Model dependence of the neutron-skin thickness on the symmetry energy

The model dependence in the correlations of the neutron-skin thickness in heavy nuclei with various symmetry energy parameters is analyzed by using several families of systematically varied microscopic mean field models. Such correlations show a varying degree of model dependence once the results for all the different families are combined. Some mean field models associated with similar values of the symmetry energy slope parameter at saturation density $L$, and pertaining to different families, yield a greater-than-expected spread in the neutron-skin thickness of the $^{208}$Pb nucleus. The effective value of the symmetry energy slope parameter $L_{\rm eff}$, determined by using the nucleon density profiles of the finite nucleus and the density derivative $S^\prime(\rho)$ of the symmetry energy starting from about saturation density up to low densities typical of the surface of nuclei, seems to account for the spread in the neutron-skin thickness for the models with similar $L$. The differences in the values of $L_{\rm eff}$ are mainly due to the small differences in the nucleon density distributions of heavy nuclei in the surface region and the behavior of the symmetry energy at subsaturation densities.


I. INTRODUCTION
The terrestrial nuclei are mostly asymmetric (i.e., N = Z), except for the light nuclei with proton number Z ≤ 28. At the other extreme, the matter in the compact astrophysical objects like neutron stars is highly asymmetric [1]. The asymmetry in the finite nuclei primarily arises due to the balance between the Coulomb energy and the nuclear symmetry energy. The conditions of β−equilibrium and charge neutrality render the matter in a neutron star to be highly asymmetric or predominantly composed of neutrons [2]. The densities at the center of nuclei are close to the normal saturation density ρ 0 (0.16 fm −3 ), whereas the densities at the center of neutron stars are predicted to be typically a few times ρ 0 . Thus, the accurate knowledge of the nuclear symmetry energy over a wide range of densities is indispensable to understand a variety of phenomena in finite nuclei as well as in neutron stars.
The details of the density dependence of the nuclear symmetry energy remain hard to isolate, though progress in this direction has been made in the last few years (see for instance Refs. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the experimental and theoretical works quoted therein). The density dependence of the nuclear symmetry energy around saturation is governed to leading order by its density derivative expressed * Electronic address: chiranjib.mondal@saha.ac.in † Electronic address: bijay.agrawal@saha.ac.in where S(ρ) is the symmetry energy at a density ρ. The macroscopic nuclear droplet model (DM) of Myers and Swiatecki [21,22] suggests that various symmetry energy parameters and the neutron-skin thickness in a heavy nucleus are related to one another. The neutron skin thickness is defined as the difference between the rms radii for the density distributions of the neutrons and protons in the nucleus: ∆r np ≡ r 2 1/2 n − r 2 1/2 p .
Nuclear mean-field models predict a nearly linear correlation of ∆r np of a heavy nucleus such as 208 Pb with the slope of the equation of state of neutron matter at a subsaturation density around 0.1 fm −3 [23,24], with the density derivative of the symmetry energy L [3,4,6,7,16,25,26], and with the surface symmetry energy in a finite nucleus [4,6,27]. The correlation of a finite nucleus property such as ∆r np with a bulk property of infinite nuclear matter such as L can be interpreted as basically due to the dependence of ∆r np on the surface symmetry energy. In a local density approximation the surface symmetry energy can be correlated with L, and this fact therefore implies the correlation between ∆r np and L. Macroscopic approaches such as the DM [21,22] often provide insightful guidance into the global features of many of these correlations [6,7,14], as it will be briefly recalled in the next section.
