Nuclear symmetry energy probed by neutron skin thickness of nuclei

We describe a relation between the symmetry energy coefficients c_sym(rho) of nuclear matter and a_sym(A) of finite nuclei that accommodates other correlations of nuclear properties with the low-density behavior of c_sym(rho). Here we take advantage of this relation to explore the prospects for constraining c_sym(rho) of systematic measurements of neutron skin sizes across the mass table, using as example present data from antiprotonic atoms. The found constraints from neutron skins are in harmony with the recent determinations from reactions and giant resonances.

We describe a relation between the symmetry energy coefficients csym(ρ) of nuclear matter and asym(A) of finite nuclei that accommodates other correlations of nuclear properties with the lowdensity behavior of csym(ρ). Here we take advantage of this relation to explore the prospects for constraining csym(ρ) of systematic measurements of neutron skin sizes across the mass table, using as example present data from antiprotonic atoms. The found constraints from neutron skins are in harmony with the recent determinations from reactions and giant resonances. A wealth of measured data on densities, masses and collective excitations of nuclei has allowed to resolve basic features of the equation of state (EOS) of nuclear matter, like the density ρ 0 ≈ 0.16 fm −3 , energy per particle a v ≈ −16 MeV, and incompressibility K v ≈ 230 MeV [1] at saturation. However, the symmetry properties of the EOS due to differing neutron and proton numbers remain more elusive to date. The quintessential paradigm is the density dependence of the symmetry energy [1,2,3,4,5,6,7,8,9,10]. The accurate characterization of this property entails profound consequences in studying the neutron distribution in stable and exotic nuclei and neutron-rich matter [2,3,4]. It impacts on heavy ion reactions [5,6,7,8,9], nuclear astrophysics [3,4,10], and on diverse areas such as tests of the Standard Model via atomic parity violation [11].
In this work we show that c sym (ρ) of the EOS equals at ρ ≈ 0.1 fm −3 the value of the symmetry energy coefficient a sym (A) of heavy finite nuclei, universally in mean field theories. The observed correlations of S [2,3,4,5,6,7] and of the excitation energy of the GDR [14] with the density dependence of c sym can be deduced naturally from this relation. We resort to the nuclear droplet model (DM) [12] to work out the analytical formulas. The result derived for S is applied to investigate limits to the slope and curvature of c sym from neutron skins measured for 26 stable nuclei, from 40 Ca to 238 U, in antiprotonic atoms [20]. A main point is ascertaining how far uniformly measured neutron skins over the periodic table may help constrain the density dependence of c sym . We provide first evidence that the constraints from skins are in consonance with the recent observations from reactions and giant resonances, though the probed densities and energies are not necessarily the same.
The symmetry energy coefficient a sym (A) of finite nuclei is smaller than the bulk value J. Given a nuclear force, the DM allows one to extract a sym (A) as [12,21] The so-called surface stiffness Q measures the resistance of the nucleus against separation of neutrons from protons to form a neutron skin. One can obtain Q of nu- clear forces by asymmetric semi-infinite nuclear matter (ASINM) calculations [12,21,22]. The contribution of a sym (A) to the nucleus energy is a sym (A) ( [12] is neglected here. Let us mention that (1) may be derived also from the notion of surface symmetry energy [4,19]. The neutron skin thickness of nuclei is obtained as in the DM [12,23]. The quantity t gives the distance between the neutron and proton mean surface locations: where in the second line we have introduced the surface symmetry term a ss (A) = [J −a sym (A)]A 1/3 using Eq. (1). The second term in Eq. (2) is due to Coulomb repulsion, and S sw = 3/5 5(b 2 n − b 2 p )/(2R) is a correction caused by an eventual difference in the surface widths b n and b p of the neutron and proton density profiles.
