Origin of the neutron skin thickness of 208Pb in nuclear mean-field models

We study whether the neutron skin thickness (NST) of 208Pb originates from the bulk or from the surface of the nucleon density distributions, according to the mean-field models of nuclear structure, and find that it depends on the stiffness of the nuclear symmetry energy. The bulk contribution to NST arises from an extended sharp radius of neutrons, whereas the surface contribution arises from different widths of the neutron and proton surfaces. Nuclear models where the symmetry energy is stiff, as typical relativistic models, predict a bulk contribution in NST of 208Pb about twice as large as the surface contribution. In contrast, models with a soft symmetry energy like common nonrelativistic models predict that NST of 208Pb is divided similarly into bulk and surface parts. Indeed, if the symmetry energy is supersoft, the surface contribution becomes dominant. We note that the linear correlation of NST of 208Pb with the density derivative of the nuclear symmetry energy arises from the bulk part of NST. We also note that most models predict a mixed-type (between halo and skin) neutron distribution for 208Pb. Although the halo-type limit is actually found in the models with a supersoft symmetry energy, the skin-type limit is not supported by any mean-field model. Finally, we compute parity-violating electron scattering in the conditions of the 208Pb parity radius experiment (PREX) and obtain a pocket formula for the parity-violating asymmetry in terms of the parameters that characterize the shape of the 208Pb nucleon densities.


I. INTRODUCTION
The study of the size and shape of the density distributions of protons and neutrons in nuclei is a classic, yet always contemporary area of nuclear physics. The proton densities of a host of nuclei are known quite well from the accurate nuclear charge densities measured in experiments involving the electromagnetic interaction [1], like elastic electron scattering. In contrast, the neutron densities have been probed in fewer nuclei and are generally much less certain.
The neutron distribution of 208 Pb, and its rms radius in particular, is nowadays attracting significant interest in both experiment and theory. Indeed, the neutron skin thickness, i.e., the neutron-proton rms radius difference ∆r np = r 2 1/2 of this nucleus has close ties with the density-dependent nuclear symmetry energy and with the equation of state of neutron-rich matter. In nuclear models, ∆r np of 208 Pb displays nearly linear correlations with the slope of the equation of state of neutron matter [2][3][4], with the density derivative L of the symmetry energy [5][6][7][8][9][10][11], and with the surface-symmetry energy of the finite nucleus [9]. At first sight, it may seem intriguing that a property of the mean position of the surface of the nucleon densities (∆r np ) is correlated with a purely bulk property of infinite nuclear matter (L). However, we have to keep in mind that ∆r np depends on the surface-symmetry energy. This quantity reduces the bulk-symmetry energy due to the finite size of the nucleus. Assuming a local density approximation, we can correlate the surface-symmetry energy with the density slope L, which determines the departure of the symmetry energy from the bulk value. The correlation of ∆r np with L then follows. Actually, these correlations can be derived almost analytically starting from the droplet model (DM) of Myers andŚwiatecki [12,13] as we showed in Refs. [9,10]. By reason of its close connections with the nuclear symmetry energy, knowing accurately ∆r np of 208 Pb can have important implications in diverse problems of nuclear structure and of heavy-ion reactions, in studies of atomic parity violation, as well as in the description of neutron stars and in other areas of nuclear astrophysics (see, e.g., Refs. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]). Since the charge radius of 208 Pb has been measured with extreme accuracy (r ch = 5.5013(7) fm [1]), the neutron rms radius of 208 Pb is the principal unknown piece of the puzzle.
The lead parity radius experiment (PREX) [29] is a challenging experimental effort that aims to determine r 2 1/2 n of 208 Pb almost model independently and to 1% accuracy by parity-violating electron scattering [29,30]. This purely electroweak experiment has been run at the Jefferson Lab very recently, although results are not yet available. The parity-violating electron scattering is useful to measure neutron densities because in the low-momentum transfer regime the Z 0 boson couples mainly to neutrons. For protons, this coupling is highly suppressed because of the value of the Weinberg angle. Therefore, from parity-violating electron scattering one can obtain the weak charge form factor and the closely related neutron form factor. From these data, the neutron rms radius can in principle be deduced [30]. This way of proceeding is similar to how the charge density is obtained from unpolarized electron scattering data [30]. The electroweak experiments get rid of the complexities of the hadronic interactions and the reaction mechanism does not have to be modeled. Thus, the analysis of the data can be both clean and model independent. There may be a certain model dependence, in the end, in having to use some neutron density shape to extract the neutron rms radius from the parity-violating asymmetry measured at a finite momentum transfer.
