∎

11institutetext: Usuf Rahaman 22institutetext: Department of Physics,
Madanapalle Institute of Technology $\&$ Science, Madanapalle-517325, India.
22email<EMAIL_ADDRESS>33institutetext: M.Ikram 44institutetext:
Department of Physics, Harsh Vidya Mandir (PG) College Raisi, Haridwar-247671,
India. 55institutetext: Ishfaq A. Rather 66institutetext: Institut für
Theoretische Physik, Goethe Universität, 60438 Frankfurt am Main, Germany.
77institutetext: Anisul Ain Usmani 88institutetext: Department of Physics,
Aligarh Muslim University, Aligarh-202002, India

# Microscopic Investigation of Ground State Properties and Shape Evolution in
Osmium Isotopes

Usuf Rahaman  M.Ikram  Ishfaq A. Rather  Anisul Ain Usmani

(Received: date / Accepted: date)

###### Abstract

The present study focuses on investigating the shape evolution of neutron-rich
even-even Osmium (Os) transitional nuclei within the range of neutron number N
= 82 to N = 190. The investigation is conducted using density-dependent meson-
nucleon and point-coupling models within the framework of the covariant
density functional theory (CDFT). Additionally, the results obtained from the
CDFT calculations are compared with those obtained using the relativistic
mean-field model with a non-linear meson-nucleon interaction. The potential
energy curve for Os isotopes (ranging from 158Os to 260Os) is analyzed in
order to identify phase shape transitions, such as oblate-spherical-prolate.
Furthermore, ground state bulk properties are calculated to gain insights into
the structure of Os isotopes. The self-consistent calculations reveal a clear
shape transition in the even-even Os isotopes, and overall, good agreement is
observed among the different models employed as well as with the available
experimental data.

###### Keywords:

Relativistic mean field theory; Skin thickness; neutron-rich nuclei.

##  1 Introduction

The investigation of transitional nuclei is of paramount importance for the
comprehensive understanding of nuclear structural properties, particularly
with regard to their prolate and oblate shaped configurations. These nuclei,
characterized by their dynamic and static effects, have become a focal point
of both theoretical and experimental studies. Over the past few decades,
various experiments alkh ; whel ; john ; pt1 ; pt2  and theoretical models
stev ; afana2016 ; afana2014 ; Sarriguren2008 ; rob2009 ; rod2010 , both
relativistic and non-relativistic in nature, have been employed to study
transitional nuclei with $\mathrm{Z}=72-80$. A numerous number of studies have
documented the evolution of triaxial shapes in these isotopes, further
enriching our knowledge in this field Nomura1 ; Nomura2 ; Nomura3 ; Ramos2014
.

Furthermore, extensive investigations have observed the presence of
superdeformed nuclei in this specific region, as evidenced by their non-
spherical shapes Robert1991 ; Sun2009 . The existence of deformed shapes in
nuclei disrupts the inherent spontaneous symmetry and introduces non-
sphericity, primarily governed by the quadrupole moment. Notably, certain
nuclei in this region exhibit multiple deformations at nearly identical
excitation energies, indicating the occurrence of shape co-existence. This
phenomenon allows for the exploration of nuclear oscillations between
different shapes. These distinct characteristics make the transitional nuclei
within this region highly promising candidates for in-depth structural
studies. It is worth noting that the most intriguing area within the nuclear
transition range is believed to lie around the mass number A$\sim$190, where
the transformation between prolate and oblate shapes has been thoroughly
investigated john . The region comprising transitional neutron-rich nuclei has
also captured significant attention within the nuclear physics community due
to its implications for neutron magicity magic31 ; magic32 ; magic33 .

The isotopes of osmium play a significant role in the transitional region,
where the nuclear shape undergoes a transition from spherical to
prolate/oblate configurations, and subsequently returns to a spherical shape
as the neutron number increases. Nuclei within this region provide a suitable
framework for a critical examination of shape evolution. Specifically, we
focus our investigation on the even-even osmium isotopes (158-260Os),
characterized by a high neutron count. Thus, our study expands our
understanding from the beta stable region to the neutron-drip line region. It
is important to note that pairing correlation is of utmost importance in these
loosely bound nuclei, which exhibit intriguing phenomena such as neutron halo
and skin tanihata1996 ; horo2001 ; Horowitz2001 . Consequently, an explicit
treatment of pairing correlations is essential when describing nuclei near the
drip line.

Density Functional Theory (DFT) is a highly versatile and powerful approach
employed to evaluate various physical observables of nuclei across the entire
periodic table, utilizing both relativistic and non-relativistic functionals
stev ; afana2016 ; afana2014 ; Sarriguren2008 ; rob2009 ; rod2010 ; bartel1982
; doba1984 ; chabanat1998 ; klupfel2009 ; kortel2010 ; kortel2012 ; erler2012
. This framework not only addresses the bulk properties but also enables the
investigation of transitional behavior in nuclei. Notably, few studies have
reported calculations on nuclear shape transitions employing relativistic
functionals afana2016 ; afana2014 . Additionally, self-consistent Hartree-Fock
calculations based on Skyrme energy density functionals have been conducted
Sarriguren2008 ; rob2009 ; rod2010 ; bartel1982 ; doba1984 ; chabanat1998 ;
klupfel2009 ; kortel2010 ; kortel2012 ; erler2012 . Furthermore, even-even
osmium isotopes have been investigated within the framework of Hartree-Fock
Bogoliubov using Skyrme interactions Ashok2019 . Despite the considerable
theoretical considerations in Covariant Density Functional Theory (CDFT),
there remains a scarcity of research concerning shape evolution in
transitional nuclei. Consequently, our current endeavor focuses on employing
CDFT to explore the spherical to prolate/oblate shape transitions in osmium
transitional nuclei. Our primary objective is to attain a more comprehensive
and systematic understanding of these nuclear isotopes within both
relativistic and non-relativistic functionals. The calculations entail
Relativistic Mean-Field (RMF) theory 106 ; gambhir1990 ; 7 ; walecka1974 ;
BB77  incorporating nonlinear meson-nucleon NL3* nl3s  interaction with BCS
pairing, as well as covariant density functional theory (CDFT) 13 ; 42
utilizing density-dependent meson-exchange (DD-ME) me1 ; me2  and density-
dependent point-coupling interactions (DD-PC) pc1 ; pcx  with RHB pairing.

Within this paper, we employ these methodologies to provide valuable insights
into the nuclear shape transitions exhibited in the even-even isotopic chain
of osmium, encompassing the mass range from 158Os to 260Os. As previously
indicated, this mass region offers a captivating opportunity for exploring
neutron magic numbers and investigating small islands of phase shape
transition, which could hold energetic favorability for reaction mechanisms.
This investigation yields valuable insights into various nuclear properties,
including ground state energy, quadrupole deformation, radii, nuclear skin,
and potential energy curve. It is worth noting that while triaxial symmetry
plays a crucial role in these nuclei, our study focuses exclusively on axially
deformed cases.

The organization of this paper is as follows. Section 2 provides a
comprehensive overview of the theoretical formalism employed, along with
detailed explanations of the calculations employed in this study. The results
of these calculations, along with their pertinent discussions, are presented
in Section 3. Finally, Section 4 concludes the paper with a summary of the key
findings.