We first illustrate the aforesaid correlation of S of heavy nuclei with L in Fig. 1(a). It depicts the quantal self-consistent value of S in 208 Pb against L for multiple Skyrme, Gogny, and covariant models of different nature [2,3,4,5,6,7,18,21,24]. In Fig. 1(b) we show that a similar correlation exists with the ratio L/J, which is proportional to γ if a scaling (ρ/ρ 0 ) γ holds for c sym (ρ). And in Fig. 1(c) we show that the close dependence of S on J − a sym (A) predicted by the DM is borne out in the quantal S value, using forces where we have computed Q in ASINM. It reassures one that the DM expression incorporates the proper elements for the study. Many of the given nuclear interactions are accurately fitted to experimental binding energies, single-particle data, and charge radii of a variety of nuclei. However, their isospin structure is not sufficiently firmed up as seen e.g. in the differing predictions for S( 208 Pb). There is thus a need to deepen our knowledge of isospin-sensitive observables like S and of their constraints on c sym (ρ). We bring into notice a genuine relation between the symmetry energy coefficients of the EOS and of nuclei: c sym (ρ) equals a sym (A) of a heavy nucleus like 208 Pb at a density ρ ≈ 0.1 fm −3 . Indeed, the relation holds similarly down to medium mass numbers, at lower ρ values and a little more spread. Table I exemplifies this fact with several nuclear models, where we show the density fulfilling c sym (ρ) = a sym (A) for A = 208, 116, and 40. We find that this density can be parametrized as with c fixed by ρ 208 = 0.1 fm −3 (which gives ρ 116 ≈ 0.093 fm −3 and ρ 40 ≈ 0.08 fm −3 for the models of Table I).
The relation "c sym (ρ) = a sym (A)" can be very helpful to elucidate other correlations of isospin observables with c sym (ρ) and to gain deeper insights into them. For example, it allows one to replace a sym (A) in Eq. (3) for a heavy nucleus by c sym (ρ) ≃ J − Lǫ The imprint of the density content of the symmetry energy on the neutron skin appears now explicitly. The leading proportionality of (5) with L explains the observed linearity of S of a heavy nucleus with L in nuclear models [2,4,7]. The correction with K sym does not alter the situation as ǫ ∼ 1/9 is small. One can use Eq. (5) in other mass regions by calculating ǫ from ρ A of Eq. (4). We have checked numerically in multiple forces that the results closely agree with Eq. (3) for the 40 ≤ A ≤ 238 stable nuclei given in Fig. 2.
With the help of Eq. (5) for t (using ρ A to compute ǫ), we next analyze constraints on the density dependence of the symmetry energy by optimization of (2) to experimental S data. We employ c sym (ρ) = 31.6(ρ/ρ 0 ) γ MeV [6,7,8,9] and take as experimental baseline the neutron skins measured in 26 antiprotonic atoms [20] (see Fig. 2). These data constitute the largest set of uniformly measured neutron skins over the mass table till date. With allowance for the error bars, they are fitted linearly by S = (0.9±0.15)I +(−0.03±0.02) fm [20]. This systematics renders comparisons of skin data with DM formulas, which by construction average the microscopic shell effect, more meaningful [26]. We first set b n = b p (i.e., S sw = 0) as done in the DM [12,23,26] and in the analysis of data in Ref. [19]. Following the above, we find L = 75 ± 25 MeV (γ = 0.79 ± 0.25). The range ∆L = 25 MeV stems from the window of the linear averages of experiment. The L value and its uncertainty obtained from neutron skins with S sw = 0 is thus quite compatible with the quoted constraints from isospin diffusion and isoscaling observables in HIC [6,7,8]. On the other hand, the symmetry term of the incompressibility of the nuclear EOS around equilibrium (K = K v +K τ δ 2 ) can be estimated using information of the symmetry energy as K τ ≈ K sym −6L [5,6,7]. The constraint K τ = −500±50 MeV is found from isospin diffusion [6,7], whereas our study of neutron skins leads to K τ = −500 +125 −100 MeV. A value K τ = −550 ± 100 MeV seems to be favored by the giant monopole resonance (GMR) measured in Sn isotopes as is described in [13]. Even if the present analyses may not be called definitive, significant consistency arises among the values extracted for L and K τ from seemingly unrelated sets of data from reactions, ground-states of nuclei, and collective excitations.
