Effects of Isovector Spin–Orbit Interaction on the Charge-Weak Form Factor Difference in ⁴⁸Ca, ²⁰⁸Pb, ⁹⁰Zr and ⁶²Ni

Abstract

The nucleon spin–orbit interaction is a cornerstone of nuclear structure theory, yet its isospin dependence remains insufficiently constrained within modern nuclear energy density functional (EDF) theory. It was recently shown that, within the framework of extended Skyrme EDFs, the charge-weak form factor difference $\Delta F_{\mathrm{CW}}$ in ⁴⁸Ca exhibits remarkable sensitivity to the effective isovector spin–orbit (IVSO) interaction, whereas $\Delta F_{\mathrm{CW}}$ in ²⁰⁸Pb is much less sensitive to this channel. Extending this analysis to other nuclei, we find that ⁹⁰Zr, with its ten spin–orbit unpaired $1{\mathrm{g}}_{9/2}$ neutrons, displays a $\Delta F_{\mathrm{CW}}$ sensitivity to the IVSO strength similar to that of ⁴⁸Ca, arising from modifications to the central mean-field potential rather than the one-body spin–orbit potential. In contrast, ⁶²Ni, like ²⁰⁸Pb, remains largely insensitive to the IVSO interaction. This structure-driven distinction suggests an experimental strategy: future parity-violating electron scattering measurements, e.g., the MREX experiment at the MESA facility, on ⁴⁸Ca and ⁹⁰Zr would help constrain the effective IVSO strength, while measurements on ²⁰⁸Pb and ⁶²Ni can provide a cleaner probe of the density dependence of the symmetry energy with reduced IVSO sensitivity.

1. Introduction

The spin–orbit (SO) interaction plays a crucial role across many branches of physics—from elementary quark–gluon dynamics [1,2] to macroscopic phenomena in condensed matter and optics [3,4]—and has long been recognized as a fundamental ingredient of nuclear structure [5,6,7,8]. Historically, it was first introduced in atomic physics in the 1920s to explain the fine structure of atomic spectra [5,9]. In 1949, Mayer [10] and Jensen [11] incorporated a strong nucleon SO interaction into the nuclear shell model, successfully explaining the emergence of magic numbers in nuclei. This achievement marked a milestone in nuclear theory, establishing SO interaction as a key mechanism for understanding nuclear structure, stability, and shell evolution.

Since nuclei are composed of protons and neutrons, a natural question then arises: is the SO interaction identical for protons and neutrons in the nuclei? One of its most direct manifestations is the spin–orbit splitting, namely, the splitting of a single-particle level with a given orbital angular momentum l ($l\ne 0$) into two sublevels with different total angular momenta $j=l\pm 1/2$. For nuclei close to the $\beta$-stability line, the isospin asymmetry (measuring difference between neutron and proton numbers) is small, leading to similar magnitudes of spin–orbit splitting for protons and neutrons. In contrast, the nuclei far from stability, particularly those with a large neutron excess, exhibit marked differences between proton and neutron spin–orbit splittings [12]. This observation suggests the isospin-dependent nature of the SO interaction, i.e., an isovector spin–orbit (IVSO) interaction. However, since the single-nucleon energy levels cannot be directly measured and the impact of SO interaction has a complex interplay with, e.g., tensor force [13] and nucleon effective masses [14], the quantitative strength of the IVSO channel remains model dependent and insufficiently constrained at the energy-density-functional (EDF) level [15].

In modern nuclear EDF theory, the origin and predictive content of spin–orbit couplings depend on how the functional is constructed. In covariant mean-field models, a strong SO potential emerges naturally from the Dirac structure of the scalar and vector mean fields, while the isospin dependence is weak in conventional $\sigma$-$\omega$-$\rho$ parametrizations [16,17]. Additional isovector degrees of freedom, such as the isovector-scalar $\delta$ meson and exchange terms in covariant EDFs, can modify the effective isospin dependence of the SO channel [18]. In nonrelativistic Skyrme EDFs, the SO term is usually introduced phenomenologically [15], and the conventional zero-range two-body form gives an IVSO strength only one third of the isoscalar one [19]. A different baseline is provided by EDFs constrained by pion-nucleon dynamics, where the analytical form and density-dependent couplings are derived from the many-body perturbation theory and few-nucleon interactions [20,21]. These calculations generate nontrivial density-dependent SO couplings and predict an IVSO strength about twice as large as that in conventional Skyrme EDFs [20]. Subsequent studies of chiral forces found a sizable enhancement of the isoscalar SO interaction [21] due to the three-body contributions but did not provide a corresponding analysis of the IVSO channel. This model dependence highlights the need for effective experimental probes of the IVSO channel.

Recent studies have identified the charge-weak form factor difference $\Delta F_{\mathrm{CW}}$ in ⁴⁸Ca as a sensitive probe of the IVSO interaction [22,23,24]. In particular, the model-independent determination of $\Delta F_{\mathrm{CW}}$ in ⁴⁸Ca by the Calcium Radius Experiment (CREX) [25] at Jefferson Lab (JLab), together with the PREX-II result in ²⁰⁸Pb [26], has highlighted the difficulty of accommodating both measurements simultaneously within conventional EDFs [27,28,29,30,31,32,33,34,35,36,37] as well as in ab initio calculations [25,38]. Within the extended Skyrme EDF framework, a stronger IVSO channel has been found to improve the simultaneous accommodation of the CREX and PREX-II $\Delta F_{\mathrm{CW}}$ data, while also highlighting the distinct sensitivities of different nuclei to the IVSO interaction [22]. Very recently, a study based on covariant density-dependent point-coupling EDFs with an isovector tensor coupling reported a similar phenomenological trend [39].

In this work, through an analysis of the single-particle levels in ⁴⁸Ca and ²⁰⁸Pb, we demonstrate that the conventional shell closures, including the magic numbers, are well reproduced and retain their robustness even in the presence of a strong IVSO interaction. Extending this analysis, we identify a structural correspondence between ⁹⁰Zr and ⁴⁸Ca, as well as between ⁶²Ni and ²⁰⁸Pb, based on their analogous spin–orbit saturation properties. This analogy implies that the charge-weak form factor difference $\Delta F_{\mathrm{CW}}=F_C-F_W$, and thereby the neutron skin, exhibits significant sensitivity to the IVSO strength in ⁹⁰Zr while remaining largely insensitive in ⁶²Ni. The broader implications of this nucleus-dependent sensitivity are discussed in what follows.

