Systematic trends of neutron skin thickness versus relative neutron excess

Available experimental neutron skin thicknesses of even-even stable Ca, Ni, Sn, Pb, and Cd isotopes are evaluated, and separate trends of neutron skin thickness versus relative neutron excess $\delta=(N-Z)/A$ are firstly observed for different isotopic chains. This phenomenon is quantitatively reproduced by the deformed Skyrme Hartree-Fock $+$ BCS model with SLy4 force.


I. INTRODUCTION
Nucleus is a quantum many-body system consisting of neutrons and protons. The root-mean-square (rms) radii of neutron and proton, which characterize the spatial matter density distributions of neutrons and protons in a nucleus, are fundamental properties of the nucleus [1,2]. Nuclear charge rms radii determined by different experimental methods were evaluated in [3], and the rms values with precisions better than 0.01 fm were reported. These valuable quantities are usually used to test and constrain microscopic theories, for instance, by which odd-even staggering of charge radii of exotic copper isotopes [4] and shape-staggering effects in mercury isotopes [5] have been well explained by dedicated theoretical calculations. It is noted that the proton distribution rms radius can be deduced from nuclear charge rms radius [6], and therefore, the precision of proton rms radius is also high. Different from the proton rms radii which are related to the wellknown electromagnetic interaction, the determinations of neutron distribution rms radii are model-dependent and are much more complicated [7]. Consequently, uncertainties of extracted neutron rms radii are relatively large and depend on the model uncertainties. Nonetheless, these quantities are still sensitive for probing nuclear structure [8,9].
Neutron skin thicknesses, ∆r np , defined as the difference of neutron and proton rms radii of a nucleus, are indispensable in nuclear reaction and nuclear astrophysics researches [10,11]. A variety of microscopic [12][13][14][15] and macroscopic [16][17][18][19] models were developed to describe ∆r np . We emphasize that theories play a decisive role in constraining the parameters of the equa- * tuxiaolin@impcas.ac.cn † yueke@impcas.ac.cn tion of state (EOS) of isospin asymmetric nuclear matter [10]. For example, a linear correlation between the ∆r np and the slope of symmetry energy at the saturation density was deduced through microscopic mean-field calculations [10,20].
Experimentally, by using the ∆r np data with large uncertainties, a linear dependence of ∆r np on the relative neutron excess, δ = (N − Z)/A, was reported with a fitting goodness (χ 2 ) of 0.6, see Fig. 4 in [21]. This result has extensively been used to constrain theories [18,[22][23][24][25][26] and to predict the nuclear ∆r np values as well. For instance, the predicted ∆r np of 133 Cs is employed in the study of atomic parity violation for testing the standard model of elementary particle physics at low energies [27].
We know that the nuclear structure is reflected in the nucleon distribution radii [9,28,29]. For example, larger neutron and proton radii were observed in deformed nuclei [8,30]. Such deformation-related effects may alter the assumed linear behavior of nucleon radii for the isotopic chains [21]. If the uncertainties of neutron skin thicknesses are improved, what can be observed on the linear trend reported in [21]? In this work, available ∆r np data of even-even stable Ca, Ni, Sn, Pb, and Cd isotopes are evaluated in order to study the systematic behavior of ∆r np versus δ.

II. EXPERIMENTAL DATA EVALUATION
In order to study the systematic behavior of ∆r np along with δ for different isotopic chains, ∆r np of eveneven stable Ca, Ni, Sn, Pb, and Cd isotopes are evaluated. Taking the rms radius, r 2 m 1/2 , of point-matter distribution, the ∆r np is deduced via [31] ∆r np = r 2 where r 2 n 1/2 and r 2 p 1/2 are point-neutron and pointproton distribution rms radii, respectively. The pointproton rms radius is related to the nuclear charge rms radius as r 2 p = r 2 ch − 0.64 in our analysis. Small corrections on r 2 p 1/2 resulting from spin-orbit term etc.
were taken into account in some adopted data [32]. Compared to the uncertainties of neutron distribution rms radii, the difference are very small and thus the corrections are neglected in the present analysis. Since the folded matter distribution rms radii, r 2 m 1/2 , such as for the Cd isotopes [33], contain the finite size of the nucleon, the corresponding point-matter rms radii are deduced via r 2 m 1/2 = r 2 m − 0.64 [31]. The evaluated neutron skin thicknesses are weighted averages. Table I in Appendix lists the ∆r np values evaluated in this work. The precisions of ∆r np have been improved.

