Nuclear Matter and Finite Nuclei: Relativistic Thomas–Fermi Approximation Versus Relativistic Mean-Field Approach

Abstract

The Thomas–Fermi approximation is a powerful method that has been widely used to describe atomic structures, finite nuclei, and nonuniform matter in supernovae and neutron-star crusts. Nonuniform nuclear matter at subnuclear density is assumed to be composed of a lattice of heavy nuclei surrounded by dripped nucleons, and the Wigner–Seitz cell is commonly introduced to simplify the calculations. The self-consistent Thomas–Fermi approximation can be employed to study both a nucleus surrounded by nucleon gas in the Wigner–Seitz cell and an isolated nucleus in the nuclide chart. A detailed comparison is made between the self-consistent Thomas–Fermi approximation and the relativistic mean-field approach for the description of finite nuclei, based on the same nuclear interaction. These results are then examined using experimental data from the corresponding nuclei.

1. Introduction

The Thomas–Fermi approximation has a long history in the study of atomic structures, finite nuclei, and nonuniform matter appearing in astrophysical environments. In its early development, the Thomas–Fermi approximation for finite nuclei was typically employed within nonrelativistic theoretical frameworks [1,2,3]. In 1977, Boguta and Rafelski [4] combined the Thomas–Fermi approximation with a relativistic Hartree approach, enabling investigations of finite nuclear densities. In Boguta and Bodmer [5], the authors applied both a relativistic Hartree approach and Thomas–Fermi approximation to study the nuclear surface properties of semi-infinite nuclear matter. Both approaches yielded remarkably similar average nuclear properties. Their studies revealed that, within the Thomas–Fermi approximation for finite nuclei, the average surface properties were consistent with those for semi-infinite nuclear matter. This consistency motivated the combination of Thomas–Fermi approximation and a relativistic Hartree approach for the study of finite nuclei. Walecka et al. [6,7] further integrated the Thomas–Fermi approximation within a covariant density functional theory, discussing the selection of model parameters and performing detailed calculations for ⁴⁰Ca and ²⁰⁸Pb. The Thomas–Fermi approximation has also been used to study hot nuclei. Zhang et al. [8] developed a description of hot nuclei within a self-consistent Thomas–Fermi (STF) approximation using the covariant density functional theory for nuclear interactions. They used a subtraction procedure to isolate the nucleus from nucleon gas inside a Wigner–Seitz cell, and then calculated the temperature dependence of nuclear symmetry energy. The surface energy and nucleon distribution are determined in a self-consistent manner within the STF approximation by solving the relevant coupled equations.

In recent years, the Thomas–Fermi approximation combined with covariant density functional theory has been applied to calculate the equation of state (EOS) for nonuniform nuclear matter existing in supernovae and neutron-star crusts, owing to its computational simplicity and effectiveness [9,10,11]. The nonuniform nuclear matter was assumed to be composed of a lattice of heavy nuclei, surrounded by free nucleons, while the Wigner–Seitz cell is commonly introduced to simplify the calculations. One of the most commonly used EOS in astrophysical simulations, often referred to as Shen EOS [12,13,14], employs the Thomas–Fermi approximation with parameterized nucleon distributions inside the Wigner–Seitz cell. At low temperatures and subnuclear densities, complex inhomogeneous matter, known as nuclear pasta phases, may appear, which is likely to occur in neutron-star crusts. Avancini et al. [15,16] employed the STF approximation to study the properties of pasta phases. Their studies showed that the STF approximation can describe not only an isolated nucleus but also a nucleus surrounded by nucleon gas within the Wigner–Seitz cell.

The relativistic quantum field theory for nuclear many-body systems, proposed by Walecka et al. [6,7], is nowadays considered as a covariant form of density functional theory. Over the past few decades, the covariant density functional theory has undergone significant development and achieved great success in studies on various nuclear phenomena [9,11,17,18]. This theory establishes a complete relativistic interaction system by introducing scalar and vector mesons as mediators of nucleon–nucleon interactions. In this work, we refer to its application to finite nuclei as the relativistic mean-field (RMF) approach. The RMF approach has been successfully applied to study the ground-state properties of finite nuclei, such as binding energies, density distributions, and single-particle spectra [19]. By incorporating proper treatment of pairing correlations, the RMF approach can be systematically extended to investigate nuclei across the nuclide chart, which reflects its great capability for reproducing experimental observations.

As two fundamental approaches for describing finite nuclei, the quantitative differences between the STF approximation and the RMF approach deserve to be examined. Within the STF approximation, the nucleon field operators are expanded on a plane-wave basis, treating the nucleon system as a relativistic Fermi gas [6,10,20]. This approximation requires only self-consistent solutions of meson field equations to obtain the source distributions directly, without the need to explicitly handle the Dirac spinor of nucleons. By employing this semi-classical treatment of nucleons, it avoids the complicated calculations involved in solving the Dirac equation in a central potential, which is particularly suitable for studying heavy nuclei within nonuniform matter existing in astrophysical contexts. In contrast, the RMF approach adopts a more microscopic method by solving the Dirac equation of nucleons together with the meson field equations in a self-consistent manner. The nucleon field can be explicitly decomposed into radial and angular components, with nucleon energy levels obtained by solving the radial Dirac equation [17,21]. This method accurately provides important physical quantities such as the intensity of spin-orbit interactions and single-nucleon spectra [22], making it particularly suitable for detailed studies of finite nuclei. Generally, the parameter set in the RMF approach is determined by fitting to the saturation properties of infinite nuclear matter and/or selected experimental data of finite nuclei. In this work, we first employ the STF approximation to calculate various properties of finite nuclei using several typical RMF parameter sets, and then conduct a detailed comparison between the STF approximation and the RMF approach for describing finite nuclei.

The paper is organized as follows. In Section 2, we briefly introduce the theoretical framework for finite nuclei, including the STF approximation and RMF approach. Furthermore, we provide a new parameter set, TM1m*, and compare it with other parameter sets. In Section 3, we present the numerical results for finite nuclei using several parameter sets. Additionally, we conduct a detailed comparison between the STF approximation and the RMF approach for describing finite nuclei. Section 4 is devoted to the conclusions.

2. Materials and Methods

We first provide a brief description of the covariant density functional theory used for a nuclear many-body system. Subsequently, the details of the STF approximation and the RMF approach for finite nuclei are discussed in two separate subsections.

In the covariant density functional theory, nucleons interact via the exchange of isoscalar scalar and vector mesons ($\sigma$ and $\omega$) and isovector vector meson ($\rho$). When describing a finite nuclear system, the electromagnetic field ($A^{\mu }$) cannot be neglected. We employ the covariant density functional theory, including nonlinear terms for the $\sigma$ and $\omega$ mesons, as well as an $\omega$-$\rho$ coupling term [10,23]. The nonlinear terms for the $\sigma$ meson address the issue of the excessive incompressibility of nuclear matter [5], while the addition of the nonlinear term for the $\omega$ meson reproduces the density-dependent nucleon self-energy obtained from the relativistic Brueckner–Hartree–Fock theory [24]. The $\omega$-$\rho$ coupling term is introduced to modify the density dependence of nuclear symmetry energy [25]. The Lagrangian density of the covariant density functional theory takes the form