2. Methods

2.1. Extended Skyrme EDF Within Hartree-Fock-Bogoliubov (HFB) Framework

In the present work, we use a nonrelativistic nuclear EDF based on an extended Skyrme interaction [40] supplemented by a zero-range tensor force [41]. The extended Skyrme interaction $v_{eSky}$ differs from the conventional Skyrme interaction $v_{Sky}$ by including momentum-dependent many-body terms $v^{\prime }$, namely,

$$\begin{array}{c}\hfill v_{eSky}=v_{Sky}+v^{\prime }.\end{array}$$

(1)

The full interaction used in this work consists of the extended Skyrme part $v_{eSky}$ together with the zero-range tensor force $V_T$ introduced below.

The conventional Skyrme interaction $v_{Sky}$ reads (see, e.g., Ref. [42]):

$$\begin{array}{c}\hfill \begin{array}{cc}& v_{Sky}=t_0\left(1+x_0{P}_{\sigma }\right)\delta \left(\mathit{r}\right)+{\displaystyle \frac{1}{2}}{t}_1\left(1+x_1{P}_{\sigma }\right)\left[{\mathit{k}}^{\prime 2}\delta \left(\mathit{r}\right)+\delta \left(\mathit{r}\right){\mathit{k}}^2\right]\hfill \\ & +t_2\left(1+x_2{P}_{\sigma }\right){\mathit{k}}^{\prime }·\delta \left(\mathit{r}\right)\mathit{k}+{\displaystyle \frac{1}{6}}{t}_3\left(1+x_3{P}_{\sigma }\right){\left[\rho \left(\mathit{R}\right)\right]}^{\alpha }\delta \left(\mathit{r}\right)\hfill \\ & +\mathrm{i}{W}_0\left({\widehat{\mathit{\sigma }}}_1+{\widehat{\mathit{\sigma }}}_2\right)·\left[{\mathit{k}}^{\prime }\times \delta \left(\mathit{r}\right)\mathit{k}\right],\hfill \end{array}\end{array}$$

(2)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v^{\prime }& ={\displaystyle \frac{1}{2}}{t}_4\left(1+x_4{P}_{\sigma }\right)\left[{\mathit{k}}^{\prime 2}\rho {\left(\mathit{R}\right)}^{\beta }\delta \left(\mathit{r}\right)+\delta \left(\mathit{r}\right)\rho {\left(\mathit{R}\right)}^{\beta }{\mathit{k}}^2\right]\hfill \\ & +t_5\left(1+x_5{P}_{\sigma }\right){\mathit{k}}^{\prime }·\rho {\left(\mathit{R}\right)}^{\gamma }\delta \left(\mathit{r}\right)\mathit{k}.\hfill \end{array}\end{array}$$

(3)

Here, $\mathit{r}={\mathit{r}}_1-{\mathit{r}}_2$, $\mathit{R}={\displaystyle \frac{1}{2}}\left({\mathit{r}}_1+{\mathit{r}}_2\right)$, $\mathit{k}={\displaystyle \frac{1}{2\mathrm{i}}}\left({\nabla }_1-{\nabla }_2\right)$ is the relative momentum, with ${\mathit{k}}^{\prime }$ being its conjugate acting on the left; the $\widehat{\mathit{\sigma }}$ are Pauli spin operators, $P_{\sigma }={\displaystyle \frac{1}{2}}\left(1+{\widehat{\mathit{\sigma }}}_1·{\widehat{\mathit{\sigma }}}_2\right)$ is the spin exchange operator, and $\rho \left(\mathit{R}\right)={\rho }_n\left(\mathit{R}\right)+{\rho }_p\left(\mathit{R}\right)$ is the total local density, with ${\rho }_n$ and ${\rho }_p$ being neutron and proton densities, respectively. $t_0\sim t_5$, $x_0\sim x_5$, $\alpha$, $\beta$ and $\gamma$ are adjustable “Skyrme parameters”, and $W_0$ is the SO interaction parameter. For simplicity, both $\beta$ and $\gamma$ are set to unity as in Ref. [43].

For the tensor force, we use the following zero-range form [41]

$$\begin{array}{cc}\hfill V_T& ={\displaystyle \frac{1}{2}}T\{\left[\left({\widehat{\mathit{\sigma }}}_1·{\mathit{k}}^{\prime }\right)\left({\widehat{\mathit{\sigma }}}_2·{\mathit{k}}^{\prime }\right)-{\displaystyle \frac{1}{3}}{\mathit{k}}^{\prime 2}\left({\widehat{\mathit{\sigma }}}_1·{\widehat{\mathit{\sigma }}}_2\right)\right]\delta \left(\mathit{r}\right)\hfill \\ \hfill & +\delta \left(\mathit{r}\right)\left[\left({\widehat{\mathit{\sigma }}}_1·\mathit{k}\right)\left({\widehat{\mathit{\sigma }}}_2·\mathit{k}\right)-{\displaystyle \frac{1}{3}}{\mathit{k}}^2\left({\widehat{\mathit{\sigma }}}_1·{\widehat{\mathit{\sigma }}}_2\right)\right]\}\hfill \\ \hfill & +U\left\{\left({\widehat{\mathit{\sigma }}}_1·{\mathit{k}}^{\prime }\right)\delta \left(\mathit{r}\right)\left({\widehat{\mathit{\sigma }}}_2·\mathit{k}\right)-{\displaystyle \frac{1}{3}}\left({\widehat{\mathit{\sigma }}}_1·{\widehat{\mathit{\sigma }}}_2\right)\left[{\mathit{k}}^{\prime }·\delta \left(\mathit{r}\right)\mathit{k}\right]\right\},\hfill \end{array}$$

(4)

where T and U quantify the magnitude of the tensor force in states of even and odd relative motion, respectively. In practical calculations, the parameters ${\alpha }_T$ and ${\beta }_T$ are employed in place of U and T, where ${\alpha }_T$ and ${\beta }_T$ are expressed as:

$${\alpha }_T={\displaystyle \frac{5}{12}}U,{\beta }_T={\displaystyle \frac{5}{24}}(T+U).$$

(5)

The explicit zero-range tensor force contributes to the EDF through the ${\alpha }_T$- and ${\beta }_T$-dependent ${\mathit{J}}^2$ and ${\tilde{\mathit{J}}}^2$ terms in Equation (10).

In addition, for calculations of open-shell nuclei, we take into account the standard mixed volume-surface density-dependent contact pairing force of the form [44]

$$V_p(\mathit{r},{\mathit{r}}^{\prime })=V_0\left[1-{\displaystyle \frac{\rho \left(\mathit{r}\right)}{2{\rho }_0}}\right]\delta (\mathit{r}-{\mathit{r}}^{\prime }),$$

(6)

where $V_0$ is the pairing strength and ${\rho }_0=0.16{\mathrm{fm}}^{-3}$.