III. SYSTEMATIC TRENDS AND DISCUSSIONS
A linear relationship between ∆r np and δ was reported in [21], but the used data have large statistical errors. The ∆r np of even-even stable Ca, Ni, Sn, Pb isotopes were determined by many experiments, and consequently, the evaluated ∆r np values have less uncertainties as shown in Table I of Appendix. Moreover, ground states of even-even nuclei with magic proton numbers have spherical shapes, and hence the influence of deformation on the neutron skin thickness is minimal. The evaluated neutron skin thicknesses for even-even Ca, Ni, Sn, Pb isotopes are thought to be reliable to study the systematic correlations between ∆r np and δ.
The correlation between ∆r np and δ is shown in Fig. 1. Although an overall linear relationship of ∆r np versus δ is observed, the normalized chi value from a linear fit to all the data in Fig. 1 is χ n =1.32. This value is apparently outside of the expected 1σ range of χ n = 1 ± 0.17. Thanks to the improved precisions of ∆r np , the curves of ∆r np versus δ for Ca, Ni, Sn, and Pb isotopic chains are separated from each other, as demonstrated in Fig. 1. We note that the overall linear relationship reported in [21] is in fact composed of several individual curves for different isotopic chains.
It is known that there is a strong correlation between ∆r np and nucleon separation energy [34]. In general, a larger proton separation energy, S p , results in a larger neutron skin thickness, and a larger neutron separation energy, S n , leads to a smaller neutron skin thickness [34]. Due to different separation energies, ∆r np would be distinguishable for nuclides with the same relative neutron excess. Figure 2 shows the S p /S n ratios as a function of δ. One see that the S p /S n ∼ δ plot has a similar pattern as that of ∆r np ∼ δ in Fig. 1. Neutron skin thickness can be calculated by both microscopic [12][13][14][15] and macroscopic [16][17][18][19] models. In the present work, the deformed Hartree-Fock (HF) plus BCS method based on the SLy4 Skyrme force are used to calculate the neutron skin thickness. Details on the deformed Skyrme HF+BCS model are referred to [12,35].
Theoretical neutron skin thickness is extracted via ∆r np = r 2 n 1/2 − r 2 ch − 0.64, where neutron and charge rms radii are calculated by the HF+BCS [12]. Figure 3(a) shows the comparison of evaluated and calculated ∆r np values. One see that the evaluated ∆r np data are practically reproduced by the theoretical calculations, and the theory yields separate ∆r np ∼ δ curves for different isotopic chains. On the other hand, we re-calculated the ∆r np values using the experimental charge rms radii and the theoretical neutron radii. The results are given in Fig. 3(b). It is worth noting that both calculations yield consistent results, and noticeably a better agreement is achieved for Pb isotopic chain by using the experimental nuclear charge radii.
We would like to point out that the unevaluated experimental ∆r np values for Sn isotopes locate in-between the SLy4 and RMF predictions (see Fig. 4 in [12]). This indicates that the theoretical models can not be effectively constrained by the unevaluated data. However, as shown in Fig. 3, the high precision of our evaluated ∆r np values makes it possible to constrain the theoretical models.  [12]. (b) Same as (a) but using the experimental charge rms radii from [3].
