Critical Impact of Isospin Asymmetry in Elucidating Magicity Across Isotonic Chains of Different Mass Regions Using Relativistic Energy Density Functional

Abstract

This study provides a comprehensive examination of the surface properties—particularly the symmetry energy and its contributing components—of isotonic chains across various mass ranges, including light, medium, heavy, and superheavy nuclei. We establish a correlation between nuclear symmetry energy and isospin asymmetry in different mass regions along isotonic chains with magic and semi-magic neutron numbers of N = 20, 40, 82, 126, and 172. Our approach integrates the coherent density fluctuation model within the relativistic mean-field (RMF) framework, utilizing both the non-linear NL3 and density-dependent DD-ME2 parameter sets. The methodology employs the Brueckner energy density functional in conjunction with our recently developed relativistic energy density functional (relativistic-EDF). The relativistic parameterization of the EDF at local density facilitates a consistent exploration of isospin-dependent surface properties across the nuclear landscape. In the present work, we successfully reproduce established shell closures and demonstrate that the relativistic approach yields significantly improved predictions for recognized magic numbers, particularly Z = 28 and 50. Additionally, we present compelling evidence for the presence of novel shell and sub-shell closures, specifically at Z = 34, 58, 92, and 118. These findings contribute to a nuanced understanding of nuclear surface properties while serving as a benchmark for future investigations and validations of nuclear models.

1. Introduction

Over the years, significant strides have been made in investigating various $\beta$-stable regions of the nuclear landscape, yielding researchers a thorough understanding of their bulk and intrinsic properties due to nuclear interaction [1]. However, our current understanding of exotic nuclei remains limited. In developing radioactive ion beams, researchers have effectively delved into various exotic nuclei [2,3,4], serving as crucial benchmarks for the validation of existing theoretical models. However, exploring a broad spectrum of nuclei in different exotic regions of the nuclear chart continues to be constrained by cost, time, and technological limitations [4,5]. Moreover, conducting experimental astrophysical studies, including those on neutron stars and dark matter, poses significant challenges within the laboratory frame of reference. Addressing these limitations requires the utilization of an observable capable of accurately predicting the properties of nuclei near the drip line. Therefore, in conjunction with experimental efforts, the development of a reliable theoretical formalism capable of comprehensively studying the entire nuclear landscape becomes imperative [6,7,8,9,10]. Such a theoretical framework would enable researchers to forecast the properties and behaviors of exotic nuclei not yet explored in experiments. By integrating experimental data with theoretical models, scientists can achieve a more holistic understanding of the nuclear landscape, potentially uncovering new phenomena that could revolutionize our understanding of nuclear physics. With ongoing advancements in experimental techniques and theoretical frameworks, there is anticipation for the unveiling of even more secrets of the exotic nuclei lying beyond our current knowledge.

Since the inception of nuclear physics, diverse theoretical frameworks such as the Skyrme [8] and relativistic mean-field (RMF) [6,11] formalisms have gained widespread recognition for their effectiveness in investigating finite nuclei and astrophysical objects such as neutron stars and dark matter. The RMF theory, which involves averaging the fields of nucleon–nucleon interactions alongside the self-interaction and cross-coupling of mesons, has demonstrated significant nucleon–nucleon potential through meson degrees of freedom. Notably, the RMF formalism inherently incorporates the spin-orbit force while accurately portraying $\beta$-stable nuclei and examining the characteristics of exotic nuclei and their responses to extreme conditions [6,11,12]. Integrating the spin-orbit force offers a more precise depiction of nuclear structure and reaction dynamics. Remarkably, as we approach the drip line, the influence of isospin asymmetry becomes increasingly prominent in shaping nuclear properties, a factor often overlooked by traditional observables [10,13,14,15,16,17,18]. Thus, in our pursuit, alongside the use of the RMF formalism, the integration of isospin-dependent observables can yield significantly enhanced results. The nuclear symmetry energy, which is contingent upon isospin asymmetry, delineates an energy distinction between pure symmetric nuclear and neutron matter. It exerts influence over terrestrial matter and astrophysical objects, including neutron stars, typically represented through a density-based series expansion. This parameter plays a pivotal role in the equation of state (EOS) for dense matter, working in tandem with the energy of symmetric matter [13,17].

Numerous investigations have endeavored to elucidate the density dependence of symmetry energy and its role in the EOS for asymmetric nuclear matter and neutron stars [19,20,21]. Determining the symmetry energy necessitates a proper translation of infinite nuclear matter quantities described in momentum space to finite nuclei described in coordinate space. The coherent density fluctuation model (CDFM) breaks the nucleus into fluctons, which are small, spherical units, and facilitates an accurate translation of nuclear matter quantities from momentum space to coordinate space [10,13,14,15,16,18,22]. Furthermore, it enhances the prediction accuracy of shell/sub-shell closures, yielding consistent outcomes for various nuclei, including Pb nuclei [17,23,24]. The CDFM has been employed to compute the weight function in nuclei using the relativistic Fermi gas-scaling function [25,26]. Lepton scattering mechanisms further utilize this scaling function, as seen in Refs. [27,28]. Moreover, the CDFM elucidates the significance of nucleon momentum and density distribution in explaining superscaling in lepton–nucleon scattering [26]. The use of EOS has witnessed significant advancements over the past four decades, owing to one-boson exchange nucleon–nucleon potentials and relativistic extensions [6,29]. Exploring nuclear matter stability at high densities has underscored the importance of understanding the fluctuations of nuclear symmetry energy with density [21].