Based on the above extended Skyrme interaction, tensor force and pairing force, we derive the EDF within the HFB framework. Under the assumption of time-reversal invariance, the energy of a nucleus can be written as the integral of a purely local EDF $\mathcal{E}$. For details on the HFB calculation based on Skyrme EDF, see e.g., Ref. [45]. In the following, we focus on the potential-energy density ${\mathcal{E}}_{eSky}$ associated with the extended Skyrme interaction and tensor force, which can be expressed in terms of local density ${\rho }_t$, kinetic-energy density ${\tau }_t$, and vector spin-current density ${\mathit{J}}_t$ of neutrons ($t=n$) and protons ($t=p$) defined by

$$\begin{array}{ccc}\hfill {\rho }_t\left(\mathit{r}\right)& =& \sum _i{v}_i^2{\left|{\phi }_i\left(\mathit{r}\right)\right|}^2,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\tau }_t\left(\mathit{r}\right)& =& \sum _i{v}_i^2{\left|\nabla {\phi }_i\left(\mathit{r}\right)\right|}^2,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\mathit{J}}_t\left(\mathit{r}\right)& =& -\mathrm{i}\sum _i{v}_i^2{\phi }_i^{+}\left(\mathit{r}\right)\nabla \times \widehat{\mathit{\sigma }}{\phi }_i\left(\mathit{r}\right).\hfill \end{array}$$

(9)

Here, ${\phi }_i$ is the wave function of canonical state i, $v_i$ is the corresponding occupation probability, and the sum runs over all states for the same species t. By further defining the isoscalar densities

$$\rho ={\rho }_n+{\rho }_p,\tau ={\tau }_n+{\tau }_p,\mathit{J}={\mathit{J}}_n+{\mathit{J}}_p,$$

and isovector densities

$$\tilde{\rho }={\rho }_n-{\rho }_p,\tilde{\tau }={\tau }_n-{\tau }_p,\tilde{\mathit{J}}={\mathit{J}}_n-{\mathit{J}}_p,$$

the ${\mathcal{E}}_{eSky}$ can be written as

$$\begin{array}{ccc}\hfill {\mathcal{E}}_{eSky}& =& {\displaystyle \frac{B_0+B_3{\rho }^{\alpha }}{2}}{\rho }^2-{\displaystyle \frac{B_0^{\prime }+B_3^{\prime }{\rho }^{\alpha }}{2}}{\tilde{\rho }}^2+(B_1+B_4{\rho }^{\beta }+B_5{\rho }^{\gamma })\rho \tau \hfill \\ & & -(B_1^{\prime }+B_4^{\prime }{\rho }^{\beta }+B_5^{\prime }{\rho }^{\gamma })\tilde{\rho }\tilde{\tau }+{\displaystyle \frac{2B_2+(2\beta +3)B_4{\rho }^{\beta }-B_5{\rho }^{\gamma }}{4}}{(\nabla \rho )}^2\hfill \\ & & -{\displaystyle \frac{2B_2^{\prime }+3B_4^{\prime }{\rho }^{\beta }-B_5^{\prime }{\rho }^{\gamma }}{4}}{(\nabla \tilde{\rho })}^2-{\displaystyle \frac{\beta B_4^{\prime }}{2}}{\rho }^{\beta -1}\tilde{\rho }\nabla \rho ·\nabla \tilde{\rho }\hfill \\ & & +{\displaystyle \frac{C_1+C_2{\rho }^{\beta }+C_3{\rho }^{\gamma }}{2}}{\mathit{J}}^2+{\displaystyle \frac{C_1^{\prime }+C_2^{\prime }{\rho }^{\beta }+C_3^{\prime }{\rho }^{\gamma }}{2}}{\tilde{\mathit{J}}}^2\hfill \\ & & +{\displaystyle \frac{b_{\mathrm{IS}}}{2}}\nabla \rho ·\mathit{J}+{\displaystyle \frac{b_{\mathrm{IV}}}{2}}\nabla \tilde{\rho }·\tilde{\mathit{J}}+{\displaystyle \frac{{\alpha }_T+{\beta }_T}{4}}{\mathit{J}}^2+{\displaystyle \frac{{\alpha }_T-{\beta }_T}{4}}{\tilde{\mathit{J}}}^2.\hfill \end{array}$$

(10)

The coefficients $B_i$, $B_i^{\prime }$, $C_i$ and $C_i^{\prime }$ are uniquely related to the Skyrme parameters $t_i$ and $x_i$ by

$$\begin{array}{cccc}\hfill B_0& ={\displaystyle \frac{3}{4}}{t}_0,\hfill & \hfill B_0^{\prime }& ={\displaystyle \frac{1}{2}}{t}_0\left({\displaystyle \frac{1}{2}}+x_0\right),\hfill \\ \hfill B_1& ={\displaystyle \frac{3}{16}}{t}_1+{\displaystyle \frac{5}{16}}{t}_2+{\displaystyle \frac{1}{4}}{t}_2{x}_2,\hfill & \hfill B_1^{\prime }& ={\displaystyle \frac{1}{8}}\left[t_1\left({\displaystyle \frac{1}{2}}+x_1\right)-t_2\left({\displaystyle \frac{1}{2}}+x_2\right)\right],\hfill \\ \hfill B_2& ={\displaystyle \frac{9}{32}}{t}_1-{\displaystyle \frac{5}{32}}{t}_2-{\displaystyle \frac{1}{8}}{t}_2{x}_2,\hfill & \hfill B_2^{\prime }& ={\displaystyle \frac{1}{16}}\left[3t_1\left({\displaystyle \frac{1}{2}}+x_1\right)+t_2\left({\displaystyle \frac{1}{2}}+x_2\right)\right],\hfill \\ \hfill B_3& ={\displaystyle \frac{1}{8}}{t}_3,\hfill & \hfill B_3^{\prime }& ={\displaystyle \frac{1}{12}}{t}_3\left({\displaystyle \frac{1}{2}}+x_3\right),\hfill \\ \hfill B_4& ={\displaystyle \frac{3}{16}}{t}_4,\hfill & \hfill B_4^{\prime }& ={\displaystyle \frac{1}{8}}{t}_4\left({\displaystyle \frac{1}{2}}+x_4\right),\hfill \\ \hfill B_5& ={\displaystyle \frac{5}{16}}{t}_5+{\displaystyle \frac{1}{4}}{t}_5{x}_5,\hfill & \hfill B_5^{\prime }& =-{\displaystyle \frac{1}{8}}{t}_5\left({\displaystyle \frac{1}{2}}+x_5\right),\hfill \\ \hfill C_1& ={\eta }_{\mathrm{tls}}{\displaystyle \frac{1}{8}}\left[t_1\left({\displaystyle \frac{1}{2}}-x_1\right)-t_2\left({\displaystyle \frac{1}{2}}+x_2\right)\right],\hfill & \hfill C_1^{\prime }& ={\eta }_{\mathrm{tls}}{\displaystyle \frac{1}{16}}(t_1-t_2),\hfill \\ \hfill C_2& ={\eta }_{\mathrm{tls}}{\displaystyle \frac{1}{8}}{t}_4\left({\displaystyle \frac{1}{2}}-x_4\right),\hfill & \hfill C_2^{\prime }& ={\eta }_{\mathrm{tls}}{\displaystyle \frac{1}{16}}{t}_4,\hfill \\ \hfill C_3& =-{\eta }_{\mathrm{tls}}{\displaystyle \frac{1}{8}}{t}_5\left({\displaystyle \frac{1}{2}}+x_5\right),\hfill & \hfill C_3^{\prime }& =-{\eta }_{\mathrm{tls}}{\displaystyle \frac{1}{16}}{t}_5.\hfill \end{array}$$