##  2 Theoretical Framework

###  2.1 Covariant density functional theory

In this paper, we employ two classes of covariant density functional models,
namely the density-dependent meson-exchange (DD-ME) model and the density-
dependent point-coupling (DD-PC) model. These models differ primarily in their
treatment of interaction range. The DD-ME model incorporates a finite
interaction range, whereas the DD-PC model utilizes a zero-range interaction
supplemented by an additional gradient term in the scalar-isoscalar channel.
By employing these distinct models, we aim to comprehensively explore the
impact of interaction range on the results and implications of our study.

####  2.1.1 The meson-exchange model

Within the meson-exchange model, the nucleus is depicted as a system
comprising Dirac nucleons that interact through the exchange of mesons with
finite masses, resulting in interactions of finite range. 42 ; me1 ; me2 ;
typel1999 ; 52 ; 111 . These mesons, namely the isoscalar-scalar $\sigma$
meson, the isoscalar-vector $\omega$ meson, and the isovector-vector $\rho$
meson, constitute the essential set of meson fields required for a
comprehensive quantitative description of nuclei. The meson-exchange model is
characterized by the utilization of a standard Lagrangian density gambhir1990
that incorporates vertices with medium-dependent interactions:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{\psi}\left[\gamma(i\partial-g_{\omega}\omega-
g_{\rho}\vec{\rho}\,\vec{\tau}-eA)-m-g_{\sigma}\sigma\right]\psi$ (1)
$\displaystyle+$
$\displaystyle\frac{1}{2}(\partial\sigma)^{2}-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{4}{\vec{R}}_{\mu\nu}{\vec{R}}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\vec{\rho}^{\,2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$

In equation (1), the symbol $\psi$ represents the Dirac spinors, while $m$
denotes the mass of the nucleon. The variable $e$ corresponds to the charge of
the proton, which is zero for neutrons. The quantities $m_{\sigma}$,
$m_{\omega}$, and $m_{\rho}$ represent the masses of the $\sigma$ meson,
$\omega$ meson, and $\rho$ meson, respectively. The corresponding coupling
constants for the interaction of these mesons with the nucleons are denoted as
$g_{\sigma}$ , $g_{\omega}$, and $g_{\rho}$. These coupling constants, along
with the unknown meson masses, serve as parameters in the Lagrangian equation.
The symbols $\Omega^{\mu\nu}$, ${\vec{R}}^{\mu\nu}$ and $F^{\mu\nu}$ refer to
the field tensors associated with the vector fields $\omega$, $\rho$ and the
proton, respectively.

$\Omega^{\mu\nu}=\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu},$ (2)
${\vec{R}}^{\mu\nu}=\partial^{\mu}{\vec{\rho}}^{\nu}-\partial^{\nu}{\vec{\rho}}^{\mu},$
(3) $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.$ (4)

The initial formulation of this linear model was proposed by Walecka
walecka1974 . However, this simplistic model lacks the capability to provide a
quantitative depiction of nuclear systems BB77 ; Pannert1987  due to its
linear interaction terms in the meson fields. Notably, it exhibits a
considerably exaggerated incompressibility for infinite nuclear matter BB77
and insufficiently small nuclear deformations gambhir1990 . To address these
limitations and achieve a more realistic portrayal of complex nuclear system
properties, one can incorporate either a nonlinear self-coupling or density
dependence in the coupling constants.

For the inclusion of nonlinear self-coupling, an additional term needs to be
incorporated into the Lagrangian:

$U(\sigma)\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}.$
(5)

This model has demonstrated successful applications in various studies
reinhard1986 ; lalazissis1997 ; toddrutel2005 .

When it comes to density-dependent coupling constants, the dependence is
defined as:

$g_{i}(\rho)=g_{i}(\rho_{\rm sat})f_{i}(x)\quad{\rm for}\quad
i=\sigma,\omega,\rho$ (6)

Here, $i$ can represent any of the three mesons: $\sigma$, $\omega$ and
$\rho$. Notably, there are no nonlinear terms present for the $\sigma$ meson,
which implies that $g_{2}=g_{3}=0$. The density dependence or $\sigma$ and
$\omega$ mesons are determined by the function:

$f_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}}.$ (7)

For the $\rho$ meson, the density dependence is given by:

$f_{\rho}(x)=\exp(-a_{\rho}(x-1)).$ (8)

In the above equations, x represents the ratio between the baryonic density
$\rho$ at a specific location and the baryonic density at saturation
$\rho_{\rm sat}$ in symmetric nuclear matter. To ensure consistency, the
parameters in Eq. (7) are constrained as follows:

$f_{i}(1)=1,\,\,\,\,f_{\sigma}^{{}^{\prime\prime}}(1)=f_{\omega}^{{}^{\prime\prime}}(1),\,\,and\,\,f_{i}^{{}^{\prime\prime}}(0)=0.$
(9)

These constraints effectively reduce the number of independent parameters for
the density dependence to three. In this study, the DD-ME1 me1  and DD-ME2 me2
density-dependent meson-exchange relativistic energy functionals are utilized,
and their details can be found in Table 1.

Table 1: The parameters of the DD-ME1 me1  and DD-ME2 me2  parameterizations of the Lagrangian Parameter | DD-ME1 me1  | DD-ME2 me2
---|---|---
$m$ | 939 | 939
$m_{\sigma}$ | 549.5255 | 550.1238
$m_{\omega}$ | 783 | 783
$m_{\rho}$ | 763 | 763
$g_{\sigma}$ | 10.4434 | 10.5396
$g_{\omega}$ | 12.8939 | 13.0189
$g_{\rho}$ | 3.8053 | 3.6836
$a_{\sigma}$ | 1.3854 | 1.3881
$b_{\sigma}$ | 0.9781 | 1.0943
$c_{\sigma}$ | 1.5342 | 1.7057
$d_{\sigma}$ | 0.4661 | 0.4421
$a_{\omega}$ | 1.3879 | 1.3892
$b_{\omega}$ | 0.8525 | 0.924
$c_{\omega}$ | 1.3566 | 1.462
$d_{\omega}$ | 0.4957 | 0.4775
$a_{\rho}$ | 0.5008 | 0.5647

####  2.1.2 Point-coupling model

The Lagrangian associated with the density-dependent point coupling model pc1
; nikolaus1992  can be expressed as

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{\psi}\left(i\gamma\cdot\partial-m\right)\psi$ (10)
$\displaystyle-$
$\displaystyle\frac{1}{2}\alpha_{S}(\hat{\rho})\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right)-\frac{1}{2}\alpha_{V}(\hat{\rho})\left(\bar{\psi}\gamma^{\mu}\psi\right)\left(\bar{\psi}\gamma_{\mu}\psi\right)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\alpha_{TV}(\hat{\rho})\left(\bar{\psi}\vec{\tau}\gamma^{\mu}\psi\right)\left(\bar{\psi}\vec{\tau}\gamma_{\mu}\psi\right)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\delta_{S}\left(\partial_{v}\bar{\psi}\psi\right)\left(\partial^{v}\bar{\psi}\psi\right)-e\bar{\psi}\gamma\cdot
A\frac{(1-\tau_{3})}{2}\psi.$

The Lagrangian encompasses various components crucial for the density-
dependent point coupling model. It incorporates the free-nucleon Lagrangian,
point coupling interaction terms, and the coupling between protons and the
electromagnetic field. The derivative terms present in Eq. (10) serve to
account for the dominant effects originating from finite-range interactions,
which hold significance in the context of nuclei. Similar to DD-ME models,
this particular formulation incorporates interactions involving isoscalar-
scalar, isoscalar-vector, and isovector-vector couplings. In this
investigation, the density-dependent point-coupling interactions, namely DD-
PC1 pc1  and DD-PCX pcx , were adopted. These interactions are presented in
Table 2.