It is worth noting that initially, the CDFM formalism was developed to incorporate the non-relativistic Skyrme–Hartree–Fock densities and the widely popular Brueckner prescription of EDF [10,13]. Later, one of the authors worked to use RMF densities as input while using Brueckner-EDF to successfully study the various isotopic chains [14]. However, a notable limitation of Brueckner’s prescription is its inability to accurately replicate the empirical saturation point of symmetric nuclear matter, which is $E/A\approx$−16 MeV at $\rho \approx 0.2$ fm⁻³ instead of $\rho \approx 0.15$ fm⁻³, which is known as the Coester band problem [24]. This limitation restricts its accuracy in exploring nuclear properties, particularly for higher-mass regions. Recently to provide much more consistent results, along with RMF densities, we provided a novel, robust relativistic parameterization of EDF (relativistic-EDF) [17] for widely employed non-linear NL3 and recently developed density-dependent DD-ME2 force parameter sets, which can successfully be used in the study of finite nuclei across the entire nuclear landscape. The NL3 and DD-ME2 parameter sets faithfully reproduce experimental data across a broad spectrum of nuclei with minimal deviations [7,11].

In the present work, we examine isotonic chains with magic neutron numbers of N = 20, 40, 82, 126, and 172, offering a comprehensive examination of the relationship between surface properties and their derivatives with respect to isospin asymmetry ($(N-Z)/A$) and the atomic number (Z). Moreover, we compare the novel relativistic-EDF with the widely used Brueckner implementation of EDF within the CDFM. The CDFM takes the input density (in the present case, the RMF densities, namely, the NL3 and DD-ME2 parameter sets) and, based on the EDF (relativistic-EDF and Brueckner-EDF), calculates the surface properties in the given mass regions. Our investigation yields insights into nuclear surface properties, which are crucial within the domain of nuclear physics. Additionally, our findings establish a benchmark for forthcoming studies and validations of nuclear models.

This paper is organized as follows: Section 2 briefly discusses the theoretical formalism, especially the relativistic mean-field formalism, and the application of the coherent density fluctuation model. We present a discussion of calculations and results in Section 3, followed by the conclusion in Section 4.

2. Theoretical Formalism

The relativistic mean-field (RMF) formalism constitutes a microscopic approach to solving the many-body problem through the use of interacting meson fields [30,31,32]. It is widely used in study of finite nuclei and infinite nuclear matter, including neutron star systems for high density and isopsin asymmetry [9,14,15,16,30,31,32]. It can be categorized as the relativistic interpretation of Hartree–Fock–Bogoliubov theory. The RMF Lagrangian density, which is derived from the Walecka Lagrangian with several modifications, can be expressed as follows [9,14,15,16,30,31,32]:

$$\begin{array}{ccc}\hfill \mathcal{L}& =& \overline{\psi }\{i{\gamma }^{\mu }{\partial }_{\mu }-M\}\psi +{\displaystyle \frac{1}{2}}{\partial }^{\mu }\sigma {\partial }_{\mu }\sigma -{\displaystyle \frac{1}{2}}{m}_{\sigma }^2{\sigma }^2-{\displaystyle \frac{1}{3}}{g}_2{\sigma }^3\hfill \\ & -& {\displaystyle \frac{1}{4}}{g}_3{\sigma }^4-g_{\sigma }\overline{\psi }\psi \sigma -{\displaystyle \frac{1}{4}}{\mathsf{\Omega }}^{\mu \nu }{\mathsf{\Omega }}_{\mu \nu }+{\displaystyle \frac{1}{2}}{m}_w^2{\omega }^{\mu }{\omega }_{\mu }-g_w\overline{\psi }{\gamma }^{\mu }\psi {\omega }_{\mu }-{\displaystyle \frac{1}{4}}{\overrightarrow{B}}^{\mu \nu }.{\overrightarrow{B}}_{\mu \nu }\hfill \\ & +& {\displaystyle \frac{1}{2}}{m}_{\rho }^2{\overrightarrow{\rho }}^{\mu }.{\overrightarrow{\rho }}_{\mu }-g_{\rho }\overline{\psi }{\gamma }^{\mu }\overrightarrow{\tau }\psi ·{\overrightarrow{\rho }}^{\mu }-{\displaystyle \frac{1}{4}}{F}^{\mu \nu }{F}_{\mu \nu }-e\overline{\psi }{\gamma }^{\mu }{\displaystyle \frac{\left(1-{\tau }_3\right)}{2}}\psi A_{\mu }.\hfill \end{array}$$

(1)

In Equation (1), the symbols $g_{\sigma }$, $g_{\rho }$, and $g_{\omega }$ represent the coupling constants of the $\sigma$, $\rho$, and $\omega$ mesons, respectively. These mesons have masses of $m_{\sigma }$, $m_{\rho }$, and $m_{\omega }$, respectively. Mesons operate as mediators in the interactions between nucleons, which have a mass of M and are described by the Dirac spinor ($\psi$). The symbols $\tau$ and ${\tau }_3$ represent the isospin and its third component, respectively. Field tensors $F^{\mu \nu }$, ${\mathsf{\Omega }}_{\mu \nu }$, and ${\overrightarrow{B}}^{\mu \nu }$ are provided in the following form:

$$\begin{array}{c}\hfill F^{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }\\ \hfill {\mathsf{\Omega }}_{\mu \nu }={\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }\\ \hfill {\overrightarrow{B}}^{\mu \nu }={\partial }_{\mu }{\overrightarrow{\rho }}_{\nu }-{\partial }_{\nu }{\overrightarrow{\rho }}_{\mu }.\end{array}$$

(2)

By expanding the upper and lower components of the Dirac spinors and the boson fields, the field equations for the nucleons and the mesons can be obtained from the given Lagrangian density. The RMF formalism allows for the density dependence of the meson–nucleon coupling [11,33]. This coupling is parameterized in the phenomenological approach [11,33,34,35], where the mesons couple with the nucleon fields according to the following expressions:

$$\begin{array}{c}\hfill g_i\left(\rho \right)=g_i\left({\rho }_{sat}\right)f_i\left(x\right){|}_{i=\sigma ,\omega },\end{array}$$