The Lead Radius Experiment (PREX) [28,29] has recently measured the neutron skin thickness ∆r np of 208 Pb. This experiment is performed via parity-violating electron scattering [30] and provides the first purely electroweak, model independent measurement of the weak charge form factor, closely connected to the neutron distribution of the 208 Pb nucleus [30]. By measuring the weak form factor of 208 Pb at momentum transfer q ≈ 0.475 fm −1 , PREX was able to determine ∆r np = 0.33 +0. 16 −0.18 fm [29]. Recently, a follow-up measurement of PREX has been proposed which intends to measure the neutron-skin thickness in the 208 Pb nucleus with an accuracy of 0.06 fm [31]. The hadronic probes are also used to estimate the neutron distribution in nuclei [32][33][34][35][36]. In this case, the strong interaction needs to be modeled and, therefore, deducing the neutron radius from these experiments can imply various theoretical uncertainties, which in some cases are difficult to estimate. The analyses from recent hadronic experiments have led to varying values of the neutron skin thickness of 208 Pb, ∆r np = 0.16 ± 0.02(stat)±0.04(syst) fm [35] and ∆r np = 0.211 +0.054 −0.063 fm [33]. A very recent measurement of coherent pion photo-production [37] provides a value ∆r np = 0.15 ± 0.03 fm for 208 Pb. Also a neutron skin thickness ∆r np = 0.165 ± 0.009(exp)±0.013(theor)±0.021(est) fm has been extracted recently from comparison of theory with the measured electric dipole polarizability in 208 Pb [10,19,[38][39][40].
Ongoing efforts are underway to perform an accurate and model independent measurement of the neutron-skin thickness in the 208 Pb nucleus. At the same time, it may not be straightforward for theory to extract various symmetry energy parameters from the neutron-skin thickness in a model-independent fashion. Starting from the seminal papers of more than a decade ago [3,23,24,41], the focus has mainly been on the linear correlation between the neutron-skin thickness and the slope parameter L of the symmetry energy. The correlation is satisfied to a large degree in the microscopic calculations with mean field models but it is not perfect and a certain model dependence appears in the results (see for example the plots in Refs. [3,4,6,7,14,24,41]). By model dependence we mean here that different mean-field models may predict similar values for the L parameter but different neutron skin thickness in a heavy nucleus. As it may be seen for example from Fig. 2 and Table II of Ref. [14], some models deviate from the linear correlation. This analysis was done by using different unbiasedly selected meanfield models. We would like to complement the earlier analysis with the one based on families of systematically varied models, in an attempt to identify the sources for the model dependence in the correlations.
In the present work we revisit the correlations of ∆r np with various symmetry energy parameters. The plausible causes for the existence of a model dependence in these correlations are investigated. The correlations are evaluated by using five different families of systematically varied microscopic mean-field models. Three out of these five families correspond to relativistic energy density functionals [42,43] and the remaining two families correspond to a non-relativistic functional [44]. We also predict the neutron skin thickness of the neutron-rich nucleus 132 Sn which has not been measured yet.
The paper is organized as follows. The geometrical definitions employed to decompose the neutron-skin thickness into bulk and surface contributions [14,45] are briefly outlined in Sec. II. We also provide in this section some results derived from the macroscopic DM suggesting possible connections between the neutron-skin thickness and various symmetry energy parameters. In Sec. III, the results for the correlations of the neutron-skin thickness in the 208 Pb and 132 Sn nuclei with the symmetry energy parameters obtained for several families of the systematically varied models are presented. The plausible causes for the model dependence in such correlations are investigated in detail. The main conclusions are presented in Sec. IV.

II. NEUTRON-SKIN THICKNESS AND SYMMETRY ENERGY PARAMETERS
From a geometrical point of view, the neutron skin thickness in a nucleus may be thought as originated by two different effects. One effect is due to the separation between the mean sharp surfaces of the neutron and proton density distributions. Since this effect corresponds to a different extent of the bulk region of the neutron and proton densities, we refer to it as the bulk contribution to the neutron skin thickness. The other effect is due to the different surface widths of the neutron and proton densities, which we call the surface contribution to the neutron skin thickness. To compute the bulk and surface contributions to the neutron skin thickness in a nucleus requires a proper definition of these quantities based on the nuclear densities. In this respect we follow closely the method described by Hasse and Myers [46] and which we applied in Refs. [14,45].
