aa r X i v : . [ nu c l - t h ] J un Nuclear skin and the curvature of the symmetry energy
Ad. R. Raduta and F. Gulminelli National Institute for Physics and Nuclear Engineering (IFIN-HH), RO-077125, Bucharest-Magurele, Romania Universit´e de Caen Normandie, ENSICAEN, LPC, UMR6534, F-14050 Caen, France (Dated: November 13, 2018)The effect of correlations between the slope and the curvature of the symmetry energy on ground state nu-clear observables is studied within the extended Thomas-Fermi approximation. We consider different isovectorprobes of the symmetry energy, with a special focus on the neutron skin thickness of
Pb. We use a recentlyproposed meta-modelling technique to generate a large number of equation of state models, where the empiricalparameters are independently varied. The results are compared to a set of calculations using 17 different Skyrmeinteractions. We show that the curvature parameter plays a non-negligible role on the neutron skin, while theeffect is reduced in Skyrme functionals because of the correlation with the slope parameter.
I. INTRODUCTION
The determination of the nuclear matter equation of state(EoS) is an extremely lively issue in modern nuclear physicsand astrophysics. The biggest uncertainties concern high den-sity and strongly asymmetric matter, where the EoS determi-nation is of outermost importance for the understanding ofa large variety of astrophysical phenomena involving com-pact stars [1, 2]. Observational measurements of neutron starmass and radii start to provide compelling constraints to thebehavior of high density matter [3, 4], including the very re-cent multi-messenger observation of a neutron star merger [5],where the EoS has a direct impact on the gravitational waveform mainly through the tidal polarizability parameter [6]. Inthis context, tight constraints coming from controlled nuclearexperiments are extremely important, particularly concerningthe isovector part of the EoS, the so-called symmetry energy[7, 8]. A huge literature is devoted to the determination of thesymmetry energy at saturation ( E sym ) and its slope ( L sym ) bycomparing selected isovector observables to EoS models is-sued from different energy density functionals (EDF) [9, 10].These studies have convincingly shown that a strong linearcorrelation exists between the L sym parameter and the neutronskin thickness [11, 12]. This latter can be measured directlyfrom parity violating electron scattering [13] and pion pho-toproduction [14], or probed via various isovector modes ofcollective excitations [15–20]. A good correlation is also typ-ically observed with the E sym parameter [21, 22] and qualita-tively explained by the fact that the behavior of the symmetryenergy is, to a first order approximation, linear in density inthe subsaturation regime [23]. However, this correlation issomewhat blurred when different families of mean field mod-els are compared [24], showing that some residual model de-pendence exists. The careful study of Ref. [24] shows that thisdifference can be ascribed to different nucleon density distri-butions in the surface region. In turn, this can be due both todifferent surface properties of the functionals, or to differentbehaviors of the symmetry energy at subsaturation, that is todeviations from the linear approximation.To progress on this issue, it is important to assess the roleof the curvature of the symmetry energy ( K sym ) on isovectorprobes such as the nuclear skin. Little attention was paid tothis parameter in the literature until recently [25, 26], mainly due to the fact that it cannot be easily varied within a specificEoS model, because the functional form of the EoS imposes acorrelation with the low order parameters E sym and L sym . Still,if K sym is of secondary role for nuclear structure observables,it is the main source of uncertainty when extrapolating thelaboratory constraints to the high density domain relevant forneutron star physics [27].To perform this study, we use a recently proposed meta-modelling approach to the EoS [28], where a large numberof different EoS models can be generated without any a-prioricorrelation among the different empirical parameters. Follow-ing Ref. [29], ground state observables are calculated withinthe Extended Thomas Fermi (ETF) approximation, with theaddition of a gradient term as an effective parameter repre-senting the different surface properties of the different mod-els.We show that the K sym parameter plays a non-negligible