(3)

where

$$\begin{array}{c}\hfill f_i\left(x\right)=a_i{\displaystyle \frac{1+b_i{(x+d_i)}^2}{1+c_i{(x+d_i)}^2}},\end{array}$$

(4)

and

$$\begin{array}{c}\hfill g_{\rho }=g_{\rho }\left({\rho }_{sat}\right)e^{a_{\rho }(x-1)}.\end{array}$$

(5)

Comprehensive explanations of the functional expressed as $x=\rho /{\rho }_{sat}$ and the parameterization process can be found in Refs. [11,14,33,35]. The formula expressed as $E_{c.m.}={\displaystyle \frac{3}{4}}\left(41A^{-1/3}\right)$ is commonly employed to approximate the correction for the energy of center-of-mass motion in the context of a harmonic oscillator. This study considers two distinct pairing methods to investigate the impact of pairing and model interdependencies on the properties of nuclear matter at local density. More comprehensive descriptions of relativistic parameterization and formalism can be found in Refs. [9,12,14,31,32]. This study focuses on characterizing the nuclear bulk properties of open-shell nuclei by considering the constant-gap BCS approach with NL3 [7], and Bogoliubov transformation with the DD-ME2 parameter [11] while taking into account pairing correlations.

2.1. Brueckner Approach and Relativistic Energy Density Parameterization at Local Density

The expression pertaining to the energy density of infinite and isotropic nuclear matter derived from Brueckner’s functional is as follows [36,37]:

$$\begin{array}{c}\hfill V\left(x\right)=A{V}_0\left(x\right)+V_c-V_{CO},\end{array}$$

(6)

where

$$\begin{array}{ccc}\hfill V_0\left(x\right)& =& 37.53\left[{(1+\delta )}^{5/3}+{(1-\delta )}^{5/3}\right]{\rho }_0^{2/3}\left(x\right)+b_1{\rho }_0\left(x\right)+b_2{\rho }_0^{4/3}\left(x\right)+b_3{\rho }_0^{5/3}\left(x\right)\hfill \\ & +& {\delta }^2\left[b_4{\rho }_0\left(x\right)+b_5{\rho }_0^{4/3}\left(x\right)+b_6{\rho }_0^{5/3}\left(x\right)\right],\hfill \end{array}$$

(7)

with $b_1=-741.28,b_2=1179.89,b_3=-467.54,b_4=148.26,b_5=372.84,b_6=-769.57$. Here, $V_0\left(x\right)$ is the energy per nucleon $\left(\mathrm{MeV}\right)$ in $\mathrm{NM}$, which explains the neutron–proton asymmetry. The term $V_C$ corresponds to the Coulomb energy of protons within a flucton given as $V_C={\displaystyle \frac{3}{5}}{\displaystyle \frac{Z^2{e}^2}{x}},$ and the Coulomb exchange energy is $V_{CO}=0.7386Z{e}^2{\left(3Z/4\pi x^3\right)}^{1/3}.$ The crucial part of the present calculation involves the conversion of nuclear matter quantities from momentum space $\left(\rho \right)$ to coordinate space $\left(r\right)$ in local density approximation. According to Refs. [10,13,36,37], the nuclear symmetry energy can be obtained using Equations (9) and (10) at local density as follows:

$$\begin{array}{ccc}\hfill S^{\mathrm{NM}}\left(x\right)& =& 41.7{\rho }_0^{2/3}\left(x\right)+b_4{\rho }_0\left(x\right)+b_5{\rho }_0^{4/3}\left(x\right)+b_6{\rho }_0^{5/3}\left(x\right).\hfill \end{array}$$

(8)

To overcome the Coester band problem [24], we developed a method for parameterizing the relativistic energy density functional (relativistic-EDF) in a recent study [17]. The work reported in Ref. [17] involved the formulation of an analytical equation for the binding energy per nucleon using relativistic parameter sets, namely the non-linear NL3 and density-dependent DD-ME2. The method was applied to analyze certain doubly magic nuclei surface characteristics, providing consistent results. This methodology defined the nucleus as an aggregation of minute spherical fragments distinguished by their local density. The energy density functional is represented in a parameterized form as follows [17]:

$$\begin{array}{ccc}\hfill \epsilon \left(\rho \right)& =& C_k{\rho }_0^{2/3}\left(x\right)+b_1{\rho }_0\left(x\right)+b_2{\rho }_0^{5/3}\left(x\right)+b_3{\rho }_0^{8/3}\left(x\right)+b_4{\rho }_0^{10/3}\left(x\right)+b_5{\rho }_0^4\left(x\right)\hfill \\ & +& {\alpha }^2(a_1{\rho }_0^{2/3}\left(x\right)+a_2{\rho }_0^{7/3}\left(x\right)+a_3{\rho }_0^{8/3}\left(x\right)).\hfill \end{array}$$

(9)

where $C_k=37.53[{(1+\alpha )}^{{\displaystyle \frac{5}{3}}}+{(1-\alpha )}^{{\displaystyle \frac{5}{3}}}]$ and $\alpha$ correspond to the neutron–proton asymmetry. The fitting procedure includes three primary components: the first kinetic energy term ($C_k$) derived from the Thomas–Fermi approximation [36,37,38]. The terms $b_i$ and $a_i$ determine the curve shape and accuracy of fit concerning the neutron–proton asymmetry, respectively. The coefficients of $b_i$ and $a_i$ are given in

Table 1. The values of the coefficients for the DD-ME2 and NL3 parameter sets are listed below for the relativistic parameterization of the energy density functional [17].

