 | 0.002 | 0.104(+30-45)
112Cd |  $3_{1}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.0001 | 0.0004 | 0.003(1)
116Cd |  $4_{2}^{+}$$\rightarrow$$4_{1}^{+}$ | 0.432 | 0.001 | 0.11(+7-11)
118Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.495 | 0.010 | \-
120Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.599 | 0.0004 | \-
122Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.605 | 0.003 | \-
124Cd |  $1_{1}^{+}$$\rightarrow$$0_{1}^{+}$ | 0.059 | 0.0002 | \-
126Cd |  $2_{2}^{+}$$\rightarrow$$2_{1}^{+}$ | 0.493 | 0.0001 | \-

In this section, we have discussed reduced transition probabilities,
quadrupole, and magnetic moments for even Cd isotopes. We have explained the
behavior of $B(E2;0^{+}_{1}\rightarrow 2^{+}_{1})$ transition for Cd isotopes
from $N=50$ to $N=82$. The $B(E2)$ values of the Cd and Sn isotopic chain show
similar trends Siciliano . In the experimental data of $B(E2)$ values of the
Cd chain, there is a dip near $N=62$, as shown in Fig. 9. With our
interactions, the neutron orbitals are $d_{5/2}$, $s_{1/2}$, $d_{3/2}$,
$g_{7/2}$ and $h_{11/2}$ in the order of increasing single-particle energies,
respectively. The asymmetry near $N=62$ is because $d_{5/2}$, $s_{1/2}$ and
$d_{3/2}$ orbitals lie lower than remaining $g_{7/2}$ and $h_{11/2}$ orbitals.
Before $N=62$, $d_{5/2}$ orbital is dominating and after that $h_{11/2}$
orbital is dominating. Our results from the G-matrix interaction are in good
agreement with the experimental trend. We have used three sets of effective
charges $(e_{\pi},e_{\nu})$ for calculating $B(E2;0^{+}_{1}\rightarrow
2^{+}_{1})$ values with the G-matrix interaction. The effective charges (1.7,
1.1) are taken from Ref. Boelaert . Another two sets (1.4, 1.1) and (1.4,
0.85) are taken from Ref. Zuker . We can notice that our results with the
effective charges (1.4, 1.1) are comparable with the experimental trend before
the mid-shell. With the effective charges (1.4, 0.85), our results show good
agreement with the experiment in the neutron-rich region of Cd-isotopes. With
$\Delta\text{NNLO}_{\text{GO}}$ interaction, the shape of the inverted
parabola is very compressed.

We have also calculated $B(E2)$ transitions for other states, reported in
Table LABEL:BE2. In 98Cd, $B(E2;4^{+}_{1}\rightarrow 6^{+}_{1})$ value is
reproduced well and equals to 0.018 $e^{2}b^{2}$ with
$\Delta\text{NNLO}_{\text{GO}}$ and the G-matrix ($e_{p}=1.7$, $e_{n}=1.1$)
interactions. Park $et$ $al.$ Park  studied the $6^{+}_{1}$ isomeric state in
a decay spectroscopy experiment with half-life, $T_{1/2}=13(2)$ ns. With the
G-matrix ($e_{p}=1.7$, $e_{n}=1.1$) interaction, the calculated half-life of
the $6^{+}_{1}$ state is 1.92 ns. In the case of even 106-112Cd isotopes, the
$B(E2;2^{+}_{1}\rightarrow 4^{+}_{1})$ values calculated from the G-matrix
($e_{p}=1.7$, $e_{n}=1.1$) interaction are showing good agreement with the
experimental data NNDC .

In most cases, $B(E2)$ values apart from $B(E2;0^{+}_{1}\rightarrow
2^{+}_{1})$ are in good agreement with the G-matrix interaction. But, with
$\Delta\text{NNLO}_{\text{GO}}$ interaction, we need to include the effects of
$NNN$ forces to reproduce better results.