The macroscopic compressible liquid-drop model gives a formula of neutron skin thickness expressed as [19] where E s denotes the symmetry energy, L the slope of the symmetry energy at saturation density ρ 0 , and K 0 the incompressibility of symmetric nuclear matter. σ 0 and C s represent the coefficients of symmetric matter surface tension and surface-asymmetry, respectively. More details are given in [19]. The separated curves for different isotopic chains can be also obtained by the formula of macroscopic model, see the inset in Fig. 1. Let us now discuss the deformed nuclei Cd and Te. Figure 4 shows ∆r np of the Cd, Sn, and Te isotopes as a function of δ. The ∆r np data for Te isotopes were taken from [36]. The fitted curve for the spherical nuclei in Fig. 1 and the HF+BCS theoretical calculations [12] are also shown for comparison. The theoretical ∆r np values of Cd and Te are located above and below the calculated curve of the Sn isotopic chain, respectively, and their difference are very small for the Cd, Sn, and Te isotopic chains, see Fig. 4. However, the experimental ∆r np values for the Cd and Te isotopes are generally smaller than the experimental ones of the Sn isotopes. Compared to the global linear fit, systematic lower ∆r np values for the Te isotopes were also reported in [37]. If only the contribution due to the quadrupole deformation is considered, the ∆r np for the deformed nucleus can be related to the proton and neutron radii of an assumed spherical shape via [14,28] ∆r np = r 2 where r 2 n(p) According to Eq. (3), a larger neutron skin thickness is expected for a deformed nucleus, assuming nucleus has the same deformation parameters for the neutron and proton distributions. However, owing to the strong Coulomb repulsion of protons, the calculations in [38,39] showed that in general the neutron matter distribution is more spherical than the proton matter distribution. Hence, the quadrupole deformation for the proton distribution is larger than that for the neutron distribution [38,40,41]. This difference has been observed for Cd and Te isotopes [42]. As a result, smaller ∆r np is expected comparing to the corresponding spherical nuclei. Thus, different deformations for neutron and proton distributions may be a reason for the deviations in Fig. 4.
Including the deformed nuclei in the global linear fit, the goodness of the fit becomes evidently worse. The normalized χ n is obtained to be 1.55 from the global fit to the data of the Ca, Ni, Sn, Pb, Cd, and Te isotopes. This χ n value is significantly outside the expected 1σ range of χ n = 1 ± 0.14. Due to large uncertainties, improved experimental data for Cd and Te isotopes are needed to confirm or disprove the present results.
Neutron skin thicknesses play an important role in constraining the EOS parameters. The SLy4 Skyrme force reproduces the separate trends observed in this work. With the SLy4 Skyrme force, the deduced EOS parameters E s , L and K 0 at saturation density are 32.00 MeV, 45.94 MeV, and 229.91 MeV [43], respectively. These parameters are consistent with recent Bayesian analysis by using ∆r np of Sn isotopes [44], also in agreement with various new analyses based on neutron star data since GW170817 [45]. However, a value of 106(37) MeV for L was determined recently [46] by the ∆r np of 0.283(71) fm for 208 Pb [47], which was deduced by model-independent parity violation electron scattering. Compared to our evaluated value for 208 Pb, the deviation is 0.116(72) fm. The ∆r np of 48 Ca from the CREX experiment would help to clarify the difference [48]. As mentioned in [45], these interesting tensions inspirit the community to make further researches.

IV. SUMMARY
Experimental neutron skin thicknesses, ∆r np , for the even-even stable Ca, Ni, Sn, Pb, and Cd isotopes have been evaluated. Systematic trends of the evaluated ∆r np as a function of relative neutron excess δ are investigated. Separate curves of ∆r np versus δ for different isotopic chains are observed from analysis of the evaluated data. This behavior has been practically reproduced by the microscopic and macroscopic models. Comparing to the experimental data of Sn isotopes, the ∆r np values of Cd and Te isotopes are systematically smaller. This might be understood by taking into account the different deformations of proton and neutron distributions in these nuclei.