In order to determine the position of the neutron and proton effective surfaces one can define different radii. In particular, one can define the central radius C as Another option for the mean position of the surface is the equivalent radius R, which is the radius of a uniform sharp distribution whose density equals the bulk value of the actual density and has the same number of particles: Finally, one can also define the equivalent rms radius Q that describes a uniform sharp distribution with the same rms radius as the given density: The radii C, R, and Q are related by the expressions [46] (6) where b is the surface width of the density profile defined as which provides a measure of the extent of the surface of the nucleus. The neutron skin thickness, which is defined through the rms radii, can be expressed by and using Eq.(6) reads: which clearly separates the bulk and surface contributions as and ∆r surf np ≡ 3 5 In Eqs. (9) and (11), we have neglected O b 4 /R 3 and higher-order terms since they represent a small correction [14] to ∆r np -of less or around a 1-2%-that will leave our conclusions unchanged. In order to extract the bulk and surface contributions to the neutron skin thickness from the quantal proton and neutron densities obtained within the Skyrme Hartree-Fock or the relativistic mean-field models, we proceed as in Refs. [14,45]. That is, we fit the self-consistent quantal proton and neutron densities by two-parameter Fermi (2pF) distributions where q = n, p. The parameters ρ 0,q , C q and a q are adjusted to reproduce the nucleon numbers as well as the values for the second and fourth moments of the actual density distributions, i.e., r 2 q and r 4 q . Once this fit is done, we can express Eqs. (9)-(11) for the neutron skin thickness in terms of the parameters C q and a q taking into account Eq.(6) and the fact that for a 2pF distribution b = πa/ √ 3. Therefore, the bulk and surface contributions to the neutron skin thickness can be written as up to terms of order O a 4 /C 3 . It should be mentioned that, the ∆r np values calculated from the actual densities obtained self consistently match very well with the ones calculated by summing Eqs. (13) and (14) after applying our prescription to determine the parameters of the Fermi function. Some insight about possible correlations between the neutron skin thickness and different observables related to the symmetry energy is provided by the DM [22]. Within this model, which neglects shell correction effects, the neutron skin thickness is expressed by where e 2 Z/70J is a correction due to the Coulomb interaction, R = r 0 A 1/3 is the nuclear radius, and b n and b p are the surface widths of the neutron and proton density profiles. The quantity t in (15) represents the distance between the location of the neutron and proton mean surfaces and therefore is proportional to the bulk contribution to the neutron skin thickness. In the DM its value is given by with where I = (N − Z)/A, J is the bulk symmetry energy at saturation, and Q stiff is the surface stiffness. For each mean field model, the parameters r 0 and J can be obtained from calculations in infinite nuclear matter and Q stiff from calculations performed in semi-infinite nuclear matter [7,47,48]. Within the DM, the symmetry energy coefficient of a finite nucleus of mass number A is given by Replacing a sym (A) in Eq. (16), the separation distance between the mean surfaces of neutrons and protons can be recast as The link between a property in finite nuclei such as a sym (A) and some symmetry energy parameters in infinite nuclear matter may be obtained from the observation [6] that for a heavy nucleus there is a subsaturation density, which for 208 Pb is around 0.1 fm −3 , such that the symmetry energy coefficient in the finite nucleus a sym (A) equals the symmetry energy in nuclear matter S(ρ) computed at that density. This relation is roughly independent of the mean field model used to compute it. Around the saturation density ρ 0 the symmetry energy can be expanded as Consequently, the distance t can be finally expressed approximately as [6] (21) Equations (19) and (21) suggest correlations between the bulk neutron skin thickness in finite nuclei and some isovector indicators such as J − a sym (A), a sym (A)/J and L, which will be discussed in detail along this paper. To compute the average symmetry energy of a finite nucleus with the DM (Eq. (18)) requires the knowledge of the surface stiffness Q stiff , which in turn requires semi-infinite nuclear matter calculations [7]. An efficient procedure to circumvent this, is to evaluate a sym (A) within a local density approximation as [9] a sym (A) = 4π where is the local isospin asymmetry and ρ(r) is the sum of the neutron and proton densities. This approximation works very well for medium heavy 132 Sn or heavy 208 Pb nuclei [49].

III. RESULTS AND DISCUSSIONS
The neutron-skin thickness and several symmetry energy parameters are calculated using five different families of systematically varied models, namely, the SAMi-J [10,50], DDME [51], FSV, TSV and KDE0-J models. The energy density functional associated with DDME, FSV, and TSV corresponds to an effective Lagrangian density typical of the relativistic mean-field models, whereas SAMi-J and KDE0-J are based on the standard form of the Skyrme force.