The trend of the $11/2^{-}$ quadrupole moments for odd 111-129Cd isotopes have
been studied in previous works Yordanov ; Yang . In 106-116Cd isotopes, the
experimental data are available for $Q(2^{+}_{1})$ values. We have used shell-
model interpreted data of $Q(2^{+}_{1})$ Ekstrom  for 102,104Cd. We have
compared our results with the experimental data for even Cd isotopic chain as
shown in Fig. 10. One can notice that the $Q(2^{+}_{1})$ values show nearly
constant behavior for 102-106Cd isotopes. Results from
$\Delta\text{NNLO}_{\text{GO}}$ interaction are comparable for these isotopes.
Experimentally, there is a dip of $Q(2^{+}_{1})$ value at 108Cd. After that,
the trend of $Q(2^{+}_{1})$ is almost constant up to 116Cd. For 98,100Cd
isotopes, $Q(2^{+}_{1})$ values are very close to each other with all
interactions. But, they have not been observed in the experiment yet.
Experimental data are not available for the electric quadrupole moment of
$2^{+}_{1}$ state in neutron-rich even Cd isotopes. In the future, our results
may be helpful in comparing upcoming experimental results in Cd isotopes.

For 102Cd, the experimental $Q(8^{+}_{1})$ value is close to our result with
the G-matrix interaction ($e_{p}=1.7$, $e_{n}=1.1$). We have also calculated
electric quadrupole moments for other states, as reported in Table 3.

For magnetic moments of $2^{+}_{1}$ state, the experimental data are available
only for 106-116Cd isotopes like quadrupole moments. The trend of our
calculated results for $\mu(2^{+}_{1})$ values is not similar to the
experimental data as shown in Fig. 11. But, some $\mu(2^{+}_{1})$ values are
close to the experimental data calculated with the G-matrix interaction. In
the case of 100,102Cd isotopes, the experimental $\mu(8^{+}_{1})$ show good
agreement with our results. For 106Cd, the $\mu(4^{+}_{1})$ value is two times
higher than the experimental value with the G-matrix interaction. With
$\Delta\text{NNLO}_{\text{GO}}$ interaction, this value is overpredicted. In
Table 3, we have also reported magnetic moments for other states where the
experimental data are not available.

For the even Cd isotopic chain, the experimental data of $B(M1)$ values are
available only for a few states. We have reported $B(M1)$ transitions for
these isotopes in Table 4. In case of 106Cd, $B(M1;2^{+}_{2}\rightarrow
2^{+}_{1})$ value is reproduced well with the G-matrix interaction. For
$\Delta\text{NNLO}_{\text{GO}}$ interaction, our calculated $B(M1)$ values are
higher than the experimental data.

Figure 12:  Comparison of the experimental excitation energy of the
$2^{+}_{1}$ states for Cd isotopic chain with our results.

###  IV.3 Variation in the excitation energy of $2^{+}_{1}$ states from $N=50$
to $N=82$

In this section, we have explained the nature of excitation energy of the
$2^{+}_{1}$ state for the Cd isotopic chain. We can notice that there is a
small change at $N=62$ in the variation of the experimental excitation energy
of $2^{+}_{1}$ state, as shown in Fig. 12. The cause may be a sub-shell gap in
the single-particle energies of neutron orbitals. When neutrons start
occupying in the valence space, first $d_{5/2}$, $s_{1/2}$ and $d_{3/2}$
orbitals prefer to filling. As we further increase the neutrons around $N=62$,
the dominance of $h_{11/2}$ orbital will be started. So, the change in
variation of $2^{+}_{1}$ state energy is related to the change in the filling
of neutron orbitals. At $N=82$, there is the neutron shell closure. Hence, the
energy of the excited $2^{+}_{1}$ state suddenly increases at $N=82$.

The trend for excited $2^{+}_{1}$ energy states for $N=50-82$ reproduced well
with the G-matrix interaction. With $\Delta\text{NNLO}_{\text{GO}}$
interaction, these energy states lie higher than the experimental data. We
need some modifications to reproduce a better trend in the variation of
$2^{+}_{1}$ state energies.