	DD-ME2	NL3
$b_1$	−627.40397	−631.22898
$b_2$	2032.92832	2177.46092
$b_3$	−9038.76463	−11,541.44105
$b_4$	19,143.35052	26,104.23289
$b_5$	−12,352.13859	−17,045.86031
$a_1$	80.43433	49.40867
$a_2$	−433.81712	2741.54540
$a_3$	446.43825	−3019.76641

. The nuclear matter symmetry energy ($S^{NM}$) is expressed as follows:

$$\begin{array}{ccc}\hfill S^{NM}\left(x\right)& =& 41.7{\rho }_0^{2/3}\left(x\right)+a_1{\rho }_0^{2/3}\left(x\right)+a_2{\rho }_0^{7/3}\left(x\right)+a_3{\rho }_0^{8/3}\left(x\right).\hfill \end{array}$$

(10)

2.2. Coherent Density Fluctuation Model

The coherent density fluctuation model (CDFM) is based on the $\delta$-function limit of the generator coordinate method [10,13,39,40,41,42,43], which involves a coherent superposition of the one-body density matrix (OBDM) (${\rho }_x(\mathbf{r},{\mathbf{r}}^{\prime })$) for spherical pieces of nuclear matter called $fluctons$ densities. The mixed density, which may be expressed in a general form, is directly related to the Fourier transformation of the momentum distribution:

$$\begin{array}{c}\hfill \rho (\mathbf{r},{\mathbf{r}}^{\prime })={\int }_0^{\infty }{\left|\mathcal{F}\left(x\right)\right|}^2{\rho }_x(\mathbf{r},{\mathbf{r}}^{\prime })dx.\end{array}$$

(11)

The weight function expressed as ${\left|\mathcal{F}\left(x\right)\right|}^2$ represents the different uniform distributions in the average density distribution. Moreover, ${\rho }_x$ represents the density matrix for A nucleons that are evenly distributed across a sphere of radius x. The spherical flucton density (${\rho }_o\left(x\right)$) is expressed as $3A/4\pi x^3$. Furthermore,

$$\begin{array}{cc}\hfill {\rho }_x(\mathbf{r},{\mathbf{r}}^{\prime })=& 3{\rho }_0\left(x\right){\displaystyle \frac{J_1(k_F\left(x\right)|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)}{{(k}_F\left(x\right)|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)}}.\end{array}$$

(12)

In this context, $J_1$ denotes a Bessel function of the first order, whereas $k_F\left(x\right)$ is the term used to describe the Fermi momentum of the nucleons. The Fermi momentum is defined as follows:

$$\begin{array}{c}\hfill k_F\left(x\right)={\left({\displaystyle \frac{3{\pi }^2}{2}}{\rho }_o\left(x\right)\right)}^{1/3}={\left({\displaystyle \frac{9\pi A}{8}}\right)}^{1/3}.\end{array}$$

(13)

It is crucial to highlight that Equation (11) pertains to a broad CDFM statement that the density distribution of nuclear matter swings about the average distribution while preserving spherical symmetry and homogeneity.

Within the CDFM, for the Wigner distribution function corresponding to the OBDM we have

$$\begin{array}{c}\hfill W(\mathbf{r},\mathbf{k})={\int }_0^{\infty }dx{\left|\mathcal{F}\left(x\right)\right|}^2{W}_x(\mathbf{r},\mathbf{k}),\end{array}$$

(14)

where $W_x(\mathbf{r},\mathbf{k})$ = ${\displaystyle \frac{4}{8{\pi }^3}}\theta (x-|\mathbf{r}\left|\right)\theta (k_F\left(x\right)-\left|\mathbf{k}\right|)$. Correspondingly to W (r,k), the density ($\rho \left(r\right)$) can be represented as follows:

$$\begin{array}{ccc}\hfill \rho \left(r\right)& =& \int d\mathbf{k}W(\mathbf{r},\mathbf{k})\hfill \\ & =& {\int }_0^{\infty }{dx\left|\mathcal{F}\left(x\right)\right|}^2{\displaystyle \frac{3A}{4\pi x^3}}\theta (x-|\mathbf{r}\left|\right).\hfill \end{array}$$

(15)

The mass number is normalized as $\int \rho (\mathbf{r})d\mathbf{r}=A$. By employing $\delta$-function approximation to the Hill–Wheeler integral equation, one may derive a differential equation for the weight function (${\left|\mathcal{F}\left(x\right)\right|}^2$) in the generator coordinate [41,42,44]. Instead of solving the differential equation, we employed a convenient method to determine ${\left|\mathcal{F}\left(x\right)\right|}^2$ as follows:

$$\begin{array}{c}\hfill {\left|\mathcal{F}\left(x\right)\right|}^2=-{\displaystyle \frac{1}{{\rho }_o\left(x\right)}}{\displaystyle \frac{d\rho \left(r\right)}{dr}}{|}_{r=x},\end{array}$$

(16)

with normalization as ${\int }_0^{\infty }dx{\left|\mathcal{F}\left(x\right)\right|}^2=1$ [42,44].