To date, the existing constraints on neutron radii, skins, and neutron distributions of nuclei have mostly used strongly interacting hadronic probes. Unfortunately, the measurements of neutron distributions with hadronic probes are bound to have some model dependence because of the uncertainties associated with the strong force. Among the more frequent experimental techniques we may quote nucleon elastic scattering [31,32], the inelastic excitation of the giant dipole and spin-dipole resonances [33,34], and experiments in exotic atoms [35][36][37][38]. Recent studies indicate that the pygmy dipole resonance may be another helpful tool to constrain neutron skins [21,27].
The extraction of neutron radii and neutron skins from the experiment is intertwined with the dependence of these quantities on the shape of the neutron distribution [35][36][37][38][39]. The data typically do not indicate unambiguosly, by themselves, if the difference between the peripheral neutron and proton densities that gives rise to the neutron skin is caused by an extended bulk radius of the neutron density, by a modification of the width of the surface, or by some combination of both effects. In the present work we look for theoretical indications on this problem and study whether the origin of the neutron skin thickness of 208 Pb comes from the bulk or from the surface of the nucleon densities according to the mean-field models of nuclear structure. The answer turns out to be connected with the density dependence of the nuclear symmetry energy in the theory.
We described in Ref. [40] a procedure to discern bulk and surface contributions in the neutron skin thickness of nuclei. It can be applied to both theoretical and experimental nucleon densities as it only requires to know the equivalent sharp radius and surface width of these densities, which one can obtain by fitting the actual densities with two-parameter Fermi (2pF) distributions. The 2pF shape is commonly used to characterize nuclear densities and nuclear potentials in both theoretical and experimental analyses. The doubly magic number of protons and neutrons in 208 Pb ensures that deformations do not influence the results and spherical density distributions describe the nuclear surface very well. We perform our calculations with several representative effective nuclear forces, namely, nonrelativistic interactions of the Skyrme and Gogny type and relativistic mean-field (RMF) interactions. The free parameters and coupling constants of these nuclear interactions have usually been adjusted to describe data that are well known empirically such as binding energies, charge radii, single-particle properties, and several features of the nuclear equation of state. However, the same interactions predict widely different results for the size of neutron skin of 208 Pb and, as we will see, for its bulk or surface nature. We also study the halo or skin character [35][36][37][38][39] of the nucleon densities of 208 Pb in mean-field models. Finally, we perform calculations of parity-violating electron scattering on 208 Pb. We show that if 2pF nucleon densities are assumed, the parityviolating asymmetry as predicted by mean-field models can be approximated by a simple and analytical expression in terms of the central radius and surface width of the neutron and proton density profiles. This suggests that an experiment such as PREX could allow to obtain some information about the neutron density profile of the 208 Pb nucleus in addition to its neutron rms radius.
The rest of the article proceeds as follows. In Sec. II, we summarize the formalism to decompose the neutron skin thickness into bulk and surface components. The results obtained in the nuclear mean-field models are presented and discussed in Sec. III. A summary and the conclusions are given in Sec. IV.

II. FORMALISM
The analysis of bulk and surface contributions to the neutron skin thickness of a nucleus requires proper definitions of these quantities based on nuclear density distributions. We presented in Ref. [40] such a study, and we summarize only its basic points here.
One can characterize the size of a nuclear density distribution ρ(r) through several definitions of radii, and each definition may be more useful for a specific purpose (see Ref. [41] for a thorough review). Among the most common radii, we have the central radius C: the equivalent sharp radius R: i.e., the radius of a uniform sharp distribution whose density equals the bulk value of the actual density and has the same number of particles; and the equivalent rms radius Q: which describes a uniform sharp distribution with the same rms radius as the given density. These three radii are related by expansion formulas [41]: Here, b is the surface width of the density profile: which provides a measure of the extent of the surface of the density. Relations (5) usually converge quickly because b/R is small in nuclei, especially in heavy-mass systems. Nuclear density distributions have oscillations in the inner bulk region and a meaningful average is needed to determine the density values ρ(0) and ρ(bulk) appearing in the above equations. This can be achieved by matching the original density with a 2pF distribution: In 2pF functions the bulk density value corresponds very closely to the central density, and the latter coincides to high accuracy with the ρ 0 parameter if exp (−C/a) is negligible. The surface width b and the diffuseness parameter a of a 2pF function are related by b = (π/ √ 3)a. As discussed in Ref. [41], the equivalent sharp radius R is the quantity of basic geometric importance of the C, Q, and R radii. This is because a sharp distribution of radius R has the same volume integral as the density of the finite nucleus and differs from it only in the surface region. We illustrate this fact in Fig. 1 using as example the neutron density of 208 Pb of a mean-field calculation. We can see that the mean-field density is clearly overestimated in the whole nuclear interior by a sharp sphere of radius C. The equivalent rms radius Q fails also, by underestimating it. Only the equivalent sharp radius R is able to reproduce properly the bulk part of the original density profile of the nucleus. Therefore, R appears as the suitable radius to describe the size of the bulk of the nucleus.
