Constraints on neutron skin thickness in 208Pb and density-dependent symmetry energy
aa r X i v : . [ nu c l - t h ] A p r Constraints on neutron skin thickness in
Pb anddensity-dependent symmetry energy
Jianmin Dong, ∗ Wei Zuo, † and Jianzhong Gu ‡ Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China China Institute of Atomic Energy,P. O. Box 275(10), Beijing 102413, China
Abstract
Accurate knowledge about the neutron skin thickness ∆ R np in Pb has far-reaching implica-tions for different communities of nuclear physics and astrophysics. Yet, the novel Lead RadiusExperiment (PREX) did not yield stringent constraint on the ∆ R np recently. We employ a morepracticable strategy currently to probe the neutron skin thickness of Pb based on a high linearcorrelation between the ∆ R np and J − a sym , where J and a sym are the symmetry energy (coefficient)of nuclear matter at saturation density and of Pb. An accurate J − a sym thus places a strongconstraint on the ∆ R np . Compared with the parity-violating asymmetry A PV in the PREX, thereliably experimental information on the J − a sym is much more easily available attributed to awealth of measured data on nuclear masses and on decay energies. The density dependence of thesymmetry energy is also well constrained with the J − a sym . Finally, with a ‘tomoscan’ method, wefind that one just needs to measure the nucleon densities in Pb starting from R m = 7 . ± . R np in hadron scattering experiments, regardless of its interior profile that ishampered by the strong absorption. PACS numbers: 21.65.Ef, 21.10.Gv, 21.65.Cd, 21.60.Jz ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION The nuclear physics overlaps and interacts with astrophysics not only expands its researchspace but also promotes the development of fundamental physics. A great of attention hasbeen paid to the equation of state (EOS) of isospin asymmetric nuclear matter in boththe two fields as the development of radioactive beam facilities and astronomical observa-tion facilities over the past decade. The symmetry energy that characterizes the isospindependence of the EOS, is a quantity of critical importance due to its many-sided roles innuclear physics [1–7] and astrophysics [8–14]. Although great efforts have been made andconsiderable progresses have been achieved both theoretically and experimentally, its densitydependence ultimately remains unsolved because of the incomplete knowledge of the nuclearforce as well as the complexity of many-body systems. Nevertheless, many important andleading issues in nuclear astrophysics require the accurate knowledge about it ungently atpresent.The symmetry energy S ( ρ ) of nuclear matter is usually expanded around saturationdensity ρ as S ( ρ ) = J + L (cid:18) ρ − ρ ρ (cid:19) + K sym (cid:18) ρ − ρ ρ (cid:19) + ..., (1)where J = S ( ρ ) is the symmetry energy at ρ . The slope parameter L = 3 ρ∂S ( ρ ) /∂ρ | ρ andcurvature parameter K sym = 9 ρ ∂ S/∂ρ | ρ characterize the density-dependent behavior ofthe symmetry energy around ρ . Extensive independent studies have been performed toconstrain the slope L , but the uncertainty is still large [15–18].It has been established that the slope parameter L is strongly correlated linearly with theneutron skin thickness ∆ R np of heavy nuclei [19–21]. Although the theoretical predictions on L and ∆ R np are extremely diverse, this linear correlation is universal in the realm of widelydifferent mean-field models [22]. Accordingly, a measurement of ∆ R np with a high accu-racy is of enormous significance to constrain the density-dependent behavior of S ( ρ ) around ρ . Actually, many experimentalists have been concentrating on it with different methodsincluding the x-ray cascade of antiprotonic atoms [23], pygmy dipole resonance [24, 25], pro-ton elastic scattering [26], proton inelastic scattering [27] and electric dipole polarizability[28]. However, systematic uncertainties associated with various model assumptions are un-avoidable. The parity-violating electron elastic scattering measurement in the parity radiusexperiment (PREX) at the Jefferson Laboratory combined with the fact that the parity-2iolating asymmetry A PV is strongly correlated with the neutron rms radius, determinedthe ∆ R np to be 0 . +0 . − . fm with a large central value compared to other measurements andanalyses [29]. Although it was suggested that ruling out a thick neutron skin in Pb seemspremature [30], in any case, the large uncertainty seems to be not of much help to explorethe symmetry energy and other interesting issues. In this work, a more practicable strategycompared with the PREX at current is introduced to probe the ∆ R np of Pb together withnuclear matter symmetry energy. A new insight into the neutron skin is also provided.