rolein the nuclear skin as well as in the differences of the protonradii of mirror nuclei and that the uncertainty on this param-eter partially blurs the correlation with the symmetry energyslope.The paper is organized as follows: the different energyfunctionals are briefly reviewed in section II, as well as theETF approximation used to calculate nuclear observables. InSection III, after discussing the overall performance of theETF approximation on the Pb isotopic chain, we show ourmain results concerning the correlations between the differ-ent isovector observables and the empirical EoS parameters.Finally conclusions are drawn in section IV. II. FORMALISMA. Skyrme EDF
The most extensive calculations of nuclear observables andtheir correlations with EoS parameters have been performedusing Skyrme EDF [30].The nuclear Skyrme energy density is expressed in terms oflocal nucleon densities n q ( rrr ) , kinetic energy densities τ q ( rrr ) and spin-orbit densities JJJ q ( rrr ) defined by [31] n q ( rrr ) = ∑ ν , s | φ ν ( rrr , s , q ) | n q ν , τ q ( rrr ) = ∑ ν , s | ∇φ ν ( rrr , s , q ) | n q ν , JJJ q ( rrr ) = ( − i ) ∑ ν , s , s ′ φ ∗ ν ( rrr , s ′ , q ) ∇φ ν ( rrr , s , q ) × h s ′ | σ | s i n q ν , (1)where φ ν ( rrr , s , q ) represent the single-particle wave functionswith orbital and spin numbers ν and s , q = n , p indexes thenucleonic species and n q ν are the occupation numbers. Thefunctional form of the EDF is generated by a mean-field cal-culation with an effective zero range momentum dependentpseudo-potential, augmented of a density dependent term.Standard pseudo-potentials, as the ones considered hereafter,depend on 10 parameters. The values of these parametersare typically determined by fits of experimental ground-stateproperties of spherical magic and semi-magic nuclei (e.g.binding energy, root mean square (rms) radius of the chargedistribution, spin-orbit splitting, isotope shifts, surface thick-ness, breathing mode energy, etc.) and/or properties of sym-metric nuclear matter (energy E sat and density n sat at sat-uration, compression modulus K sat , symmetry energy E sym )and/or equation of state of pure neutron matter as predictedby ab-initio models. These parameters vary largely from oneSkyrme model to another. Properties of nuclear matter (NM)can be expressed analytically in terms of the same parameters[30].In the following, 17 Skyrme EDFs will be employed:SKa [32], SKb [32], Rs [33], SkMP [34], SLy2 [35],SLy9 [35], SLy4 [36], SLy230a [37], SkI2 [38], SkI3 [38],SkI4 [38], SkI5 [38], SkI6 [39], SKOp [40], SK255 [41],SK272 [41] and KDE0v1 [42]. The extent to which theyfulfill various constraints that have been obtained from ex-periment or microscopic calculations during the last decade[43] has been thoughtfully investigated in Ref. [44] in thecontext of unified equations of state for neutron star mat-ter. Their values of saturation density of symmetric nuclearmatter (SNM), energy per particle and compression modulusof symmetric saturated matter span relatively narrow ranges0 . ≤ n sat ≤ . − , − . ≤ E sat ≤ − .
52 MeV,222 . ≤ K sat ≤ . . ≤ E sym ≤ . . ≤ L sym ≤ . − . ≤ K sym ≤ . E sat , n sat , K sat , E sym , L sym , the higher order parameters can be analyt-ically expressed as a function of those fixed quantities. Inparticular, Skyrme EDFs show a clear correlation between theslope L sym and the curvature K sym of the symmetry energy atsaturation, which are a-priory independent EoS parameters. K sym (MeV) L sy m ( M e V ) metamodelingSkyrme C fin (MeV) m s a t * / m FIG. 1: Correlations between EoS parameters. Numbers onl.h.s. of each plot correspond to Pearson correlationcoefficients between the parameters plotted on the axis.Upper (lower) values: meta-model (Skyrme).This correlation, which obviously affects the extrapolationof the EoS to super-saturation densities, is graphically illus-trated in the top panel of Fig. 1 (open circles). Its Pear-son correlation coefficient[72] is C ( K sym , L sym ) = .
87. It wasrecently shown that this correlation is observed in a largeclass of functionals and might therefore be physically founded[25, 26], even if its origin is not fully understood.Another interesting non-trivial correlation is found be-tween the effective nucleon mass at saturation, m ∗ sat , andthe isoscalar-like finite size parameter C fin (see section IIBand Ref. [29]). This correlation is illustrated in the bot-tom panel of Fig. 1. Its Pearson correlation coefficient is C ( m ∗ sat , C fin ) = .