$$\begin{array}{ccc}\hfill \mathcal{L}& =& \sum _{i=p,n}{\overline{\psi }}_i\left\{i{\gamma }_{\mu }{\partial }^{\mu }-(M+g_{\sigma }\sigma )-{\gamma }_{\mu }\left[g_{\omega }{\omega }^{\mu }+{\displaystyle \frac{g_{\rho }}{2}}{\tau }_a{\rho }^{a\mu }+{\displaystyle \frac{e}{2}}(1+{\tau }_3)A^{\mu }\right]\right\}{\psi }_i\hfill \\ & & +{\displaystyle \frac{1}{2}}{\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -{\displaystyle \frac{1}{2}}{m}_{\sigma }^2{\sigma }^2-{\displaystyle \frac{1}{3}}{g}_2{\sigma }^3-{\displaystyle \frac{1}{4}}{g}_3{\sigma }^4-{\displaystyle \frac{1}{4}}{W}_{\mu \nu }{W}^{\mu \nu }+{\displaystyle \frac{1}{2}}{m}_{\omega }^2{\omega }_{\mu }{\omega }^{\mu }+{\displaystyle \frac{1}{4}}{c}_3{\left({\omega }_{\mu }{\omega }^{\mu }\right)}^2\hfill \\ & & -{\displaystyle \frac{1}{4}}{R}_{\mu \nu }^a{R}^{a\mu \nu }+{\displaystyle \frac{1}{2}}{m}_{\rho }^2{\rho }_{\mu }^a{\rho }^{a\mu }+{\Lambda }_{\mathrm{v}}\left(g_{\omega }^2{\omega }_{\mu }{\omega }^{\mu }\right)\left(g_{\rho }^2{\rho }_{\mu }^a{\rho }^{a\mu }\right)-{\displaystyle \frac{1}{4}}{F}_{\mu \nu }{F}^{\mu \nu },\hfill \end{array}$$

(1)

where $W^{\mu \nu }$, $R^{a\mu \nu }$, $F^{\mu \nu }$ represent the antisymmetric field tensors for ${\omega }^{\mu }$, ${\rho }^{a\mu }$, $A^{\mu }$, respectively. In the mean-field approximation, the meson fields are treated as classical fields and the meson field operators are replaced by their expectation values, namely, $\sigma =\langle \sigma \rangle$, $\omega =\langle {\omega }^0\rangle$, $\rho =\langle {\rho }^{30}\rangle$. The equations of motion for these mesons, derived from the Lagrangian density, (1) are written as

$$\begin{array}{c}-{\nabla }^2\sigma +m_{\sigma }^2\sigma +g_2{\sigma }^2+g_3{\sigma }^3=-g_{\sigma }\left(n_p^s+n_n^s\right),\hfill \end{array}$$

(2)

$$\begin{array}{c}-{\nabla }^2\omega +m_{\omega }^2\omega +c_3{\omega }^3+2{\Lambda }_{\mathrm{v}}\left(g_{\omega }^2\omega \right)\left(g_{\rho }^2{\rho }^2\right)=g_{\omega }\left(n_p+n_n\right),\hfill \end{array}$$

(3)

$$\begin{array}{c}-{\nabla }^2\rho +m_{\rho }^2\rho +2{\Lambda }_{\mathrm{v}}\left(g_{\omega }^2{\omega }^2\right)\left(g_{\rho }^2\rho \right)={\displaystyle \frac{g_{\rho }}{2}}\left(n_p-n_n\right),\hfill \end{array}$$

(4)

$$\begin{array}{c}-{\nabla }^{2}A=e{n}_p,\hfill \end{array}$$

(5)

where $n_i^s$ and $n_i$ denote, respectively, the scalar density and vector density of species i. In a finite nuclear system, the meson fields are assumed to exhibit spherically symmetric spatial profiles. This assumption allows for the existence of spatial derivatives of these meson fields, and quantities such as densities and mean fields are dependent on the radial position r.

2.1. Self-Consistent Thomas–Fermi Approximation for Finite Nuclei

In the STF approximation, the variations of the meson fields are sufficiently slow, which allows baryons to be treated as moving in locally constant fields [7]. Consequently, the source terms of meson fields can be considered to have local spatial dependence. The local scalar and vector densities are given by

$$\begin{array}{c}{n}_i^s\left(r\right)={\displaystyle \frac{1}{{\pi }^2}}{\int }_0^{k_i^F}{\displaystyle \frac{M^{\ast }}{\sqrt{k^2+M^{\ast 2}}}}{k}^{2}dk,\hfill \end{array}$$

(6)

$$\begin{array}{c}{n}_i\left(r\right)={\displaystyle \frac{1}{{\pi }^2}}{\int }_0^{k_i^F}{k}^{2}dk={\displaystyle \frac{{\left[k_i^F\left(r\right)\right]}^3}{3{\pi }^2}},\hfill \end{array}$$

(7)

where $k_i^F$ is the Fermi momentum of species i and $M^{\ast }=M+g_{\sigma }\sigma$ is the effective nucleon mass. The numbers of protons and neutrons inside a nucleus can be determined from the local vector densities,

$$\begin{array}{ccc}& & N_p={\int }_{\mathrm{nucleus}}{n}_p\left(r\right)d^{3}r,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & N_n={\int }_{\mathrm{nucleus}}{n}_n\left(r\right)d^{3}r.\hfill \end{array}$$

(9)

The baryon number conservation implies that chemical potentials of protons and neutrons are spatially constant throughout the nucleus, which are given by

$$\begin{array}{ccc}& & {\mu }_p=\sqrt{{\left(k_p\right)}^2+M^{\ast 2}}+g_{\omega }\omega +{\displaystyle \frac{g_{\rho }}{2}}\rho +eA,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & {\mu }_n=\sqrt{{\left(k_n\right)}^2+M^{\ast 2}}+g_{\omega }\omega -{\displaystyle \frac{g_{\rho }}{2}}\rho .\hfill \end{array}$$

(11)

In the STF approximation for heavy nuclei, the center-of-mass corrections and the pairing correlations can be disregarded, as previously discussed in nonrelativistic Thomas–Fermi calculations [20]. Consequently, the total energy of a nucleus is given by

$$\begin{array}{cc}\hfill E_{\mathrm{total}}=E_{\mathrm{STF}}=& \int d^{3}r\left[\sum _{i=p,n}{\displaystyle \frac{1}{{\pi }^2}}{\int }_0^{k_i^F}dk{k}^2\sqrt{k^2+M^{\ast 2}}\right.\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{g}_{\sigma }\sigma \left(n_p^s+n_n^s\right)+{\displaystyle \frac{1}{2}}{g}_{\omega }\omega \left(n_p+n_n\right)+{\displaystyle \frac{1}{4}}{g}_{\rho }\rho \left(n_p-n_n\right)+{\displaystyle \frac{1}{2}}eA{n}_p\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{6}}{g}_2{\sigma }^3-{\displaystyle \frac{1}{4}}{g}_3{\sigma }^4+{\displaystyle \frac{1}{4}}{c}_3{\omega }^4+{\Lambda }_{\mathrm{v}}\left(g_{\omega }^2{\omega }^2\right)\left(g_{\rho }^2{\rho }^2\right)\right],\hfill \end{array}$$

(12)

where Equations (2)–(5) are used to simplify the calculations of the derivatives of mean fields.

In practical calculations, we first specify the number of protons and neutrons, $N_p$ and $N_n$, in the nucleus. Then, we provide initial guesses for the meson fields $\sigma \left(r\right)$, $\omega \left(r\right)$, $\rho \left(r\right)$, and the electromagnetic field $A\left(r\right)$. These initial guesses are used to determine the chemical potentials, ${\mu }_p$ and ${\mu }_n$, subject to the constraints $N_p$ and $N_n$. Once the chemical potentials are achieved, we can calculate various densities using Equations (6) and (7), which will be used to solve Equations (2)–(5) to obtain new mean fields. If the resulting fields differ from the initial guesses, the guesses are updated and the processes are repeated until the fields converge to the guesses.