II. NEUTRON SKIN THICKNESS ∆ R np PROBED BY THE J − a sym The neutron skin thickness of nuclei is given as ∆ R np = q (cid:2) r J ( J − a sym ( A )) A / ( I − I c ) − e Z/ (70 J ) (cid:3) + S sw in the nuclear droplet model [31, 32]with isospin asymmetry I , nuclear radius R = r A / and a correction I c = e Z/ (20 J R )due to the Coulomb interaction. Z , A are the proton and mass numbers, respectively. S sw is a correction caused by an eventual difference in the surface widths of nucleon densityprofiles. a sym ( A ) is symmetry energy (coefficient) that has been received great interestbecause with the help of it one may obtain some information on the density dependenceof S ( ρ ) [33–35]. Centelles et al . showed that the neutron skin thickness ∆ R np correlateslinearly with J − a sym ( A ) based on different mean-field models, where the symmetryenergy (coefficient) a sym ( A ) is obtained within the asymmetric semi-infinite nuclear matter(ASINM) calculations [32]. In our previous work, instead of using the ASINM calculations,the a sym ( A ) was obtained in the framework of the Skyrme energy-density functionalapproach by directly integrating the density functional of the symmetry energy aftersubtracting Coulomb polarization effect without introducing additional assumptions [33].In the present work, the a sym ( A ) of Pb, marked as a sym , is extracted with both theSkyrme effective interactions and relativistic effective interaction Lagrangians, and the localdensity approximation is adopted by dropping the negligible non-local terms comparedto [33]. As done in Ref. [22], to prevent eventual biases, we avoid including more thantwo models of the same kind fitted by the same authors and protocol and avoid modelsproviding a charge radius of Pb away from experiment data by more than 1%.The calculated neutron skin thickness ∆ R np of Pb and J − a sym with different mean-field models are presented in Fig. 1, in which a close dependence of ∆ R np on J − a sym K D E M S k K D E S L y v S k S C S kz BS k H F B T S L y N S S kz S k P L N S S G III U - F S UN L3 - . S k S C O S k M * M S L0 M S k AS k M PS I V F S U G o l d S k a SV S k I SK T SK T M A N L - SV N L S H S k I T M F S U G o l d - N L3 N L3 * Linear Fit, r = 0.989 R np (f m ) J - a sym (MeV) N L1 FIG. 1: (Color Online) Neutron skin thickness ∆ R np in Pb against the J − a sym with differentnuclear energy-density functionals. predicted by the droplet model is displayed. By performing a two-parameter fitting, thecorrelation is given by∆ R np = (0 . ± . J − a sym ) + (0 . ∓ . , (2)with the correlation coefficient r = 0 . R np and J − a sym are in units of fmand MeV, respectively. Here the empirical saturation density ρ = 0 .
16 fm − [36] is useduniformly. If the symmetry energy is calculated at their own saturation densities fromthe mean-field models, the linear correlation vanishes due to the fact that the relativisticinteractions provide smaller saturation densities compared with the non-relativistic ones.The ∆ R np of Pb is found to have a high linear correlation with J − a sym as that with theslope L (not shown here). It is thus indisputable that the J − a sym with a high accuracy placesa stringent constrain on the ∆ R np . As the primary advantage, the reliably experimentalinformation about the J − a sym is much more easily available compared with that aboutthe parity-violating asymmetry A PV in the PREX. Recently, the symmetry energy J atsaturation density ρ has been well determined to rather narrow regions, in particular, 32 . ± . . ± .
31 MeV from the double differencesof experimental symmetry energies [38] agreeing with that of the mass systematics. Theseresults are very useful in exploring the density-dependent symmetry energy as inputs [39].4ere we adopt the union of the two values, i.e. J = 32 . ± . a sym of Pb accurately. We extract the massdependent symmetry energy a sym ( A ) = J/ (1 + κA / ) [40, 41] with β − -decay energies Q β − of heavy odd- A nuclei and with mass differences ∆ B between A ( Z −
1) and A ( Z + 1) as ourprevious calculations [42, 43] but with a new input quantity J , and then derive the a sym of Pb. The merit of these two approaches is that only the well known Coulomb energysurvives in Q β − and in ∆ B when determining the unknown a sym , where the Q β − and ∆ B are all taken from experimental data. Consequently, the a sym is extracted to be 22 . ± . J . As a result, the derived J − a sym is 10 . ± . L in our subsequent calculations.The neutron skin thickness in Pb is predicted to be ∆ R np = 0 . ± .