88. As already discussed in ref. [29], thiscorrelation is probably induced by the parameter fitting proto-col of Skyrme functionals. Indeed m ∗ sat and C fin are related tonon-local terms in the EDF which have an opposite effect onthe surface energy, and neither of them plays a role on the de-termination of EoS parameters: for a given set of EoS param-eters a similar overall reproduction of binding energies overthe nuclear chart can be obtained with compensating effectsof the non-local terms. B. Meta-modelling of the EDF
A theoretical calculation of a nuclear observable depends,besides the EoS, on the functional form assumed for the EDFas well as on the many-body technique employed. To as-sess the model dependence due to the functional form of theEDF, one should consider different families of models withsimilar values for the EoS parameters. To this aim, a meta-modelling technique was proposed in Ref. [28] and extendedto finite nuclei EDF in Ref. [29]. Varying the parametersof the meta-modelling, a large number of EoS from differ-ent families of mean-field EDF can be generated. Moreover,novel density dependencies that do not correspond to existingfunctionals but do not violate any empirical constraint, can bealso explored [28]. The inclusion of a single gradient termprovides a minimal flexible EDF for finite nuclei, with per-formances on nuclear mass and radii comparable to the onesof full Skyrme functionals [29]. The exploration of the meta-modelling parameter space thus allows a full estimation of thepossible model dependence of the extraction of EoS parame-ters from nuclear ground state observables, due to the choiceof the EDF.The potential energy per baryon is expressed as a Taylorexpansion around saturation of symmetric nuclear matter interms of the density parameter x = ( n − n sat ) / ( n sat ) , e pot ( x , δ ) = N ∑ α = ( a α + a α δ ) x α α ! u α ( x ) , (2)where the functions u α ( x ) represent a low density correctioninsuring a vanishing energy in the limit of vanishing density,without affecting the derivatives at saturation.To correctly reproduce with a limited expansion order N existing non-relativistic (Skyrme and ab-initio) and relativistic(RMF and RHF) EDFs up to total densities n = n n + n p ≈ . − , and isospin asymmetries δ = ( n n − n p ) / n rangingfrom symmetric matter δ = δ = n / dependence at low densities, as well as thecontribution of higher orders in the δ expansion, as: e kin ( x , δ ) = t FGsat ( + x ) / " ( + δ ) / mm ∗ n + ( − δ ) / mm ∗ p , (3)where t FGsat = (cid:0) h (cid:1) / ( m ) (cid:0) π / (cid:1) / n / sat is the energy pernucleon of a free symmetric Fermi gas at nuclear saturation, m stands for the nucleon mass and m ∗ q denote the effective massof the nucleons q = n , p . For more details, see model ELFc inRef. [28].In the present work, we only consider subsaturation matterand, to avoid proliferation of unconstrained parameters, welimit the expansion to N =
2, which was shown to be enoughto get a fair reproduction of nuclear masses [29]. The possi-ble influence of higher order parameter is left for future work.When only average nuclear properties (e.g. binding energiesand rms radii of neutron and proton distributions) are calcu-lated, isoscalar and isovector finite-size and spin-orbit inter-actions can be fairly well described by a single isoscalar-like density gradient term [29] of the form C fin ( ∇ n n + ∇ n p ) . Forthe sake of convenience only this isoscalar density gradientwill be considered in this work. Following Ref. [29], we alsoneglect the effective mass splitting between neutrons and pro-tons. The meta-modelling parameters are then directly linkedto the usual first and second order empirical parameters of theEoS by: a = E sat − t FGsat ( + κ sat ) (4) a = − t FGsat ( + κ sat ) (5) a = K sat − t FGsat ( − + κ sat ) (6) a = E sym − t FGsat ( + κ sat ) (7) a = L sym − t FGsat ( + κ sat ) (8) a = K sym − t FGsat ( − + κ sat ) (9)where κ sat = m / m ∗ sat − { P α } = { n sat , E sat , K sat , E sym , L sym , K sym , m ∗ sat , C fin } . For a givenmodel, the ground state nuclear energies and radii are calcu-lated in the extended Thomas Fermi approximation at secondorder, as detailed in the next section. We retain for the sub-sequent analysis only the models { P α } which provide a fairdescription of the experimental binding energies of the spher-ical magic nuclei: (40,20), (48,20), (48,28), (58,28), (88, 38),(90, 40), (114, 50), (132, 50), (208, 82) and charge radii of(40,20), (48,20), (58,28), (88, 38), (90, 40), (114, 50), (132,50), (208, 82). The absence of the nucleus (48,28) in the sec-ond list is due to the fact that its experimental charge radius isnot yet available. We recall that this set of data represents thecore of nuclear properties on which the parameters of manySkyrme interactions have been fitted. The limitation to spher-ical nuclei is obviously due to the simplifying spherical ap-proximation of most approaches, including ours. Specifically,retained EDFs correspond to sets of parameters { P α } whichprovide χ ( B ) ≤ χ ( R ch ) ≤ .
