Constraint of symmetry energy slope using neutron skins of ⁴⁸Ca, ⁶⁴Ni, ¹²⁴Sn, and ²⁰⁸Pb and its impact on neutron star radius

The equation of state (EoS) of isospin asymmetric nuclear matter plays an important role in researches of nuclear reaction, structure, and astrophysics [1]. For example, bulk properties of neutron stars are mainly governed by the EoS [1−4]. Many investigations indicate that there are significant correlations between neutron star radii and the EoS parameters, namely, the symmetry energy slope L at the saturation density $ \rho_{0} $, the nuclear matter incompressibility coefficient $ K_0 $, and the skewness parameter $ Q_0 $ [4−9]. Especially, the radii of neutron stars with density $ \rho\sim\rho_0 $ are primarily determined by the slope L [4]. Owing to the importance including but not limited to that mentioned above, many attempts are made to precisely determine the EoS parameters [7, 10]. The $ K_0 $ was constrained with a precision of about 10 percents to be 240(20) MeV by giant resonance [11, 12], which agrees with the recent results [13]. However, the deduced L values at $ \rho_{0} $ by different methodologies have a large spread from about 30 to 110 MeV, more details see review article [14]. In particular, due to lack of sensitive observables, the $ Q_0 $ is still poorly known. Theoretical values for the $ Q_0 $ are in the range of $ \sim - 1000$ to $ \sim 1000$ MeV [15].

Similar to the neutron star radii, it is well known that neutron skins formed in heavy nuclei with an excess of neutron over proton are also sensitive to the slope L at $ \rho_{0} $ [16]. Since the linear relationships were established based on different effective nucleon-nucleon interactions of the self-consistent mean-field models [16], neutron skin thicknesses $\Delta{{\rm{R}}} _{ {\rm{np}}} $ as the difference of neutron and proton distribution rms radii in nuclei, namely $\Delta{{\rm{R}}} _{ {\rm{np}}} = {{\rm{R}}} _ {\rm{n}}- {{\rm{R}}} _ {\rm{p}} $, have been widely employed to constrain the slope L. Theoretical progresses also promote the developments of novel experimental methods for neutron distribution radius measurements [17−26]. Recently, the model-dependent $\Delta{{\rm{R}}} _{ {\rm{np}}} $ data for the stable Ca, Ni, Sn, and Pb isotopes were compiled, and the evaluated $\Delta{{\rm{R}}} _{ {\rm{np}}} $ values have a high precision of around 0.02 fm [27]. These compiled $\Delta{{\rm{R}}} _{ {\rm{np}}} $ data [27] are taken from different experiments related to hadron scatterings [18, 24, 28], interaction cross sections [25, 29], giant resonances [30, 31], and antiprotonic atoms [22, 32]. Based on the compiled $\Delta{{\rm{R}}} _{ {\rm{np}}} $ data, separate trends of $\Delta{{\rm{R}}} _{ {\rm{np}}} $ versus relative neutron excess, $ \delta = (N-Z)/A $, were observed for the Ca, Ni, Sn, and Pb isotopic chains [27], where N, Z, and A are neutron, proton, and mass numbers, respectively. In addition, the parity-violating electron scattering also provides a so-called mode-independent approach to determine $\Delta{{\rm{R}}} _{ {\rm{np}}} $ in nuclei [23, 33]. With this method, the $\Delta{{\rm{R}}} _{ {\rm{np}}} $ of $ ^{48} {\rm{Ca}}$ and $ ^{208} {\rm{Pb}}$ were determined by the CREX and PREX collaboration groups [23, 33], respectively. Although the tension of the CREX-PREX neutron skins with other experimental and theoretical results was widely discussed from the perspective of physics [34−37], the possibility of statistical fluctuations was also mentioned by Refs. [38−40]

The $\Delta{{\rm{R}}} _{ {\rm{np}}} $ data from different experimental methods would be helpful to reduce the model-dependent effects of single experimental data [41]. For instance, using the $\Delta{{\rm{R}}} _{ {\rm{np}}} $ of Sn isotopes from different experiments, highly precise L values at different densities have already been obtained in Refs. [42, 43]. We note that, the compiled neutron skin data from different experimental methods reproduce the predicted $\Delta{{\rm{R}}} _{ {\rm{np}}} $ separate structure of Ca, Ni, Sn, and Pb isotope chains by the SLy4 interaction [27]. This would indicate that these compiled neutron skin data are reliable. Therefore, it is interesting to precisely constrain the symmetry energy slopes L at the saturation density using the compiled $\Delta{{\rm{R}}} _{ {\rm{np}}} $ data [27], and to address their impacts on the neutron star radii at 1.4 solar-mass ($ 1.4\,M_{\odot} $).