(11)

The $b_{\mathrm{IS}}$ and $b_{\mathrm{IV}}$ characterize the isoscalar and isovector SO strengths, respectively. For the standard zero-range two-body SO term in the Skyrme interaction, one has $b_{\mathrm{IS}}=3b_{\mathrm{IV}}=3W_0/2$, where $W_0$ is the Skyrme SO parameter. In the present work, however, following Ref. [17], $b_{\mathrm{IS}}$ and $b_{\mathrm{IV}}$ are treated as independent phenomenological EDF-level couplings, rather than as parameters of a strict short-range two-body SO interaction, in order to explore the sensitivity of finite-nucleus observables to the IVSO channel within the adopted EDF framework.

Following Ref. [22], we employ the eS53 and eS250 Skyrme EDFs constructed therein. Among them, eS250 is a fully optimized strong-IVSO functional with $(b_{\mathrm{IS}},b_{\mathrm{IV}})\approx (160,250)$ MeV·fm⁵. As shown in Ref. [22], eS250 gives $\Delta F_{\mathrm{CW}}$ predictions for ⁴⁸Ca and ²⁰⁸Pb that are consistent with the CREX and PREX-II data within their respective $1\sigma$ uncertainties. The functional eS53, with $(b_{\mathrm{IS}},b_{\mathrm{IV}})\approx (160,53)$ MeV·fm⁵ corresponding to the conventional relation $b_{\mathrm{IV}}=b_{\mathrm{IS}}/3$, is obtained from eS250 by modifying the IVSO strength while keeping the other EDF parameters unchanged. For reference, we also discuss the reference functional eS500_T from Ref. [22], which is likewise fully optimized and adopts a very strong IVSO coupling with $(b_{\mathrm{IS}},b_{\mathrm{IV}})\approx (134,500)$ MeV·fm⁵.

In Equation (10), the coefficients $C_i$ and $C_i^{\prime }$ collect the ${\mathit{J}}^2$ and ${\tilde{\mathit{J}}}^2$ contributions associated with the momentum-dependent central part of the extended Skyrme interaction, whereas the zero-range tensor force contributes separately through the ${\alpha }_T$- and ${\beta }_T$-dependent terms. In the EDF implementation, the parameter ${\eta }_{\mathrm{tls}}$ is used as a bookkeeping switch to control whether the central contribution to the ${\mathit{J}}^2$ sector is retained [14]. For the EDFs employed in this work, ${\eta }_{\mathrm{tls}}=0$ for both eS53 and eS250, whereas ${\eta }_{\mathrm{tls}}=1$ for eS500_T.

Based on the above EDF, nuclear ground-state properties are calculated by solving the HFB equations. The single nucleon Hamiltonian entering into the HFB equation can be expressed as

$$\begin{array}{c}\hfill {\widehat{h}}_t=-\nabla ·{\displaystyle \frac{{\hslash }^2}{2m_t^{*}}}\nabla +U_t+\mathrm{i}{\mathit{W}}_t·(\widehat{\mathit{\sigma }}\times \nabla ),t=n,p,\end{array}$$

(12)

where the single-nucleon fields can be derived from the energy density $\mathcal{E}$ by

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\hslash }^2}{2m_t^{*}}}& ={\displaystyle \frac{\partial \mathcal{E}}{\partial {\tau }_t}},U_t={\displaystyle \frac{\partial \mathcal{E}}{\partial {\rho }_t}}-\nabla ·{\displaystyle \frac{\partial \mathcal{E}}{\partial \left[\nabla {\rho }_t\right]}},{\mathit{W}}_t\hfill & \hfill ={\displaystyle \frac{\partial \mathcal{E}}{\partial {\mathit{J}}_t}}.\end{array}$$

(13)

From Equation (10), one can obtain the nucleon effective mass $m_t^{*}$, i.e.,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\hslash }^2}{2m_t^{*}}}& =& {\displaystyle \frac{{\hslash }^2}{2m}}+(B_1+B_4{\rho }^{\beta }+B_5{\rho }^{\gamma })\rho -{\eta }_t(B_1^{\prime }+B_4^{\prime }{\rho }^{\beta }+B_5^{\prime }{\rho }^{\gamma })\tilde{\rho },\hfill \end{array}$$

(14)

the single nucleon potential (without Coulomb potential)