021 fm (solidsquare in Fig. 1), where the estimated error stems from the uncertainties of the J − a sym as well as Eq. (2). To reach such an error level, the A PV in the PREX should be measuredat least up to 2% accuracy, which is hardly implemented at present. This fact indicatesthe J − a sym is much more effective to probe the ∆ R np currently. The precise informationabout the ∆ R np is of fundamental importance and has far-reaching implications in neutronstar physics, such as the structure, composition and cooling. As an example, a relation of ρ c ≈ . − . R np was put forward to describe the relation between the ∆ R np of Pband the transition density ρ c from a solid neutron star crust to the liquid interior [44], wherethe ρ c is estimated to be 0 . ± .
008 fm − . The properties of the crust-core transition isof crucial importance in understanding of the pulsar glitch [45]. III. DENSITY DEPENDENCE OF THE SYMMETRY ENERGY PROBED BYTHE J − a sym Since the neutron skin thickness ∆ R np correlates linearly with both the slope L and J − a sym ,the slope L naturally correlates linearly with the J − a sym , which is displayed in Fig. 2(a).The linear relation is L = (9 . ± . J − a sym )+( − . ∓ . L and J − a sym are in units of MeV. Imposing the above obtained J − a sym , the slope parameter is estimatedto be L = 54 ±
16 MeV. Recently, the properties of nuclear matter at subsaturation density ρ ≈ . − have attracted considerable attention because it has been shown that the ∆ R np
5s uniquely fixed by the slope L ( ρ ≈ .
11 fm − ) [46] and the giant monopole resonance ofheavy nuclei is constrained by the nuclear matter EOS at this density [47]. Fig. 2(b) showsthat the slope L ( ρ = 0 . − ) (labeled L . for short) and J − a sym have a higher lineardependence L . = (4 . ± . J − a sym ) + (2 . ∓ . r = 0 . L . is evaluated to be 48 ± L ( M e V ) J - a sym (MeV) Linear Fit r = 0.995 Linear Fit r = 0.984 = 0.11 fm -3 (a) J - a sym (MeV) (b) = 0.16 fm -3 FIG. 2: (Color Online) Correlation of the slope parameter L at densities ρ = 0 .
16 fm − and ρ = 0 .
11 fm − with the J − a sym . The slope L is constrained with the J − a sym in another way for comparison. Centelles et al . found that the symmetry energy a sym of Pb is approximately equal to the nuclearmatter symmetry energy S ( ρ A ) at a reference density ρ A ≃ . − [32]. This importantrelation bridges the symmetry energies of nuclear matter and the finite nucleus. We calculatethe reference density ρ A for Pb and find that the interactions which provide the values of J and a sym agreeing with the ones extracted from experimental information, give ρ A ≃ . − = 0 . ρ . It should be noted that the a sym does not equal the symmetry energy atthe mean density of Pb as a result of the extremely inhomogeneous isospin asymmetrydistribution in the nucleus as shown in [33]. Since the accurate value of the reference density ρ A is of crucial importance for determining the slope parameter L [42, 43], we further examineit. Instead of the DDM3Y-shape expression used before [42, 43], Eq. (1) is employed directlyto describe the density dependent symmetry energy to reduce the uncertain factors as far6s possible. The K sym term that contributes weakly to the symmetry energy nearby ρ isestimated with the relation K sym = 39 + 5 L − J [48] obtained from the DDM3Y-shapeexpression without loss of accuracy. In terms of J − S ( ρ A ) = 10 . ± . ρ A = 0 . ρ ,the slope L at the saturation density ρ is predicted to be 53 ±
10 MeV according to Eq.(1), which is in excellent agreement with that from Fig. 2(a). At the density of ρ = 0 . − , the slope L . = 49 ± ± ρ A = 0 . ρ . As an important conclusion, the a sym = S ( ρ = 0 . ρ ) ≃ . L . and L values, the curvature parameter is evaluated to be K sym = − ±
70 MeV. Currently, the symmetry energy at suprasaturation densities isextremely controversial. It was indicated that the three bulk parameters J , L and K sym well characterize the symmetry energy at densities up to ∼ ρ while higher order termscontribute negligibly small [49]. If true, the symmetry energy S ( ρ ) at high densities up to ∼ ρ turns out to be not stiff, as shown in Fig. 3. The symmetry energy at 2 ρ is estimatedto be S (2 ρ ) = 42 ±
10 MeV. In short, to characterize the symmetry energy at high densities,the accurate knowledge about its density dependence at the saturation density is crucial.