We have obtained the different families of systematically varied parameter sets so that they explore different values of the symmetry energy parameters around an optimal value, while reasonably keeping the quality of the best fit. The values of the neutron-skin thickness in a heavy nucleus like 208 Pb vary over a wide range within the families due to the variations of the symmetry energy parameters. The parameter sets for the FSV, TSV and KDE0-J families are obtained in the present work. The effective Lagrangian density employed for the FSV family is similar to that for the FSU model [52]. In addition to the coupling of ρ meson to the nucleons as conventionally employed, the presence of a crosscoupling between the ω and ρ mesons in the FSU model enables one to vary the symmetry energy, and accordingly the symmetry energy slope parameter L, over a wide range without significantly affecting the quality of the fit to the bulk properties of the finite nuclei. The TSV family is obtained using the effective Lagrangian density as introduced in Ref. [53] in which the ρ−meson and its coupling to the σ−meson govern the isovector part of the interactions between the nucleons. The ω − ρ cross coupling in the FSV family and the σ − ρ cross coupling in the TSV family produce different behaviors in the density dependence of the symmetry energy, because the source term for the ω-field is governed by the baryon density and that for the σ-field is governed by the scalar density. The experimental data employed to determine the TSV and FSV families are the total binding energies for the 16 O, 40,48 Ca, 68 Ni, 90 Zr, 100,132 Sn, 208 Pb nuclei, and the root mean square charge radii for the 16 O, 40,48 Ca, 90 Zr, 208 Pb nuclei. The energy density functional for the KDE0-J family calculated within the Skyrme ansatz is taken from the KDE0 force of Ref. [54]. The model parameters are constrained to yield the nuclear matter incompressibility coefficient in the range of 225-250 MeV. The calculated values of the total binding energy and the charge radius for the 208 Pb nucleus obtained for all the models considered deviate from the experimental data only within 0.25% and 0.8%, respectively.

A. Correlation plots associated with isovector indicators
As we discussed in the previous Section, the DM is a useful guideline to suggest the kind of correlations that we can expect between the neutron skin thickness and the symmetry energy parameters. As shown in Ref. [14], these correlations are mainly due to the bulk term of Eq.(15) rather than to the surface contribution to ∆r np . In the bulk part of ∆r np , the quantity (J − a sym (A)) /J determines the ratio of the surface symmetry to volume symmetry energies, see Eq. (19); the close relation of different isovector observables in finite nuclei with the ratio of the surface and volume symmetry energies has been observed in several studies, cf. for example Refs. [11,27] and references therein. The values of r 0 for the various models considered in the present work display only a small variation indicating that the total neutron-skin thickness ∆r np of a given heavy nucleus may be correlated to the ratio (J − a sym (A)) /J, or also to the difference (J − a sym (A)) provided the value of J does not show a large variation as compared to (J − a sym (A)).
In Fig. 1 At this point, it is interesting to address the constraints on the neutron-skin thickness that may be deduced from  [57] using the experimental binding energy differences. Furthermore, the effect of the Coulomb interaction on the surface asymmetry and the effect of the surface diffuseness on the Coulomb energy were taken into account. The value of J = 32.5 ± 2.5 MeV as used in the present work has a quite reasonable overlap with the ones extracted either from a version of the finiterange droplet model (FRDM) that performs very well in reproducing the experimental mass systematics [58], by analyzing the experimental data on the electric dipole polarizability in 68 Ni, 120 Sn and 208 Pb nuclei [19], from specific manipulation of the semi-empirical mass formula [59], through analysis of the properties of semi-infinite nuclear matter [60], or by analyzing pygmy dipole resonance data on 68 Ni and 132 Sn nuclei [61]. This value of J also overlaps with the conclusions provided in recent papers [13,62].
It is desirable to check the degree of consistency between the results for different heavy nuclei, in particular between 208 Pb and 132 Sn which would allow to predict the neutron skin thickness of the nucleus 132 Sn assumed that the one of 208 Pb is known. In the left panel of Fig. 3, we plot ∆r np for the 132 Sn nucleus against that for the 208 Pb nucleus. Similarly, the results for ∆r bulk np and ∆r surf np are plotted in the middle and right panels of Fig. 3, respectively. It is observed that the values of ∆r np , ∆r bulk np and ∆r surf np for the 132 Sn nucleus are very well correlated with the corresponding values in the 208 Pb nucleus. This is in harmony with earlier work [8]. Hence, the information provided by the neutron skin of two heavy nuclei on the isovector channel of the nuclear effective interaction is mutually inclusive. Such an observation allows us to predict ∆r np = 0.260 ± 0.050 fm for 132 Sn nucleus by using the above estimated value for 208 Pb of ∆r np = 0.197 ± 0.047 fm.