Table 2: The parameters of the DD-PC1 pc1 , and DD-PCX pcx  parameterization in the Lagrangian. Parameter | DD-PC1 pc1  | DD-PCX pcx
---|---|---
$\it{m}$ (MeV) | 939 | 939
$a_{s}$ (fm2) | -10.04616 | -10.979243836
$b_{s}$ (fm2) | -9.15042 | -9.03825091
$c_{s}$ (fm2) | -6.42729 | -5.31300882
$d_{s}$ | 1.37235 | 1.37908707
$a_{v}$ (fm2) | 5.91946 | 6.430144908
$b_{v}$ (fm2) | 8.8637 | 8.870626019
$d_{v}$ | 0.65835 | 0.655310525
$b_{tv}$ (fm2) | 1.83595 | 2.963206854
$d_{tv}$ | 0.64025 | 1.309801417
$\delta_{s}$ (fm4) | -0.8149 | -0.878850922

###  2.2 Axially deformed relativistic mean-field model

The relativistic mean field (RMF) theory has emerged as a highly successful
approach in addressing the nuclear many-body problem and elucidating various
nuclear phenomena across the entire range of the periodic table, from the
proton drip line to the neutron drip line region S92 ; gambhir1990 ; R96 ; 7 ;
BB77 ; yadav ; skpatra ; Lalazissis1998 . One notable advantage of the RMF
theory, compared to its non-relativistic counterpart, is its inherent ability
to naturally reproduce the spin-orbit splitting. This feature plays a pivotal
role in explaining the formation of closed shell structures in atomic nuclei
sharma ; warrier .

The foundation of the RMF theory lies in a fundamental Lagrangian density that
encompasses nucleons interacting with $\sigma-$, $\omega-$ and $\rho-$meson
fields. Additionally, the inclusion of the photon field $A_{\mu}$ is crucial
for adequately accounting for the Coulomb interaction among protons. The
Lagrangian density of the relativistic mean field theory can be expressed as
gambhir1990 ; 7 ; BB77 ; R96 ; S92 ,

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{\psi_{i}}\\{i\gamma^{\mu}\partial_{\mu}-M\\}\psi_{i}+{\frac{1}{2}}\partial^{\mu}\sigma\partial_{\mu}\sigma-{\frac{1}{2}}m_{\sigma}^{2}\sigma^{2}-{\frac{1}{3}}g_{2}\sigma^{3}$
(11) $\displaystyle-$
$\displaystyle{\frac{1}{4}}g_{3}\sigma^{4}-g_{s}\bar{\psi_{i}}\psi_{i}\sigma-{\frac{1}{4}}\Omega^{\mu\nu}\Omega_{\mu\nu}+{\frac{1}{2}}m_{w}^{2}V^{\mu}V_{\mu}$
$\displaystyle-$ $\displaystyle
g_{w}\bar{\psi}_{i}\gamma^{\mu}\psi_{i}V_{\mu}-{\frac{1}{4}}\vec{B}^{\mu\nu}\vec{B}_{\mu\nu}+{\frac{1}{2}}m_{\rho}^{2}{\vec{R}^{\mu}}{\vec{R}_{\mu}}-{\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}$
$\displaystyle-$ $\displaystyle
g_{\rho}\bar{\psi}_{i}\gamma^{\mu}\vec{\tau}\psi_{i}\vec{R^{\mu}}-e\bar{\psi}_{i}\gamma^{\mu}\frac{\left(1-\tau_{3i}\right)}{2}\psi_{i}A_{\mu}.$

Here, the masses of the nucleons, ${\sigma}$-, ${\omega}$\- and
${\rho}$-mesons are represented by $M$, $m_{\sigma}$, $m_{\omega}$ and
$m_{\rho}$, respectively. The fields associated with the ${\sigma}$-,
${\omega}$\- and ${\rho}$-mesons are denoted by ${\sigma}$, $V_{\mu}$ and
$R_{\mu}$. The coupling constants for the ${\sigma}$-, ${\omega}$\- and
${\rho}$-mesons, as well as the electromagnetic interaction, are given by
$g_{s}$, $g_{\omega}$, $g_{\rho}$ and $e^{2}/4{\pi}$=1/137, respectively.
Additionally, the self-interaction coupling constants for ${\sigma}$-mesons
are denoted by $g_{2}$ and $g_{3}$.

By employing the classical variational principle, the field equations
governing the behavior of nucleons and mesons can be derived. Specifically,
the Dirac equation for nucleons, which encapsulates their dynamics, can be
written as

$\\{-i\alpha\bigtriangledown+V(r_{\perp},z)+\beta
M^{\dagger}\\}\psi_{i}=\epsilon_{i}\psi_{i}.$ (12)

Where

$M^{\dagger}=M+S(r_{\perp},z)=M+g_{\sigma}\sigma^{0}(r_{\perp},z),$ (13)

and

$V(r_{\perp},z)=g_{\omega}V^{0}(r_{\perp},z)+g_{\rho}\tau_{3}R^{0}(r_{\perp},z)+e\frac{(1-\tau_{3})}{2}A^{0}(r_{\perp},z).$
(14)

The Klein-Gordon equations for mesons are given by

$\displaystyle\\{-\bigtriangleup+m^{2}_{\sigma}\\}\sigma^{0}(r_{\perp},z)$
$\displaystyle=$ $\displaystyle-g_{\sigma}\rho_{s}(r_{\perp},z)$ (15)
$\displaystyle-$ $\displaystyle
g_{2}\sigma^{2}(r_{\perp},z)-g_{3}\sigma^{3}(r_{\perp},z),$
$\displaystyle\\{-\bigtriangleup+m^{2}_{\omega}\\}V^{0}(r_{\perp},z)$
$\displaystyle=$ $\displaystyle g_{\omega}\rho_{v}(r_{\perp},z),$ (16)
$\displaystyle\\{-\bigtriangleup+m^{2}_{\rho}\\}R^{0}(r_{\perp},z)$
$\displaystyle=$ $\displaystyle g_{\rho}\rho_{3}(r_{\perp},z),$ (17)
$\displaystyle-\bigtriangleup A^{0}(r_{\perp},z)$ $\displaystyle=$
$\displaystyle e\rho_{c}(r_{\perp},z).$ (18)

The scalar and vector densities for the $\sigma$\- and $\omega$-fields are
denoted as $\rho_{s}$ and $\rho_{v}$, respectively. These densities represent
the distribution and magnitude of the $\sigma$-meson and $\omega$-meson fields
within the nuclear environment. Mathematically, these densities are expressed
as

$\displaystyle\rho_{s}(r)$ $\displaystyle=$
$\displaystyle\sum_{i=n,p}\bar{\psi}_{i}(r)\psi_{i}(r)\;,$
$\displaystyle\rho_{v}(r)$ $\displaystyle=$
$\displaystyle\sum_{i=n,p}\psi^{\dagger}_{i}(r)\psi_{i}(r)\;.$ (19)

The vector density $\rho_{3}(r)$ for $\rho$-field and charge density
$\rho_{c}(r)$ are expressed by

$\displaystyle\rho_{3}(r)$ $\displaystyle=$
$\displaystyle\sum_{i=n,p}\psi^{\dagger}_{i}(r)\gamma^{0}\tau_{3i}\psi_{i}(r)\;,$
$\displaystyle\rho_{c}(r)$ $\displaystyle=$
$\displaystyle\sum_{i=n,p}\psi^{\dagger}_{i}(r)\gamma^{0}\frac{(1-\tau_{3i})}{2}\psi_{i}(r)\;.$
(20)

To describe the fundamental characteristics of nuclei in their ground state, a
static solution is obtained by solving the equations of motion. These
equations, which are nonlinear and coupled, capture the interactions and
dynamics of nucleons and mesons within the nuclear system. The solution is
achieved through a self-consistent approach, where the equations are solved
iteratively to ensure internal consistency.