L = 54 16 MeV S () ( M e V ) L=54MeVL=70MeVL=38MeV / FIG. 3: (Color Online) Density dependent symmetry energy at high densities. V. FURTHER EXPLORATION ON THE MEASUREMENT OF THE ∆ R np Based on the above discussions on the neutron skin thickness ∆ R np and symmetry energy, wemake an exploration on the measurement of the ∆ R np in Pb. To grasp richer informationon the ∆ R np , we formulate it as an integral of a distribution function∆ R np = p < r n > − q < r p > = Z ∞ f ( r ) dr, (3)where f ( r ) = 4 πr (cid:0) ρ n N − ρ p Z (cid:1) / (cid:16)p < r n > + p < r p > (cid:17) is defined as the radial distributionfunction which is actually determined by the nucleon densities and reflects the detailedinformation about the neutron skin. p < r n > + p < r p > ≃ . f ( r ) in Pb as a function of distance r generated bythe SLy5 interaction as an example. It is a misleading idea to consider the neutron skinmerely originating from the nuclear surface. The area enclosed by the x-axis and the curve f ( r ) (colored regions) is exactly the neutron skin thickness ∆ R np . We name this new methodthat dissects the ∆ R np with a distribution function as ‘tomoscan’ picturesquely here. As anew concept in nuclear physics, it could also be used to analyze other intriguing issues, suchas the halo structure in exotic nuclei. The region of r < R contributes negatively whilethat of r > R contributes positively to the ∆ R np . Thus, there exists a distance R m belowwhich (0 ≤ r < R m ) the contributions (red shaded regions) cancel each other out, and hencethe ∆ R np can be calculated by the neutron and proton density distributions just startingfrom R m (blue filled region).The calculated values of R m with different interactions are marked in Fig. 4(b). The R m is found to be model dependent, which should be further constrained. The interactionsgenerating smaller (larger) ∆ R np tend to yield slightly larger (smaller) R m . As we mentionedabove, one important conclusion of this work is that the a sym = S ( ρ = 0 . ρ ) ≃ . J ≃ . R m = 7 . ± .
04 fm (colored solid symbols),where the error bar of ± .
04 fm just leads to an uncertainty of the ∆ R np by about ± . ± .
005 fm for the ∆ R np is so small that the obtained R m valueshould not be regarded as model dependent any more. This result leads to an intriguing8 .1 0.2 0.37.57.67.77.87.98.0 SLy5 R R np (fm) R m = 7.62 fm LNS1 (c)(b) r (fm) SkaKDE SLy4 R m (f m ) f ( r ) r (fm) SLy5 (a) e rr o r acc u m u l a ti on (f m ) FIG. 4: (Color Online) (a) Radial distribution function f ( r ) of the neutron skin thickness in Pb. The contributions from the two parts in the red shaded regions cancel each other out. Thearea under the curve of f ( r ) starting from R m (blue filled region) is equal to the neutron skinthickness ∆ R np . (b) Calculated R m values with different energy density functionals. The coloredsolid symbols are from the interactions generating the reference density ρ A ≃ . ρ , a sym ≃ . J ≃ . R np measurement in hadron scatteringexperiments as a function of distance r , where the nucleonic density distributions are from TablesIII and IV in Ref. [26]. conclusion: one just needs to measure the rather dilute matter located in the nuclear surfaceto determine the neutron skin thickness of Pb, namely, only measures the nucleon densitiesfrom r = R m = 7 . ± .
04 fm to about r = 12 fm. Thus, the measurement of the∆ R np would be substantially simplified in hadron scattering experiments which have beenhampered by the strong absorption in the nuclear interior. We stress that, contrary to theusual understanding, the nuclear surface properties are in fact not well constrained by thenuclear mean-field models obtained by fitting nuclear masses and charge radii. For instance,both the SLy5 and NL3 interactions give R m = 7 .
62 fm, but they provide a substantialdifference in the ∆ R np amounting to 0 .