As recalled above, and discussed in the literature (cf., in particular, Ref. [14] and references therein), we expect that the correlation between the neutron-skin thickness and (J − a sym (A)) /J leads to a correlation between the neutron-skin thickness and the symmetry energy slope parameter L. In Fig. 4, we display the variation of L as a function of ∆r np (left), ∆r bulk np (middle) and ∆r surf np (right panel) for the 208 Pb nucleus in the analyzed families of models. Using the constraint on ∆r np ( 208 Pb) obtained in Fig. 2, the bound on the value of L comes out to be L = 64 ± 23 MeV; displayed as the shaded region of left panel in Fig. 4. The correlation coefficients of L with ∆r np and with ∆r bulk np are lower than in the case of the correlations displayed in Figs. 1 and 2, suggesting that the neutron-skin thickness is slightly better correlated with J − a sym (A) or the ratio a sym (A)/J than with the slope parameter L. This might be a feature of the families we have chosen and does not necessarily apply to the situation in which one employs a large set of unbiasedly selected models [14]. As above, the ∆r np -L correlation is weaker in comparison to the ∆r bulk np -L correlation, in qualitative agreement with Ref. [14].
The "arrow"marks in Fig. 4 indicate the five models, each from a different family, with L varying in a narrow range of 62.1 MeV to 67.0 MeV. For these five models, there happens to be a spread in ∆r np of almost 0.05 fm which is larger than expected. In comparison, the equation of the linear fit of the results of all models in the left panel of Fig. 4 gives a variation in the value of ∆r np ( 208 Pb) with the change of L as, δ(∆r np ) ≃ 0.002 δL, so that a change in L of 5 MeV implies an average change in ∆r np of about 0.01 fm only, which is smaller than the observed spread of 0.05 fm in the five models mentioned above. The DM supports a similar conclusion, as it can be seen from Eq. (21) that the DM predicts an average variation of ∆r np ( 208 Pb) with L approximately as, δ(∆r np ) ≃ 0.003 δL. The two mentioned models from the TSV and SAMi-J families have L = 67 MeV and L = 63.2 MeV, respectively, and yield in 208 Pb smaller values of ∆r np ≃ 0.18 fm, whereas the two models from the FSV and DDME families have L = 64.8 MeV and L = 62.1 MeV, respectively, and give rise to larger values of ∆r np ≃ 0.22 fm. The model from KDE0-J family with L = 65.7 MeV yields an intermediate value of ∆r np ( 208 Pb) ≃ 0.19 fm. Actually, it comes as an intriguing fact that the extracted values of ∆r np differ by ∼ 0.05 fm for the two models of the FSV and TSV families with similar L, although the parameters for these two families are obtained by using exactly the same kind of fitting protocol. In the next subsection, we aim to search for plausible interpretations for such differences in the neutron skin thickness corresponding to models with similar L values.

B. Systematic differences between the families of functionals
In an attempt to understand the issues raised at the end of the previous subsection, we make a detailed comparison between the results for the five models belonging to different families but yielding almost the same values for L. We first take a closer look in Fig. 5 into the values of the symmetry energy S(ρ) (lower panel) and its density derivative 3ρ 0 S ′ (ρ) (upper panel) as a function of density for these models. The behavior of S(ρ) as a function of density seemingly appears to be similar for the five models. But the values of 3ρ 0 S ′ (ρ) show significant differences in the low density region (ρ < 0.10 fm −3 ). Furthermore, one may note that the TSV and SAMi-J models corresponding to ∆r np ( 208 Pb) ∼ 0.18 fm and the KDE0-J model with ∆r np ( 208 Pb) ∼ 0.19 fm display a relatively similar behavior in the density dependence of S ′ (ρ). The same is true for the FSV and DDME models corresponding to ∆r np ( 208 Pb) ∼ 0.22 fm.