$$\begin{array}{ccc}\hfill U_t& =& B_0\rho -{\eta }_t{B}_0^{\prime }\tilde{\rho }+B_1\tau -{\eta }_t{B}_1^{\prime }\tilde{\tau }+{\displaystyle \frac{\alpha +2}{2}}{B}_3{\rho }^{\alpha +1}-{\displaystyle \frac{B_3^{\prime }}{2}}(\alpha \tilde{\rho }+2{\eta }_t\rho ){\rho }^{\alpha -1}\tilde{\rho }\hfill \\ & & +(\beta +1)B_4{\rho }^{\beta }\tau +(\gamma +1)B_5{\rho }^{\gamma }\tau -(B_4^{\prime }\beta {\rho }^{\beta }+B_5^{\prime }\gamma {\rho }^{\gamma })\tilde{\rho }\tilde{\tau }-{\eta }_t(B_1^{\prime }+B_4^{\prime }{\rho }^{\beta }+B_5^{\prime }{\rho }^{\gamma })\tilde{\tau }\hfill \\ & & -{\displaystyle \frac{\beta (2\beta +3)B_4{\rho }^{\beta -1}-B_5\gamma {\rho }^{\gamma -1}}{4}}{(\nabla \rho )}^2-{\displaystyle \frac{2B_2+(2\beta +3)B_4{\rho }^{\beta }-B_5{\rho }^{\gamma }}{2}}{\nabla }^2\rho \hfill \\ & & -{\displaystyle \frac{3\beta B_4^{\prime }{\rho }^{\beta -1}-\gamma B_5^{\prime }{\rho }^{\gamma -1}}{4}}\nabla \tilde{\rho }·(2{\eta }_t\nabla \rho +\nabla \tilde{\rho })-{\displaystyle \frac{2B_2^{\prime }+3B_4^{\prime }{\rho }^{\beta }-B_5^{\prime }{\rho }^{\gamma }}{2}}{\eta }_t{\nabla }^2\tilde{\rho }\hfill \\ & & +{\displaystyle \frac{\beta B_4^{\prime }}{2}}{\rho }^{\beta -1}{(\nabla \tilde{\rho })}^2+{\displaystyle \frac{\beta B_4^{\prime }}{2}}{\rho }^{\beta -1}\tilde{\rho }{\nabla }^2\tilde{\rho }+{\displaystyle \frac{\beta (\beta -1)B_4^{\prime }}{2}}{\rho }^{\beta -2}\tilde{\rho }{(\nabla \rho )}^2{\eta }_t\hfill \\ & & +{\displaystyle \frac{\beta B_4^{\prime }}{2}}{\rho }^{\beta -1}\tilde{\rho }\left({\nabla }^2\rho \right){\eta }_t+{\displaystyle \frac{\beta C_2{\rho }^{\beta -1}+\gamma C_3{\rho }^{\gamma -1}}{2}}{\mathit{J}}^2+{\displaystyle \frac{\beta C_2^{\prime }{\rho }^{\beta -1}+\gamma C_3^{\prime }{\rho }^{\gamma -1}}{2}}{\tilde{\mathit{J}}}^2\hfill \\ & & -{\displaystyle \frac{b_{\mathrm{IS}}}{2}}\nabla ·\mathit{J}-{\eta }_t{\displaystyle \frac{b_{\mathrm{IV}}}{2}}\nabla ·\tilde{\mathit{J}},\hfill \end{array}$$

(15)

and the SO potential

$$\begin{array}{ccc}\hfill {\mathit{W}}_t& =& {\displaystyle \frac{b_{\mathrm{IS}}}{2}}\nabla \rho +{\eta }_t{\displaystyle \frac{b_{\mathrm{IV}}}{2}}\nabla ({\rho }_n-{\rho }_p)+{\displaystyle \frac{{\alpha }_J+{\beta }_J}{2}}\mathit{J}+{\eta }_t{\displaystyle \frac{{\alpha }_J-{\beta }_J}{2}}({\mathit{J}}_n-{\mathit{J}}_p),\hfill \end{array}$$

(16)

where ${\eta }_t=1$ and $-1$ for $t=n$ and p, respectively. The parameters ${\alpha }_J$ and ${\beta }_J$ in the SO potentials are defined by

$$\begin{array}{c}\hfill {\alpha }_J={\alpha }_C+{\alpha }_T,{\beta }_J={\beta }_C+{\beta }_T,\end{array}$$

(17)

with the central-exchange parameters given by

$$\begin{array}{c}\hfill {\alpha }_C=2C_1+2C_2{\rho }^{\beta }+2C_3{\rho }^{\gamma },{\beta }_C=2C_1^{\prime }+2C_2^{\prime }{\rho }^{\beta }+2C_3^{\prime }{\rho }^{\gamma }.\end{array}$$

(18)

As can be seen from Equations (15) and (16), the coupling constants $b_{\mathrm{IS}}$ and $b_{\mathrm{IV}}$ appear in both the central potential $U_t$ and the SO potential ${\mathit{W}}_t$. In contrast, the tensor force contributes only to ${\mathit{W}}_t$ and does not affect $U_t$.

2.2. Charge and Weak Form Factors

The normalized nuclear form factors, namely, the charge form factor $F_C\left(q\right)$ and the weak charge form factor $F_W\left(q\right)$, are calculated by folding the point-nucleon form factors $F_t\left(q\right)$ and SO current form factors $F_t^{ls}\left(q\right)$ ($t=n,p$) with the corresponding intrinsic nucleon electromagnetic $G_{E/M,t}$ and weak $G_{E/M,t}^{\left(W\right)}$ form factors [46]. The point-nucleon and SO current form factors are defined from the local density ${\rho }_t$ and spin-current density ${\mathit{J}}_t$ as

$$\begin{array}{ccc}\hfill F_t\left(q\right)& =& \int d^3\mathit{r}{e}^{\mathrm{i}\mathit{q}·\mathit{r}}{\rho }_t\left(\mathit{r}\right),\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill F_t^{ls}\left(q\right)& =& \int d^3\mathit{r}{e}^{\mathrm{i}\mathit{q}·\mathit{r}}\nabla ·{\mathit{J}}_t\left(\mathit{r}\right).\hfill \end{array}$$

(20)

The charge and weak-charge form factors are then given by

$$\begin{array}{ccc}\hfill F_C\left(q\right)& =& {\displaystyle \frac{1}{Z}}\sum _{t=p,n}\left[G_{E,t}\left(q\right)F_t\left(q\right)+G_{M,t}\left(q\right)F_t^{\left(ls\right)}\left(q\right)\right],\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill F_W\left(q\right)& =& \sum _{t=p,n}{\displaystyle \frac{\left[G_{E,t}^{\left(W\right)}\left(q\right)F_t\left(q\right)+G_{M,t}^{\left(W\right)}\left(q\right)F_t^{\left(ls\right)}\left(q\right)\right]}{Z{Q}_p^{\left(W\right)}+N{Q}_n^{\left(W\right)}}},\hfill \end{array}$$

(22)

where $N\left(Z\right)$ is the neutron (proton) number, and $Q_p^{\left(W\right)}=0.0713$ and $Q_n^{\left(W\right)}=-0.9888$ are proton and neutron weak charges, respectively. The electromagnetic form factors $G_{E/M,t}$ are constructed from the isospin-coupled Sachs form factors with the inclusion of the relativistic Darwin correction. Following Ref. [46], the weak intrinsic nucleon form factors $G_{E/M,t}^{\left(W\right)}$ are obtained from the standard neutral-current decomposition and are expressed in terms of the proton, neutron, and strange electromagnetic form factors (see, e.g., Ref. [47] for details):

$$\begin{array}{ccc}\hfill G_{E/M,n}^{\left(W\right)}& =& Q_n^{\left(W\right)}{G}_{E/M,p}+Q_p^{\left(W\right)}{G}_{E/M,n}+Q_n^{\left(W\right)}{G}_{E/M,s},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill G_{E/M,p}^{\left(W\right)}& =& Q_p^{\left(W\right)}{G}_{E/M,p}+Q_n^{\left(W\right)}{G}_{E/M,n}+Q_n^{\left(W\right)}{G}_{E/M,s},\hfill \end{array}$$