As known, the symmetry energy slope L at the saturation density can be sensitively constrained by the $\Delta{{\rm{R}}} _{ {\rm{np}}} $ of heavy nuclei with large neutron-to-proton values [43, 44]. Therefore, for each isotope chain tabulated in Ref. [27], we only adopt the $\Delta{{\rm{R}}} _{ {\rm{np}}} $ data of magic nucleus with the maximum neutron-to-proton value, namely $ ^{48} {\rm{Ca}}$, $ ^{64} {\rm{Ni}}$, $ ^{124} {\rm{Sn}}$, and $ ^{208} {\rm{Pb}}$, to determine the slope L. It would also be helpful to reduce the effects of possible $\Delta{{\rm{R}}} _{ {\rm{np}}} $ isotope-dependent deviation on the L extraction through balancing the contribution of each isotope chain. To constrain the L in this work, the $\Delta{{\rm{R}}} _{ {\rm{np}}} $ are analyzed via the chi-square $ \chi _{j}^{2} $ defined here as

where $ k = 1, 2......M $, stands for $ ^{48} {\rm{Ca}}$, $ ^{64} {\rm{Ni}}$, $ ^{124} {\rm{Sn}}$, and $ ^{208} {\rm{Pb}}$, respectively. $ i = 1, 2......N $, with N the number of adopted neutron skin thicknesses for nucleus k. $ \Delta {\rm{R}}_{ {\rm{np}}}^{ {\rm{th}}}(k, j) $ ($ j = 1, 2......28 $) is the theoretical neutron skin thickness of nucleus k corresponding to the effective interaction j. The randomly adopted effective interactions are Z, Es, E, Zs, Zs* [45], SIII [46], SkP [47], SkS1 [48], SkT6, SkT7 [49], RATP [50], SGII [51], SkS2 [48], SkM$ ^{*} $ [52], SLy4 [53], SkM [54], DD-ME2 [55], SkS3 [48], TW99 [56], PKO2, PKO3 [57], PKDD [58], PKO1 [59], PK1 [58], NL3 [60], NL-Z2 [61], NL-Z [62], and NL1 [63]. They are labeled with numbers from 1 to 28, respectively, as shown in Fig. 1. Each effective interaction corresponds to a specific L value. These interactions cover a large L range from about $ - $50 to 140 MeV. $\Delta{{\rm{R}}} _{ {\rm{np}}}^{ {\rm{exp}}}(k, i) $ and $ \delta\Delta {{\rm{R}}} _{ {\rm{np}}}^{ {\rm{exp}}}(k, i) $ are the ith experimental neutron skin thickness and error for nucleus k, respectively.

Figure 1.  (color online) The slope L versus $\chi ^{2}$ obtained by the compiled forty-nine model-dependent $\Delta{\rm{R}}_{\textrm{np}}$ of $^{48}{\rm{Ca}}$, $^{64}{\rm{Ni}}$, $^{124}{\rm{Sn}}$, and $^{208}{\rm{Pb}}$, see Table I in Ref. [27] for details on the $\Delta{\rm{R}}_{\textrm{np}}$ data. The dashed line denotes the quadratic polynomial (P2) fit with a R-Square value of 0.95. The filled red circle with error bar represents the extracted L result at the minimum $\chi ^{2}$. The used interaction parameter sets are labeled with numbers from 1 to 28, see text for details.