The symmetry energy (S) for a finite nucleus can be computed based on the CDFM formalism as follows [10,13,14,44,45]:

$$\begin{array}{c}\hfill S={\int }_0^{\infty }dx{\left|\mathcal{F}\left(x\right)\right|}^2{S}^{NM}\left(x\right).\end{array}$$

(17)

Following the Danielewicz prescription [46], S can be mathematically represented using the concepts of volume symmetry energy ($S_V$) and surface symmetry energy ($S_S$):

$$\begin{array}{c}\hfill S={\displaystyle \frac{S_V}{1+{\displaystyle \frac{S_S}{S_V}}{A}^{-1/3}}}={\displaystyle \frac{S_V}{1+{\displaystyle \frac{1}{\kappa A^{1/3}}}}},\end{array}$$

(18)

where $\kappa$ represents the ratio of $S_V$ to $S_S$ and is calculated according to Refs. [47,48] as follows:

$$\begin{array}{c}\hfill \kappa ={\displaystyle \frac{3}{r_0{\rho }_0}}{\int }_0^{\infty }dx{\left|\mathcal{F}\left(x\right)\right|}^{2}x{\rho }_o\left(x\right)\left[{\displaystyle \frac{S^{NM}\left({\rho }_o\right)}{S^{NM}\left(x\right)}}-1\right].\end{array}$$

(19)

Here, $r_0$ refers to the radius of the nuclear volume per nucleon, ${\rho }_0$ is the equilibrium density of nuclear matter [47,49], and $S^{NM}\left({\rho }_o\right)$ is the symmetry energy at equilibrium density [47,48].

Following Refs. [46,47], the volume and surface symmetry energy can be expressed as follows:

$$\begin{array}{c}\hfill S_V=S\left(1+{\displaystyle \frac{1}{\kappa A^{1/3}}}\right),\end{array}$$

(20)

and

$$\begin{array}{c}\hfill S_S={\displaystyle \frac{S}{\kappa }}\left(1+{\displaystyle \frac{1}{\kappa A^{1/3}}}\right),\end{array}$$

(21)

respectively.

It is important to note that the current methodology for calculating the volume and surface symmetry energy is accurate for finding the shell and sub-shell closure in the domain of finite nuclei, which is the primary focus of our study. However, for extremely high mass numbers, particularly in the limit of $A\to \infty$, the precision of the current formalism may decrease. To perform precise calculations of components of symmetry energy in the realm of infinite nuclear matter, recently, an alternative approach was proposed within the CDFM framework by incorporating non-relativistic inputs in the weight function [50]. This method ensures accurate behavior of the terms in the denominator of the equation relating S with $S_V$ and $S_S$ as A approaches infinity.

In this alternative method, as $A\to \infty$, the $S_S/S_V$ ratio approaches zero more accurately; thus, S converges to $S_V$. This contrasts with our current approach, where the surface coefficient ($S_S\left(A\right)$) decreases at a slower rate than $A^{-1/3}$ as $A\to \infty$. Consequently, this alternative method provides a higher value of $S_S$ while yielding a lower value of $S_V$, although the impact on finite nuclei is minimal. It is important to note that both formalisms consistently provide the same evidence of shell and sub-shell closures in finite nuclei, which is the primary objective of this study.

As a comparison of the formalism, within the LDM framework [51,52], the symmetry energy is expressed as follows:

$$\begin{array}{c}\hfill S=J-\left({\displaystyle \frac{9J^2}{4Q}}\right)A^{-1/3},\end{array}$$

(22)

where J is the bulk symmetry energy and Q is the surface stiffness coefficient. The ratio of the surface-to-volume contributions is given by ${\displaystyle \frac{9J}{4Q}}$.

Our approach, although rooted in a different formalism, with minor alteration, helps to recover a consistent limit for $A\to \infty$, similar to the LDM. The primary difference lies in the calculation of $\kappa$, which reflects the integrated effects of nuclear density fluctuations, offering a more dynamic interpretation of the components of the symmetry energy.

3. Calculations and Results

The current computation proceeds in two stages. Initially, the relativistic mean-field (RMF) equations are solved self-consistently for the requisite number of boson ($N_B$) and fermion ($N_F$) shells. Given the diverse regions covered within the nuclear chart, achieving convergence of ground-state solutions necessitates a range of $N_B=N_F=$ 12 to 14. Gauss–Hermite integration requires 20 mesh points, while Gauss–Laguerre integration requires 24 mesh points. Recently, the impact of deformation on symmetry energy computation was examined in Ref. [53]. This study demonstrated that for significant deformations, such as ${\beta }_2\approx$ 0.6, the estimated symmetry energy exhibits a relative difference of approximately 0.4 MeV. Since nuclei typically possess deformations (${\beta }_2$) of less than 0.6, using spherical density is deemed appropriate for computation [10,13]. These properties serve as a well-established method for exploring potential shell and/or sub-shell closures within the exotic region of the nuclear chart. The isospin properties, which encompass the symmetry energy and its derivatives, are closely intertwined with the surface characteristics of the nuclear density distributions. To determine the symmetry energy pertinent to finite nuclei, we used density distributions of various isotonic chains across different mass regions, which include conventional magic numbers of N = 20, 40, 82, 126, and 172. These density profiles are derived from the well-established RMF approach using non-linear NL3 and the relativistic Hartree–Bogoliubov approach for density-dependent DD-ME2 parameter sets. The density profile is subsequently integrated into the coherent density fluctuation model (CDFM) with Brueckner’s energy density functional (EDF) and relativistic-EDF approaches to calculate the weight function. The calculated weight functions are then employed to estimate the effective symmetry energy for the specified isotonic chains.

It is worth noting that the density of a highly asymmetric system significantly affects the finite nuclear characteristics. As we move away from the $\beta$-stability line, the influence of isospin asymmetry becomes increasingly prominent in shaping nuclear properties, a factor often overlooked by traditional observables [10,15,17,18,50,54,55]. In Figure 1, we first observe that the Brueckner-EDF for the NL3 and DD-ME2 parameter sets shows a sharp dip below $x=4$ fm. Interestingly, the relativistic-EDF using the DD-ME2 parameter set shows precise reproduction of symmetry energy properties in the central part of the nucleus. However, although the relativistic-EDF using the NL3 parameter set shows improvement over Brueckner’s prescription by improving the range of such a dip, i.e., below $x=3$ fm instead of $x=4$ fm (as observed in Brueckner’s prescription), its behavior is closely associated with its parametrization as a function of the density and is one of the stiffest equations of state widely studied [17,56,57]. From this, we can conclude that the relativistic-EDF using the DD-ME2 parameter set provides consistent results in all the density regimes, whereas relativistic-EDF using the NL3 parameter set marginally improves the range for precise calculation of nuclear matter properties over the Brueckner-EDF.