As the neutron skin thickness (1) is defined through rms radii, it can be expressed with Q: Recalling from (5) that Q ≃ R + 5 2 (b 2 /R), we have a natural distinction in ∆r np between bulk (∝ R n − R p ) and surface contributions. That is to say, with ∆r bulk np = independent of surface properties, and ∆r surf np = 3 5 of surface origin. The nucleus may develop a neutron skin by separation of the bulk radii R of neutrons and protons or by modification of the width b of the surfaces of the neutron and proton densities. In the general case, both effects are expected to contribute. We note that Eq. (11) coincides with the expression of the surface width contribution to the neutron skin thickness provided by the DM of Myers andŚwiatecki [12,13] if we set in Eq.
The next-order correction to Eq. (11) can be easily evaluated for 2pF distributions (cf. Ref. [41] for the higher-order corrections to the expansions (5)) and gives ∆r surf,corr This quantity is usually very small-indeed, we neglected it in [40]. In the case of 208 Pb, we have found in all calculations with mean-field models that it is between −0.0025 fm and −0.004 fm, and thus can be obviated for most purposes. But because in the present work we deal with some detailed comparisons among the models, we have included (12) in the numerical values shown for the surface contribution ∆r surf np in the later sections. It is to be mentioned that there is no universal method to do the parametrization of the neutron and proton densities with 2pF functions. A popular prescription is to use a χ 2 minimization of the differences between the density to be reproduced and the 2pF profile, or of the differences between their logarithms. These methods may somewhat depend on conditions given during minimization (number of mesh points, limits, etc.). As in [40], we have preferred to extract the parameters of the 2pF profiles by imposing that they reproduce the same quadratic r 2 and quartic r 4 moments of the self-consistent mean-field densities, and the same number of nucleons. These conditions allow us to determine in a unique way the equivalent 2pF densities and pay attention to a good reproduction of the surface region of the original density because the local distributions of the quantities r 2 ρ(r) and r 4 ρ(r) are peaked at the peripheral region of the nucleus. An example of this type of fit is displayed in Fig. 1 by the dash-dotted line. It can be seen that the equivalent 2pF distribution nicely averages the quantal oscillations at the interior and reproduces accurately the behavior of the mean-field density at the surface.
As is well known, nonrelativistic and relativistic models differ in the stiffness of the symmetry energy. Note that by soft or stiff symmetry energy we mean that the symmetry energy increases slowly or rapidly as a function of the nuclear density around the saturation point. Of course, the soft or stiff character can depend on the explored density region; for example, it is possible that a symmetry energy that is soft at nuclear densities becomes stiff at much higher densities [49], or that a model with a stiff symmetry energy at normal density has a smaller symmetry energy at low densities [50]. The density dependence of the nuclear symmetry energy c sym (ρ) around saturation is frequently parametrized through the slope L of c sym (ρ) at the saturation density: The pressure of pure neutron matter is directly proportional to L [51] and thus the L value has important implications for both neutron-rich nuclei and neutron stars. The symmetry energy of the Skyrme and Gogny forces analyzed in this work displays, as usual in the nonrelativistic models, from the very soft to the moderately stiff density dependence at nuclear densities (see Table I for the L parameter of the models). On the contrary, the majority of the relativistic parameter sets have a stiff or very stiff symmetry energy around saturation. The exception to the last statement in our case are the covariant parameter sets FSUGold and DD-ME2 that have a milder Pb calculated with the self-consistent densities of several nuclear mean-field models (∆r s.c. np ) and its partition into bulk and surface contributions defined in Sec. II, as well as the relative weight of these bulk and surface parts. The models have been set out according to increasing ∆r s.c.
np . The density slope L of the symmetry energy of the models is also listed. In order to help distinguish relativistic and nonrelativistic models, we have preceded the relativistic ones with an r in this symmetry energy than the typical RMF models. FSUGold achieves this through an isoscalar-isovector nonlinear meson coupling [46] and DD-ME2 because of having density-dependent meson-exchange couplings [47].