Table 5: Comparison between theoretical and experimental $B(E3;0^{+}_{1}$$\rightarrow$$3^{-}_{1})$ (in $e^{2}b^{3}$) values in cadmium isotopes. In our calculations we have taken ($e_{p}$, $e_{n}$)$=$(1.7, 1.1). For the G-matrix interaction, another two sets of effective charges (1.4, 1.1) and (1.4, 0.85) are also used. Isotope | $J_{i}^{\pi}\rightarrow J_{f}^{\pi}$ | $\Delta\text{NNLO}_{\text{GO}}$ | G-matrix | Exp.
---|---|---|---|---
|  |  | (1.7, 1.1) | (1.4, 1.1) | (1.4, 0.85) |
106Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.021 | 0.001 | 0.001 | 0.007$\times 10^{-1}$ | 0.16(4)
108Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.030 | 0.003$\times 10^{-1}$ | 0.003$\times 10^{-1}$ | 0.002$\times 10^{-1}$ | 0.142(21)
110Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.041 | 0.002 | 0.002 | 0.001 | 0.120(23)
112Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.055 | 0.096$\times 10^{-4}$ | 0.096$\times 10^{-4}$ | 0.057$\times 10^{-4}$ | 0.120(25)
114Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.060 | 0.675$\times 10^{-3}$ | 0.675$\times 10^{-3}$ | 0.403$\times 10^{-3}$ | 0.166(27)
116Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.060 | 0.001 | 0.902$\times 10^{-3}$ | 0.538$\times 10^{-3}$ | 0.100(11)
118Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.058 | 0.003 | 0.003 | 0.002 | \-
120Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.055 | 0.004$\times 10^{-1}$ | 0.004$\times 10^{-1}$ | 0.002$\times 10^{-1}$ | \-
122Cd |  $0_{1}^{+}$$\rightarrow$$3_{1}^{-}$ | 0.050 | 0.677$\times 10^{-4}$ | 0.007$\times 10^{-2}$ | 0.004$\times 10^{-2}$ | \-

###  IV.4 Octupole collective $3^{-}$ states in even-even Cd isotopes

The study of $3^{-}$ states are important for the understanding of the
octupole vibrational mode of a nucleus. Low-lying $3^{-}$ octupole states are
found in 106-122Cd isotopes. They decay via electromagnetic transitions to
$0^{+}$ and $2^{+}$ states. In Fig. 13, we have compared the experimental
trend of the $3^{-}$ state in the Cd chain with our results. We can see that
the results with $\Delta\text{NNLO}_{\text{GO}}$ interaction are very close to
the experimental data. The $3^{-}_{1}$ states calculated with the G-matrix
interaction deviate in the mid-shell.

In 106Cd, the octupole $3^{-}_{1}$ state is experimentally observed at the
energy of 2.378 MeV. The location of this state is comparable with
$\Delta\text{NNLO}_{\text{GO}}$ interaction. For
$\Delta\text{NNLO}_{\text{GO}}$ interaction, the wave function corresponding
to $3^{-}_{1}$ state is
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{4}d_{5/2}^{4})$
with 16.9% probability. With the G-matrix interaction, this state lies higher
than the experimental data. Here, the $3^{-}_{1}$ state is formed by the
coupling of one unpaired proton in $p_{3/2}$ orbital and one unpaired proton
in $g_{9/2}$ orbital with $\Delta\text{NNLO}_{\text{GO}}$ interaction. Thus,
the seniority $\nu=2$.

Figure 13:  Comparison of the experimental excitation energy of the octupole
collective $3^{-}_{1}$ states for 106-122Cd isotopes with our results.