10 fm. The mimi-mum values here obtained for standard deviation of massesand charge radii are 2.7 MeV and, respectively, 2 . · − fm.As usual in the literature, the chi-square function is defined as χ ( X ) = ∑ Ni = (cid:0) X ETF ( i ) − X exp ( i ) (cid:1) / N . The accepted values ofstandard deviation on mass are typically one order of magni-tude larger than the lowest value in the literature, 0.5 MeV,which corresponds to more than 2350 nuclei and has been ob-tained in the framework of a Hartree-Fock-Bogoliubov (HFB)mass model [45].The variation domain of each parameter is obtained by con-sidering the dispersion of the corresponding values in a largenumber of relativistic and non-relativistic mean-field models,see Ref. [28]. The precise frontiers of this domain dependon the number of models considered and their selection crite-ria, and is therefore somewhat arbitrary. However, a variationof the borders of the parameter space might affect the overalldispersion in the predictions of the meta-model, but not theTABLE I: Average and standard variation of the different parameters of the phenomenological EDF, calculated based on 51Skyrme interactions and 15 relativistic mean-field interactions (see table IV in Ref. [28]). Parameter { P α } n sat E sat K sat E sym L sym K sym m ∗ sat / m C fin (fm − ) (MeV) (MeV) (MeV) (MeV) (MeV) (MeVfm )Average h{ P α }i σ α ± ± ± ± ± ± ± ± quality of the correlations among parameters and observables,which is the scope of the present work.The domain considered for each parameter P α is reportedin Table I in terms of average value and standard deviation.Good/poor experimental constraints on n sat , E sat and E sym onone hand and K sat , L sym and K sym on the other hand lead tonarrow/wide variation domains of these variables.As a first application of the meta-modelling, we can investi-gate the model dependence of the correlations among empiri-cal parameters observed in the previous section for the SkyrmeEDFs.The only significant correlation that was found in the differ-ent models generated by the meta-modelling technique afterapplication of the mass and radius filter, is the one between m ∗ sat and C fin , as shown by solid squares in the bottom panelof Fig. 1. The value of the correlation coefficient, C = . L sym and K sym emerges from the meta-modelling after application of themass constraint, see solid squares in the top panel of Fig. 1.This suggests that the origin of that correlation observed indifferent functionals [25, 26] is not due to the constraint ofmass reproduction. C. The Extended Thomas-Fermi approximation withparametrized density profiles
For a given EDF model, average properties of atomic nu-clei can be reasonably well described within the ExtendedThomas-Fermi (ETF) approximation [46]. In this work, wewill limit ourselves to the second order expansion in ¯ h andto parametrized density profiles in spherical symmetry, suchas to limit the number of variational parameters. Because ofthese approximations, the degree of reproduction of experi-mental data is not comparable to the one of dedicated fullyquantal HFB calculations [45], and more realistic calculationswill definitely have to be performed in order to determine EoSparameters in a fully quantitative way. Still, the complete ex-ploration of the parameter space is not affordable with thesemore sophisticated many body techniques, and we believe thatan ETF meta-modelling is sufficient to extract the correlationsbetween EoS parameters and the neutron skin. In the ETF framework, the energy of an arbitrary distribu-tion of nucleons with densities { n n ( rrr ) , n p ( rrr ) } is given by thevolume integral of the energy density according to: E tot = Z drrr ( e nuc [ n n , n p ] + e Coul [ n p ]) , (10)where the first term stands for the nuclear energy and the sec-ond for the electrostatic contribution.At second-order in the ¯ h expansion, the nuclear energy den-sity functional writes e nuc [ n n , n p ] = ∑ q = n , p ¯ h m ∗ q τ q + e TF , (11)where e TF is the Thomas-Fermi approximation of the chosennuclear EDF model, which can depend on local densities n q as well as on density gradients ∇ n q and currents JJJ q and τ q is the (local and non-local) density dependent correction aris-ing from the second order ¯ h expansion of the kinetic energydensity operator.The Coulomb energy density is expressed as [47], e Coul [ n p ] = e n p ( rrr ) Z n p ( rrr ′′′ ) | rrr − rrr ′′′ | drrr ′′′ − e (cid:18) π (cid:19) / n / p ( rrr ) , (12)where the Slater approximation has been employed to esti-mate the exchange Coulomb energy density.The ground state is determined by energy minimizationusing parametrized neutron and proton distributions. For ageneric nucleus with N neutrons and Z protons and under thesimplifying approximation of spherical symmetry, these arecustomarily parametrized as Wood-Saxon (WS) density pro-files, n WSq ( r ) = n bulk , q + exp (cid:2)(cid:0) r − R WSq (cid:1) / a q (cid:3) , (13)where n bulk , q is linked to the central density of the q = n , p distribution, and R WSq and a q respectively stand for radius anddiffuseness parameters. With the extra condition of particlenumber conservation, Z = π Z ∞ dr r n p ( r ) , N = π Z ∞ dr r n n ( r ) (14)only four variables out of six are independent. In the varia-tional calculation of the ground state, we make the choice ofvariating { n bulk , q , a q ; q = n , p } , while R WSq are obtained fromEq. (14).The only experimental observables related to the distribu-tion of matter are the root mean squared (rms) radius of thecharge distribution and, with larger error bars, neutron skinthickness. Rms radius of the charge distribution is defined asthe rms radius of the proton distribution corrected for the in-ternal charge distribution of the proton S p =0.8 fm, h r ch i / = (cid:2) h r p i + S p (cid:3) / . (15)Neutron skin thickness is defined as the difference in theneutron-proton rms radii, ∆ r np = h r n i / − h r p i / , (16)and, as demonstrated in Ref. [24], it can be decomposed withgood accuracy into a bulk contribution, ∆ r bulknp = r "(cid:0) R WSn − R WSp (cid:1) + π a n R WSn − a p R WSp ! , (17)and a surface contribution, ∆ r sur fnp = r