In Table I we display the neutron skin thickness of 208 Pb obtained from the self-consistent densities of the various interactions (denoted as ∆r s.c. np ). It is evident that the nuclear mean-field models predict a large window of values for this quantity. The nonrelativistic models with softer symmetry energies point toward a range of about 0.1-0.17 fm. Most of the relativistic models, having a stiff symmetry energy, point toward larger neutron skins of 0.25-0.3 fm. In between, the relativistic models DD-ME2 and FSUGold predict a result close to 0.2 fm and the Skyrme interactions that have relatively stiffer symmetry energies fill in the range between 0.2 and 0.25 fm.
Before proceeding, we would like to briefly survey some of the recent results deduced for ∆r np in 208 Pb from experiment. For example, the recent analysis in Ref. [52] of the data measured in the antiprotonic 208 Pb atom [35,36] gives ∆r np = 0.16 ± (0.02) stat ± (0.04) syst fm, including statistical and systematic errors. Another recent study [53] of the antiprotonic data for the same nucleus leads to ∆r np = 0.20 ± (0.04) exp ± (0.05) th fm, where the theoretical error is suggested from comparison of the models with the experimental charge density. These determinations are in consonance with the average value of the hadron scattering data for the neutron skin thickness of 208 Pb, namely, ∆r np ∼ 0.165 ± 0.025 fm (taken from the compilation of hadron scattering data in Fig. 3 of Ref. [36]). We may also mention that the constraints on the nuclear symmetry energy derived from isospin diffusion in heavy ion collisions of neutron-rich nuclei suggest ∆r np = 0.22 ± 0.04 fm in 208 Pb [54]. Following Ref. [55], the same type of constraints exclude ∆r np values in 208 Pb less than 0.15 fm. A recent prediction based on measurements of the pygmy dipole resonance in 68 Ni and 132 Sn gives ∆r np = 0.194 ± 0.024 fm in 208 Pb [27]. Finally, we quote the new value ∆r np = 0.211 +0.054 −0.063 fm determined in [56] from proton elastic scattering. Thus, in view of the empirical information for the central value of ∆r np and in view of the ∆r s.c.
np values predicted by the theoretical models in Table I, it may be said that those interactions with a soft (but not very soft) symmetry energy, for example, HFB-17, SLy4, SkM*, DD-ME2, or FSUGold, agree better with the determinations from experiment. Nevertheless, the uncertainties in the available information for ∆r np are rather large and one cannot rule out the predictions by other interactions. If the PREX experiment [29,30] achieves the purported goal of accurately measuring the neutron rms radius of 208 Pb, it will allow to pin down more strictly the constraints on the neutron skin thickness of the mean-field models.
B. Bulk and surface contributions to ∆rnp of 208 Pb in nuclear models and the symmetry energy We next discuss the results for the division of the neutron skin thickness of 208 Pb into bulk (∆r bulk np ) and surface (∆r surf np ) contributions in the nuclear mean-field models, following Sec. II. We display this information in Table I. It may be noticed that the value of ∆r bulk np plus ∆r surf np (quantities obtained from Eqs. (10)- (12)) agrees excellently with ∆r s.c.
np (neutron skin thickness obtained from the self-consistent densities). One finds that the bulk contribution ∆r bulk np to the neutron skin of 208 Pb varies in a window from about 0.03 fm to 0.22 fm. The surface contribution ∆r surf np is comprised approximately between 0.07 fm and 0.085 fm in the nonrelativistic forces, and between 0.085 fm and 0.105 fm in the relativistic ones. Thus, whereas the bulk contribution to the neutron skin thickness of 208 Pb changes largely among the different mean-field models, the surface contribution remains confined to a narrower band of values. Table I shows that the size of the neutron skin thickness of 208 Pb is divided into bulk and surface contributions in almost equal parts in the nuclear interactions that have soft symmetry energies (say, L ∼ 20-60). This is the case of multiple nonrelativistic interactions and When the symmetry energy becomes softer, the bulk part tends to be smaller. Indeed, we see that in the models that have a very soft symmetry energy (L 20), which we may call "supersoft" [57], the surface contribution takes over and it is responsible for the largest part (∼ 75%) of ∆r np of 208 Pb. At variance with this situation, in the models with stiffer symmetry energies (L 75) about two thirds of ∆r np of 208 Pb come from the bulk contribution, as seen in the Skyrme forces of stiffer symmetry energy and in all of the relativistic forces that have a conventional isovector channel (G2, TM1, NL3, etc.). We therefore note that in a heavy neutronrich nucleus with a sizable neutron skin such as 208 Pb, the nuclear interactions with a soft symmetry energy predict that the contribution to ∆r np produced by differing widths of the surfaces of the neutron and proton densities (b n = b p ) is similar to, or even larger than, the effect from differing extensions of the bulk of the nucleon densities (R n = R p ). On the contrary, the nuclear interactions with a stiff symmetry energy favor a dominant bulk nature of the neutron skin of 208 Pb, and then the largest part of ∆r np is caused by R n = R p . We collect in Table II the found equivalent sharp radii R n and R p and surface widths b n and b p of the densities of 208 Pb in the present mean-field models.