The G-matrix and $\Delta\text{NNLO}_{\text{GO}}$ interactions are able to
reproduce octupole $3^{-}_{1}$ state in 108Cd. The wave functions
corresponding to $3^{-}_{1}$ state are
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{6}d_{5/2}^{4})$ and
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{4}g_{7/2}^{5}h_{11/2}^{1})$
with probabilities 12.4% and 15.8% for $\Delta\text{NNLO}_{\text{GO}}$ and the
G-matrix interactions, respectively. In the case of
$\Delta\text{NNLO}_{\text{GO}}$ interaction, $3^{-}_{1}$ state is arising from
the coupling of single unpaired protons in $g_{9/2}$ and $p_{3/2}$ orbitals.
For the G-matrix interaction, this state is formed by the coupling of one
unpaired neutron in $h_{11/2}$ orbital with one unpaired neutron in $g_{7/2}$
orbital. In both cases, the seniority is conserved and equal to 2.

For 110Cd, the octupole $3^{-}_{1}$ state is showing good agreement with the
experimental data for $\Delta\text{NNLO}_{\text{GO}}$ and the G-matrix
interactions. For $\Delta\text{NNLO}_{\text{GO}}$ interaction, configuration
of $3^{-}_{1}$ state is
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{6}d_{5/2}^{6})$
with 11.8% probability. The neutron occupancies in $g_{7/2}$, $d_{5/2}$,
$d_{3/2}$, $s_{1/2}$ and $h_{11/2}$ orbitals are 5.250, 4.276, 0.514, 0.191
and 1.769, respectively. This $3^{-}_{1}$ state is formed by the coupling of
single unpaired protons in $g_{9/2}$ and $p_{3/2}$ orbitals. Thus, the
seniority $\nu=2$.

In case of 112,114,116Cd isotopes, the configurations of $3^{-}_{1}$ state
from $\Delta\text{NNLO}_{\text{GO}}$ interaction are
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{6}d_{5/2}^{6}h_{11/2}^{2})$,
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{6}d_{5/2}^{6}h_{11/2}^{4})$
and
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{4})$
with 8.5%, 6.1% and 4.7% probabilities, respectively. For these three
isotopes, this octupole $3^{-}$ state arises from the coupling of one unpaired
proton in $p_{3/2}$ orbital and one unpaired proton in $g_{9/2}$ orbital.
Thus, the seniority is 2.

For 118Cd, $3^{-}_{1}$ state is very compressed with G-matrix interaction.
With $\Delta\text{NNLO}_{\text{GO}}$ interaction, this state is 0.322 MeV
higher than the experimental level with configuration
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{6})$
and 4.7% probability. The neutron occupancies are 6.813, 4.672, 1.328, 0.701,
6.485 in $g_{7/2}$, $d_{5/2}$, $d_{3/2}$, $s_{1/2}$ and $h_{11/2}$ orbitals,
respectively. Here, $h_{11/2}$ orbital starts to dominate in the formation of
the $3^{-}_{1}$ state.

In case of 120Cd, location of $3^{-}_{1}$ state is 0.363 MeV higher than the
experimental data with $\Delta\text{NNLO}_{\text{GO}}$ interaction. For this
interaction, the wave function corresponding to $3^{-}_{1}$ state is
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{8}d_{5/2}^{6}h_{11/2}^{8})$
with maximum probability 4.4%.

For 122Cd, the wave functions corresponding to octupole $3^{-}_{1}$ state are
$\pi(p_{3/2}^{3}p_{1/2}^{2}g_{9/2}^{9})\otimes\nu(g_{7/2}^{8}d_{5/2}^{6}d_{3/2}^{2}h_{11/2}^{8})$
and
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{6}g_{7/2}^{6}d_{3/2}^{3}h_{11/2}^{9})$
with probabilities 5.3% and 27.3% for $\Delta\text{NNLO}_{\text{GO}}$ and the
G-matrix interactions, respectively. Here, $\nu h_{11/2}$ orbital is dominant
for both interactions with maximum occupancies of 8.998
($\Delta\text{NNLO}_{\text{GO}}$) and 8.912 (G-matrix interaction),
respectively.