35 5 π a n R WSn − a p R WSp ! . (18)It is worthwhile to notice that each of these contributions de-pends on both WS radii and diffusivities of neutron and protondistribution. III. RESULTSA. Performance of the ETF approximation on experimentaldata
In order to visualize the overall performance of the ETF ap-proximation, we consider in this section a single nuclear EDFmodel, namely the SLy4 [36] functional. We remind that thelot of data on which SLy4 [36] has been constrained includesbinding energies and rms radii of doubly magic nuclei and theequation of state of pure neutron matter of Ref. [48]. Thelast constraint guarantees a correct behavior at high isospinasymmetry.In terms of average standard deviation on masses and radii,we obtain for the considered pool of spherical nuclei χ ( B ) = . χ ( R ch ) = . · − fm.The results of total, i.e. nuclear plus electro-static, energy minimization in the 4-dimensional space { n bulk , n , n bulk , p , a n , a p } are plotted in Figs. 2 and 3 for theisotopic chain of Pb as a function of the isospin asymmetry, I = − Z / A . Two different methods are used to calculatethe Coulomb energy. In one case it is calculated by account-ing for the diffusivity of the proton distribution via eq. (12)(”self-consistent”). In the second, a uniformly charge distribu-tion approximation is employed, which leads to 0 . Z / A / (”approx.”). The top and middle panels of Fig. 2 present theevolution of each of the four variational parameters as a func-tion of I . The bottom panel presents the I -dependence of the self-consistentapprox. n bu l k , n n bu l k , p ( f m - ) a n , a p ( f m ) R n W S , R p W S ( f m ) FIG. 2: ETF results corresponding to the ground state of Pbisotopes, for a representative EDF model (SLy4 [36]).Variational parameters n bulk , n , n bulk , p (top panel), a n , a p (middle panel) and R WSn and R WSp (bottom panel), are plottedas a function of total isospin asymmetry. The results obtainedby considering the diffusivity of the charge distribution(”self-consistent”, Eq. (12)) are confronted with thosecorresponding to the uniformly charged sphereapproximation (”approx.”).WS radii on neutron and proton distributions, obtained fromparticle number conservation. We can notice the importanteffect of a self-consistent treatment of Coulomb in the deter-mination of the density profiles. In particular, the obtainedbulk densities and diffuseness parameters are in good agree-ment with fits of HF density profiles with the same EDF [49],which comforts us on the quality of the approximation.We can also see that WS radii of neutron and proton distri-butions have similar values, though strongly dependent on I .This might suggest that the skin is mainly a surface effect forthis calculation. However, this interpretation is not correct be-cause the equivalent sharp radius R q = (cid:0) R drr n q ( r ) (cid:1) / n bulk , q is different from the WS radius parameter, R WSq , and effec-tively depends on the diffuseness of the profile [24]. More-over, as explicitly worked out in Ref. [50], the diffusenessparameter itself depends in a highly non trivial way both onthe gradient terms of the EDF and on the bulk properties of -8-7.75-7.5-7.25-7 exp.HFself-consistentapprox. E / A ( M e V ) self-consistentapprox. < r n2 > / , < r c h2 > / ( f m ) bulkskin D r np ( f m ) FIG. 3: ETF results corresponding to the ground state of Pbisotopes and SLy4 [36]. Binding energy per nucleon (toppanel), rms radii of neutron and charge distributions (middlepanel) and neutron skin thickness are plotted as a function oftotal isospin asymmetry. When available, experimentalmasses [51] and charge radii [52] are plotted as well. Forneutron skin thicknesses of
Pb the following experimentaldata are illustrated: 0 . ± . . + . − . fm[16] and 0 . ± ( . ) exp ± ( . ) model ± ( . ) strange fm [13, 53]. As in Fig. 2, two methods for calculating theCoulomb energy are considered. Neutron skin thicknessdecomposition into bulk and surface contributions accordingto eqs. (17, 18) is represented on the bottom panel for thecase in which the Coulomb energy is calculatedself-consistently (open symbols).matter.Fig. 3 illustrates the total binding energy per nucleon(top panel), rms radius of charge distribution (middle panel)and neutron skin thickness (lower panel) as a function of I .When available experimental data for binding energies [51]and charge radii [52] are plotted as well. For neutron skinthickness of Pb we display data from Refs. [13, 15, 16, 53].Self-consistent calculation of the Coulomb energy leads toa fair agreement with experimental data though a system-atic over binding is obtained for nuclei with I < .