In Figure 2, we illustrate the nuclear density alongside the weight function (${\left|\mathcal{F}\left(x\right)\right|}^2$) for the ³⁴Si and ⁷⁰Zn nuclei. The resemblance between the density profile and the corresponding weight function is notable. Specifically, significant values of the weight function align with the density in the surface region, suggesting these attributes as surface characteristics. To gain a comprehensive understanding of the influence exerted by ${\left|\mathcal{F}\left(x\right)\right|}^2$ on the calculation of symmetry energy (S) and to delineate the suitable density range for finite nucleus calculations, we impose a physical constraint linked to the weight function (${\left|\mathcal{F}\left(x\right)\right|}^2$) [10,14,15,50]. The value of ${\left|\mathcal{F}\left(x\right)\right|}^2$ given in Equation (16) is negative when the radial derivative of the density of the nucleus is positive. Moreover, the symmetry energy at local density ($S^{NM}\left(x\right)$) is negative for some values $x<x_{min}$ and $x>x_{max}$. However, in the physical realm, the symmetry energy cannot have a negative value. Moreover, practically, a small value of x corresponds to a large value of the density (${\rho }_o\left(x\right)$), which is much higher than the saturation density. In the equation of symmetry energy (17), to avoid non-physical scenarios where the symmetry energy might become negative, we exclude the terms corresponding to $x<x_{min}$ and $x>x_{max}$. In other words, $x_{min}$ refers to the point where the symmetry energy at local density ($S^{NM}\left(x\right)$) changes sign from a negative to positive value, i.e., where the isospin instability starts. This method was initially used in the calculations involving Brueckner’s prescriptions [10,14].

However, very recently, a new method involving the half-width method was introduced in Ref. [50]. Typically, the core density of a nucleus, denoted as ${\rho }_c$, ranges from approximately 0.10 to 0.16 fm⁻³. The weight function (${\left|\mathcal{F}\left(x\right)\right|}^2$) reaches its peak around $\rho \left(R_{1/2}\right)\approx 0.05$ – $0.08{\mathrm{fm}}^{-3}$. In this region, where $\rho \approx 0.5{\rho }_0$, the values of $S^{NM}\left(x\right)$ are crucial for the calculations. The central density (${\rho }_0$) of the nucleus generally ranges between $0.10{\mathrm{fm}}^{-3}$ and $0.16{\mathrm{fm}}^{-3}$. Consequently, the peak of the weight function (${\left|F\left(x\right)\right|}^2$) corresponds to a density ($\rho \left(R_{1/2}\right)$) in the range of $0.05{\mathrm{fm}}^{-3}$ to $0.08{\mathrm{fm}}^{-3}$. In Figure 2, the maximum of ${\left|F\left(x\right)\right|}^2$ is observed at $\rho =0.05{\mathrm{fm}}^{-3}$, and within its width, the density ($\rho$) spans from $0.12{\mathrm{fm}}^{-3}$ to $0.01{\mathrm{fm}}^{-3}$. This density range is critical for accurately determining the symmetry energy within the CDFM formalism. Following the half-width method, we set the lower limit of integration as the lower value of the radius, which is equivalent to the left point of the half-width ($\mathsf{\Gamma }$), as shown in Figure 2. This new method helps in the precise calculation of the nuclear properties. In the present work, we made use of this new method for the calculation of the surface properties.

Using the CDFM formalism, we calculate the symmetry energy (S), the volume symmetry energy ($S_V$), and the surface symmetry energy ($S_S$) using Equations (17)–(21). Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 depict the plots of symmetry energy and its surface and volume components as functions of atomic number (Z) for isotonic chains with magic and semi-magic neutron numbers, namely $N=20$, 40, 82, 126, and 172, using the NL3 and DD-ME2 parameters. The pronounced kinks/peaks in the figures of symmetry energy and its components are particularly noteworthy, as they symbolize the region of increased stability compared to the neighboring nuclei. Next, we take a closer look at each figure and corroborate the results with the experimental shell closures while ascertaining novel regions of shell and/or sub-shell closure.

Figure 3a–c present the variations of S, $S_V$, and $S_S$ as a function of atomic number Z for N = 20 isotones. We observe that the values of the terms S, $S_V$, and $S_S$ have ranges of 22.6 $\le S$ ≤ 31.2, 25.8 $\le S_V$ ≤ 41.0, and 18.51 $\le S_S$ ≤ 27.4 MeV, respectively, for the relativistic-EDF, whereas the their values under Brueckner’s approach lie within the ranges of 10.4 $\le S$ ≤ 23.7, 20.1 $\le S_V$ ≤ 35.7, and 13.5 $\le S_S$ ≤ 23.5 MeV, respectively. It is worth noting that the ranges of S, $S_V$, and $S_S$ are near the expected the theoretical limits [19,46,47]. The figure shows that the NL3 force parameter produces marginally higher values than the DD-ME2 parameter for both the relativistic and Brueckner prescriptions. This difference is due to the NL3 equation of state being stiffer than the DD-ME2 [56]. As expected, based on the discussed formalism, volume and surface components mirror the overall trend observed in the symmetry energy. The values calculated based on the Brueckner approach yield values smaller than those of its relativistic counterpart. In particular, the magnitude of $S_V$ is greater than that of S. It is worth noting for readers that in very dense neutron matter, namely neutron stars, the contribution of the volume component to the symmetry energy is greater than that of the surface component. The volume component contributes to nearly 90% of the total symmetry energy [19,46]. Furthermore, there is a marked increase in the surface properties for $Z\ge 14$ relative to preceding isotones under both the relativistic and Brueckner frameworks. An increase in stability with mass number is observed along isotonic chains possessing low neutron magic numbers, applicable to both frameworks. The nucleus with $Z=18$ and $N=20$, corresponding to ³⁸Ar, exhibits maximal stability along this chain. This assertion is corroborated by traditional observables, such as the binding energy per nucleon, as reported in Refs. [58,59] (data not presented here). As this work is geared towards exploring isospin-dependent surface properties of nuclei and their remarkable achievement in validating the experimental shell closures, we focus our discussion on the variation of isospin-dependent properties along the isotonic chains in different mass regions.