(24)

where the strange-quark electromagnetic form factors $G_{E/M,s}$ are taken as

$$\begin{array}{ccc}\hfill G_{E,s}\left(q\right)& =& {\rho }_s{\displaystyle \frac{{\hslash }^2{q}^2/\left(4c^2{m}_N^2\right)}{1+4.97{\hslash }^2{q}^2/\left(4c^2{m}_N^2\right)}},\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill G_{M,s}\left(q\right)& =& {\displaystyle \frac{{\kappa }_s}{{\left(1+{\displaystyle \frac{q^2}{12{\kappa }_s}}{r}_{M,s}^2\right)}^2}}.\hfill \end{array}$$

(26)

Here, ${\rho }_s=-0.24$ [48,49], ${\kappa }_s=-0.017$ [50], and $r_{M,s}^2=-0.015{\mathrm{fm}}^2$ [50] are adopted for the strange electric coupling, strange magnetic moment, and strange magnetic radius, respectively. The center-of-mass correction to $F_C$ and $F_W$ is approximately taken into account by renormalizing the nucleon mass $m_N$ to $(1-1/A)m_N$ in the HFB calculation (see, e.g., Ref. [15] for details).

3. Results

3.1. Single-Particle Energy Spectrum of ⁴⁸Ca and ²⁰⁸Pb

As shown in Ref. [22], compared with the conventional-IVSO reference eS53, the eS250 functional with $(b_{\mathrm{IS}},b_{\mathrm{IV}})\approx (160,250)$ MeV·fm⁵ provides a representative strong-IVSO case: it selectively shifts the calculated $\Delta F_{\mathrm{CW}}$ of ⁴⁸Ca toward the CREX value while leaving the corresponding ²⁰⁸Pb prediction essentially unchanged and still consistent with PREX-II within the reported $1\sigma$ uncertainty. For this EDF, the description of ground-state nuclear properties, such as binding energies, charge radii, and spin–orbit splittings of single-particle levels, as well as selected excited-state observables including the giant monopole resonance in ²⁰⁸Pb and the electric dipole polarizability ${\alpha }_{\mathrm{D}}$, remains in good agreement with experimental data [22]. In addition, the pure neutron matter (PNM) equation of state (EOS) predicted by this EDF is consistent with microscopic chiral effective field theory (ChEFT) [22].

Moreover, owing to the enhanced IVSO interaction, which further promotes spin–orbit splitting, the single-particle levels and shell structure of many nuclei are affected, with potential implications for magic numbers. A further examination of the single-particle spectra of ²⁰⁸Pb and ⁴⁸Ca, as shown in Figure 1, reveals that eS250 reproduces the magic shell closures as well as the conventional IVSO-strength EDF eS53 [22]. However, when the IVSO interaction is further enhanced using the EDF eS500_T with extreme strength from Ref. [22], the single-particle energies in ⁴⁸Ca and ²⁰⁸Pb exhibit certain shifts compared to those obtained with eS53 and eS250. Nevertheless, the magic number structure remains well reproduced, although it slightly violates the usual energy level ordering of the neutron $1{\mathrm{p}}_{3/2}$ and $1{\mathrm{p}}_{1/2}$ states in ⁴⁸Ca. It is worth emphasizing that when $b_{\mathrm{IV}}$ is set to as high as 500 MeV·fm⁵ as in the eS500_T, such an extremely large IVSO strength significantly alters the spin–orbit splittings. To partially compensate for these effects, the tensor force is included in the eS500_T EDF (see Supplemental Material (SM) in Ref. [22]). Moreover, the eS500_T fails to reproduce the ${\alpha }_{\mathrm{D}}$ in ²⁰⁸Pb and violates the PNM EOS from microscopic ChEFT calculations. This indicates that a $b_{\mathrm{IV}}$ value of 500 MeV·fm⁵ is excessively large, whereas $b_{\mathrm{IV}}\approx 250$ MeV·fm⁵ represents a more suitable strong-IVSO choice within the adopted EDF family.

Figure 1. Single-particle energy levels for ⁴⁸Ca and ²⁰⁸Pb. Panel (a) shows neutron levels in ⁴⁸Ca; panels (b–d) display proton levels in ⁴⁸Ca, neutron levels in ²⁰⁸Pb, and proton levels in ²⁰⁸Pb, respectively.

3.2. Sensitivity of $\Delta F_{\mathrm{CW}}$ to the IVSO Interaction for ⁴⁸Ca, ²⁰⁸Pb, ⁹⁰Zr, and ⁶²Ni

3.2.1. ⁴⁸Ca and ²⁰⁸Pb

As elucidated in Ref. [22], the sensitivity of ⁴⁸Ca to the IVSO interaction originates from the eight neutrons occupying the $1{\mathrm{f}}_{7/2}$ orbital. This distinction between ⁴⁸Ca and ²⁰⁸Pb can be traced to the IVSO energy term ${\displaystyle \frac{b_{\mathrm{IV}}}{2}}\tilde{\mathit{J}}·\nabla \tilde{\rho }$ in Equation (10). As shown in Ref. [22], in ⁴⁸Ca, all proton spin–orbit partners are filled, while the neutron $1{\mathrm{f}}_{7/2}$ orbital is fully occupied and its $1{\mathrm{f}}_{5/2}$ partner is empty. This configuration produces a large $\tilde{J}$, enabling the IVSO interaction to modify the central mean field and induce global density rearrangements that alter the contribution of every orbital to $\Delta F_{\mathrm{CW}}$. In ²⁰⁸Pb, by contrast, the spin–orbit unsaturated orbitals ($1{\mathrm{h}}_{11/2}$ for protons and $1{\mathrm{i}}_{13/2}$ for neutrons) yield comparable positive contributions to $J_{\mathrm{p}}$ and $J_{\mathrm{n}}$, resulting in a small $\tilde{J}$ and hence a weak IVSO effect. A detailed orbital-by-orbital analysis of $F_{\mathrm{C}}$ and $F_{\mathrm{W}}$ for ⁴⁸Ca and ²⁰⁸Pb, comparing the eS53 and eS250 EDFs, can be found in the lower panel of Figure S5 in the SM of Ref. [22]. As shown there, the contributions in ⁴⁸Ca exhibit clear differences across all orbitals when varying the IVSO interaction strength, with the most pronounced variation appearing in the $1{\mathrm{d}}_{3/2}$ rather than $1{\mathrm{f}}_{7/2}$ orbital—a consequence of the mean-field rearrangement induced by the IVSO interaction. In contrast, the orbital contributions in ²⁰⁸Pb remain almost unchanged, confirming its insensitivity to the IVSO interaction.