For the octupole $3^{-}_{1}$ state, we can notice that the seniority remains
constant ($\nu=2$) with $\Delta\text{NNLO}_{\text{GO}}$ interaction in the Cd
chain.  From this interaction, the $3^{-}_{1}$ states are showing
$\pi(p_{3/2}^{-1}g_{9/2}^{-1})$ character for 106-122Cd isotopes. We have also
calculated $B(E3;0^{+}_{1}\rightarrow 3^{-}_{1})$ transitions in 106-122Cd
isotopes and compared them with the experimental data Fewell  as reported in
Table 5. With $\Delta\text{NNLO}_{\text{GO}}$ interaction, our calculated
$B(E3;0^{+}_{1}\rightarrow 3^{-}_{1})$ values are slightly lower than the
experimental data. The G-matrix calculated results are very small.

Table 6: Comparison of theoretical and experimental $B(E2)$ transitions (in $e^{2}b^{2}$) of $8^{+}_{1}$ isomeric states in Cd isotopes. The calculated half-lives are also reported. In our calculations we have taken $e_{p}=1.7e$ and $e_{n}=1.1e$. For the G-matrix interaction, another set of effective charge (1.4, 1.1) is also used Zuker . |  |  | $B(E2)$ |  $T_{1/2}$
---|---|---|---|---
Isotope | $J_{i}^{\pi}\rightarrow J_{f}^{\pi}$ | $\Delta\text{NNLO}_{\text{GO}}$ | G-matrix | Exp. | $\Delta\text{NNLO}_{\text{GO}}$ | G-matrix | Exp.
|  |  | (1.7, 1.1) | (1.4, 1.1) |  |  | (1.7, 1.1) | (1.4, 1.1) |
98Cd | $8^{+}_{1}\rightarrow 6^{+}_{1}$ | 0.005 | 0.005 | 0.0034 | 0.0038(4) | 0.39 $\mu$s | 0.81 $\mu$s | 1.19 $\mu$s | 0.154(16) $\mu$s
100Cd | $8^{+}_{1}\rightarrow 6^{+}_{1}$ | 0.008 | 0.0002 | 0.0001 | 0.000042(5) | 2.19 $\mu$s | 4.33$\times 10^{-3}\mu$s | 5.78$\times 10^{-3}\mu$s | 0.073 $\mu$s
| $8^{+}_{1}\rightarrow 6^{+}_{2}$ | 0.0196$\times 10^{-4}$ | 0.0079 | 0.0061 | 0.0061(+33-28) | 79.85 $\mu$s | 10.03 $\mu$s | 13.101 $\mu$s | 0.413 $\mu$s
102Cd | $8^{+}_{1}\rightarrow 6^{+}_{1}$ | 0.009 | 0.007 | 0.006 | 0.000017(3) | 12.2 $\mu$s | 97.5$\times 10^{-6}\mu$s | 1.17$\times 10^{-4}\mu$s | 0.093 $\mu$s
| $8^{+}_{1}\rightarrow 6^{+}_{2}$ | 0.838$\times 10^{-4}$ | 0.004 | 0.0029 | - | 6.65 $\mu$s | 0.12 $\mu$s | 0.15 $\mu$s | \-
| $8^{+}_{1}\rightarrow 6^{+}_{3}$ | 0.192$\times 10^{-4}$ | 0.0006 | 0.0005 | - | 0.426 $\mu$s | 49.3 $\mu$s | 55.25 $\mu$s | 0.067 $\mu$s
104Cd | $8^{+}_{1}\rightarrow 6^{+}_{1}$ | 0.0083 | 0.1087 | 0.096 | 0.0015(5) | 2.37 $\mu$s | 3.12$\times 10^{-6}\mu$s | 3.55$\times 10^{-6}\mu$s |  8.6$\times 10^{-4}\mu$s
| $8^{+}_{1}\rightarrow 6^{+}_{2}$ | 0.0004 | 0.0001 | 0.005$\times 10^{-2}$ | 0.0002(1) | 621.45 $\mu$s | 37.07$\times 10^{-3}\mu$s | 63.5$\times 10^{-3}\mu$s |  11.43$\times 10^{-3}\mu$s
130Cd | $8^{+}_{1}\rightarrow 6^{+}_{1}$ | 0.006 | 0.006 | 0.004 | - | 0.33 $\mu$s | 1.12 $\mu$s | 1.65 $\mu$s | 0.220(30) $\mu$s