To assess the influence of the correction S sw in (2) we compute the surface widths b n and b p in ASINM [22]. This yields the b n(p) values of a finite nucleus if we relate the asymmetry δ 0 in the bulk of ASINM to I by δ 0 (1 + x A ) = I + x A I C [21,22,23]. In doing so, we find that Eq. (2) reproduces trustingly S (and its change with I) of self-consistent Thomas-Fermi calculations of finite nuclei made with the same nuclear force. Also, S sw is very well fitted by S sw = σ sw I. All slopes σ sw of the forces of Fig. 1(c) lie between σ min sw = 0.15 fm (SGII) and σ max sw = 0.31 fm (NL3). We then reanalyze the experimental neutron skins including S min sw and S max sw in Eq. (2) to simulate the two conceivable extremes of S sw according to mean field models. The results are shown in Fig. 3. Our above estimates of L and K τ could be shifted by up to −25 and +125 MeV, respectively, by nonzero S sw . This is on the soft side of the HIC [6,7,8] and GMR [13] analyses of the symmetry energy, but closer   The crosses express the L and Kτ ranges compatible with the uncertainties in the skin data. The shaded regions depict the constraints on L and Kτ from isospin diffusion [6,7] and on Kτ as determined in [13] from the GMR of Sn isotopes.
to the alluded predictions from nucleon emission ratios [9], the GDR [14], and nuclear binding systematics [17]. One should mention that the properties of c sym (ρ) derived from terrestrial nuclei have intimate connections to astrophysics [3,4,10]. As an example, we can estimate the transition density ρ t between the crust and the core of a neutron star [3,10] as ρ t /ρ 0 ∼ 2/3 + (2/3) γ K sym /2K v , following the model of Sect. 5.1 of Ref. [10]. The constraints from neutron skins hereby yield ρ t ∼ 0.095±0.01 fm −3 . This value would not support the direct URCA process of cooling of a neutron star that requires a higher ρ t [3,10]. The result is in accord with ρ t ∼ 0.096 fm −3 of the microscopic EOS of Friedman and Pandharipande [27], as well as with ρ t ∼ 0.09 fm −3 predicted by a recent analysis of pygmy dipole resonances in nuclei [15]. We would like to close with a brief comment regarding the GDR. As mentioned, Ref. [14] very interestingly constrains c sym (0.1) from the GDR of 208 Pb. The anal-ysis notes that the mean excitation energy of the GDR depends on g(A) = J/{1 + 5 3 a ss (A)A −1/3 /J} [4,14] and shows numerically that the values of g(208) and c sym (0.1) are correlated in Skyrme forces. Inserting a ss (A) given below Eq. (3), one has g(A) = J/{1 + 5 3 [J − a sym (A)]/J}. Immediately, the equivalence a sym (208) ≈ c sym (0.1) explains why g(208) has a dependence on c sym (0.1), gives it analytically, and validates it for any type of mean field model [28]. One could extend it to other A values through Eq. (4). In conclusion, the discussed relation of c sym (ρ) with a sym (A) can be much valuable to link different problems depending upon a sym (A) of nuclei to the symmetry properties of the EOS. Summarizing, we have described a generic relation between the symmetry energy in finite nuclei and in nuclear matter at subsaturation. It plausibly encompasses other prime correlations of nuclear observables with the density content of the symmetry energy. We take advantage of this relation to explore constraints on c sym (ρ) from neutron skins measured in antiprotonic atoms [20]. We discuss the L and K τ values that skins favor vis-à-vis most recent observations from reactions and giant resonances. The difficult experimental extraction of neutron skins limits their potential to constrain c sym (ρ). Interestingly, we learn that in spite of present error bars in the data of [20], the size of the final uncertainties in L or K τ is comparable to the other analyses. This highlights the value of having skin data consistently measured across the mass table, and calls for pursuing extended measurements of neutron radii and skins with "conventional" hadronic probes. Combined with a precision extraction of R n of 208 Pb through electroweak probes [29], they would contribute to cast uniquely tight constraints on c sym (ρ).