As we have had the opportunity to see, the neutron skin thickness of a heavy nucleus is strongly influenced by the density derivative L of the symmetry energy. Indeed, one easily suspects from Table I that ∆r np of 208 Pb is almost linearly correlated with L in the nuclear meanfield models, which Fig. 2 confirms for the present interactions. The correlation of the neutron skin thickness of 208 Pb with L has been amply discussed in the literature [5][6][7][8][9][10][11], as it implies that an accurate measurement of the former observable could allow to tightly constrain the density dependence of the nuclear symmetry energy. In particular, we studied the aforementioned correlation in Ref. [9] where it is shown that the expression of the neutron skin thickness in the DM of Myers andŚwiatecki [12,13] can be recast to leading order in terms of the L parameter. To do that, we use the fact that in all meanfield models the symmetry energy coefficient computed at ρ ≈ 0.10 fm −3 is approximately equal to the DM symmetry energy coefficient in 208 Pb which includes bulkand surface-symmetry contributions [9]. In the standard DM, where the surface widths of the neutron and proton densities are taken to be the same [12,13], the neutron skin thickness is governed by the ratio between the bulksymmetry energy at saturation J ≡ c sym (ρ 0 ) and the surface stiffness coefficient Q of the DM [9, 10] (the latter is not to be confused with the equivalent rms radius Q of Eq. (4)). The DM coefficient Q measures the resistance of the nucleus against the separation of the neutron and proton surfaces to form a neutron skin. We have shown [9,10] in mean-field models that the DM formula for the neutron skin thickness in the case where one assumes b n = b p , undershoots the corresponding values computed by the semiclassical extended Thomas-Fermi method in finite nuclei and, therefore, a nonvanishing surface contribution is needed to describe more accurately the meanfield results. However, this surface contribution has a more involved dependence on the parameters of the interaction and does not show a definite correlation with the J/Q ratio (see Fig. 4 of Ref. [10]). Now, we wondered to which degree the correlation with L of the neutron skin thickness of 208 Pb holds in its bulk and surface parts extracted from actual mean-field densities. From our discussion of the indications provided by the DM, we can expect this correlation to be strong in the bulk part and weak in the surface part. Indeed, the plots of ∆r bulk np and ∆r surf np against L in Fig. 2 show that the bulk part displays the same high correlation with L as the total neutron skin (the linear correlation factor is of 0.99 in both cases), whereas the surface part exhibits a mostly flat trend with L. The linear fits in Fig. 2 of the neutron skin thickness of 208 Pb and of its bulk part have also quite similar slopes. One thus concludes that the linear correlation of ∆r np of 208 Pb with the density content of the nuclear symmetry energy arises mainly from the bulk part of ∆r np . In other words, the correlation arises from the change induced by the density dependence of the symmetry energy in the equivalent sharp radii of the nucleon density distributions of 208 Pb rather than from the change of the width of the surface of the nucleon densities.   The value of about 0.1 fm that the surface contribution to ∆r np takes in 208 Pb can be understood as follows starting from Eq. (11). Taking into account that in 2pF distributions fitted to mean-field densities R n ∼ R p ∼ 1.16A 1/3 fm and b n + b p ∼ 1.8 fm (see Table  II), Eq. (11) can be approximated as Given that b n − b p ∼ 0.2 fm for 208 Pb on the average in mean-field models (see Table II), one finds ∆r surf np ∼ 0.1 fm, rather independently of the model used to compute it. It is interesting why the range of variation of b n with respect to b p is not larger in nuclear models, in view of the fact that R n − R p can take more different values. As discussed in Ref. [30], this constraint is imposed on the models most likely by the mass fits. For example, a model having nucleon densities with very small or very large surface widths (i.e., very sharp or very extended surfaces) would provoke a large change in the surface energy of the nucleus, but that hardly would be successful to reproduce the known nuclear masses.