17. Com-plete HF calculations from Ref. [29], performed in spherical symmetry, are also shown. Aside a residual deviation whichcan be ascribed to the choice of the functional and/or beyondmean-field effects, HF calculation describe very well the ex-perimental data. Concerning the ETF calculations, we can seethat missing higher ¯ h orders and the use of a parametrizeddensity profiles lead to a deviation with respect to the experi-mental data which is larger then the one of the HF calculation.The energy error is however very small for Pb and neigh-boring nuclei. This justifies the method described in IIB andemployed to build meta-modelling EDF based on best fit ofproperties of spherical nuclei.The performances of the ETF approximation whenCoulomb is consistently included in the variation, can bejudged also from the agreement of rms radii of charge distri-butions with experimental data. As one may see in the middlepanel of Fig. 3, the overall accord is good. The most im-portant deviations, of the order of 0.05 fm, are obtained for I > .
17. This deviation is comparable to the one obtainedwith complete ETF or DFT calculations in the absence of de-formation [54, 55], and can be ascribed to the choice of thefunctional and/or to beyond mean field effects. Neutron skinthickness presents a linear dependence on I irrespective howCoulomb was calculated. As easy to anticipate, the consistentdisplacement of neutron and proton distributions, due to theCoulomb repulsion, leads to values of the neutron skin thick-ness lower than those obtained in the simplifying approxima-tion. It is interesting to remark that while the Coulomb ef-fect decreases the neutron skin, the different diffuseness ofthe proton and neutron density profiles tends to increase it. Asa consequence, the two effects partially cancel and the globalresult is close to our previous calculations [29], where botheffects were neglected in order to obtain analytic approxima-tions. The bottom panel depicts also the bulk and surfacecontribution to the skin thickness [24], calculated accordingto eqs. (17, 18). One notices that, for Pb, they contributeequally to the total thickness while in neutron-richer (neutron-poorer) isotopes it is the bulk (surface) term that dominates.Given the relatively low L sym =46 MeV value of SLy4 [36], thisresult is in good agreement with the droplet model (DM) cal-culations of Ref. [56], where the dominance of bulk/surfacecontributions was shown to be linked to the value of L sym . B. Correlations between nuclear observables and parametersof nuclear matter
The correlation between the neutron skin thickness of
Pband L sym has been reported in the past years in many differ-ent studies based on density functionals [11, 12, 24], semi-classical approaches [56, 57], as well as DM [56].More recently, the existence of other correlations with var-ious isovector modes of collective excitation was suggested,namely electric dipole polarizability [16–18], isovector giantdipole resonance (IVGDR) [58], isovector giant quadrupoleresonance (IVGQR) [20], pygmy dipole resonance (PDR)[19, 58, 59], anti-analog giant dipole resonance (AGDR) [60–62]. A correct description of these modes demands a dynam-ical treatment in the framework of linear response theory and metamodelingSkyrme D r m i rr o r ( f m ) a D ( P b ) ( f m ) D r np ( Pb) (fm) D ( P b ) ( M e V ) FIG. 4: Correlations between neutron skin thickness in
Pband differences in the proton radii of mirror nuclei R p ( Ni ) − R p ( Ca ) (top), electric dipole polarizability of Pb (middle) and IVGDR energy constant of
Pb(bottom). Results corresponding to Skyrme andmeta-modelling are represented with open circles and,respectively, solid squares. Skyrme predictionscorresponding to differences in the proton radii of mirrornuclei R p ( Ni ) − R p ( Ti ) , R p ( Ni ) − R p ( Cr ) , R p ( Ni ) − R p ( Fe ) are also plotted in the top panel.Numbers on l.h.s. of each plot correspond to Pearsoncorrelation coefficients between the observables plotted onthe axis. Upper (lower) values: meta-modelling (Skyrme).is beyond the purpose of this work. However, simplified ex-pressions were proposed. An example in this sense is givenby Ref. [63] which relates the electric dipole polarizability ofa nucleus of mass number A and isospin asymmetry I , α D = π e A h r i E sym (cid:18) + E sym − a sym E sym (cid:19) . (19)with the ground state symmetry energy in the local density approximation [24] a sym ( A ) = π AI Z ∞ dr r n ( r ) δ ( r ) e sym ( n ( r )) , (20)where e sym = ( / ) ∂ e ( n , δ ) / ∂δ | δ = represents the localsymmetry energy. Another example is offered by Ref. [64]which expresses the IVGDR energy constant in terms of sym-metry energy, saturation density and surface stiffness coeffi-cient, Q sti f f , as, D = D ∞ / q + E sym A − / / Q sti f f , (21)where D ∞ = q h E sym / (cid:0) mr (cid:1) and r = / ( π n sat ) . Thesurface stiffness coefficient measures the resistance of theasymmetric semi-infinite nuclear matter against separation ofneutrons and protons to form a skin and is typically per-formed within HF or ETF approaches. Such calculationsshowed some sensitivity of Q sti f f to the calculation proce-dure [65, 