12 fm. In other words, it is exactly the ambiguityof the nuclear surface profile that leads to the large uncertainty of the ∆ R np , because theradial distribution function f ( r ) relies on the fourth power of distance r according to Eq.(3), causing a drastic amplification of the error as r increases. Fig. 4(c) illustrates the erroraccumulation of the ∆ R np in hadron scattering experiments for different regions, which isobtained by analyzing the data in Ref. [26] combined with the ‘tomoscan’ method. The9rror accumulation at distance r is defined as the error generated by the region from thenuclear center to r . It indicates that the error also primarily originates from the surfacestructure. Therefore, the surface profiles must receive particular attention and be measuredwith a much higher accuracy. V. SUMMARY
We have developed alternative methods in the present study to explore the neutron skinthickness ∆ R np of Pb and density dependence of symmetry energy. The main conclusionsare summarized as follows. i) We have established a high linear correlation between the ∆ R np and J − a sym on the basis of widely different nuclear energy-density functionals. Accordingly,an accurate J − a sym value sets a significant constrain on the ∆ R np , which turns out tobe a much more effective probe than the parity-violating asymmetry A PV in the currentPREX. ii) The symmetry energy (coefficient) a sym of Pb was extracted accurately withthe experimental β − -decay energies of heavy odd- A nuclei and with the experimental massdifferences. Given that the symmetry energy J has been well determined recently, the ∆ R np in Pb was thus predicted to be 0 . ± .
021 fm robustly. This conclusion would besignificantly meaningful to discriminate between the models and predictions relevant for thedescription of nuclear properties and neutron stars. iii) With the above derived J − a sym ,the values of the slope L of the symmetry energy at the densities of ρ = 0 .
16 fm − and ρ = 0 .
11 fm − which are of great concern, are predicted to be 54 ±
16 MeV and 48 ± R np of Pb, can be applied to explore someintriguing problems in nuclear astrophysics. In particular, the derived a sym and S ( ρ A ) serveas important calibrations for a reliable construction of new effective interactions in nuclearmany-body models. iv) The symmetry energy at suprasaturation densities up to ∼ ρ waspredicted to be not stiff. v) With the firstly proposed ‘tomoscan’ method, we concludedthat to obtain the ∆ R np one needs to only measure the nucleon densities in Pb from R m = 7 . ± .
04 fm as the densities in the range of r < R m have no contribution to the∆ R np . Thus, the measurement on the ∆ R np is significantly simplified in hadron scatteringexperiments which have been hampered by the strong absorption in the nuclear interior.Incidentally, the ‘tomoscan’ method could be employed to analyze the halo structure in exoticnuclei. vi) It has been widely believed that the nuclear surface structure is well constrained10n nuclear energy-density functionals and in experimental measurements. However, withinthe ‘tomoscan’ concept, we have showed that it is not true but a complete illusion. To graspthe ∆ R np , one must especially concentrate on the dilute matter located in nuclear surfacewhich results in the dominant uncertainty. Acknowledgment
This work was supported by the National Natural Science Foundation of China under GrantsNo. 11405223, No. 11175219, No. 10975190 and No. 11275271, by the 973 Program ofChina under Grant No. 2013CB834405, by the Knowledge Innovation Project (KJCX2-EW-N01) of Chinese Academy of Sciences, by the Funds for Creative Research Groups ofChina under Grant No. 11321064, and by the Youth Innovation Promotion Association ofChinese Academy of Sciences. [1] V. Baran, M. Colonna, V. Greco, and M. Di Toro, Phys. Rep. , 335 (2005).[2] B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. , 113 (2008).[3] P. Danielewicz, R. Lacey, and W. G. Lynch, Science , 1592 (2002).[4] A. W. Steiner, M. Prakash, J. Lattimer, and P. J. Ellis, Phys. Rep. , 325 (2005).[5] J. M. Pearson, N. Chamel, A. F. Fantina, and S. Goriely, Eur. Phys. J. A , 43 (2014).[6] N. Wang, M. Liu, and X. Wu, Phys. Rev. C , 044322 (2010).[7] J. Dong, W. Zuo, and W. Scheid, Phys. Rev. Lett. , 012501 (2011).[8] H.-T. Janka, K. Langanke, A. Marek, G. Mart´ınez-Pinedo, and B. M¨uler, Phys. Rep. , 38(2007).[9] J. M. Lattimer and M. Prakash, Phys. Rep. , 121 (2000); Phys. Rep. , 109 (2007).[10] K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk, Phys. Rev. Lett. , 161102(2010).[11] A. W. Steiner and A. L. Watts, Phys. Rev. Lett. , 181101 (2009).[12] D. H. Wen, B. A. Li, and P. G. Krastev, Phys. Rev. C , 025801 (2009).[13] H. Sotani, K. Nakazato, K. Iida, and K. Oyamatsu, Phys. Rev. Lett. , 201101 (2012).[14] L. F. Roberts et al. , Phys. Rev. Lett. , 061103 (2012).
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