Next, Figure 4a–c provide graphs of the variation of surface properties as a function of Z for a isotonic chain with a (semi) magic neutron number of N = 40. We observe that the relativistic-EDF has ranges of 26.3 $\le S$ ≤ 30.6, 31.2 $\le S_V$ ≤ 35.7, and 22 $\le S_S$ ≤ 25.2 MeV, while the Brueckner approach with the given parameter sets has ranges of 15.6 $\le S$ ≤ 23.6, 21.9 $\le S_V$ ≤ 28.3, and 14.5 $\le S_S$ ≤ 19.7 MeV. Here, we find that the values corresponding to N = 40 isotones increase marginally compared to those corresponding to N = 20. Figure 4 reveals a distinct peak at the proton magic number of $Z=28$ with the relativistic prescription, aligning with the established proton magic number. Furthermore, the relativistic parameter sets exhibit substantial discontinuity at $Z=34$. However, such discontinuities are not prominent when employing the non-relativistic Brueckner framework.

Figure 5a–c show the variation of surface properties for an isotonic chain with a magic neutron number of N = 82. We observe that the values of the terms S, $S_V$, and $S_S$ have ranges of 26.3 $\le S$ ≤ 31.1, 35.5 $\le S_V$ ≤ 41.1, and 23.8 $\le S_S$ ≤ 27.3 MeV, respectively, for the relativistic-EDF, whereas under the Brueckner approach, their ranges are 17.9 $\le S$ ≤ 23.7, 28 $\le S_V$ ≤ 35.7, and 18.6 $\le S_S$ ≤ 23.5 MeV, respectively. Interestingly, the values of S, $S_V$, and $S_S$ also increase here compared to N = 40 and 20 isotones. Thus, the magnitude/range of surface properties increases with increasing mass numbers for different isotones, at least in light- to medium-mass nuclei. A significant peak at $Z=58$ is discernible within the relativistic framework, suggesting potential shell and/or sub-shell closures. Conversely, Brueckner’s approach does not indicate a peak in the vicinity of this region. The discovery of Z = 58 represents a significant milestone in exploring potential shell closures in this region. We eagerly anticipate future experimental validation along the isotonic chain.

Next, in Figure 6a–c, we present the surface properties of a neutron magic number of N = 126. The surface properties have ranges of 24.6 $\le S$ ≤ 27.5, 27.6 $\le S_V$ ≤ 30.8, and 19.7 $\le S_S$ ≤ 22.1 MeV for the relativistic-EDF, while under Brueckner’s approach, the values lie in the ranges of 18.8 $\le S$ ≤ 22.8, 24 $\le S_V$ ≤ 28.1, and 17.7 $\le S_S$ ≤ 20.3 MeV, respectively. However, contrary to what was observed in the range of surface properties while moving across the light- to medium-mass region, here, we observe that the minimum and maximum values of the three surface properties (S, $S_V$, and $S_S$) decrease sharply and are lower than those observed for the N = 20 isotones. The observed behavior can be attributed to the interplay of nuclear shell structure, nuclear mass, and neutron–proton asymmetry. Moreover, we observe a significant peak at Z = 92 in the relativistic prescription, with minor discontinuity near the N = 80 regions, indicating the possible existence of a shell closure. Moreover, Brueckner’s prescription does not provide clear evidence of a peak in the concerned region.

In Figure 7a–c, the variation of surface properties corresponds to the predicted neutron magic number of N = 172. We observe that the values of S, $S_V$, and $S_S$ lie within the ranges of 23.2 $\le S$ ≤ 25.6, 25.8 $\le S_V$ ≤ 28.3, and 18.5 $\le S_S$ ≤ 20.1 MeV, respectively, for the relativistic-EDF prescription, while under Brueckner’s prescription, the values fall in the ranges of 19.8 $\le S$ ≤ 21.9, 22.1 $\le S_V$ ≤ 24.3, and 16.2 $\le S_S$ ≤ 18.3 MeV, respectively. On the basis of the data, the surface properties exhibit their minimum values in this specific region. By integrating this observation with the previously discussed results across different mass regions (ranging from light to heavy), it can be inferred that the surface properties initially increase with the neutron number (along the isotonic chain) up to the intermediate-mass region. This is followed by a pronounced decrease in the heavy and superheavy regions. Here, we reiterate that the variation in surface properties is significant in understanding the influences of nuclear shell structure, nucleus mass, and neutron–proton asymmetry. More discussion related to the variation of surface properties as a function of neutron–proton asymmetry in different mass regions is provided below. In Figure 7, for the reported neutron magic number of N = 172, the relativistic prescription provides a minor peak at Z = 118. As expected according to the limitation of Brueckner’s prescription to resolve the Coester band problem (which becomes significant for higher-mass nuclei, especially for ²⁰⁸Pb), we only find a minor hint of discontinuity near Z = 118 to 120 regions following the Brueckner formalism. The region around Z = 118 and 120 has attracted widespread attention in the experimental community with respect to discovering the next island of stability.