66] as well as significant correlations with thesymmetry energy and its first and second order derivatives[57, 67]. Different approximation formulas have been pro-posed. Some of them express Q sti f f in terms of a num-ber of nuclear matter parameters and are based on fits ofHF or ETF calculations performed using different EDFs.Within the Liquid Drop Model, Ref. [46] calculates Q sti f f from calculations of finite nuclei disregarding the Coulombinteraction. In the present work we adopt the expression, Q sti f f = E sym A − / / / ( E sym / a asym − ) , obtained by equat-ing the ground state symmetry energy given by eq. (20) withthe corresponding DM expression [24]. For the case of Pbits accuracy is of the order of 10%, which leads to a rela-tive error of 2% on the IVGDR energy constant of
Pb cal-culated according to eq. (21). This small uncertainty onlymarginally affects the correlation between the macroscopi-cally derived IVGDR energy constant and various propertiesassociated with the finite nuclei or the nuclear matter. How-ever, more important distortions might come from the natureof the approximation itself, namely the use of macroscopicexpressions in case of dynamical quantities. Such distortionsapply to both α D and D .Another interesting observable, potentially linked to theisovector EoS parameters, is given by the difference betweenthe proton radii R p = h r p i / of mirror nuclei [68, 69]. Thisobservable has the interesting feature of being directly acces-sible from a variational calculation without any extra modelassumption. Moreover, it is much more accessible experimen-tally than the neutron skin, which demands the measurementof the neutron distribution.The correlation between the proton radii differences in mir-ror nuclei and electric dipole polarizability on one hand andneutron skin thicknesses on the other hand has been addressedin Refs. [23, 68–70]. Ref. [64] focused on the nuclear sym-metry energy dependence of the IVGDR energies by consider-ing a series of Skyrme interaction potentials. The correlationsbetween neutron skin thickness, electric dipole polarizabilityand IVGDR energy constant of Pb and proton radii differ-ence for A =
48 mirror nuclei are investigated in Fig. 4 for metamod. metamod., 0metamod., 100metamod., -100Skyrme D r np ( P b ) ( f m ) L sym (MeV) D r m i rr o r ( N i , C a ) ( f m ) FIG. 5: Correlations between L sym and ∆ r np ( Pb ) (toppanel) and L sym and R p ( Ni ) − R p ( Ca ) (bottom panel). Inaddition to meta-modelling EDF plotted in the previousfigures here we consider also meta-modelling EDF with fixedvalues of K sym = , , −
100 MeV. The numbers on the l.h.smention the Pearson correlation coefficients in the followingorder: meta-modelling with freely varying K sym ,meta-modelling with K sym = , , −
100 MeV and Skyrme.both meta-modelling EDF and Skyrme functional. For com-pleteness, Skyrme predictions corresponding to differences inthe proton radii of A = , ,
54 and ∆ r np ( Pb ) are alsoplotted in the top panel. In the case ∆ r mirror vs. ∆ r np ( Pb ) .meta-modelling EDF lead to a strong correlation, with a Pear-son correlation coefficient of 0.98. A moderate correlationis obtained for D ( Pb ) vs. ∆ r np ( Pb ) . A poor correla-tion is found between the dipole polarizability and neutronskin thickness. Skyrme functionals provide very similar re-sults. Very strong correlations are obtained only between ∆ r np ( Pb ) and proton radii differences in mirror nuclei with A = , , ,
54. This result is in agreement with Refs.[68, 69]. The correlation between electric dipole polarizabil-ity and ∆ r np ( Pb ) is loose, in agreement with Ref. [70].Ref. [70] has actually evidenced that a much better correla-tion holds between ∆ r np and ( α D E sym ) , as expected from eq.(19). Finally Skyrme functionals lead to medium strength cor-relations between IVGDR energy constant and neutron skinthickness of Pb. This result can be understood consideringthe L sym - and K sym - dependence of the D quantity via Q sti f f .We now turn to test the sensitivity of the observables to the different isovector parameters of the EoS. In a previ-ous work [29], a full Bayesian analysis of the correlationmatrix was performed, though with a more simplified ver-sion of the ETF meta-modelling, which did not include theself-consistent treatment of Coulomb nor the definition of ( n bulk , n , n bulk , p , a n , a p ) as independent variational variables. Inthat study, it was shown that the neutron skin is only sensitiveto the L sym parameter. The present calculations, with a moresophisticated treatment of the ETF meta-modelling, confirmthe results of our previous work.The correlation between the neutron skin in Pb and the L sym parameter is shown in the top panel of Fig. 5. The lowervalue of the correlation coefficient with respect to the resultsof Ref. [29] can be understood from the fact that the differ-ence between the neutron and proton diffusivity was neglectedin Ref. [29]. This value is also lower than the one correspond-ing to Skyrme functionals, as well as to the ones reported bymost analyses in the literature using specific energy function-als [23, 24, 56, 57, 71]. The higher dispersion of the meta-modelling is due to the fact that the different EoS parametersare fully independent in the meta-modelling approach. As al-ready observed in Ref. [29], though the EoS parameters are allinfluential in the calculation of nuclear masses and radii, theconstraint on those quantities does not generate correlationsamong the EoS parameters because compensations can freelyoccur.To demonstrate this statement, we have generated modelswith arbitrary fixed values of K sym fulfilling the same crite-ria imposed to the global set of models, see Section IIB. Theresulting correlations are shown in Fig. 5 for three cases K sym = − , ,