###  IV.5 $8^{+}$ isomeric states in Cd chain

Many theoretical and experimental investigations have been performed to study
isomeric states in the different mass regions of the nuclear chart Srivastava
; Bhoy ; Bhoy1 ; Srivastava1 ; Garg ; Axel ; Wimmer . The $8^{+}$ isomers have
been observed experimentally in 98-104,130Cd isotopes. In this section, we
briefly discussed the $B(E2)$ transitions, neutron occupancies, configurations
of the $8^{+}$ isomeric states, and their g-factor values in the Cd chain.

Figure 14:  Neutron occupancies in $8^{+}$ isomeric states for 100,102,130Cd
isotopes.

In 98Cd, the wave function corresponding to $8^{+}_{1}$ isomeric state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})$ for the G-matrix interaction with 100%
probability. Here, the $8^{+}_{1}$ state is formed by one proton pair breaking
of the $g_{9/2}$ orbital. So, the seniority $\nu=2$. Our result for
$B(E2;8^{+}_{1}\rightarrow 6^{+}_{1})$ transition is comparable to the
experimental data. The experimental and the G-matrix calculated (with
$e_{p}=1.4$, $e_{n}=1.1$) $B(E2;8^{+}_{1}\rightarrow 6^{+}_{1})$ values are
0.0038(4) $e^{2}b^{2}$ and 0.0034 $e^{2}b^{2}$, respectively. As reported in
Table 6, the small value of $B(E2;8^{+}_{1}\rightarrow 6^{+}_{1})$ support
$8^{+}_{1}$ state is an isomeric state with half-life 0.39 $\mu$s, calculated
from $\Delta\text{NNLO}_{\text{GO}}$ interaction and corresponding
experimental half-life is 0.154(16) $\mu$s NNDC .

The experimentally observed $8^{+}_{1}$ isomeric state in 100Cd (with a total
half-life 62(6) ns NNDC ) is reproduced well with the G-matrix interaction.
For the G-matrix interaction, the wave function corresponding to this state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{2})$ with 55.0% probability
and the neutron occupancies for $8^{+}_{1}$ state are 1.185, 0.079, 0.573,
0.143 and 0.020 in $d_{5/2}$, $s_{1/2}$, $g_{7/2}$, $d_{3/2}$ and $h_{11/2}$
orbitals, respectively. With the G-matrix interaction, $8^{+}_{1}$ state
arises because of one pair breaking in $g_{9/2}$ proton orbital, thus $\nu=2$.
From the G-matrix interaction, the configuration of $6^{+}_{1}$ state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})_{0^{+}}\otimes\nu(g_{7/2}^{1}d_{5/2}^{1})_{6^{+}}$
with 82.4% probability, which shows predominant neutron character of the
$6^{+}_{1}$ state. But, the configuration of $6^{+}_{2}$ state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})_{6^{+}}\otimes\nu(d_{5/2}^{2})_{0^{+}}$ with
73.3% probability. Here, the $6^{+}_{2}$ state is formed by one proton pair
breaking in $g_{9/2}$ orbital. So, the $6^{+}_{2}$ state has more resemblance
to the $8^{+}_{1}$ isomer in 100Cd. For this isotope, there are a difference
in $B(E2;8^{+}_{1}\rightarrow 6^{+}_{1})$ values of our calculated result
(with the G-matrix interaction) and the experimental data. But, the
$B(E2;8^{+}_{1}\rightarrow 6^{+}_{2})$ transition is comparable with the
experimental value from this interaction. Clark $et$ $al.$ Clark  reported the
experimental branching ratios of $B(E2)$ transitions in $8^{+}_{1}$ state in
100Cd. We have used their data to find the partial half-lives of $8^{+}_{1}$
state as reported in Table 6.