First of all, the compiled 49 model-dependent $\Delta{{\rm{R}}} _{ {\rm{np}}} $ of $ ^{48} {\rm{Ca}}$, $ ^{64} {\rm{Ni}}$, $ ^{124} {\rm{Sn}}$, and $ ^{208} {\rm{Pb}}$ [27] are adopted to constrain the L. These data are available in Table I in Ref. [27], which are obtained by various experiments, see Ref. [27] and references cited therein for details. The obtained $ \chi ^{2} $ as a function of L is plotted in Fig. 1. In the $ \chi ^{2} $ calculations, see Eq. (1), only the reported experimental errors of neutron skins are considered. The scattering of the $ \chi ^{2} $ data points would be mainly caused by the difference of interaction models. Subsequently, the obtained $ \chi ^{2} $ as a function of L is fitted by the quadratic polynomial (P2) function, namely, $ \chi ^{2} = aL^2+bL+c $. Then, a L of 50(6) MeV is deduced at the minimum $ \chi ^{2} $, $ \chi _{ {\rm{min}}}^{2} $, via the P2 fit function. The final error of L includes the uncertainties from fitting parameters and statistics. The L uncertainty caused by fitting parameters is mainly originated from the model difference. As shown in Fig. 1, the obtained $ \chi _{ {\rm{min}}}^{2} $ value for the 49 $\Delta{{\rm{R}}} _{ {\rm{np}}} $ is about 60. As a result, our normalized $ \chi_n $ of 1.1$ \pm $0.1 is close to 1 within the error bar, which means that there are no significant unidentified systematic uncertainties. Therefore, similar to Ref. [42], the statistical error with a confidence level of 68.3% for the slope L is obtained using the $ \chi _{ {\rm{min}}}^{2} +1 $ method.

Furthermore, we also constrain the slope L using the so-called model-independent $\Delta{{\rm{R}}} _{ {\rm{np}}} $ of $ ^{48} {\rm{Ca}}$ and $ ^{208} {\rm{Pb}}$ determined by the CREX and PREX experiments [23, 33], as shown in Fig. 2. The obtained L of 45(32) MeV is very consistent with the result of 50(6) MeV determined by the compiled model-dependent $\Delta{{\rm{R}}} _{ {\rm{np}}} $ of $ ^{48} {\rm{Ca}}$, $ ^{64} {\rm{Ni}}$, $ ^{124} {\rm{Sn}}$, and $ ^{208} {\rm{Pb}}$ [27]. The difference of their centre values is only about 5 MeV, which would show the consistency of $\Delta{{\rm{R}}} _{ {\rm{np}}} $ from different experiments.

Figure 2.  (color online) Same as Fig. 1, but $\chi ^{2}$ are obtained by the two model-independent $\Delta{\rm{R}}_{\textrm{np}}$ from the CREX and PREX experiments [23, 33].

Finally, see Fig. 3, the slope L is constrained to be 50(6) MeV using all fifty-one $\Delta{{\rm{R}}} _{ {\rm{np}}} $ values, including the model-dependent and -independent data used in Fig. 1 and Fig. 2. Figure 4 shows a comparison of the obtained L in this work with the reported data from different observables and methodologies, see review article [14] and references cited therein for more details. Our L result constrained by the abundant $\Delta{{\rm{R}}} _{ {\rm{np}}} $ data from various experiments has a high precision. Especially, our L value is consistent with the so-called world-averaged result, namely, a weighted mean value of 57.2(22) MeV obtained by these literature data in Fig. 4. Their difference is only about 7.2(64) MeV.

Figure 3.  (color online) Same as Fig. 1, but $\chi ^{2}$ are determined by all data used in Fig. 1 and Fig. 2.

Figure 4.  (color online) Comparison of the obtained L in this work with the reported results from various methods, which were taken from Ref. [14]. Gray area indicates the uncertainty of our L.

We further revisit the effects of our L on the neutron star radii $ R_{1.4} $ at 1.4 solar-mass ($ 1.4\,M_{\odot} $). To easily understand the effects of the EoS parameters, the neutron star radii R can be empirically expressed as [4]

where $ \delta = (\rho_n-\rho_p)/\rho $ stands for the asymmetry parameter, with $ \rho_n $ and $ \rho_p $ the neutron and proton densities, respectively. The $ R_{1.4} $-L correlation deduced from various EoS parameter sets is relatively weak [4], and there is a model-dependent $ R_{1.4} $ spread of around 2 km at one given L [4, 15]. Consequently, as indicated by Eq. 2, the L value alone is insufficient to characterize the $ R_{1.4} $.