In this particular study, an axially deformed harmonic oscillator basis is
employed for both fermions and bosons. The basis size is chosen as
$N_{F}=N_{B}=20$, indicating the number of states included in the basis for
fermions and bosons, respectively.

One of the important quantities of interest in this study is the calculation
of radii. These radii are determined from the corresponding densities obtained
from the solution. The densities provide information about the spatial
distribution and size of the nuclear system, and the radii serve as
quantitative measures of these distributions. The rms radii of proton
($r_{p}$), neutron ($r_{n}$) and nuclear matter ($r_{m}$) are given by,

$\displaystyle\langle r_{p}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\int r_{p}^{2}d^{3}r\rho_{p}\;,$
$\displaystyle\langle r_{n}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N}\int r_{n}^{2}d^{3}r\rho_{n}\;,$
$\displaystyle\langle r_{m}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{A}\int r_{m}^{2}d^{3}r\rho\;,$ (21)

The rms charge radius of a nucleus can be determined by utilizing the proton
rms radius and accounting for the finite size of the proton. This is achieved
by employing the relation $r_{c}=\sqrt{r_{p}^{2}+0.64}$.

To extract the quadrupole deformation parameter ($\beta_{2}$) of the nucleus,
the calculated quadrupole moments of neutrons and protons are used as

$Q=Q_{n}+Q_{p}=\sqrt{\frac{16\pi}{5}}\left(\frac{3}{4\pi}AR^{2}\beta_{2}\right),$
(22)

where R = 1.2$A^{1/3}$.
The total energy of the nuclear system is composed of various contributions.

$E_{\text{total}}=E_{\text{part}}+E_{\sigma}+E_{\omega}+E_{\rho}+E_{c}+E_{\text{pair}}+E_{c.m.},$
(23)

Firstly, the sum of single particle energies of the nucleons is denoted as
$E_{\text{part}}$. Additionally, there are contributions from the meson
fields, namely $E_{\sigma}$, $E_{\omega}$, and $E_{\rho}$. The Coulomb field
contributes to the energy as $E_{c}$, and the pairing energy is represented by
$E_{\text{pair}}$. Furthermore, the center-of-mass energy is denoted as
$E_{\text{cm}}$.

The non-linear NL3* parameter set nl3s , which is a modified version of the
successful NL3 parameter set lalazissis1997 , is utilized throughout the
calculations. This parameter set provides the necessary parameters for the
model being employed. For a detailed understanding of the formalism and
numerical techniques used, references such as Refs. gambhir1990 ; patra  and
other relevant literature should be consulted.

In this study, pairing correlations are treated using the BCS (Bardeen-Cooper-
Schrieffer) approach Dobaczewski1984 . Although the BCS approach may not be
appropriate for light neutron-rich nuclei, the nuclei considered in this study
are not in that category, ensuring the reliability of the results obtained
within the framework of the relativistic mean field (RMF) theory.

Table 3: The parameters of the NL3* nl3s  parameterization of the Lagrangian Parameter | NL3* nl3s
---|---
$m$ | 939
$m_{\sigma}$ | 502.5742
$m_{\omega}$ | 782.6
$m_{\rho}$ | 763
$g_{\sigma}$ | 10.0944
$g_{\omega}$ | 12.8065
$g_{\rho}$ | 4.5748
$g_{2}$ | -10.8093
$g_{3}$ | -30.1486

##  3 Results and Discussion

###  3.1 Binding energy

The binding energy is a crucial parameter for studying the stability of atomic
nuclei. The binding energies of ground states for even-even osmium isotopes
160-264Os is shown in Table LABEL:tab:Ostab1 and plotted in Fig. 1. These
energies are calculated within the CDFT with effective density-dependent
interactions DD-ME1 me1 , DD-ME2 me2 , DD-PC1 pc1 , and DD-PCX pcx  and also
with RMF+BCS model using nonlinear NL3* nl3s .

Additionally, Fig. 2 depicts the calculated binding energies for these
isotopes. Since experimental data for neutron-rich drip-line nuclei is
limited, it is necessary to compare the results obtained from different
theoretical approaches 19 ; 39  in order to assess the model’s predictability
and reliability. Comparisons were made with other theoretical frameworks, the
HFB + THO approach in conjunction with various nuclear interactions, including
Skyrme SLy4, SkP, and SkM* 19 , alongside CHFB+5DCH calculations predicated on
the Gogny D1S interaction 39 , which exhibited excellent agreement with both
RMF and CDFT outcomes, extending towards the neutron drip-line region. The
calculated binding energies exhibit a consistent agreement across the entire
isotopic range. Particularly, the results obtained using the SkP functional
demonstrate a strong alignment with our findings, extending even to the
neutron dripline region. Conversely, the SkM* investigations indicate that
isotopes in the neutron-rich region are slightly more tightly bound than our
predictions, whereas the SLy4 and CHFB data suggest the opposite trend. The
results obtained within the study range displayed good consistency among
various relativistic and non-relativistic functionals, as well as with
available experimental data 43 ; wang2012 ; 22 ; Audi2003 .

In general, the binding energy per nucleon (BE/A) increases as the number of
neutrons (or the mass number A) increases, reaching a maximum value at A = 182
for nearly all interactions. This trend is consistent with experimental
observations. Consequently, it is concluded that the isotope 182Os is the most
stable among the entire isotopic chain.