Table 7: The calculated g-factor (in $\mu_{N}$) for the experimentally observed $8^{+}$ isomeric states in Cd isotopes compared to the experimental data. The shell-model calculations are performed with $g^{\pi}_{l}=1.0$, $g^{\nu}_{l}=0.0$; $g^{\pi}_{s}=3.91$, $g^{\nu}_{s}=-2.678$; $e_{\pi}=1.7e$, $e_{\nu}=1.1e$. Isotope | $J^{\pi}$ | $\Delta\text{NNLO}_{\text{GO}}$ | G-matrix | Exp.
---|---|---|---|---
98Cd | $8^{+}$ | 1.323 | 1.323 | \-
100Cd | $8^{+}$ | 1.278 | 1.269 | 1.24(6)
102Cd | $8^{+}$ | 1.235 | 1.140 | 1.29(3)
104Cd | $8^{+}$ | 1.240 | 0.392 | \-
130Cd | $8^{+}$ | 1.323 | 1.323 | \-

For 102Cd, the configuration of $8^{+}_{1}$ isomeric state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(g_{7/2}^{2}d_{5/2}^{2})$ from the
G-matrix interaction with 31.7% probability. With this interaction,
$8^{+}_{1}$ state is formed by one pair breaking in $g_{9/2}$ proton orbital
like 98,100Cd isotopes, thus the seniority $\nu=2$. For the G-matrix
interaction, the neutron occupancies are 1.178, 2.361, 0.264, 0.159, and
0.039, corresponding to $g_{7/2}$, $d_{5/2}$, $d_{3/2}$, $s_{1/2}$ and
$h_{11/2}$ orbitals, respectively. For the experimental partial half-lives of
$8^{+}_{1}$ state, we have used the experimental branching ratios of $B(E2)$
transitions in $8^{+}_{1}$ state from the Ref. Lieb .

In 104Cd, the $8^{+}_{1}$ isomeric state is observed with half-life of
0.8(+2-1) ns Blachot . For the G-matrix interaction, the configuration of this
state is $\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{4}g_{7/2}^{2})$ with
39.6% probability. This isomeric state comes from one proton pair breaking in
$g_{9/2}$ orbital. Hence, seniority $\nu=2$. We have used the experimental
data of branching ratios from the Ref. Sukhoruchkin  to calculate the partial
half-lives of $8^{+}_{1}$ state.

In 130Cd, the $8^{+}_{1}$ isomeric state lies slightly higher with an energy
difference of 0.174 MeV with the G-matrix interaction. For this interaction,
the configuration of $8^{+}_{1}$ state is
$\pi(p_{1/2}^{2}g_{9/2}^{8})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}g_{7/2}^{8}d_{3/2}^{4}h_{11/2}^{12})$
with 100% probability. Here, this $8^{+}_{1}$ isomeric state is also arising
because of one pair breaking of $g_{9/2}$ proton orbital and $\nu=2$. With
$\Delta\text{NNLO}_{\text{GO}}$ interaction, the half-lives of $8^{+}_{1}$
isomeric state is 0.33 $\mu$s. This value is close to the experimental half-
life (0.220(30) $\mu$s NNDC ) of $8^{+}_{1}$ state.

In the vicinity of the doubly magic nuclei 132Sn, microsecond isomers are
studied in previous work Pinston . Scherillo $et$ $al.$ Scherillo
investigated microsecond isomers in the neutron-rich region of the Cd isotopes
using the LOHENGRIN mass spectrometer. But, the expected $8^{+}$ isomeric
states were not observed. Our shell-model calculations predict the half-lives
of the $8^{+}_{1}$ states in the ns range for 126,128Cd. The half-lives with
our calculations are much shorter, which may explain the non-observation of
microsecond isomers in 126,128Cd isotopes.