C. Discussion of the shape of the neutron density profiles The use of 2pF functions in order to represent the nuclear densities by approximate distributions is also quite common in the experimental investigations. The parameters of the proton 2pF distribution can be assumed known in experiment, by unfolding from the accurately measured charge density [40]. However, the shape of the neutron density is more uncertain, and even if the neutron rms radius is determined, it can correspond to different shapes of the neutron density. Actually, the shape of the neutron density is a significant question in the extraction of nuclear information from experiments in exotic atoms [35][36][37][38] and from parity-violating electron scattering [39]. To handle the possible differences in the shape of the neutron density when analyzing the experimental data, the so-called "halo" and "skin" forms are frequently used [35][36][37][38][39]. In the "halo-type" distribution the nucleon 2pF shapes have C n = C p and a n > a p , whereas in the "skin-type" distribution they have a n = a p and C n > C p . To complete our study, we believe worth discussing the predictions of the theoretical models for the parameters of the 2pF shapes in 208 Pb. We compile in Table III the central radii C n and C p and the diffuseness parameters a n and a p of the 2pF nucleon density profiles of 208 Pb obtained from the mean-field models of Table I. We see that C n of neutrons spans a range of approximately 6.7-6.85 fm in the nonrelativistic interactions and that it is of approximately 6.8-7 fm in the relativistic parameter sets. In the case of the proton density distribution, the value of C p is smallest (∼6.65 fm) in the two Gogny forces, it is about 6.67-6.71 fm in the Skyrme forces, and it is in a range of 6.7-6.77 fm in the RMF models. Then, we note that not only C n of neutrons but also C p of protons is generally smaller in the nonrelativistic forces than in the relativistic forces. The total spread in C p among the models (about 0.12 fm) is, though, less than half the spread found in C n (about 0.3 fm). Indeed, the accurately known charge radius of 208 Pb is an observable that usually enters the fitting protocol of the effective nuclear interactions.
If we inspect the results for the surface diffuseness of the density profiles of 208 Pb in Table III, we see that a n of neutrons lies in a window of 0.53-0.59 fm (with the majority of the models having a n between 0.545 and 0.565 fm). The nonrelativistic interactions favor a n 0.555 fm, whereas the RMF sets favor a n 0.555 fm. This indicates that the fall-off of the neutron density of 208 Pb at the surface is in general faster in the interactions with a soft symmetry energy than in the interactions with a stiff symmetry energy. The surface diffuseness a p of the proton density spans in either the nonrelativistic or the relativistic models almost the same window of values (0.43-0.47 fm; with the majority of the models having a p between 0.445 and 0.465 fm). This fact is in contrast to the other 2pF parameters discussed so far. Actually, the a p value of the proton density can be definitely larger in some nonrelativistic forces than in some relativistic forces (for example, in the case of SkM* and NL3). One finds that the total spread of a n and a p within the analyzed models is quite similar: about 0.05 fm in both a n and a p . This spread corresponds roughly to a 10% variation compared to the mean values of a n and a p . It is remarkable that while among the models C n has a significantly larger spread than C p , the surface diffuseness a n of the neutron density has essentially the same small spread as the surface diffuseness a p of the proton density. As we have discussed at the end of Sec. III B, this is likely imposed by the nuclear mass fits. It means that our ignorance about the neutron distribution in 208 Pb does not seem to produce in the mean-field models a larger uncertainty for a n of neutrons than for a p of protons, and that most of the uncertainty goes to the value of C n .
The difference C n − C p of the central radii of the nucleon densities of 208 Pb turns out to range approximately between 0. and 0.2 fm. It is smaller for soft symmetry energies and larger for stiff symmetry energies. We realize that the limiting situation of a halo-type distribution where the nucleon densities of 208 Pb have C n = C p and a n > a p is actually attained in the nuclear mean-field models with a very soft symmetry energy (like in HFB-8 or MSk7 where C n − C p even is slightly negative). The difference a n − a p of the neutron and proton surface diffuseness in 208 Pb is comprised between nearly 0.08 and 0.1 fm in the nonrelativistic forces and between nearly 0.1 and 0.12 fm in the RMF forces. This implies that no interaction predicts a n − a p of 208 Pb as close to vanishing as C n − C p is in some forces. Thus, the limiting situation where the nucleon densities in 208 Pb would have a n = a p and C n > C p is not found in the nuclear mean-field models. Indeed, we observe in Table III that if C n − C p becomes larger in the models, also a n − a p tends overall to become larger. In order to help visualize graphically the change in the mean-field nucleon densities of 208 Pb from having a nearly vanishing C n − C p or a large C n − C p , we have plotted in Fig. 3 the example of the densities of the MSk7 and NL3 interactions. On the one hand, we see that both models MSk7 and NL3 predict basically the same proton density, as expected. On the other hand, the difference between having C n ≈ C p in MSk7 and C n > C p in NL3 can be appreciated in the higher bulk and the faster fall-off at the surface of the neutron density of MSk7 compared with NL3.