Table 4: Comparison of the total binding energies (in MeV) of the ground states for 158-266Os isotopes calculated using different effective interactions (DD-ME1 me1 , DD-ME2 me2 , DD-PC1 pc1 , DD-PCX pcx , and NL3* nl3s ) and with available experimental data 22 ; Audi2003 ; wang2012 ; 43 . The results are also compared with calculations using HFB models. 19 ; 39  Nuclei | DD-ME1 | DD-ME2 | DD-PC1 | DD-PCX | NL3* | EXP | SkP | SkM* | SLy4 | CHFB
---|---|---|---|---|---|---|---|---|---|---
156Os | 1190.90 | 1188.57 | 1188.82 | 1186.16 | 1195.96 | – | – | – | – | –
158Os | 1221.14 | 1218.82 | 1219.25 | 1216.94 | 1224.09 | – | – | – | – | –
160Os | 1242.88 | 1240.44 | 1240.66 | 1237.69 | 1246.55 | – | – | – | – | 1245.14
162Os | 1265.02 | 1262.52 | 1262.49 | 1259.15 | 1268.13 | 1262.47 | 1252.25 | 1261.62 | 1261.88 | 1265.90
164Os | 1286.78 | 1284.22 | 1284.14 | 1280.62 | 1288.56 | 1284.66 | 1273.42 | 1282.98 | 1283.02 | 1286.93
166Os | 1307.78 | 1305.18 | 1305.20 | 1301.58 | 1310.12 | 1305.81 | 1293.73 | 1303.71 | 1303.37 | 1308.28
168Os | 1328.00 | 1325.39 | 1325.49 | 1321.88 | 1329.82 | 1326.52 | 1313.80 | 1323.77 | 1322.99 | 1328.20
170Os | 1347.54 | 1344.97 | 1345.12 | 1341.43 | 1348.75 | 1346.59 | 1333.75 | 1343.28 | 1341.93 | 1347.64
172Os | 1366.34 | 1363.84 | 1364.13 | 1360.42 | 1368.62 | 1366.05 | 1353.94 | 1362.30 | 1361.27 | 1366.82
174Os | 1386.15 | 1383.84 | 1384.43 | 1380.52 | 1387.63 | 1384.95 | 1373.24 | 1380.82 | 1380.18 | 1385.35
176Os | 1404.50 | 1402.21 | 1402.98 | 1398.95 | 1405.72 | 1403.19 | 1391.97 | 1399.12 | 1398.38 | 1403.27
178Os | 1422.06 | 1419.87 | 1420.77 | 1416.50 | 1423.13 | 1420.78 | 1410.08 | 1416.62 | 1415.36 | 1420.48
180Os | 1439.12 | 1437.10 | 1438.14 | 1433.66 | 1439.83 | 1437.74 | 1427.62 | 1432.87 | 1431.77 | 1436.86
182Os | 1455.14 | 1453.24 | 1454.54 | 1450.31 | 1455.48 | 1454.13 | 1444.31 | 1448.76 | 1447.59 | 1452.62
184Os | 1470.02 | 1468.38 | 1469.83 | 1465.75 | 1470.21 | 1469.92 | 1460.47 | 1464.21 | 1462.69 | 1467.84
186Os | 1484.24 | 1482.67 | 1484.35 | 1480.52 | 1484.21 | 1484.81 | 1475.92 | 1479.21 | 1477.43 | 1482.32
188Os | 1497.97 | 1496.49 | 1498.37 | 1494.75 | 1497.51 | 1499.09 | 1490.77 | 1493.75 | 1491.72 | 1496.23
190Os | 1511.24 | 1509.79 | 1512.04 | 1508.03 | 1510.40 | 1512.80 | 1505.47 | 1507.80 | 1505.03 | 1510.03
192Os | 1524.36 | 1523.05 | 1525.58 | 1521.39 | 1523.30 | 1526.12 | 1520.10 | 1521.30 | 1517.78 | 1523.00
194Os | 1537.08 | 1535.96 | 1538.79 | 1534.63 | 1535.59 | 1538.81 | 1532.93 | 1534.25 | 1530.09 | 1535.38
196Os | 1548.13 | 1546.99 | 1549.80 | 1545.42 | 1546.58 | 1550.80 | 1546.97 | 1546.91 | 1542.18 | 1547.07
198Os | 1559.81 | 1558.63 | 1562.16 | 1557.54 | 1557.50 | 1562.42 | 1559.72 | 1559.34 | 1554.55 | 1558.74
200Os | 1571.93 | 1570.99 | 1574.68 | 1570.10 | 1569.11 | 1573.60 | 1572.78 | 1571.95 | 1567.04 | 1569.47
202Os | 1584.15 | 1583.26 | 1586.67 | 1582.11 | 1580.48 | 1584.08 | 1585.78 | 1584.52 | 1579.54 | 1578.59
204Os | 1588.93 | 1587.96 | 1591.75 | 1586.29 | 1585.59 | – | 1592.33 | 1591.39 | 1583.78 | 1585.21
206Os | 1593.82 | 1592.82 | 1596.97 | 1590.65 | 1590.68 | – | 1599.08 | 1598.10 | 1588.35 | 1590.07
208Os | 1599.51 | 1598.52 | 1602.80 | 1595.96 | 1596.84 | – | 1605.54 | 1604.32 | 1593.12 | 1595.06
210Os | 1605.24 | 1604.25 | 1608.80 | 1601.36 | 1602.87 | – | 1611.54 | 1610.04 | 1597.95 | 1599.77
212Os | 1610.77 | 1609.79 | 1614.62 | 1606.59 | 1608.61 | – | 1617.86 | 1615.30 | 1602.71 | 1604.35
214Os | 1615.99 | 1615.04 | 1620.27 | 1611.64 | 1613.94 | – | 1624.44 | 1620.69 | 1607.33 | 1608.89
216Os | 1621.26 | 1620.51 | 1626.77 | 1617.35 | 1619.07 | – | 1631.21 | 1625.98 | 1612.50 | 1613.49
218Os | 1627.01 | 1626.37 | 1633.04 | 1623.12 | 1626.20 | – | 1638.37 | 1631.13 | 1617.79 | 1618.23
220Os | 1632.42 | 1631.93 | 1638.97 | 1628.56 | 1632.46 | – | 1645.16 | 1636.12 | 1623.10 | 1622.56
222Os | 1637.33 | 1636.88 | 1644.47 | 1633.31 | 1637.72 | – | 1651.58 | 1640.93 | 1627.78 | 1626.58
224Os | 1641.91 | 1641.56 | 1649.75 | 1637.93 | 1642.84 | – | 1657.50 | 1645.53 | 1631.78 | 1630.19
226Os | 1646.11 | 1645.91 | 1654.68 | 1642.38 | 1647.76 | – | 1662.68 | 1649.70 | 1635.18 | 1633.41
228Os | 1649.73 | 1649.59 | 1658.58 | 1645.67 | 1651.93 | – | 1667.79 | 1653.45 | 1637.98 | 1636.26
230Os | 1652.87 | 1652.81 | 1662.17 | 1648.72 | 1655.84 | – | 1672.43 | 1656.77 | 1640.77 | 1638.72
232Os | 1655.57 | 1655.63 | 1665.50 | 1651.54 | 1659.12 | – | 1676.94 | 1659.83 | 1643.70 | 1640.82
234Os | 1657.97 | 1658.14 | 1668.65 | 1654.15 | 1661.80 | – | 1681.53 | 1662.57 | 1646.51 | 1642.62
236Os | 1660.15 | 1660.46 | 1671.60 | 1656.67 | 1664.31 | – | 1685.71 | 1664.93 | 1647.93 | 1644.08
238Os | 1661.99 | 1662.41 | 1673.95 | 1658.61 | 1666.48 | – | 1688.90 | 1667.03 | 1649.45 | 1645.18
240Os | 1663.52 | 1664.06 | 1675.72 | 1659.90 | 1668.69 | – | 1691.94 | 1668.93 | 1650.62 | 1645.92
242Os | 1664.84 | 1665.50 | 1677.23 | 1660.92 | 1671.16 | – | 1694.86 | 1670.64 | 1651.56 | 1646.40
244Os | 1665.83 | 1666.62 | 1678.42 | 1661.63 | 1673.39 | – | 1697.80 | 1672.13 | 1652.42 | 1646.64
246Os | 1666.37 | 1666.70 | 1679.32 | 1659.78 | 1673.96 | – | 1699.14 | 1673.52 | 1653.30 | 1646.64
248Os | 1667.41 | 1667.86 | 1680.58 | 1660.86 | 1676.05 | – | 1702.17 | 1674.74 | 1653.70 | 1646.45
250Os | 1668.14 | 1668.70 | 1681.87 | 1661.85 | 1677.29 | – | 1704.83 | 1675.79 | 1653.35 | 1646.40
252Os | 1668.73 | 1669.40 | 1682.47 | 1662.74 | 1678.96 | – | 1707.58 | 1676.59 | 1653.85 | 1645.91
254Os | 1669.50 | 1670.41 | 1683.59 | 1663.19 | 1680.47 | – | 1710.32 | 1677.30 | 1654.60 | 1645.38
256Os | 1670.38 | 1671.36 | 1685.00 | 1664.01 | 1682.30 | – | 1712.08 | 1678.31 | 1654.66 | 1644.68
258Os | 1671.52 | 1672.69 | 1686.70 | 1665.42 | 1684.30 | – | 1714.25 | 1679.77 | 1655.17 | 1643.74
260Os | 1672.75 | 1674.18 | 1688.43 | 1666.65 | 1684.52 | – | 1717.44 | – | – | 1641.23
262Os | 1669.92 | 1671.30 | 1684.88 | 1662.41 | 1682.07 | – | 1703.71 | – | – | 1638.69
264Os | 1667.03 | 1668.37 | 1681.31 | 1658.08 | 1680.20 | – | 1703.65 | – | – | –
266Os | 1664.10 | 1665.39 | 1677.72 | 1653.66 | 1678.60 | – | – | – | – | –
Figure 1: Total binding energy for even-even Os nuclei, obtained within
different relativistic interactions.  Figure 2: Binding energy per nucleon for
even-even Os nuclei, obtained within different relativistic interactions.