2.2. Relativistic Mean-Field Approach for Finite Nuclei

In the covariant density functional theory, no-sea approximation is imposed. The nucleon field operator can be expanded in terms of single-particle wave functions, which is written as

$$\begin{array}{c}\hfill {\psi }_i={\left[\sum _{\alpha }{A}_{\alpha }{\varphi }_{\alpha }\right]}_i,\end{array}$$

(13)

where $A_{\alpha }$ denotes the annihilation operator for nucleons and ${\varphi }_{\alpha }$ represents the single-particle wave function, with $\alpha$ being a set of quantum numbers. For a spherically symmetric nuclear system, the set of quantum numbers, $\alpha$, includes the conventional quantum numbers for angular momentum and parity. The nucleon wave function ${\varphi }_{\alpha }$ can then be divided into radial and angular parts,

$$\begin{array}{c}\hfill {\varphi }_{\alpha }=\left(\begin{array}{c}\mathrm{i}\left[G_{n\kappa }\left(r\right)/r\right]{\Phi }_{\kappa m}\\ -\left[F_{n\kappa }\left(r\right)/r\right]{\Phi }_{-\kappa m}\end{array}\right),\end{array}$$

(14)

where n is the principal quantum number, $\kappa$ is determined by the angular momentum quantum numbers j and l, and m is the magnetic quantum number. The upper and lower components have opposite values of $\kappa$, and they are normalized by ${\int }_0^{\infty }\left({\left|G_{\alpha }\left(r\right)\right|}^2+{\left|F_{\alpha }\left(r\right)\right|}^2\right)dr=1$. They satisfy the coupled equations

$${\displaystyle \frac{d}{dr}}{G}_{\alpha }\left(r\right)+{\displaystyle \frac{k}{r}}{G}_{\alpha }\left(r\right)=M^{\ast }{F}_{\alpha }\left(r\right)-\left[g_{\omega }\omega +{\displaystyle \frac{g_{\rho }}{2}}{\tau }_3\rho +{\displaystyle \frac{e}{2}}(1+{\tau }_3)A\right]F_{\alpha }\left(r\right)+E_{\alpha }{F}_{\alpha }\left(r\right),$$

(15)

$${\displaystyle \frac{d}{dr}}{F}_{\alpha }\left(r\right)-{\displaystyle \frac{k}{r}}{F}_{\alpha }\left(r\right)=M^{\ast }{G}_{\alpha }\left(r\right)+\left[g_{\omega }\omega +{\displaystyle \frac{g_{\rho }}{2}}{\tau }_3\rho +{\displaystyle \frac{e}{2}}(1+{\tau }_3)A\right]G_{\alpha }\left(r\right)-E_{\alpha }{G}_{\alpha }\left(r\right).$$

(16)

When the nucleon wave function ${\varphi }_{\alpha }$ is obtained, we can calculate various densities in the source terms of meson field equations, such as

$$\begin{array}{ccc}& & n_i^s\left(r\right)={\left[\sum _{\alpha }^{\mathrm{occ}}{w}_{\alpha }{\overline{\varphi }}_{\alpha }{\varphi }_{\alpha }\right]}_i={\left[\sum _{\alpha }^{\mathrm{occ}}{w}_{\alpha }{\displaystyle \frac{2j_{\alpha }+1}{4\pi r^2}}\left({\left|G_{\alpha }\left(r\right)\right|}^2-{\left|F_{\alpha }\left(r\right)\right|}^2\right)\right]}_i,\hfill \end{array}$$

(17)

$$\begin{array}{c}{n}_i\left(r\right)={\left[\sum _{\alpha }^{\mathrm{occ}}{w}_{\alpha }{\overline{\varphi }}_{\alpha }{\gamma }_0{\varphi }_{\alpha }\right]}_i={\left[\sum _{\alpha }^{\mathrm{occ}}{w}_{\alpha }{\displaystyle \frac{2j_{\alpha }+1}{4\pi r^2}}\left({\left|G_{\alpha }\left(r\right)\right|}^2+{\left|F_{\alpha }\left(r\right)\right|}^2\right)\right]}_i.\hfill \end{array}$$

(18)

Here, we incorporate the pairing correlations based on the BCS theory to extend the study beyond closed-shell nuclei [19,21], introducing the occupation probability $w_{\alpha }$ for each nucleon state. When the number of protons or neutrons corresponds to a magic number, the occupation probability equals one for occupied states and zero for unoccupied states. Otherwise, the occupation probability is evaluated by

$$\begin{array}{c}\hfill w_{\alpha }={\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{E_{\alpha }-\lambda }{\sqrt{{\left(E_{\alpha }-\lambda \right)}^2+{\Delta }^2}}}\right),\end{array}$$

(19)

where $E_{\alpha }$ is the single-particle energy for a quantum state. The gap energy $\Delta$ is taken to be $\Delta =11.2\mathrm{MeV}/{\left(N_p+N_n\right)}^{1/2}$, as given in [24]. The Fermi energy $\lambda$ is determined by the condition

$$\begin{array}{c}\hfill {\left[\sum _{\alpha }{w}_{\alpha }\right]}_i=N_i.\end{array}$$

(20)

The energy correction arising from pairing correlations is given by

$$\begin{array}{c}\hfill E_{\mathrm{pair}}=-\Delta \sum _{i=p,n}\sum _{\alpha }{\left[\sqrt{w_{\alpha }\left(1-w_{\alpha }\right)}\right]}_i.\end{array}$$

(21)

In addition, we need to make the center-of-mass corrections on the total energy. For the sake of computational convenience, we employ the center-of-mass correction derived from the nonrelativistic harmonic oscillator potential, which is expressed as

$$\begin{array}{c}\hfill E_{\mathrm{ZPE}}={\displaystyle \frac{3}{4}}\times 41{\left(N_p+N_n\right)}^{-1/3}\left(\mathrm{MeV}\right).\end{array}$$

(22)

To ensure completeness, we give an expression for the total energy of the system,

$$\begin{array}{c}\hfill E_{\mathrm{total}}=E_{\mathrm{RMF}}+E_{\mathrm{pair}}-E_{\mathrm{ZPE}},\end{array}$$

(23)

where $E_{\mathrm{RMF}}$ is calculated in the RMF approach as

$$\begin{array}{cc}\hfill E_{\mathrm{RMF}}=& \sum _{i=p,n}\sum _{\alpha }^{\mathrm{occ}}{\left[w_{\alpha }\left(2j_{\alpha }+1\right)E_{\alpha }\right]}_i\hfill \\ \hfill & +\int d^{3}r\left[-{\displaystyle \frac{1}{2}}{g}_{\sigma }\sigma \left(n_p^s+n_n^s\right)-{\displaystyle \frac{1}{2}}{g}_{\omega }\omega \left(n_p+n_n\right)-{\displaystyle \frac{1}{4}}{g}_{\rho }\rho \left(n_p-n_n\right)-{\displaystyle \frac{1}{2}}eA{n}_p\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{6}}{g}_2{\sigma }^3-{\displaystyle \frac{1}{4}}{g}_3{\sigma }^4+{\displaystyle \frac{1}{4}}{c}_3{\omega }^4+{\Lambda }_{\mathrm{v}}\left(g_{\omega }^2{\omega }^2\right)\left(g_{\rho }^2{\rho }^2\right)\right].\hfill \end{array}$$

(24)

The total energy expression in the RMF approach incorporates baryon field contributions calculated by solving the Dirac equation, whereas the Thomas–Fermi approximation employs momentum integration combined with source terms from Equations (2)–(5). This fundamental methodological difference ultimately manifests as discrepancies in the resulting total energies.