From this perspective, one can understand why an effective IVSO interaction can change the $\Delta F_{\mathrm{CW}}$ predictions for ⁴⁸Ca and ²⁰⁸Pb in different ways within the extended Skyrme EDF framework. If one instead maintains the IVSO interaction at its conventional strength (i.e., $b_{\mathrm{IV}}=b_{\mathrm{IS}}/3$) and attempts to adjust the tensor force strengths ${\alpha }_T$ and ${\beta }_T$, one finds that the CREX and PREX-II $\Delta F_{\mathrm{CW}}$ values cannot be simultaneously described by such tensor-force adjustments, despite the fact that the tensor force and the IVSO interaction often appear similar in certain respects, such as their ability to modify the ordering of single-particle levels as noted in Ref. [22]. The key distinction between the two can be seen from Equations (15) and (16). The IVSO interaction contributes to both the central potential U and the spin–orbit potential W, whereas the tensor force contributes only to the spin–orbit potential W. It is precisely this modification of the central potential U by the IVSO interaction that significantly alters the contributions of all orbitals to $F_{\mathrm{C}}$ and $F_{\mathrm{W}}$ [22].

3.2.2. ⁹⁰Zr and ⁶²Ni

Following the same line of reasoning used in Ref. [22] to analyze the contribution of single-particle orbitals to the IVSO term in the energy density (Equation (10)), one can infer that nuclei with a spin–orbit pattern analogous to ⁴⁸Ca, such as ⁹⁰Zr, should also exhibit sensitivity to the IVSO interaction in their $\Delta F_{\mathrm{CW}}$ values. Conversely, nuclei similar to ²⁰⁸Pb, such as ⁶²Ni, are expected to be insensitive to the IVSO interaction. Applying the analysis framework of Ref. [22], one finds that in ⁹⁰Zr, the proton orbitals are spin–orbit saturated, whereas the neutron orbitals are spin–orbit unsaturated, with ten neutrons occupying the $1{\mathrm{g}}_{9/2}$ orbital playing a role analogous to that of the eight neutrons in the $1{\mathrm{f}}_{7/2}$ orbital of ⁴⁸Ca. In contrast, ⁶²Ni exhibits spin–orbit unsaturation in both proton and neutron channels: eight protons occupy the $1{\mathrm{f}}_{7/2}$ orbital, and eight neutrons occupy the same orbital. Consequently, the IVSO contribution to the energy density, which is proportional to $\tilde{J}$, becomes negligible.

To illustrate this point, we perform an analysis for ⁹⁰Zr and ⁶²Ni analogous to that presented in the lower panel of Figure S5 in the SM of Ref. [22] for ⁴⁸Ca and ²⁰⁸Pb. As these nuclei are open-shelled, pairing effects must be taken into account. We again compare results obtained with the eS53 and eS250 EDFs, with the proton and neutron pairing strengths determined by fitting the pairing gaps of ¹⁴⁴Sm and ¹²⁰Sn [51], respectively, using eS250. The resulting orbital-by-orbital contributions to $F_{\mathrm{C}}$ and $F_{\mathrm{W}}$ are presented in Figure 2a–d. For ⁹⁰Zr, as shown in Figure 2a,c, each orbital exhibits noticeable differences in its contribution when comparing eS53 and eS250. The most pronounced difference in the contribution to $F_{\mathrm{C}}$ appears in the $1{\mathrm{d}}_{3/2}$ orbital, while for $F_{\mathrm{W}}$, the largest difference is observed in the $1{\mathrm{f}}_{5/2}$ orbital. For ⁶²Ni, as shown in Figure 2b,d, the orbital contributions to both $F_{\mathrm{C}}$ and $F_{\mathrm{W}}$ are almost identical between eS53 and eS250, confirming our expectation. Meanwhile, we note that the differences in the orbital contributions to $F_{\mathrm{C}}$ and $F_{\mathrm{W}}$ between eS53 and eS250 are not as pronounced for ⁹⁰Zr as they are for ⁴⁸Ca, nor are they as negligible for ⁶²Ni as they are for ²⁰⁸Pb. This behavior can be attributed, on the one hand, to the diminished influence of the IVSO interaction in heavier nuclei relative to lighter ones and, on the other hand, to the increased number of orbitals contributing in heavier nuclear systems.

Figure 2. Orbital contributions to $F_{\mathrm{C}}$ and $F_{\mathrm{W}}$ obtained with the eS53 and eS250 EDFs. Panels (a,c) show the contributions to $F_{\mathrm{C}}$ and $F_{\mathrm{W}}$ for ⁹⁰Zr, respectively. Panels (b,d) show the corresponding results for ⁶²Ni. The analyzed orbitals (from left to right) include $1{\mathrm{s}}_{1/2}$, $1{\mathrm{p}}_{3/2}$, $1{\mathrm{p}}_{1/2}$, $1{\mathrm{d}}_{5/2}$, $1{\mathrm{d}}_{3/2}$, $2{\mathrm{s}}_{1/2}$, $1{\mathrm{f}}_{7/2}$, $1{\mathrm{f}}_{5/2}$, $2{\mathrm{p}}_{3/2}$, $2{\mathrm{p}}_{1/2}$, and $1{\mathrm{g}}_{9/2}$, both for ⁹⁰Zr and ⁶²Ni.

A complementary perspective on this sensitivity is provided by the momentum-transfer dependence of $\Delta F_{\mathrm{CW}}$. Figure 3 displays this quantity for ⁹⁰Zr and ⁶²Ni in panels (c) and (d), respectively, with panels (a) and (b) showing the corresponding results for ⁴⁸Ca and ²⁰⁸Pb for comparison. The comparison reveals that both ⁴⁸Ca and ⁹⁰Zr exhibit sensitivity to the IVSO interaction. For ⁹⁰Zr, this sensitivity is most pronounced at $q\approx 0.68$ fm⁻¹. In contrast, ⁶²Ni and ²⁰⁸Pb are both insensitive to the IVSO interaction.

Figure 3. Momentum-transfer dependence of $\Delta F_{\mathrm{CW}}$ for various nuclei. Panels (a–d) show the results for ⁴⁸Ca, ²⁰⁸Pb, ⁹⁰Zr, and ⁶²Ni, respectively, as functions of the momentum transfer q (in fm⁻¹).