In summary, we conclude that in 208 Pb the nuclear mean-field models favor the halo-type distribution with C n ≈ C p and a n > a p if they have a very soft ("supersoft") symmetry energy, they favor a mixed-type distribution if they have mild symmetry energies, and a situation where C n is clearly larger than C p if the symmetry energy is stiff, but that the pure skin-type distribution where a n − a p = 0 in 208 Pb is not supported (not even a n − a p ≈ 0) by the mean-field models. Although the experimental evidence available to date on the neutron skin thickness of 208 Pb is compatible with the ranges of the C n − C p and a n − a p parameters considered in our study, it is not to be excluded that the description of a precision measurement in 208 Pb may need of nucleon densities with C n −C p or a n −a p values not fitting Table III. However, a sizable deviation (such as a n − a p = 0) could mean that there is some missing physics in the isospin channel of present mean-field interactions, because once these interactions are calibrated to reproduce the observed binding energies and charge radii of nuclei they typically lead to the ranges of Table III.

D. Application to parity-violating electron scattering
Parity-violating electron scattering is expected to be able to accurately determine the neutron density in a nucleus since the Z 0 boson couples mainly to neutrons [29,30]. Specifically, the PREX experiment [29] aims to provide a clean measurement of the neutron radius of 208 Pb. In this type of experiments one measures the parity-violating asymmetry where dσ ± /dΩ is the elastic electron-nucleus cross section. The plus (minus) sign accounts for the fact that electrons with a positive (negative) helicity state scatter from different potentials (V ± (r) = V Coulomb (r) ± V weak (r) for ultra-relativistic electrons). Assuming for simplicity the plane wave Born approximation (PWBA) and neglecting nucleon form factors, the parity-violating asymmetry at momentum transfer q can be written as [30] A PWBA where sin 2 θ W ≈ 0.23 for the Weinberg angle and F n (q) and F p (q) are the form factors of the point neutron and proton densities. Because F p (q) is known from elastic electron scattering, it is clear from (16) that the largest uncertainty to compute A LR comes from our lack of knowledge of the distribution of neutrons inside the nucleus. PREX intends to measure A LR in 208 Pb with a 3% error (or smaller). This accuracy is thought to be enough to determine the neutron rms radius with a 1% error [29,30].
To compute the parity-violating asymmetry we essentially follow the procedure described in Ref. [30]. For realistic results, we perform the exact phase shift analysis of the Dirac equation for electrons moving in the potentials V ± (r) [58]. This method corresponds to the distorted wave Born approximation (DWBA). The main input needed for solving this problem are the charge and weak distributions. To calculate the charge distribution, we fold the mean-field proton and neutron point-like densities with the electromagnetic form factors provided in [59]. For the weak distribution, we fold the nucleon pointlike densities with the electric form factors reported in [30] for the coupling of a Z 0 to the protons and neutrons. We neglect the strange form factor contributions to the weak density [30]. Because the experimental analysis may involve parametrized densities, in our study we use the 2pF functions extracted from the self-consistent densities of the various models. The difference between A LR calculated in 208 Pb with the 2pF densities and with the self-consistent densities is anyway marginal at most. In Fig. 4, A DWBA LR obtained with the Fermi distributions listed in Table III is plotted against the values of C n − C p (lower panel) and a n − a p (upper panel). To simulate the kinematics of the PREX experiment [29], we set the electron beam energy to 1 GeV and the scattering angle to 5 • , which corresponds to a momentum transfer in the laboratory frame of q = 0.44 fm −1 .