2.3. Parameter Sets

Recent studies by Schneider et al. [26] and Yasin et al. [27] have shown that a larger effective mass provides a more favorable environment for the shock evolution and explosion of core-collapse supernova. However, their studies are limited to nonrelativistic Skyrme models. To investigate the effects of the effective nucleon mass within a relativistic framework, we refitted the TM1e [23] parameter set in the RMF approach, which is referred to as the TM1m model in our previous work [28]. The new TM1m model features a high effective mass ratio $M^{\ast }/M\sim 0.8$ at saturation density, while maintaining the same saturation properties as the TM1e model. Using the TM1m model, we constructed a new equation of state (EOS5) for uniform nuclear matter at densities higher than ∼${10}^{14}\mathrm{g}/{\mathrm{cm}}^3$, which is applicable to complex astrophysical phenomena, such as core-collapse supernovae and binary neutron star mergers.

At subsaturation densities and low temperatures, nucleons may form nonuniform matter including heavy nuclei and light clusters to lower the free energy of the system. To investigate nonuniform matter, both liquid-drop models and Thomas–Fermi approximations are widely employed. In these models, the parameters are relevant to their predictions for the ground-state properties of finite nuclei. However, the results of the TM1m model for finite nuclei are not in good agreement with relevant experimental data. In the present work, we incorporate the RMF calculations for finite nuclei, as described in Section 2.2, into the parameter adjustment process. We adjust the new parameter set to match the energies per nucleon and charge radii of ⁹⁰Zr, ¹¹⁶Sn, ¹²⁴Sn, ¹⁹⁶Pb, ²⁰⁸Pb, ²¹⁴Pb [29,30]. The energy per nucleon is given by

$$\begin{array}{c}\hfill E/A=E_{\mathrm{total}}/A-M,\end{array}$$

(25)

where $A=N_p+N_n$ denotes the mass number of the nucleus. The adopted charge radius is calculated using the formula given in [24]:

$$\begin{array}{c}\hfill R_c^2=\left[R_p^2+{\left(0.862\mathrm{fm}\right)}^2-{\left(0.336\mathrm{fm}\right)}^2{\displaystyle \frac{N_n}{N_p}}\right].\end{array}$$

(26)

We obtain a new parameter set, referred to as TM1m*, which features a large effective mass and improved agreement with the ground-state properties of finite nuclei.

The parameter sets of TM1m*, TM1m, TM1e, and their original TM1 models are presented in

Table 1. Parameter sets of TM1m*, TM1m, TM1e, TM1, and NL3 models.

	TM1m*	TM1m	TM1e	TM1	NL3
$m_{\sigma }$ (MeV)	463.680	511.198	511.198	511.198	508.194
$g_{\sigma }$	7.15454	7.9353	10.0289	10.0289	10.217
$g_{\omega }$	8.59221	8.6317	12.6139	12.6139	12.868
$g_{\rho }$	11.52164	11.5130	13.9714	9.2644	8.948
$g_2$ (${\mathrm{fm}}^{-1}$)	−7.86677	−11.5163	−7.2325	−7.2325	−10.431
$g_3$	40.55692	54.8872	0.6183	0.6183	−28.885
$c_3$	0	0.00025	71.3075	71.3075	0
${\Lambda }_{\mathrm{v}}$	0.09465	0.0933	0.0429	0	0

, while the corresponding saturation properties are listed in

Table 2. Nuclear matter properties of TM1m*, TM1m, TM1e, TM1, NL3, and SFHo models. The saturation density and energy per particle are denoted by $n_0$ and $E/A$, respectively. The incompressibility is represented by K, the symmetry energy by $E_{\mathrm{sym}}$, the slope of the symmetry energy by L, and the effective mass ratio by $M^{\ast }/M$.

	TM1m*	TM1m	TM1e	TM1	NL3	SFHo
$n_0({\mathrm{fm}}^{-3})$	0.145	0.145	0.145	0.145	0.148	0.158
$E/A\left(\mathrm{MeV}\right)$	−16.2	−16.3	−16.3	−16.3	−16.2	−16.2
$K\left(\mathrm{MeV}\right)$	282	281	281	281	272	245
$E_{\mathrm{sym}}\left(\mathrm{MeV}\right)$	31.4	31.4	31.4	36.9	37.4	31.6
$L\left(\mathrm{MeV}\right)$	40	40	40	111	118	47
$M^{\ast }/M$	0.794	0.793	0.634	0.634	0.595	0.761

. The original TM1 model has a rather large slope parameter of the symmetry energy ($L=111$ MeV), which predicts too large radii for neutron stars as compared to the estimations from astrophysical observations [14]. In contrast, a small value of $L=40$ MeV has been achieved in the TM1e, TM1m, and TM1m* models by introducing an additional $\omega$-$\rho$ coupling term. The main difference between TM1e and TM1m lies in the effective mass, with TM1e having $M^{\ast }/M\sim 0.63$ and TM1m having $M^{\ast }/M\sim 0.8$. Both TM1m and TM1m* have relatively large effective masses, whereas their saturation properties are almost identical. The difference between TM1m and TM1m* is attributed to distinct fitting procedures. Specifically, the ground-state properties of finite nuclei are incorporated into the fitting process for the TM1m* parameter set to improve its description of finite nuclei. In contrast, only saturation properties of nuclear matter were considered in the fitting process for the TM1m parameter set. Therefore, the TM1m* model is able to provide better descriptions for finite nuclei than the TM1m model. Notably, the effective mass ratio $M^{\ast }/M\sim 0.8$ in TM1m and TM1m* is an extreme large value within the covariant density functional theory [28]. In addition, we consider two well-known RMF models, NL3 [31] and SFHo [32], in our calculations for comparison. As shown in Table 2, the saturation properties of NL3 are similar to those of the TM1 model, particularly in terms of the symmetry energy and its slope. The NL3 model has been shown to perform well in describing nuclear properties [33,34]. The SFHo model is commonly used in astrophysical simulations [35,36]. It is characterized by multiple coupling terms for the $\rho$ meson, and its effective mass ratio $M^{\ast }/M\sim 0.76$ is comparable to that of the TM1m and TM1m* models. There are alternative RMF models, such as the DDRMF models with density-dependent couplings [37,38], which have proven to be very powerful in describing nuclear matter and finite nuclei. Furthermore, the DDRMF model including tensor couplings can increase the effective nucleon masses to be about $M^{\ast }/M\sim 0.67$ [38]. For simplicity, we do not consider the DDRMF models in the present calculations.

In Figure 1, we display the energy per nucleon of symmetric nuclear matter and neutron matter as a function of baryon number density $n_B$. It is shown that both the TM1m* and TM1m models exhibit consistent behavior of symmetric nuclear matter, as do the TM1e and TM1 models. The curves of TM1m* and TM1m for symmetric nuclear matter are slightly lower than those of TM1e and TM1, which is because the larger effective masses in the TM1m* and TM1m models lead to smaller kinetic energy contributions [28]. In pure neutron matter, the TM1m* and TM1m models still exhibit identical behavior, as they possess the same symmetry energy characteristics. In contrast, the TM1e and TM1 models present different behaviors in pure neutron matter because of their distinct symmetry energy slopes. The TM1 model shows a more rapid increase with increasing density, while similar behavior is also observed in the NL3 model, due to their large slope values. At densities lower than ∼$0.2{\mathrm{fm}}^{-3}$, the curves for symmetric nuclear matter and pure neutron matter in the NL3 model are identical to those in the TM1 model, owing to their nearly identical saturation properties. However, as the density increases, the NL3 curves rise rapidly. This is because the vector potential in the NL3 model grows rapidly [39,40]. The TM1 model incorporates a quartic vector self-interaction, which reduces the vector potential and yields relatively slower growth. The curve of the SFHo model for symmetric nuclear matter lies below those of TM1m* and TM1m, primarily because the SFHo model has a lower incompressibility than that of TM1m* and TM1m. The value of the incompressibility directly affects the theoretical predictions of giant monopole resonance (GMR) characteristics [31,41].