In Fig. 14, we have shown the occupancy of neutron orbitals corresponding to
$8^{+}_{1}$ isomeric states in 100-104,130Cd isotopes with
$\Delta\text{NNLO}_{\text{GO}}$ and the G-matrix interactions. For 100-104Cd
isotopes, $d_{5/2}$ and $g_{7/2}$ neutron orbitals are dominant corresponding
to $8^{+}_{1}$ isomeric states. In the case of 130Cd, there is a shell closure
at $N=82$.

The g-factor for $8^{+}_{1}$ isomeric states in Cd isotopes are reported in
Table 7. The experimental data of the g-factors for $8^{+}_{1}$ isomeric
states in Cd isotopes are available only for 100-102Cd. With our calculations,
the g-factor is almost constant for $8^{+}_{1}$ isomeric states in
98-104,130Cd isotopes (except in the case of 104Cd with the G-matrix
interaction), which supports the additivity rule for nuclear magnetic moments.
The measurement of the g-factor explains the purity of a configuration, and
the contribution of the component corresponding to a particular state Bhoy .
We expect that the main configuration of an isomeric state will be pure. The
$8^{+}_{1}$ isomeric states in 98-104,130Cd isotopes are formed by the pair
breaking in the same proton orbital. We have calculated magnetic moment
corresponding to $g_{9/2}$ orbital $(\mu(g_{9/2}))$ using formula
$\mu=[g_{l}l+g_{s}/2]\mu_{N}$. After that, we calculated the g-factor value
$g=\mu(g_{9/2})/j$, which equals 1.510. This value is close to our calculated
g-factor values with both interactions. Thus, our calculations support these
$8^{+}_{1}$ isomers show $(\pi g_{9/2})^{2}$ character.

##  V Summary and Conclusions

In the present work, we have investigated the low-lying states in 98-130Cd
isotopes. Systematic shell-model calculations have been performed for Cd
isotopic chain from $N=50$ to $N=82$ with $\Delta\text{NNLO}_{\text{GO}}$ and
the G-matrix interactions. From our calculations for 98-104Cd isotopes with
the G-matrix interactions, we can see that the energy gap between $0^{+}_{1}$
and $2^{+}_{1}$ state increases as we move from 104Cd to 98Cd. This indicates
we are approaching the collective vibrational to the closed shell region. The
behavior of $4^{+}_{1}/2^{+}_{1}$ energy ratio ($R_{4/2}$) has been studied in
the Cd chain. We have shown the trend of $B(E2;0^{+}_{1}\rightarrow
2^{+}_{1})$ transition along Cd isotopic chain and tried to explain the
asymmetry near $N=62$. We have also calculated the other electromagnetic
properties like electric quadrupole moments, $B(M1)$ transitions, and magnetic
moments for even 98-130Cd isotopes. We have studied different aspects of the
octupole collective $3^{-}_{1}$ states in Cd isotopes and compared the
$B(E3;0^{+}_{1}\rightarrow 3^{-}_{1})$ transition values with the
corresponding experimental data. The properties of experimentally observed
$8^{+}$ isomeric states in 98-104,130Cd isotopes have been investigated using
shell-model calculations. Thus, our study also supports the occurrence of E2
isomerism in $8^{+}_{1}$ states in neutron-deficient Cd isotopes. We have also
reported the g-factor values for the $8^{+}_{1}$ isomeric states in Cd
isotopes and tried to explain the purity of the configuration of these states.

##  VI Acknowledgements

We acknowledge financial support from MHRD, the Government of India, and SERB
(India), CRG/2019/000556. We would like to thank the National Supercomputing
Mission (NSM) for providing computing resources of ‘PARAM Ganga’ at the Indian
Institute of Technology Roorkee, implemented by C-DAC and supported by the
Ministry of Electronics and Information Technology (MeitY) and Department of
Science and Technology (DST), Government of India. We would also like to thank
Professor Larry Zamick for useful discussions during this work.

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