100 MeV. We can observe that the correla-tion between
Pb and L sym is greatly improved when K sym is fixed. In the case of Skyrme functionals, K sym can largelyvary but its value is positively correlated to L sym because ofthe specific function form of the density dependent term inSkyrme interactions (see Figure 1 (a)). As a consequence, theSkyrme results interpolate the more general meta-model onesand the correlation coefficient is only slightly less than thosecorresponding to meta-model EDF with fixed K sym -values.The bottom panel of Fig. 5 summarizes the analyses doneabove but for the correlation between the proton radii differ-ence in A =
48 mirror nuclei and L sym . The conclusions aresimilar: strong (poor) correlations exist in the case of Skyrmefunctionals and meta-modelling EDF with fixed K sym -values(meta-modelling EDF with freely varying K sym ).The correlations of the dipole polarizability and IVGDRenergy constant of Pb with L sym are reported in Figure 6,for the meta-modelling and for the selected Skyrme function-als. As in Fig. 5 meta-modelling EoS with fixed values of K sym = − , ,
100 MeV are also considered. As one maysee, α D ( Pb ) and D ( Pb ) show less correlation with L sym than with ∆ r np ( Pb ) , when meta-modelling EDF are em-ployed. At variance with this, Skyrme functionals providefor α D ( Pb ) and D ( Pb ) almost the same degree of cor-relation with L sym as with the neutron skin of Pb. A wordof caution is nevertheless in order. The accurate calculationof these two dynamical quantities is possible only within thelinear response theory. The Eqs. (19, 21) presently employed metamodelingSkyrme a D ( P b ) ( f m ) L sym (MeV) D ( P b ) ( M e V ) FIG. 6: Correlations between L sym and α D ( Pb ) (top) and L sym and D ( Pb ) (bottom). Numbers on r.h.s. of each plotcorrespond to Pearson correlation coefficients between theobservables plotted on the axis. Upper (lower) values:meta-modelling (Skyrme). As in Fig. 5 meta-modelling EDFwith fixed values of K sym = , , −
100 MeV are plotted aswell.rely on approximations and are, thus, expected to distort thesensitivity to nuclear matter EoS.In Ref. [29] the isoscalar and isovector parameters of themeta-modelling EDF have been determined by fits of exper-imental binding energies of symmetric nuclei with masses20 ≤ A ≤
100 and full isotopic chains of Ca, Ni, Sn and Pb.We have tested that the conclusions drawn above and the de-grees of correlation remain the same if the pool of nuclei onwhich the parameters of the EDF are determined is replacedby the one considered in Ref. [29].
IV. CONCLUSIONS
In this paper, we have explored the influence of the dif-ferent isovector empirical EoS parameters on some properties of atomic nuclei, namely neutron skin thickness, differencein proton radii of mirror nuclei, dipole polarizability and theIVGDR energy constant of
Pb.The analysis was done within a recently proposed meta-modelling technique [28, 29]. Varying the parameters of themeta-modelling, it is possible to reproduce existing relativisticand non-relativistic EDF, as well as to consider novel densitydependencies which are not explored by existing functionals.With respect to our previous work Ref. [29], we have im-proved the ETF formalism employed to extract nuclear massesand radii out of a given EDF: the Coulomb interaction is con-sistently included in the variational procedure, and the bulkdensities and diffuseness parameters of the density profilesare treated as independent variational parameters. These im-provements allow a better description of nuclear radii and thenuclear skin. The correlation between this latter observableand the slope of the symmetry energy L sym , already reportedin numerous studies in the literature with different EDFs aswell as many body techniques, is confirmed by our study.However, we show that the quality of this correlation is con-siderably worsened if we allow independent variations of thecurvature parameter K sym with respect to the slope L sym , whilethis was not observed in previous studies probably because inmost existing functionals such correlation exists. We concludethat it will be very important to constrain the curvature param-eter with dedicated studies, in order to reduce the confidenceintervals of EoS parameter and allow more reliable extrapola-tions to the higher density domain.We have shown that the condition of a reasonable repro-duction of nuclear masses and radii does not necessarily implyany strong correlation between L sym and K sym . 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