In Figure 2, the effective nucleon mass $M^{\ast }$ is plotted as a function of the baryon number density $n_B$. It is shown that the curves of the TM1e and TM1 models are identical due to their same isoscalar properties. The curves of the TM1m* and TM1m models are significantly higher than those of TM1e and TM1. The SFHo model exhibits a lower effective nucleon mass compared to TM1m* and TM1m, but it is still much higher than that of TM1 and NL3. Notably, the NL3 model exhibits the smallest effective nucleon mass among these parameterizations. For finite nuclei, the effective mass influences the spin-orbit splitting [42], which can be enhanced by introducing a tensor term into the model [38,43].

In Figure 3, we plot the symmetry energy $E_{\mathrm{sym}}$ as a function of baryon number density $n_B$. At high densities, the symmetry energy curves of the TM1 and NL3 models reach very high values, while those of other models exhibit lower values. This difference is primarily due to the distinct density dependence of the symmetry energy in these models. Specifically, the TM1 and NL3 models have significantly larger L values than other models, as shown in Table 2. This results in a more pronounced increase in symmetry energy at higher densities, which can significantly influence the behavior in neutron-rich environments, such as neutron stars. On the other hand, the TM1m* and TM1m models exhibit lower symmetry energies than TM1e at high densities. Although the TM1m*, TM1m, and TM1e models have the same symmetry energy and slope values at saturation density, the smaller effective mass and larger coupling parameters $g_{\sigma }$ and $g_{\omega }$ in the TM1e model favor more pronounced relativistic effects, which leads to a larger symmetry energy at high densities. The SFHo model has a slightly larger symmetry energy slope compared to the TM1m* and TM1m models, and therefore it predicts a larger symmetry energy than the TM1m* and TM1m models at high densities.

3. Results and Discussion

In this section, we systematically examine the ground-state properties for 19 doubly magic and semi-magic nuclei that span a wide range of the nuclide chart, including ⁴⁰Ca, ⁴⁸Ca, ⁵⁶Ni, ⁵⁸Ni, ⁶⁸Ni, ⁷⁸Ni, ⁹⁰Zr, ¹³⁰Cd, ¹⁰⁰Sn, ¹¹⁶Sn, ¹²⁴Sn, ¹³²Sn, ¹³⁴Te, ¹⁴⁴Sm, ¹⁸²Pb, ¹⁸⁴Pb, ¹⁹⁶Pb, ²⁰⁸Pb, and ²¹⁴Pb. We exclude lighter nuclei because both the RMF approach and the STF approximation are inapplicable for few-body systems. In the present work, we employ six RMF parameter sets:, namely TM1m*, TM1m, SFHo, TM1e, TM1, and NL3. Our analysis focuses on three fundamental nuclear properties, specifically, (1) energy per nucleon, (2) charge radius, and (3) nucleon density distribution.

3.1. Energy per Nucleon

In Figure 4, we systematically compare the resulting energy per nucleon obtained in the RMF approach and the STF approximation. The energy per nucleon $E/A$ for the 19 selected nuclei is presented as a function of the mass number A, together with the experimental results, AME2020 [29]. To quantify the discrepancies between theoretical predictions and experimental values, we define the energy deviation $\Delta (E/A)={(E/A)}_{\mathrm{RMF}/\mathrm{STF}}-{(E/A)}_{\mathrm{Expt}.}$, which is plotted on the same x-axis together with the energy per nucleon $E/A$. Comparing Figure 4a,b, it is shown that the deviations in the TM1m* model are significantly smaller than those in the TM1m model. This is because the ground-state properties of finite nuclei are included in the fitting process for the TM1m* parameter set, which helps to improve the description of finite nuclei within the RMF approach. The results of the SFHo model, as shown in Figure 4c, are very similar to those of the TM1m* model, due to the comparable fitting processes of both models. Comparing Figure 4d,e, the results obtained in the TM1e and TM1 models are very similar to each other, owing to their identical isoscalar properties. Figure 4f within the NL3 model also shows a similar pattern. Both the original TM1 and NL3 models employ the nuclear properties, such as the binding energies and charge radii, as inputs in the fitting procedures within the RMF framework [31,42], and therefore their RMF results, shown by the black squares, are in excellent agreement with the empirical values. Also, they have been shown to be successful in predicting various properties of finite nuclei [44,45,46]. However, within the STF approximation, the NL3 model predicts systematically higher values for the energy per nucleon than the experimental values, indicating a tendency to overestimate the nuclear binding energy. In contrast, in the TM1m*, TM1m, and SFHo models, the STF results, shown by the red circles, fall systematically below the experimental values. In the case of the TM1e and TM1 models, the STF results are in good agreement with the RMF results shown by the black squares. Our analysis shows that the six models (TM1m*, TM1m, SFHo, TM1e, TM1, and NL3) exhibit varying degrees of discrepancies between the results from the RMF approach and those from the STF approximation.

By comparing Figure 4d,e, it is evident that the TM1e and TM1 models yield obviously different results for nuclei with relatively high isospin asymmetry, such as ⁴⁸Ca, ⁷⁸Ni, ¹³⁰Cd, ¹³²Sn, and ¹³⁴Sm. It is found that the TM1e model predicts higher $E/A$ values for these nuclei compared to the TM1 model. This is because the density dependence of nuclear symmetry energy in the TM1e model differs from that in the TM1 model, as shown in Figure 3. Compared to the TM1 model, the TM1e model has a smaller symmetry energy slope, implying that the symmetry energy is smaller at supersaturation densities and larger at subsaturation densities. This difference results in a more concentrated neutron distribution and a more diffuse proton distribution at the nuclear surface [47,48]. Consequently, the overall nucleon distribution in nuclei with high isospin asymmetry becomes more concentrated in the TM1e model than in the TM1 model. This implies that nucleons are more tightly bound, thereby reducing the energy of the system. However, this effect is counteracted by increased energy contributions from meson coupling terms, particularly the additional $\omega$-$\rho$ coupling term in the TM1e model. Ultimately, the TM1e model yields a higher total energy than the TM1 model.

In order to analyze why the six models exhibit discrepancies between the results obtained using the RMF approach and those obtained using the STF approximation, we conduct a detailed comparison of the components of the energy. The total energy is decomposed into several components using Equation (12), which includes the relativistic kinetic energy term; the density terms of the meson fields $\sigma$, $\omega$, and $\rho$ and the electromagnetic field A; the nonlinear terms of the $\sigma$ and $\omega$ mesons; and the $\omega$-$\rho$ coupling term. In Table 3, we present these energy terms for ²⁰⁸Pb obtained in both the RMF and STF methods. In the RMF calculations, since the relativistic kinetic energy term cannot be directly obtained from the nucleon wave functions, we define it based on the single-particle energy as follows:

$$\begin{array}{c}\hfill E_k=\sum _{\alpha }^{\mathrm{occ}}\left[w_{\alpha }\left(2j_{\alpha }+1\right)E_{\alpha }\right]-\int d^{3}r\left[g_{\omega }\omega \left(n_p+n_n\right)+{\displaystyle \frac{1}{2}}{g}_{\rho }\rho \left(n_p-n_n\right)+eA{n}_p\right].\end{array}$$

(27)

The density term associated with the a meson is given by

$$\begin{array}{c}\hfill E_a=\int {\displaystyle \frac{1}{2}}{g}_a\langle a\rangle n_a{d}^{3}r,\end{array}$$

(28)

where the corresponding source density $n_a$ can be identified in Equation (12).