###  3.2 Quadrupole deformation

Nuclear shape and deformation are significant physical parameters that play a
crucial role in determining properties such as nuclear size, isotopic shift,
and quadrupole moments. Table LABEL:tab:Osbeta provides the quadrupole
deformation parameters $\beta_{2}$) obtained from various relativistic and
non-relativistic interactions. To provide a more clear understanding of the
shape transition, Fig. 3 presents the variation deformation parameter with
neutron number.

In order to assess the shape transition more comprehensively, we have compared
the results obtained from the Relativistic Mean Field (RMF) approach with
those obtained from non-relativistic functionals such as Skyrme SkP, SLy4, and
SkM* 19 , as well as CHFB+5DCH 39  calculations based on the Gogny D1S
interaction. These results were also compared with available experimental data
33 ; 34 . Throughout the considered isotopic chain, spanning from the neutron-
deficient to the neutron-rich region, transitions in shape from spherical to
prolate, prolate to oblate, and then oblate to spherical were observed. This
indicates a transitional behavior of osmium isotopes. In Os nuclei ranging
from N = 112 to N = 124, both the experimental data and our calculated results
based on DD-ME1, DD-ME1, DD-PC1, DD-PCX, and NL3* exhibit consistent trends.
However, deviations arise when comparing these trends with predictions derived
from HFB theory utilizing SKM* functionals. It is important to note that
nuclei near the drip-line region exhibit a spherical shape.

Table 5: Comparison of the Quadrupole Deformation Parameter ($\beta_{2}$) for the ground states of 158-266Os isotopes calculated using different effective interactions (DD-ME1, DD-ME2, DD-PC1, DD-PCX, and NL3*). The results are compared with available experimental data 22 ; wang2012 ; 43  and calculations performed using HFB models 19 ; 39  Nuclei | DD-ME1 | DD-ME2 | DD-PC1 | DD-PCX | NL3* | EXP | SkP | SkM* | SLy4
---|---|---|---|---|---|---|---|---|---
156Os | 0.067 | 0.070 | 0.043 | 0.064 | -0.077 | – | – | – | –
158Os | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | – | – | – | –
160Os | -0.065 | -0.067 | -0.058 | -0.060 | 0.000 | – | – | – | –
162Os | 0.110 | 0.111 | 0.107 | 0.114 | 0.077 | – | -0.085 | 0.074 | 0.112
164Os | 0.134 | 0.134 | 0.136 | 0.145 | -0.092 | – | 0.141 | 0.124 | 0.145
166Os | 0.157 | 0.157 | 0.158 | 0.164 | 0.139 | – | 0.166 | 0.150 | 0.165
168Os | 0.167 | 0.168 | 0.171 | 0.177 | 0.164 | – | 0.187 | 0.169 | 0.179
170Os | 0.173 | 0.174 | 0.181 | 0.190 | 0.187 | – | 0.208 | 0.186 | 0.195
172Os | 0.185 | 0.187 | 0.196 | 0.208 | 0.305 | 0.224 | 0.266 | 0.201 | 0.278
174Os | 0.327 | 0.326 | 0.326 | 0.319 | 0.323 | 0.246 | 0.284 | 0.218 | 0.294
176Os | 0.338 | 0.336 | 0.337 | 0.331 | 0.331 | – | 0.288 | 0.260 | 0.302
178Os | 0.332 | 0.331 | 0.334 | 0.332 | 0.329 | 0.246 | 0.276 | 0.265 | 0.300
180Os | 0.322 | 0.320 | 0.322 | 0.307 | 0.319 | 0.248 | 0.256 | 0.242 | 0.278
182Os | 0.303 | 0.299 | 0.302 | 0.284 | 0.304 | 0.236 | 0.240 | 0.225 | 0.251
184Os | 0.286 | 0.284 | 0.285 | 0.263 | 0.287 | 0.208 | 0.226 | -0.188 | 0.236
186Os | 0.264 | 0.262 | 0.261 | 0.240 | 0.269 | 0.200 | 0.214 | -0.176 | 0.223
188Os | 0.243 | 0.241 | 0.234 | 0.226 | 0.245 | 0.184 | 0.200 | -0.166 | 0.207
190Os | 0.190 | 0.190 | 0.191 | 0.193 | 0.202 | 0.177 | 0.184 | -0.157 | 0.187
192Os | 0.167 | 0.167 | 0.168 | 0.167 | 0.174 | 0.164 | 0.166 | -0.145 | 0.162
194Os | 0.149 | 0.149 | 0.150 | 0.148 | 0.150 | – | -0.150 | -0.129 | -0.143
196Os | 0.121 | 0.122 | 0.119 | 0.119 | 0.124 | – | -0.132 | -0.111 | -0.122
198Os | -0.092 | -0.092 | -0.094 | -0.094 | -0.095 | – | -0.104 | -0.088 | -0.096
200Os | -0.072 | -0.070 | -0.072 | -0.070 | -0.064 | – | -0.066 | 0.000 | -0.061
202Os | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | – | 0.000 | 0.000 | 0.000
204Os | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | – | 0.000 | 0.000 | 0.000
206Os | -0.055 | -0.057 | -0.050 | -0.055 | 0.071 | – | -0.033 | 0.000 | 0.024
208Os | 0.109 | 0.110 | 0.100 | 0.106 | 0.117 | – | -0.046 | 0.000 | 0.076
210Os | 0.135 | 0.135 | 0.127 | 0.130 | 0.144 | – | -0.064 | 0.000 | 0.120
212Os | 0.155 | 0.154 | 0.148 | 0.149 | 0.163 | – | 0.134 | 0.116 | 0.143
214Os | 0.170 | 0.169 | 0.182 | 0.170 | 0.178 | – | 0.165 | 0.142 | 0.166
216Os | 0.252 | 0.253 | 0.251 | 0.242 | 0.189 | – | 0.188 | 0.162 | 0.217
218Os | 0.279 | 0.278 | 0.275 | 0.268 | 0.299 | – | 0.234 | 0.180 | 0.243
220Os | 0.299 | 0.299 | 0.293 | 0.286 | 0.326 | – | 0.252 | 0.196 | 0.258
222Os | 0.305 | 0.303 | 0.300 | 0.290 | 0.329 | – | 0.262 | 0.217 | 0.266
224Os | 0.311 | 0.308 | 0.307 | 0.294 | 0.332 | – | 0.265 | 0.236 | 0.270
226Os | 0.318 | 0.315 | 0.313 | 0.300 | 0.338 | – | 0.263 | 0.240 | 0.270
228Os | 0.319 | 0.316 | 0.313 | 0.300 | 0.339 | – | 0.255 | 0.238 | 0.267
230Os | 0.318 | 0.314 | 0.312 | 0.299 | 0.340 | – | 0.245 | 0.233 | 0.263
232Os | 0.311 | 0.308 | 0.304 | 0.292 | 0.339 | – | 0.237 | 0.225 | 0.256
234Os | 0.296 | 0.293 | 0.289 | 0.275 | 0.329 | – | 0.230 | 0.216 | 0.245
236Os | 0.281 | 0.276 | 0.275 | 0.259 | 0.310 | – | 0.220 | 0.207 | 0.231
238Os | 0.266 | 0.263 | 0.263 | 0.248 | 0.292 | – | 0.208 | 0.197 | 0.217
240Os | 0.252 | 0.249 | 0.249 | 0.234 | 0.274 | – | 0.194 | 0.187 | 0.205
242Os | 0.239 | 0.236 | 0.235 | 0.219 | 0.260 | – | 0.182 | 0.177 | 0.192
244Os | 0.224 | 0.222 | 0.219 | 0.204 | 0.251 | – | 0.170 | -0.151 | 0.177
246Os | -0.150 | -0.151 | 0.186 | -0.157 | 0.229 | – | -0.162 | -0.141 | 0.162
248Os | -0.137 | -0.137 | 0.157 | -0.142 | -0.147 | – | -0.150 | -0.131 | 0.146
250Os | -0.120 | -0.121 | 0.138 | -0.130 | -0.128 | – | -0.134 | -0.120 | 0.125
252Os | -0.100 | -0.101 | 0.113 | -0.117 | -0.104 | – | -0.119 | -0.106 | -0.115
254Os | 0.083 | 0.086 | -0.097 | 0.084 | -0.094 | – | -0.102 | -0.088 | -0.100
256Os | 0.058 | 0.062 | -0.066 | 0.018 | 0.085 | – | -0.079 | 0.000 | -0.074
258Os | 0.000 | 0.000 | 0.000 | 0.000 | 0.069 | – | -0.039 | 0.000 | 0.000
260Os | 0.000 | 0.000 | 0.000 | 0.000 | 0.031 | – | 0.000 | – | –
262Os | 0.002 | 0.002 | 0.001 | 0.004 | 0.058 | – | 0.450 | – | –
264Os | 0.005 | 0.005 | 0.003 | 0.006 | 0.093 | – | 0.447 | – | –
266Os | 0.008 | 0.008 | 0.006 | 0.007 | 0.125 | – | – | – | –
Figure 3: The quadrupole deformation ($\beta_{2}$) of even-even Osmium nuclei
from N=82 to neutron drip line, as obtained using different interactions.
Figure 4: Two-neutron separation energies (upper panel), and two-neutron shell
gaps (lower panel), for even-even Osmium nuclei from N=82 to neutron N = 190,
obtained using different interactions. The other interactions and results are
presented for comparison. Figure 5: The neutron pairing energy (Epair,n ) for
even-even Osmium nuclei from N=82 to N = 190, obtained using different
interactions.