It is important to understand that the appearance of these traditional magic numbers can vary in isotonic chains due to structural effects such as shape transitions and nucleon–nucleon correlations. These effects can lead to the emergence of new shell closures and the suppression of traditional ones [60,61]. For example, the structural evolution along isotonic chains can be influenced by the interplay between monopole interactions, spin-orbit coupling, and tensor forces [60]. These interactions can modify the shell structure, leading to the appearance of new magic numbers far from stability. This explains why new shell closures such as $N=14$, 34, 58, and 92 are predicted in the isotonic chains of $N=20$, 40, 82, and 126, respectively. The study of the density dependence and isospin dependence of nuclear symmetry energy is a key area of interest in contemporary nuclear physics. The investigation of shell/sub-shell closures in the drip-line region is crucial for understanding the isospin dependence of effective nuclear symmetry in the isotonic chain, especially for the proton magic number. The neutron–proton asymmetry quantifies the degree to which a nucleus departs from symmetry, specifically in terms of the imbalance between the numbers of neutrons and protons. The neutron–proton asymmetry is associated with the isospin asymmetry parameter, denoted as $\beta$, which is defined as $\beta =(N-Z)/(N+Z)$. The neutron–proton asymmetry varies from 0 in symmetric nuclei to 1 in pure neutron matter. The changes in the symmetry energy with isospin asymmetry across various isotonic chains are shown in Figure 8 and Figure 9. The figures demonstrate that the symmetry energy increases as the asymmetry increases for nuclei with lower neutron numbers. For lighter nuclei, the increase in the difference between the number of neutrons and protons leads to an increase in the symmetry energy. In contrast, for nuclei with larger N values, the symmetry energy decreases as the asymmetry increases. These findings indicate that in the case of heavier nuclei, a larger disparity between the number of neutrons and protons results in a reduction in symmetry energy.

It is crucial to note that the use of current parameterization with the potential part at the liquid-drop-approximation (LDA) level, along with the kinetic energy density of free nucleons and the solution of corresponding Hartree equations can reproduce the general trends of the exact solution but inevitably miss some finer details due to the non-local effects inherent in the full relativistic mean-field (RMF) models, such as those induced by the mesonic fields ($\sigma$, $\omega$, $\rho$) and strong spin-orbit coupling. One potential future research direction involves generalizing the CDFM-RMF approach to incorporate Kohn–Sham (KS) orbitals [38,62,63]. The KS density functional theory [62,63] is a powerful tool for investigating many-body quantum systems by mapping the interacting problem onto a system of non-interacting particles under an effective potential. This process involves expressing the density fluctuations ($\delta \rho \left(\mathbf{r}\right)$) in terms of KS orbitals. This could be achieved by decomposing the energy density functional (EDF) into contributions from the kinetic energy, Hartree, and exchange-correlation terms, each modified by the density fluctuations as described by the CDFM [10,13,14]. Furthermore, the inclusion of relativistic effects in the KS orbitals [64,65], characterized by the Dirac equation, naturally extends the KS scheme to account for strong spin-orbit coupling and other relativistic phenomena. This is particularly relevant in the RMF formalism, where mesonic fields mediate the interactions, and the energy density functional explicitly depends on the ground-state four-current. By doing so, it is possible to better account for the quantal nature of the kinetic energy density, as well as the relativistic effects embedded in the Dirac equation. This leads to a more accurate representation of the non-local effects, improving agreement between the simplified parameterized functional and exact RMF models. Details of the application of the Kohn–Sham framework within the RMF formalism can be found in Refs. [64,65].

4. Summary and Conclusions

The emergence of nuclear physics has enabled predictions of shell closures near the $\beta$-stability region through traditional bulk properties such as binding energy, nucleon separation energy, and shell correction. However, as one moves away from the stability region towards the drip line, isospin asymmetry (neutron–proton asymmetry) starts to play a dominant role in influencing nuclear stability. Addressing isospin asymmetry poses challenges, as quantities associated with nuclear matter are typically defined in momentum space, while finite nuclei at local density are defined in coordinate space. The coherent density fluctuation model (CDFM) was developed to accurately convert symmetry energy and other nuclear matter properties from momentum space to coordinate space. Initially, the CDFM was based on the non-relativistic Brueckner approach, which faces issues like the Coester band problem. Recently, we integrated the relativistic energy density functional into the CDFM formalism, effectively resolving these challenges. This advancement allows for a comprehensive exploration of the entire nuclear landscape, validating existing experimental shell closures and predicting additional closures in the future.

The present study investigated the occurrence of new shell closures in different mass regions for various isotonic chains with magic neutron numbers of N = 20, 40, 82, 126, and 172. The relativistic prescription produces significantly improved predictions for well-established magic numbers, specifically Z = 28 and 50, while employing the widely popular non-linear NL3 and density-dependent DD-ME2 force parameters. We discovered compelling evidence supporting the presence of novel shell/sub-shell closure along various isotonic chains in different mass regions, namely Z = 14, 34, 58, 92, and 118. Moreover, the relativistic prescription of the energy density functional within the CDFM formalism yields significantly larger values than the Brueckner prescription. Furthermore, this study demonstrates that the NL3 force parameter produces slightly higher values compared to the DD-ME2 parameter set due to the stiffness of the equation of state associated with NL3. Moreover, the increase in the difference between the number of neutrons and protons in lighter nuclei increases the symmetry energy, whereas for nuclei with larger N values, the symmetry energy decreases as the asymmetry increases. The investigation results provide a nuanced understanding of isospin-dependent surface properties that vary along different isotonic chains, from light to superheavy mass regions in the nuclear chart. Moreover, identifying novel evidence of shell closures opens up new opportunities for experiments along these isotonic chains, serving a as a benchmark for future explorations and validations of theoretical nuclear models.