However, the effects of L on $ R_{1.4} $ can be evaluated by adjusting the coupling constants associated with the vector-isovector meson ρ in relativistic mean-field Lagrangian, while the coupling strengths of isoscalar mesons are kept, more details see Ref. [5]. For instance, the linear $ R_{1.4} $-L relationships were established by using such kinds of methods based on the TM1 and IUFSU EoS families [5], respectively. Similarly, we also deduce the $ R_{1.4} $-L relationships from the DD-ME2 and TW99 EoS families, respectively. The matter in the core of neutron star is composed of neutrons, protons, electrons, and muons, and is assumed to be in β-equilibrium. For the core, we describe the EoS of uniform neutron star matter using the DD-ME2 and TW99 models. For the inner and outer crust regions, the EoS of nonuniform matters are generate by the Thomas-Fermi approximation and the Baym-Pethick-Sutherland, respectively. By fitting the deduced $ R_{1.4} $-L relationships, see Fig 5, linear functions of $ R_{1.4} = 12.12+0.0189L $ and $ R_{1.4} = 11.03+0.0211L $ are obtained for the DD-ME2 and TW99 families, respectively. These linear functions would be helpful to evaluate the effects of L on the $ R_{1.4} $ determinations. The spread of 30-110 MeV for L would lead to a $ R_{1.4} $ uncertainty of about 1.6 km for the same theoretical framework, see Fig. 5. It is comparable to the model-dependent $ R_{1.4} $ uncertainty of about 2 km for the different EoS families [4, 15]. Compared to about 1.6 km $ R_{1.4} $ uncertainty caused by the L spread of 30-110 MeV, the obtained L of 50(6) MeV in this work from finite nucleus system improves the $ R_{1.4} $ precision (resulted only by L) by a factor of about 7 to be around 0.24 km at the same theoretical framework, see Fig. 5. Therefore, at the precision level of our L, the model-dependent $ R_{1.4} $ uncertainty of about 2 km becomes more obvious [4, 15]. Furthermore, as shown in Fig. 5, there is also an obvious model-dependent $ R_{1.4} $ difference of about 1 km between the DD-ME2 and TW99 EoS families. For the model-dependent $ R_{1.4} $ difference between the TM1 and IUFSU EoS families, as shown in Fig. 15 in Ref. [5], which was explained by the difference of incompressibility coefficient $ K_0 $ [5], because the $ Q_{0} $ value of $ - $285 MeV for the TM1 is very close to the one of $ - $290 MeV for the IUFSU EoS family [15]. Compared to the $ K_0 $ and L, the poorly known $ Q_0 $ has a large spread from $ \sim - 1000$ to $ \sim 1000$ MeV [15]. We note that the DD-ME2 and TW99 families have almost the same $ K_0 $, but the $ Q_0 $ values are totally different, they are 478 MeV and $ - $544 MeV, respectively [15]. Therefore, it is also interesting to further study the $ R_{1.4} $ deviation caused by the poorly known $ Q_0 $ by using the DD-ME2 and TW99 EoS families. Anyway, the $ K_0 $ and $ Q_0 $ will largely influence the high-density behaviors of EoSs and the magnitude of $ R_{1.4} $ of neutron stars.

Figure 5.  (color online) The $R_{1.4}$-L correlations obtained by the DD-ME2 and TW99 EoS families. The dashed lines show the linear fit curves. The shaded area is the range of L constrained by this work.

The symmetry energy slope L at the saturation density was constrained by neutron skins from various experiments. The determined L of 50(6) MeV is consistent with the so-called world-averaged result from different observables and methodologies. Furthermore, based on the DD-ME2 and TW99 EoS families, the linear $ R_{1.4} $-L relationships were established to study the effects of the newly constrained L on the radius determinations of 1.4 solar-mass neutron stars. Significant model-dependence of neutron star radius resulted by the different skewness parameter $ Q_0 $ was observed. Our L value effectively reduces the $ R_{1.4} $ uncertainty of neutron star caused by the L spread with a factor of about 7 to be around 0.24 km at the same theoretical framework, which is smaller than the uncertainty of about 2 km from the different EoS families.

We would like to thank J. J. Li for the useful discussions and calculations.