In the RMF approach, the various energy terms of TM1m* and TM1m are very close to each other. We note the differences in $E_k$ and $E_{\sigma }$ are related to the scalar field $\sigma$, which may be attributed to different values of $m_{\sigma }$ used in the TM1m* and TM1m parameter sets. On the other hand, the main difference between TM1m* and TM1e lies in the effective nucleon mass. Compared to the TM1e model, the TM1m* model has a larger effective mass $M^{\ast }$ and a weaker attractive potential $g_{\sigma }\sigma$. According to the analysis in the covariant density functional theory, the saturation mechanism relies on a delicate balance between the attractive $\sigma$ field and the repulsive $\omega$ field [42]. Since the negative attractive potential in the TM1m* model is relatively weak, the repulsive potential $g_{\omega }\omega$ is also reduced. These differences result in discrepancies in the relativistic kinetic energy $E_k$, as well as in the density terms $E_{\sigma }$ and $E_{\omega }$. The delicate balance among various energy contributions ensures that total energies in the TM1m and TM1e models are close to the experimentally required value. Similar distinctions between TM1m* and TM1 occur, since the results in TM1e and TM1 are nearly identical. The NL3 and TM1 models exhibit similar saturation properties and both provide good descriptions of finite nuclei. However, the most notable differences between these two models are found in the nonlinear energy terms associated with the $\sigma$ and $\omega$ mesons. This is primarily due to the differences in the coupling constants of nonlinear terms, as shown in Table 1. The NL3 model uses a negative value for $g_3$, which has a smaller impact on the ground-state properties of finite nuclei. However, a negative $g_3$ may make the $\sigma$ meson equation asymptotically unstable and cause instabilities in calculations of the equation of state at very high densities [49,50]. Additionally, the NL3 model lacks a fourth-order term for the $\omega$ meson, causing its vector repulsive potential to increase too rapidly with increasing density. This leads to a stiffer equation of state in the NL3 model, which exceeds the constraints from heavy-ion collision (HIC) experiments [51]. The maximum mass of neutron stars obtained in the NL3 model, by solving the Tolman–Oppenheimer–Volkoff (TOV) equation, is found to be as large as $2.77M_{\odot }$ [52,53]. The results obtained in the SFHo model are comparable to those in the TM1m* model, as both models exhibit similar saturation properties. The main differences lie in the attractive and repulsive potentials. The nonlinear term associated with the $\rho$ meson in the SFHo model has a small impact and can therefore be neglected.

Table 3. Analysis of various energy terms for ²⁰⁸Pb obtained using the RMF approach and the STF approximation. $E_k$ represents the relativistic kinetic energy term of nucleons. The density terms associated with the $\sigma$, $\omega$, $\rho$, and A fields are denoted by $E_{\sigma }$, $E_{\omega }$, $E_{\rho }$, and $E_A$, respectively. The nonlinear terms of the $\sigma$ and $\omega$ mesons are respectively denoted by $E_{\sigma }^{\mathrm{nl}}$ and $E_{\omega }^{\mathrm{nl}}$, while $E_{\omega \rho }^{\mathrm{cpl}}$ represents the $\omega$-$\rho$ coupling term. $E_{\mathrm{total}}$ and $E/A$ are the total energy and energy per nucleon, respectively. The energy terms are all given in MeV. The values in parentheses reflect the differences between the STF and RMF results for each energy term. The nonlinear term associated with the $\rho$ meson in the SFHo model is too small and has therefore been omitted. Additionally, the center-of-mass correction and pairing energy have been incorporated into the total energy.

		TM1m*	TM1m	SFHo	TM1e	TM1	NL3
RMF	$E_k$	166,769.60	166,089.04	162,865.60	142,452.45	143,515.32	139,231.87
	$E_{\sigma }$	15,992.34	16,354.71	18,091.24	27,858.89	27,318.40	29,462.69
	$E_{\omega }$	11,116.80	11,403.13	13,079.89	22,986.98	22,508.88	24,682.22
	$E_{\sigma }^{\mathrm{nl}}$	−1382.97	−1439.95	−1309.33	−1325.01	−1279.22	−628.92
	$E_{\omega }^{\mathrm{nl}}$	0.00	0.0005	−4.80	500.71	475.07	0.00
	$E_{\rho }$	123.10	124.82	108.20	116.78	109.43	103.85
	$E_A$	820.30	822.60	840.33	819.93	823.42	827.17
	$E_{\omega \rho }^{\mathrm{cpl}}$	39.32	40.88	29.53	60.89	0.00	0.00
	$E_{\mathrm{total}}$	193,473.30	193,390.03	193,695.47	193,466.42	193,466.11	193,673.69
	$E/A$	−7.84	−8.24	−7.83	−7.87	−7.87	−7.88
STF	$E_k$	165,995.23 (−774.37)	165,157.89 (−931.15)	162,479.36 (−386.24)	143,555.73 (1103.28)	144,583.83 (1068.51)	141,928.88 (2697.02)
	$E_{\sigma }$	16,371.93 (379.59)	16,818.50 (463.79)	18,258.43 (167.19)	27,250.99 (−607.89)	26,730.02 (−588.38)	28,076.58 (−1386.11)
	$E_{\omega }$	11,405.32 (288.52)	11,752.04 (348.91)	13,208.84 (128.95)	22,474.92 (−512.06)	22,014.45 (−494.43)	23,445.06 (−1237.16)
	$E_{\sigma }^{\mathrm{nl}}$	−1386.61 (−3.64)	−1449.11 (−9.16)	−1300.66 (8.67)	−1237.30 (87.71)	−1194.37 (84.85)	−663.57 (−34.65)
	$E_{\omega }^{\mathrm{nl}}$	0.00 (0.00)	0.00 (0.00)	−4.49 (0.31)	437.48 (−63.23)	414.95 (−60.12)	0.00 (0.00)
	$E_{\rho }$	135.23 (12.13)	139.01 (14.19)	117.64 (9.44)	124.98 (8.20)	113.85 (4.42)	105.69 (1.84)
	$E_A$	816.16 (−4.15)	816.24 (−6.37)	833.75 (−6.58)	807.26 (−12.67)	810.17 (−13.25)	808.37 (−18.80)
	$E_{\omega \rho }^{\mathrm{cpl}}$	43.59 (4.28)	46.21 (5.33)	31.28 (1.75)	64.05 (3.16)	0.00 (0.00)	0.00 (0.00)
	$E_{\mathrm{total}}$	193,380.85 (−97.64)	193,280.77 (−114.45)	193,624.16 (−76.50)	193,478.11 (6.50)	193,472.89 (1.59)	193,701.01 (22.14)
	$E/A$	−8.28	−8.77	−8.17	−7.82	−7.84	−7.75

Table 3. Analysis of various energy terms for ²⁰⁸Pb obtained using the RMF approach and the STF approximation. $E_k$ represents the relativistic kinetic energy term of nucleons. The density terms associated with the $\sigma$, $\omega$, $\rho$, and A fields are denoted by $E_{\sigma }$, $E_{\omega }$, $E_{\rho }$, and $E_A$, respectively. The nonlinear terms of the $\sigma$ and $\omega$ mesons are respectively denoted by $E_{\sigma }^{\mathrm{nl}}$ and $E_{\omega }^{\mathrm{nl}}$, while $E_{\omega \rho }^{\mathrm{cpl}}$ represents the $\omega$-$\rho$ coupling term. $E_{\mathrm{total}}$ and $E/A$ are the total energy and energy per nucleon, respectively. The energy terms are all given in MeV. The values in parentheses reflect the differences between the STF and RMF results for each energy term. The nonlinear term associated with the $\rho$ meson in the SFHo model is too small and has therefore been omitted. Additionally, the center-of-mass correction and pairing energy have been incorporated into the total energy.