To investigate whether such differences in the values of the density derivative of the symmetry energy at lower densities have an influence in the finite nuclei calculations, and motivated by Eq. (22), we determine an effective value of the slope parameter L eff , which might be more sensitive to the relative distributions of neutrons with respect to protons in finite nuclei, as follows: Here, I(r) is the local asymmetry parameter defined as, I(r) ≡ (ρ n (r) − ρ p (r))/ρ(r). If one assumes S(ρ) to be linear in density, the L eff parameter coincides with L (see Eq. (20)). However, we have seen in Fig. 5 that S(ρ) can depart significantly from linearity at low densities. MeV. Each of these models belongs to a different family (see also Table I).
Therefore, the L eff parameter as defined in Eq. (23) tries to take into account this effect. At very low densities (ρ < 0.01 fm −3 ) S(ρ) deviates largely from linearity. The integrals in the numerator and denominator of Eq. (23) are thus evaluated by integrating from the center of the nucleus, where the density ρ(r) is of the order of ρ 0 , up to the point where the density of the nucleus falls to 0.01 fm −3 , which corresponds to a radial coordinate r of about 9 fm. It is worthwhile to mention that we wanted to study the effect of S ′ (ρ) but not the quantity L(ρ) (≡ 3ρS ′ (ρ)) on the ∆r np of a heavy nucleus. That is why we kept ρ 0 outside the integral of the numerator in Eq. (23). The values of L eff along with various other properties evaluated for the five models corresponding to L ∼ 65 MeV are compared in Table I. It can be easily observed in Table I that though the values of L for these models vary only by ∼ 5 MeV, the values of ∆r np of heavy nuclei calculated from the same models can differ by ∼ 0.05 fm, which is larger than the average spread of the correlation between ∆r np and L. Interestingly, when we look at the extracted L eff parameter, the models from SAMi-J and TSV families those predict ∆r np ( 208 Pb) ∼ 0.18 fm give similar L eff ∼ 82 MeV, and the models from FSV and DDME families those predict ∆r np ( 208 Pb) ∼ 0.22 fm give similar L eff ∼ 96 MeV. The model from the KDE0-J family with ∆r np ( 208 Pb) ∼ 0.19 fm predicts L eff ∼ 91 MeV. That is, the models with larger L eff give larger ∆r np and vice versa. In fact, further inspection of Fig. 4 reveals that two members of the FSV and DDME families with MeV in the FSV model and L = 46.5 MeV in the DDME model). It turns out that these FSV and DDME models also explore similar values of L eff (83.9 MeV in FSV and 86.6 MeV in DDME) as done by the models from the SAMi-J and TSV families displayed in Table I with ∆r np ∼ 0.18 fm. In principle, one can also define L eff without the I 2 (r) terms in Eq. (23). That is why, we repeated the calculations of L eff by taking I 2 (r) to be unity in Eq. (23) and found similar trends as explained above. In Table I, concerning the properties of uniform matter, it is also noticeable that the models do not display the same value of the saturation density. For the non-relativistic functionals belonging to the SAMi-J and KDE0-J family this value is about 5-10% larger than the values explored by the relativistic functionals. This fact has some impact on the extracted values of L eff for these models (see Eq. (23)).
To have a better insight into the source of the differences between the values of L eff for the models with similar values of L at ρ 0 , we plot in Fig. 6 the total density distribution ρ(r) of 208 Pb multiplied by r 2 I 2 (r) for the models with L ∼ 65 MeV. The values of r 2 ρ(r)I 2 (r) for all the different cases are close to each other up to r ∼ 6 fm, in this region ρ(r) 0.1 fm −3 . With further increase in r, the differences in the values of r 2 ρ(r)I 2 (r) gradually become noticeable. One can argue that different behaviors in the surface region may be responsible for different values of L eff and consequently lead to different values of ∆r np in heavy nuclei like 208 Pb or 132 Sn. The question still remains whether L eff is more sensitive to the density dependence of S ′ (ρ) (upper panel of Fig. 5) or to the density distributions of nucleons inside the nucleus (Fig. 6). To unmask this, we calculated the values of L eff using S ′ (ρ) of a given model, but with the density distributions of nucleons from the five models that have L ∼ 65 MeV. We repeated this calculation for the different choices of S ′ (ρ) of these five models. The values of L eff so obtained did not show the trend as observed in Table I, where S ′ (ρ) and the density distributions of nucleons used correspond to the same model consistently.