The contrasting behavior between the pairs (⁴⁸Ca, ⁹⁰Zr) and (²⁰⁸Pb, ⁶²Ni) makes comprehensive experimental measurements on these nuclei particularly meaningful and important. ⁹⁰Zr is a stable natural isotope of zirconium with an abundance of 51.45% in nature [52], while ⁶²Ni is likewise a stable natural isotope of nickel with an abundance of 3.63% [52]. Both are suitable for experimental measurements. Owing to the insensitivity of ²⁰⁸Pb and ⁶²Ni to the IVSO interaction, parity-violating electron scattering (PVES) experiments measuring $\Delta F_{\mathrm{CW}}$ (or the neutron skin thickness) on these two nuclei would provide a combined constraint on the symmetry energy that is more precise than that obtained from PREX-II alone. Meanwhile, PVES measurements on ⁹⁰Zr, when combined with the CREX results, would enable a more accurate determination of the IVSO interaction strength $b_{\mathrm{IV}}$. Such measurements can be realistically performed at existing and upcoming facilities. The MESA accelerator in Mainz, with its planned PVES program MREX [53], is well suited for precision measurements on nuclei including ⁹⁰Zr and ⁶²Ni as well as ⁴⁸Ca and ²⁰⁸Pb. Complementary measurements at JLab, building on the successful PREX-II and CREX experiments, are also feasible, as demonstrated by recent proposals and feasibility studies [54,55]. This would have a positive and significant impact on deepening our understanding of both the symmetry energy and the IVSO interaction.

3.3. Neutron Skin R_np of ⁴⁸Ca, ²⁰⁸Pb, ⁹⁰Zr, ⁶²Ni, ²²O and ²⁴O

Compared to $\Delta F_{\mathrm{CW}}$, the neutron skin thickness $R_{\mathrm{np}}$ is a more intuitive and frequently discussed quantity, which, like $\Delta F_{\mathrm{CW}}$, characterizes the overall difference between neutron and proton distributions. Beyond ⁴⁸Ca and ⁹⁰Zr, following the analysis in Ref. [22], one can readily identify ²²O and ²⁴O as nuclei that are also sensitive to the IVSO interaction, with evidence supporting their double magic nature [56,57]. Table 1 presents the calculated $R_{\mathrm{np}}$ values for ⁴⁸Ca, ²⁰⁸Pb, ⁹⁰Zr, ⁶²Ni, ²²O, and ²⁴O obtained with the eS53 and eS250 EDFs. For ²⁰⁸Pb and ⁶²Ni, the results from the two EDFs, are nearly identical. In contrast, for ⁴⁸Ca, ⁹⁰Zr, ²²O, and ²⁴O, the eS250 predictions are smaller than those of eS53, reflecting a behavior similar to that observed for $\Delta F_{\mathrm{CW}}$. It is worth noting that the charge-exchange reaction measurement by Ref. [58] reported $\Delta R_{\mathrm{np}}=0.07\pm 0.04$ fm for ⁹⁰Zr, with the central value of 0.07 fm lying between the eS250 and eS53 predictions, indicating the need for more precise experimental determinations. Furthermore, future measurements of $R_{\mathrm{np}}$ for ²²O and ²⁴O, e.g., by charge-changing cross sections [59,60], would be highly desirable to validate the predictions of the present work.

Table 1. Neutron skin $R_{\mathrm{np}}$ (in fm) of ⁴⁸Ca, ²⁰⁸Pb, ⁹⁰Zr, ⁶²Ni, ²⁴²O and ²⁴O with eS53 and eS250.

Nucleus	eS53	eS250
48Ca	0.179	0.129
208Pb	0.199	0.195
90Zr	0.083	0.053
62Ni	0.096	0.093
22O	0.367	0.290
24O	0.510	0.430

4. Conclusions

In this work, we have systematically investigated the sensitivity of the charge-weak form factor difference $\Delta F_{\mathrm{CW}}$ to the IVSO interaction in ⁴⁸Ca, ²⁰⁸Pb, ⁹⁰Zr, and ⁶²Ni using the extended Skyrme EDFs eS53 and eS250. Relative to eS53, the strong-IVSO functional eS250 with $(b_{\mathrm{IS}},b_{\mathrm{IV}})\approx (160,250)$ MeV·fm⁵ selectively shifts the calculated $\Delta F_{\mathrm{CW}}$ of ⁴⁸Ca toward the CREX value while leaving the corresponding prediction for ²⁰⁸Pb essentially unchanged [22]. At the same time, eS250 preserves the established magic shell closures in ⁴⁸Ca and ²⁰⁸Pb and maintains a good description of global nuclear properties.

By examining the orbital-by-orbital contributions to $F_{\mathrm{C}}$ and $F_{\mathrm{W}}$, we demonstrate that the sensitivity of ⁴⁸Ca originates from the eight $1{\mathrm{f}}_{7/2}$ neutrons, which modify the central mean-field potential and thereby influence all orbitals. A similar mechanism is identified in ⁹⁰Zr, where ten $1{\mathrm{g}}_{9/2}$ neutrons play an analogous role, leading to a pronounced IVSO sensitivity in its $\Delta F_{\mathrm{CW}}$ and neutron skin thickness $R_{\mathrm{np}}$. In contrast, ²⁰⁸Pb and ⁶²Ni, characterized by near-cancellation of the isovector spin–orbit density $J_{\mathrm{n}}-J_{\mathrm{p}}$, exhibit little sensitivity to the IVSO interaction.

This structural distinction between the pairs (⁴⁸Ca, ⁹⁰Zr) and (²⁰⁸Pb, ⁶²Ni) offers a controlled experimental strategy for disentangling the IVSO interaction from the density dependence of the symmetry energy. The natural abundances and experimental accessibility of ⁹⁰Zr and ⁶²Ni make them ideal candidates for future PVES measurements. Such measurements could be performed at MREX and JLab. Once available, they can be combined with existing PREX-II and CREX observables to help separate EDF-level IVSO effects from symmetry-energy effects and thereby constrain both the density dependence of the symmetry energy and the IVSO strength $b_{\mathrm{IV}}$.

Furthermore, our predictions for the neutron skin thickness in ²²O and ²⁴O, which could be measured by charge-changing cross sections, suggest that these neutron-rich oxygen isotopes are also sensitive to the IVSO interaction, offering additional testing grounds for the proposed strong IVSO scenario. Overall, our results show that the sensitivity of $\Delta F_{\mathrm{CW}}$ to the IVSO interaction is strongly nucleus-dependent and is closely tied to the underlying spin–orbit saturation pattern. In this respect, PVES measurements across different nuclei offer a valuable way to constrain the isovector spin–orbit channel in nuclear EDFs.