		TM1m*	TM1m	SFHo	TM1e	TM1	NL3
RMF	$E_k$	166,769.60	166,089.04	162,865.60	142,452.45	143,515.32	139,231.87
	$E_{\sigma }$	15,992.34	16,354.71	18,091.24	27,858.89	27,318.40	29,462.69
	$E_{\omega }$	11,116.80	11,403.13	13,079.89	22,986.98	22,508.88	24,682.22
	$E_{\sigma }^{\mathrm{nl}}$	−1382.97	−1439.95	−1309.33	−1325.01	−1279.22	−628.92
	$E_{\omega }^{\mathrm{nl}}$	0.00	0.0005	−4.80	500.71	475.07	0.00
	$E_{\rho }$	123.10	124.82	108.20	116.78	109.43	103.85
	$E_A$	820.30	822.60	840.33	819.93	823.42	827.17
	$E_{\omega \rho }^{\mathrm{cpl}}$	39.32	40.88	29.53	60.89	0.00	0.00
	$E_{\mathrm{total}}$	193,473.30	193,390.03	193,695.47	193,466.42	193,466.11	193,673.69
	$E/A$	−7.84	−8.24	−7.83	−7.87	−7.87	−7.88
STF	$E_k$	165,995.23 (−774.37)	165,157.89 (−931.15)	162,479.36 (−386.24)	143,555.73 (1103.28)	144,583.83 (1068.51)	141,928.88 (2697.02)
	$E_{\sigma }$	16,371.93 (379.59)	16,818.50 (463.79)	18,258.43 (167.19)	27,250.99 (−607.89)	26,730.02 (−588.38)	28,076.58 (−1386.11)
	$E_{\omega }$	11,405.32 (288.52)	11,752.04 (348.91)	13,208.84 (128.95)	22,474.92 (−512.06)	22,014.45 (−494.43)	23,445.06 (−1237.16)
	$E_{\sigma }^{\mathrm{nl}}$	−1386.61 (−3.64)	−1449.11 (−9.16)	−1300.66 (8.67)	−1237.30 (87.71)	−1194.37 (84.85)	−663.57 (−34.65)
	$E_{\omega }^{\mathrm{nl}}$	0.00 (0.00)	0.00 (0.00)	−4.49 (0.31)	437.48 (−63.23)	414.95 (−60.12)	0.00 (0.00)
	$E_{\rho }$	135.23 (12.13)	139.01 (14.19)	117.64 (9.44)	124.98 (8.20)	113.85 (4.42)	105.69 (1.84)
	$E_A$	816.16 (−4.15)	816.24 (−6.37)	833.75 (−6.58)	807.26 (−12.67)	810.17 (−13.25)	808.37 (−18.80)
	$E_{\omega \rho }^{\mathrm{cpl}}$	43.59 (4.28)	46.21 (5.33)	31.28 (1.75)	64.05 (3.16)	0.00 (0.00)	0.00 (0.00)
	$E_{\mathrm{total}}$	193,380.85 (−97.64)	193,280.77 (−114.45)	193,624.16 (−76.50)	193,478.11 (6.50)	193,472.89 (1.59)	193,701.01 (22.14)
	$E/A$	−8.28	−8.77	−8.17	−7.82	−7.84	−7.75

In the STF approximation, the energy terms in the six models exhibit a similar pattern as observed in the RMF approach. Our primary focus lies on the values in parentheses, which represent the differences between the STF and RMF results for each energy term. These differences reveal what causes the same model to yield divergent outcomes under the two frameworks. The values in parentheses show significant discrepancies in the relativistic kinetic energy $E_k$, as well as in the density terms $E_{\sigma }$ and $E_{\omega }$ associated with $\sigma$ and $\omega$ mesons, all of which are related to the Dirac equation. This may be due to different methods used to handle the Dirac equation and extract the kinetic energy within the two approaches. In the RMF approach, the single-particle energy of nucleons is self-consistently obtained by solving the Dirac equation. In contrast, under the STF approximation, nucleons are treated as moving in locally constant fields, and the kinetic energy is calculated in a manner analogous to that in homogeneous nuclear matter. It is likely these models can be divided into two groups. In the TM1m*, TM1m, and SFHo models, the values in parentheses exhibit similar features with the same sign, particularly for $E_k$, $E_{\sigma }$, and $E_{\omega }$. In contrast, in the TM1e, TM1, and NL3 models, the sign of these values exhibits an opposite trend.

3.2. Charge Radius

To compare the charge radii between the RMF and STF approaches, we plot in Figure 5 the charge radii for the 19 selected nuclei as a function of the mass number A. The experimental data are taken from ref. [30], although some values are absent.

Comparing Figure 5a,b, we find that, in the TM1m* and TM1m models, the charge radii obtained using the RMF and STF approaches are nearly identical and very close to the experimental values. In the case of the SFHo model, as shown in Figure 5c, the theoretical predictions from both the RMF and STF approaches are slightly lower than the experimental values. On the other hand, in the TM1e, TM1, and NL3 models shown in Figure 5d–f, the charge radii obtained using the STF approximation are obviously larger than those in the RMF approach. This difference may be attributed to the different density distributions of the two approaches. The results in the TM1e model are slightly larger than those in the TM1 model. This might be due to the fact that TM1e has a smaller slope parameter $L=40\mathrm{MeV}$, which results in a larger proton radius.

We have calculated the charge radii using Equation (26) proposed by Sugahara and Toki [24]. Alternatively, Typel and Wolter have suggested another formula, Equation (45) in ref. [37], for calculating charge radii. We have compared the results obtained with these two formulas. The comparison shows that their mutual discrepancies are much smaller than those between theoretical predictions and experimental data. The qualitative conclusion is therefore unaffected.

Recently, more attention has been paid to the relationship between the symmetry energy slope L and the neutron-skin thickness $\Delta R_{np}$ [10,14]. Generally, a larger value of L corresponds to a higher pressure of neutron matter and a greater neutron-skin thickness. Accurate measurements of neutron-skin thickness deduced from parity-violating electron scattering suggests a small neutron-skin thickness of ⁴⁸Ca by CREX [54] and a large neutron-skin thickness of ²⁰⁸Pb by PREX-2 [55]. In

Table 4. Neutron-skin thickness (in fm) obtained using the RMF approach and the STF approximation. The experimental data for ⁴⁸Ca and ²⁰⁸Pb are taken from ref. [54] and ref. [55], respectively.

		48Ca	208Pb
RMF	TM1m*	0.1934	0.1782
	TM1m	0.1807	0.1659
	SFHo	0.1992	0.1931
	TM1e	0.1712	0.1579
	TM1	0.2274	0.2709
	NL3	0.2271	0.2808
STF	TM1m*	0.117	0.104
	TM1m	0.098	0.082
	SFHo	0.135	0.129
	TM1e	0.121	0.103
	TM1	0.184	0.211
	NL3	0.189	0.222
Experimental data		$0.121\pm 0.026$ (exp) ± 0.024 (model)	$0.283\pm 0.071$
		($0.071\sim 0.171$)	($0.212\sim 0.354$)

, we present the resulting neutron-skin thickness, $\Delta R_{np}=R_n-R_p$, for ⁴⁸Ca and ²⁰⁸Pb obtained using the RMF approach and the STF approximation, and compared with experimental data from CREX [54] and PREX-2 [55]. It is noteworthy that the TM1 and NL3 models, characterized by large values of symmetry energy slope L, predict systematically greater neutron-skin thicknesses than the other models with small L. Furthermore, the results obtained using the STF approximation are smaller than those using the RMF approach. This difference is likely attributed to the different density distributions of the two approaches. Compared to the experimental data, we find that none of the models can simultaneously predict the neutron-skin thicknesses for ⁴⁸Ca and ²⁰⁸Pb within the experimental error range. Currently, there are still large uncertainties in the neutron-skin thicknesses obtained from experimental measurements.