###  3.3 Two-neutron separation energy ($S_{2n}$), shell gap $\delta S_{2n}$
and neutron pairing energy (Epair,n)

The identification of shell closures in atomic nuclei is crucial for
understanding nuclear structure. Two important observables used in this regard
are the two-neutron separation energy($S_{2n}$) and the two-neutron shell gap
($\delta S_{2n}$). The $\delta S_{2n}$, also known as the $S_{2n}$
differential, is calculated using the following relation:

$\delta S_{2n}=\frac{S_{2n}(N,Z)-S_{2n}(N+2,Z)}{2}$

We present S2n, and $\delta S_{2n}$ for even-even 158-266Os isotopes in Fig.
4. To validate our calculations, we compare our results with available
experimental data and also with predictions from other theoretical models.
Specifically, we compare our Relativistic Mean Field (RMF) values with those
obtained from the HFB+THO model using Skyrme SLy4, SkP, and SkM* functionals
19 , as well as the CHFB+5DCH model based on the Gogny D1S interaction 39 .

In Fig. 4, we observe a sudden decrease in the two-neutron separation energy
(S2n) and pronounced kinks in the $\delta S_{2n}$ at neutron numbers N=82,
N=126, and N=184. These findings are consistent with the long-established
neutron shell closure at N=82, N=126, and N=184. Additionally, we note the
presence of kinks at N=118 and N=168 in the relativistic interactions (DD-ME1,
DD-ME2, DD-PC1, DD-PCX, NL3*), suggesting a possible neutron shell or subshell
closure at these neutron numbers. However, non-relativistic functionals such
as SkP, SkM*, SLy4, and CHFB do not exhibit such kinks at N=118 and and N=168.
Previous research has proposed the existence of several deformed subshell
closures Bassem ; Naz2018 ; Naz2019 ; afaque .

The idea of neutron shell closure based on the disappearing neutron pairing
energy Del2001 ; Sil2004 . Supporting the notion of shell closures, the
neutron pairing energy (Epair,n) vanishes at neutron numbers 82, 126, and 184,
as shown in Fig. 5. We found that Epair,n becomes zero at N = 118, matching
the expected shell behavior seen in Fig. 4 with $S_{2n}$ and $\delta S_{2n}$.
However, at N = 168, Epair,n does not reach zero, which challenges the idea of
a shell closure at this neutron number. This discrepancy suggests that the
shell structure at N = 168 is different from what is typically seen at closed
neutron shells. This highlights the complexity of nuclear shell evolution,
especially in heavy, neutron-rich nuclei, and suggests the need for further
study. These observations provide valuable insights into the nuclear structure
of Os isotopes and contribute to our understanding of shell closures in atomic
nuclei.

Figure 6: The charge radii (rc) for even-even Osmium nuclei from N=82 to
N=190, obtained using different interactions. Figure 7: The nucleon
radii($r_{n}$, and $r_{p}$) (upper-panel) and neutron skin thickness ($\Delta
r=r_{n}-r_{p}$) (lower-panel) for even-even Osmium nuclei from N=82 to the
neutron drip line, obtained using different interactions.

###  3.4 Neutron, proton and charge radii

Nuclear radii provide crucial information about the size and distribution of
neutrons and protons within a nucleus. In particular, the neutron skin
thickness, which is the difference between the neutron and proton radii,
becomes significant in neutron-rich nuclei where an excess of neutrons is
present. The neutron skin thickness is directly related to fundamental
properties of nuclear matter and helps bridge the understanding of finite
nuclear systems to infinite nuclear matter Reinhard2016 ; Hagen2015 ;
Agarwal2006 .