Thus, the values of L eff are sensitive to both the density dependence of the symmetry energy and the density distributions of nucleons inside the nucleus. To this end, we would like to point out that the differences in the values of L eff for the models with similar L parameter are mainly due to the differences in the low density behavior of S ′ (ρ) and the distributions of nucleons in the surface region of the nucleus.

IV. SUMMARY
In this work, we revisit the correlations of the neutronskin thickness in finite nuclei with various symmetry energy parameters pertaining to infinite nuclear matter. Particular attention is paid to the model dependence in such correlations that can play a role in understanding the density dependence of the nuclear symmetry energy. The finite nuclei analyzed are 208 Pb and 132 Sn. The symmetry energy parameters considered are J − a sym (A), a sym (A)/J and L, where J and L are the symmetry energy and the symmetry energy slope associated with infinite nuclear matter at the saturation density, and a sym (A) corresponds to the symmetry energy parameter in finite nuclei. Five different families of systematically varied mean-field models corresponding to different energy density functionals are employed to calculate the relevant quantities for the finite nuclei and those for the infinite nuclear matter. Consideration of recent constraints on the symmetry energy parameters (a sym (A) and J) and the present correlations suggest the values ∆r np = 0.197 ± 0.047 fm and ∆r np = 0.260 ± 0.050 fm for the neutron skin thickness in the 208 Pb and 132 Sn nuclei, respectively and L = 64 ± 23 MeV.
In general, the correlations of the neutron-skin thickness with the different symmetry energy parameters are strong within the individual families of the models. Once the results for all the different families are combined, the correlation coefficients become smaller, indicating a model dependence. The neutron skin in a nucleus entails two main components related to the geometry of the nucleon density profiles. On the one hand, there is a bulk contribution (∆r bulk np ) produced by the separation between the effective sharp surfaces of the density distributions of neutrons and protons. On the other hand, there is a surface contribution (∆r surf np ) caused by the different surface widths of the neutron and proton density profiles. The correlations of the symmetry energy parameters with the bulk part ∆r bulk np of the neutronskin thickness are less model dependent than with the total neutron-skin thickness ∆r np . Exceptionally, the bulk part of the neutron-skin thickness is correlated with J − a sym (A) and a sym (A)/J in an almost model independent manner. This fact is much compatible with the predictions of the macroscopic droplet model. We notice a model dependence in the correlations of the neutron-skin thickness with the symmetry energy slope parameter L when the results of the various families of models are considered together. By model dependence we mean that different models of different families with the same value of the slope L of the symmetry energy predict different neutron skin thickness, or vice versa. For different models having similar slope parameter L ∼ 65 MeV and belonging to the different families, a spread in ∆r np of about 0.05 fm is observed, which is large in view of the average spread of the correlation (Fig. (4)), as well as in view of the DM estimate for the change of ∆r np with L.
We have found two independent indications that the surface of the nucleus plays a key role in introducing a model dependence, or in other words, a systematic theoretical uncertainty, to the well-known linear correlation between the neutron skin thickness and L and to some other correlations that can be used to extract the parameters characterizing the density dependence of the symmetry energy. These indications are, (i) the existence of stronger correlations of various symmetry energy parameters with the bulk part of the neutron-skin thickness rather than with the total neutron-skin thickness, and (ii) the differences between the density distributions for the nucleons at the surface region for the different models corresponding to similar values of the slope parameter L.
To understand better the model dependence in the various correlations considered, the results are compared for the models belonging to different families, but yielding similar values of L. We have determined an effective value of the symmetry energy slope parameter L eff using the density distributions of nucleons and the density derivative of the symmetry energy for these models. It is found that the values of ∆r np , which differ for the models with the same L ∼ 65 MeV, are in harmony with the values of L eff . We conclude that differences in the values of L eff caused by differences in the density distributions of nucleons in the surface region and the derivative of the symmetry energy at subsaturation densities are the plausible sources for the aforesaid model dependence.