3.3. Nucleon Density Distribution

The density distributions of protons and neutrons provide crucial insights into the internal structure of finite nuclei. We now turn our attention to the discussion of density distributions for two typical nuclei, ⁴⁰Ca and ²⁰⁸Pb.

In Figure 6, we display the density distributions of protons and neutrons for ⁴⁰Ca. The curves obtained using the RMF approach exhibit pronounced fluctuations and spatially diffuse distributions, with minimal differences observed in the tail regions across different parameter sets. The density distributions of the TM1e and TM1 models show nearly identical, while the TM1m* curves differ with them only in the middle region ($2\mathrm{fm}<r<3\mathrm{fm}$). Similarly, the curves of the TM1m and NL3 models exhibit slight deviations from that of the TM1e model. However, the SFHo model yields enhanced central density compared to other models, which may be related to its larger saturation density, as shown in Table 2. This characteristic also contributes to the systematically smaller charge radii predicted by the SFHo model.

According to Equation (18), the complex structure of the density distribution obtained from the RMF approach originates from the superposition of squared wave function for all occupied states. Different parameter sets yield distinct wave functions ${\varphi }_{\alpha }$ for the same single-particle state. In open-shell nuclei, the occupation probabilities $w_{\alpha }$ of occupied states also vary across parameter sets. This inconsistency in occupation probabilities further enhances the parameter dependence of the density distributions. Furthermore, the density distributions calculated from the superposition of squared wave function for all occupied states retain the oscillatory features inherent in the individual radial wave functions, giving rise to pronounced oscillations in the RMF density distributions. These oscillations, however, substantially exceed the fluctuation amplitudes observed in the charge density distributions deduced from elastic electron scattering experiments [56].

Within the STF approximation, the density distributions exhibit relatively smooth variations and have three distinctive characteristics compared to the RMF results: (1) lower densities in the central region; (2) slower attenuation with increasing radius, eventually exceeding the RMF densities and decreasing rapidly; and (3) reduced spatial diffusion in the tail region, where greater discrepancies emerge between different parameter sets. The STF approximation produces relatively smooth distributions because it treats nucleons as a locally uniform Fermi gas subject to slowly varying mean fields. The fundamental difference between the STF approximation and the RMF approach is in how nucleons are handled, which leads to the pronounced disparities between the sets of density distributions. In terms of predictive capability, although the oscillations in the RMF densities deviate from the experimental charge density profiles, the RMF approach successfully reproduces various properties and deformation characteristics of finite nuclei. In contrast, the STF approximation yields reliable ground-state bulk properties but cannot predict microscopic quantities such as single-particle spectra.

In Figure 7, the density distributions of protons and neutrons for ²⁰⁸Pb are presented. It is shown that, for ²⁰⁸Pb, the curves obtained using the RMF and STF approaches exhibit similar features to those of ⁴⁰Ca. However, more pronounced differences among these models are observed in the curves of ²⁰⁸Pb. The curves of the TM1e and TM1 models no longer overlap, particularly in the neutron distribution. The curves of the neutron distribution show much higher sensitivity to the symmetry energy slope L than that of the proton distribution. This fact supports the theoretical prediction that the value of L predominantly affects the neutron radius. For ²⁰⁸Pb, the density distributions exhibit more pronounced oscillations compared to ⁴⁰Ca. Additionally, in Figure 7a, the proton density distribution within the STF approximation exhibits an inward indentation in the central region of the nucleus, which is mainly attributed to the Coulomb repulsion [4]. This effect occurs in heavy nuclei because their larger Coulomb repulsion causes a redistribution of protons, leading to a lower density in the central region compared to the surrounding areas.

4. Conclusions

In this work, we have made a detailed comparison between the STF approximation and the RMF approach for describing finite nuclei. We have employed the covariant density functional theory with nonlinear terms for $\sigma$ and $\omega$ mesons, as well as an $\omega$-$\rho$ coupling term. Besides considering several typical RMF parameter sets, we have provided a new TM1m* parameterization. The TM1m* features a high effective nucleon mass ratio, $M^{\ast }/M\sim 0.8$, identical to that of the TM1m model, but with better descriptions for finite nuclei. Recent studies have indicated that a high effective mass is helpful for the shock evolution and explosion processes in core-collapse supernovae. The other saturation properties of TM1m* are very similar to those of the TM1m and TM1e models. On the other hand, the TM1e and TM1 models have lower effective nucleon mass ratio, $M^{\ast }/M\sim 0.63$. The difference between TM1e and TM1 is reflected in the symmetry energy and its slope. It is well known that the slope parameter of the symmetry energy can significantly affect both the neutron-star radius and the neutron-skin thickness of finite nuclei. In addition, two popular models, NL3 and SFHo, have also been employed in our calculations for comparison. The NL3 model has been shown to perform well in describing nuclear properties. The SFHo model has been widely used in astrophysical simulations and has a large effective mass ratio, $M^{\ast }/M\sim 0.76$, similar to the TM1m* models.

To systematically analyze the differences between the STF approximation and the RMF approach for the descriptions of finite nuclei, we have calculated the ground-state properties of 19 doubly magic and semi-magic nuclei that span a wide range of the nuclide chart. The experimental data of the corresponding nuclei have been used to examine these results. Within the RMF approach, the models used could provide reasonable good results for finite nuclei, whereas the TM1m model exhibits relatively large discrepancies between theoretical predictions and experimental values. This is because the ground-state properties of finite nuclei have not been included in the fitting process for the TM1m parameter set, whereas the TM1m* parameterization considered and improved its description of finite nuclei. On the other hand, within the STF approximation, models with larger effective mass, such as the TM1m*, TM1m, and SFHo models, generally predict results lower than the experimental values. In contrast, the TM1e and TM1 models show better agreement with the experimental data, while the NL3 model exhibits slightly higher predictions. Notably, when using the same parameter set, the variations in predictions between the two methods primarily arise from different treatments for the relativistic kinetic term and the density terms related to the Dirac equation.

For the charge radius of finite nuclei, most parameter sets can provide reasonably good predictions in the RMF approach. It has been found that the TM1e model predicts slightly larger charge radii than the TM1 model, which might be attributed to the smaller slope parameter of the TM1e model. In the STF approximation, the TM1m* and TM1m parameter sets can reproduce the experimental charge radii, whereas the results from the SFHo model remain lower than the experimental values. Furthermore, in the TM1e, TM1, and NL3 models, the charge radii obtained using the STF approximation are obviously larger than those in the RMF approach.

We have also discussed the nucleon density distributions for two typical nuclei, ⁴⁰Ca and ²⁰⁸Pb. The density distributions obtained using the RMF approach exhibit pronounced fluctuations and spatially diffuse characteristics, with minor differences observed in the tail regions across different parameter sets. In contrast, the density distributions obtained through the STF approximation exhibit relatively smooth variations, showing lower values in the central regions and weaker tail diffuse. These features are particularly pronounced in the results of ²⁰⁸Pb. Furthermore, the discrepancy between the TM1e and TM1 curves demonstrates that neutron distributions are more sensitive to the symmetry energy slope L than the proton distributions.