In this subsection, we aim to investigate the correlation between nuclear
shape and the distribution of nuclear matter. We compare theoretical results
obtained from different models and also compare them with available
experimental data. The charge radii ($r_{c}$) are presented in Fig. 6, while
the root-mean-square (RMS) nucleon radii ($r_{n}$ and $r_{p}$) and the neutron
skin thickness ($\Delta r=r_{n}-r_{p}$) for the considered isotopic chains are
shown in Fig. 7. For even-even Os isotopes, the neutron radius exhibits an
increasing trend with an increasing number of neutrons. The dips observed at
neutron numbers N=82, 126, and 184 indicate the closed shell behavior of these
nuclei. Due to variations in deformation, the increments in different radii
for even-even Os isotopes are not entirely smooth. We compare our calculated
values of the RMS charge radii ($r_{c}$) with the available experimental data
angeli2004 ; 31 . The comparison reveals a good agreement between our
theoretical results and experimental measurements. Fig. 7 demonstrates that
the neutron skin thickness ($\Delta r$) monotonically increases with the
number of neutrons. This observation further supports our understanding of the
relationship between neutron-rich nuclei and their matter distribution.

###  3.5 Potential energy curve

The shape of a nucleus is one of its most essential as well as fundamental
features, described by quadrupole deformation. Most of the nuclei exhibit the
spherical as well as ellipsoid shape. In axially symmetric case, a deformed
nucleus is classified by quadrupole deformation parameter $\beta_{2}$. Here,
we employed CDFT with density-dependent effective interactions DD-ME1 me1 ,
DD-ME2 me2 , DD-PC1 pc1 , and DD-PCX pcx  to perform quadrupole-constrained
calculations for examining the Potential energy curves for even-even Os chain.
The potential energy curves for the considered isotopic chain are plotted in
Figs. 8\- 11.

Figure 8: Potential energy curves for the even-even isotopes 158-188Os as
functions of the quadrupole deformation from DD-ME1, DD-ME2, DD-PC1, and DD-
PCX force parameters. The curves are scaled with the ground state binding
energy. Figure 9: Same as Fig. 8 but for 190-212Os Figure 10: Same as Fig. 8
but for 214-236Os. Figure 11: Same as Fig. 8 but for 238-260Os.

One can see that the results do not differ from density-dependent meson
exchange model to point coupling model. It can also be noted that the location
of spherical, oblate, and prolate minima appears at the same deformation, no
matter what the force or model is used. However, the relative energies can be
slightly different depending on the models. In general, the outcomes of energy
barriers for deformations through all the effective interactions are found to
be comparable. The ground state of 158Os is revealed to be spherical
confirming the spherical form as expected at well-known magic neutron number N
= 82. As the neutron number increases, the global minima move from spherical
to prolate region. 162Os and 164Os seem to exist in both slightly prolate and
oblate shapes as the corresponding minima have an energy difference of about
1.0 MeV or less. In 166-194Os isotopes, we see oblate shapes with an excited
state at about $2.5$ MeV higher than the global minima. Additionally, Figs. 9
clearly shows that there is a very strong competition between different low-
lying configurations corresponding to different intrinsic deformations, and
this situation of a nucleus is described by shape coexistence. Here, in
considered nuclei, 196Os nucleus has two separate minima at $\beta_{2}=0.1$
(prolate) and $\beta_{2}=-0.1$ (oblate) with a small amount of energy
difference about 0.5 MeV.

It is worth mentioning that a clear shape evolution from spherical to prolate
as well as oblate shapes is observed for increasing neutron number. A shape
transition can be seen from deformed to spherical through flat minima. The
ground state of 202Os has a sharp global minimum at $\beta_{2}=0.0$ refers to
N = 126 neutron magicity. Again in the neutron-rich region, the sphericity
disappears and the global minima move to the well-deformed prolate side. The
214-238Os isotopes show well-deformed prolate shapes, and the prolate minima
lie $2.0-3.0$ MeV deeper than the oblate minima in the case of DD-ME1 and DD-
ME2 calculations while DD-PC1, DD-PCX interactions give $2.0-4.0$ MeV energy
difference for prolate to oblate as a first intrinsic excited state. The
isotopes, 240-254Os coexist as both prolate and oblate in their ground state,
while DD-PC1 and DD-PCX interactions are not affirmative regarding the
coexistence but favor prolate as a ground state. One can see that spherical
minimum for 260Os supports the N = 184 as a neutron magicity. Our results are
in qualitatively good agreement with those earlier predictions obtained in
Refs. afana2016 ; Sarriguren2008 ; Fossion2006 .

##  4 Conclusion

In this study, we provide a comprehensive analysis of the Os isotopic chain,
ranging from the beta-stable region to the neutron drip-line region, using a
covariant density functional and relativistic self-consistent mean-field
description. Various ground state bulk properties, including binding energy,
two-neutron separation energy ($S_{2n}$), two-neutron shell gap ($\delta
S_{2n}$), neutron pairing energy (Epair,n), quadrupole deformation parameter,
rms radii, and neutron skin thickness ($\Delta r$) are calculated. The results
obtained from the covariant density functional theory (CDFT) and relativistic
mean-field (RMF) approach are compared with the self-consistent Hartree-Fock-
Bogoliubov (HFB) formalism based on the Skyrme SLy4, SkP, and SkM*
interactions 19 , as well as the extended HFB theory (CHFB+5DCH) with the
Gogny D1S interaction 39 . These calculations are further compared with
available experimental data 22 ; 33 ; 34 ; 31 ; angeli2004 , and a good
agreement is observed between them.

In the isotopic series under investigation, prominent shell closures are
observed at N = 82, 126, and 184, while N = 118 is suggested to be a shell or
subshell closure. These closures are identified based on the behavior of two-
neutron separation energy, two-neutron shell gap, and neutron pairing energy.
The quadrupole deformation parameters reveal shape transitions across the
isotopic series. Additionally, significant evidence of shape coexistence is
found in in 196Os and 248-254Os, which is consistent with the HFB results.

In order to investigate the phase shape transition in the even-even osmium
isotopic series, we employ the Covariant Density Functional Theory (CDFT) with
separable pairing. Our calculations utilize different parameter sets, namely
the density-dependent DD-ME1 me1 , DD-ME2 me2 , DD-PC1 pc1 , DD-PCX pcx , and
nonlinear NL3* nl3s  interactions. The ground state configurations are
determined by locating the minima on the potential energy curve, which allows
us to study phenomena such as shape coexistence and phase shape transitions.

It is worth noting that the nuclear shape transitions primarily occur at shell
closures, indicating the transitional nature of osmium isotopes. Notably, the
meson exchange and point coupling interactions (i.e., DD-ME1, DD-ME2, DD-PC1,
DD-PCX) yield consistent behavior in the evolution of shape within the
potential energy curves. The analysis of these results leads us to the
conclusion that the potential energy curves are insensitive to the choice of
interactions used in the Density Functional Theory.

The results obtained from CDFT using various effective interactions provide a
reasonable description of the ground-state bulk properties. Furthermore, these
findings exhibit strong similarities with non-relativistic Hartree-Fock-
Bogoliubov (HFB) calculations and other theoretical predictions. Consequently,
we can confidently state that the outcomes presented in this study are
independent of the specific models and force parameters employed, thus
highlighting the authenticity and robustness of our predictions.

## Acknowledgements

One of the authors (UR) would like to thank the Department of Physics, Aligarh
Muslim University, for using the computational facility.

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