First, one can see from Fig. 4 that the mean-field calculations constrain in a rather narrow window the value of the parity-violating asymmetry in 208 Pb. The increasing trend of A LR with decreasing C n −C p indicates that A LR is larger when the symmetry energy is softer. Note that a large value A LR ≈ 7×10 −7 (at 1 GeV and 5 • ) would be in support of a more surface than bulk origin of the neutron skin thickness of 208 Pb and of the halo-type density distribution for this nucleus. Second, A DWBA LR displays in good approximation a linear correlation with C n − C p (r = 0.978), while the correlation with a n − a p is not remarkable. Nevertheless, we have found a very good description of A DWBA LR of the mean-field models-well below the 3% limit of accuracy of the PREX experiment-by means of a fit in C n − C p and a n − a p (red crosses in Fig. 4): with α = 7.33, β = −2.45 fm −1 , and γ = −3.62 fm −1 .
The parametrization (17) may be easily understood if we consider the PWBA expression of A LR given above in Eq. (16). At low-momentum transfer, the form factors F n (q) and F p (q) of the neutron and proton densities (these are point densities in PWBA) can be expanded to first order in q 2 , so that the numerator inside brackets in Eq. (16) becomes −(q 2 /6) ( r 2 n − r 2 p ). In 2pF density distributions we have r 2 q = (3/5) C 2 q + (7π 2 /5) a 2 q . Now, assuming constancy of F p (q 2 ) in the nuclear models and taking into account that C n + C p ≫ C n − C p and a n + a p ≫ a n − a p , it is reasonable to assume that the variation of A LR is dominated by the change of C n − C p and a n − a p as proposed in Eq. (17).
In the analysis of a measurement of A LR in 208 Pb through parametrized Fermi densities, one could set C p and a p to those known from experiment [40] and then vary C n and a n in (17) to match the measured value. According to the predictions of the models in Table III, it would be reasonable to restrict this search to windows of about 0-0.22 fm for C n − C p and 0.08-0.125 fm for a n − a p . Therefore, the result of a measurement of the parity-violating asymmetry together with Eq. (17) (or Fig. 4) would allow not only to estimate the neutron rms radius of 208 Pb but also to obtain some insight about the neutron density profile in this nucleus. This assumes that the experimental value for A LR will fall in, or at least will be not far from, the region allowed by the mean-field calculations at the same kinematics.

IV. SUMMARY
We have investigated using Skyrme, Gogny, and relativistic mean-field models of nuclear structure whether the difference between the peripheral neutron and proton densities that gives rise to the neutron skin thickness of 208 Pb is due to an enlarged bulk radius of neutrons with respect to that of protons or, rather, to the difference between the widths of the neutron and proton surfaces. The decomposition of the neutron skin thickness in bulk and surface components has been obtained through twoparameter Fermi distributions fitted to the self-consistent nucleon densities of the models.
Nuclear models that correspond to a soft symmetry energy, like various nonrelativistic mean-field models, favor the situation where the size of the neutron skin thickness of 208 Pb is divided similarly into bulk and surface components. If the symmetry energy of the model is "supersoft", the surface part even becomes dominant. Instead, nuclear models that correspond to a stiff symmetry energy, like most of the relativistic models, predict a bulk component about twice as large as the surface component. We have found that the size of the surface component changes little among the various nuclear mean-field models and that the known linear correlation of ∆r np of 208 Pb with the density derivative of the nuclear symmetry energy arises from the bulk part of ∆r np . The latter result implies that an experimental determination of the equivalent sharp radius of the neutron density of 208 Pb could be as useful for the purpose of constraining the density-dependent nuclear symmetry energy as a determination of the neutron rms radius.
We have discussed the shapes of the 2pF distributions predicted for 208 Pb by the nuclear mean-field models in terms of the so-called "halo-type" (C n − C p = 0) and "skin-type" (a n − a p = 0) distributions of frequent use in experiment. It turns out that the theoretical models can accomodate the halo-type distribution in 208 Pb if the symmetry energy is supersoft. However, they do no support a purely skin-type distribution in this nucleus, even if the model has a largely stiff symmetry energy. Let us mention that the information on neutron densities from antiprotonic atoms favored the halo-type over the skin-type distribution [35,36].
We have closed our study with a calculation of the asymmetry A LR for parity-violating electron scattering off 208 Pb in conditions as in the recently run PREX experiment [29], using the equivalent 2pF shapes of the models. This has allowed us to find a simple parametrization of A LR in terms of the differences C n − C p and a n − a p of the parameters of the nucleon distributions. With a measured value of the parity-violating asymmetry, it would provide a new correlation between the central radius and the surface diffuseness of the distribution of neutrons in 208 Pb, assuming the same properties of the proton density known from experiment.