Electric dipole polarizability in neutron-rich Sn isotopes as a probe of nuclear isovector properties
aa r X i v : . [ nu c l - t h ] J a n Electric dipole polarizability in neutron-rich Sn isotopes as a probe of nuclear isovectorproperties
Z. Z. Li a , Y. F. Niu a , W. H. Long a a School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China
Abstract
The determination of nuclear symmetry energy, and in particular, its density dependence, is a long-standing problem for nuclearphysics community. Previous studies have found that the product of electric dipole polarizability α D and symmetry energy atsaturation density J has a strong linear correlation with L , the slope parameter of symmetry energy. However, current uncertainty of J hinders the precise constraint on L . We investigate the correlations between electric dipole polarizability α D (or times symmetryenergy at saturation density J ) in Sn isotopes and the slope parameter of symmetry energy L using the quasiparticle random-phaseapproximation based on Skyrme Hartree-Fock-Bogoliubov. A strong and model-independent linear correlation between α D and L is found in neutron-rich Sn isotopes where pygmy dipole resonance (PDR) gives a considerable contribution to α D , attributed to thepairing correlations playing important roles through PDR. This newly discovered linear correlation would help one to constrain L and neutron-skin thickness ∆ R np sti ffl y if α D is measured with high resolution in neutron-rich nuclei. Besides, a linear correlationbetween α D J in a nucleus around β -stability line and α D in a neutron-rich nucleus can be used to assess α D in neutron-rich nuclei. Keywords:
Electric dipole polarizability, Slope parameter of symmetry energy, Neutron-skin thickness
1. Introduction
The determination of nuclear equation of state (EoS) at highdensity is a challenge for both experimental and theoretical nu-clear physics [1, 2], which is crucial for constraining currenttheoretical models [3, 4] and understanding many phenomenain astrophysics [5, 6]. The biggest uncertainty of EoS comesfrom its isovector parts, which are governed by the nuclear sym-metry energy S ( ρ ). The symmetry energy can be expanded as afunction of ε = ( ρ − ρ ) / ρ by S ( ρ ) = J + L ε + K sym ε + ... (1)where J = S ( ρ ) is the symmetry energy at saturation density ρ , while L = ρ (cid:16) ∂ S ∂ρ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ and K sym = ρ (cid:16) ∂ S ∂ρ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ corre-spond to the slope and curvature parameters at saturation den-sity, respectively.The slope parameter of symmetry energy L determines thebehavior of symmetry energy at high density, however, it variesa lot in di ff erent nuclear models. Constraints on L can be ob-tained from heavy-ion collisions [1, 7], properties of neutronstars [5, 8], and nuclear properties of ground state and excitedstates of finite nuclei [9]. For example, it is revealed that L is proportional to the neutron-skin thickness ∆ R np by dropletmodel [10, 11], which is further conformed by many micro-scopic models [12, 13]. However, the obstacle in the measure-ments of neutron radius hinders the access to high-resolution Email address: [email protected] (Y. F. Niu) neutron skin data. As an alternative, charge radii di ff erence ∆ R c between mirror nuclei is proposed as another possible wayto constrain L [14–16], which also faces di ffi culties in the mea-surements of charge radius in proton-rich nucleus.The electric dipole ( E
1) excitation in nucleus is mainly com-posed of the giant dipole resonance (GDR), which is formedby the relative dipole oscillation between neutrons and pro-tons, thus reflecting asymmetry information in nuclear EoS.The electric dipole polarizability α D , being proportional to theinverse energy-weighted sum rule of E E α D and other nuclearisovector properties have been investigated in recent years. Cal-culations performed by RPA model based on Skyrme densityfunctionals SV-min series [26] and relativistic density function-als RMF- δ -t series in Pb suggested a strong linear correlationbetween α D and neutron-skin thickness ∆ R np [27]. However,when one combines the results from a host of di ff erent nucleardensity functionals, this linear correlation is not universal any-more [28]. Starting from droplet model, and further supportedby RPA calculations based on many di ff erent Skyrme and rel- Preprint submitted to Elsevier January 22, 2021 tivistic density functionals in
Pb, the product of dipole po-larizability and symmetry energy at saturation density α D J wassuggested to be much better correlated with neutron-skin thick-ness and symmetry energy slope parameter L than α D alone is[29]. Based on this correlation, L = ± (6) expt ± (8) theor ± (12) est MeV was given by using the experimental α D value in Pb[29], and the intervals J = −
35 MeV and L = −
66 MeVwere further obtained by combining the measured polarizabil-ities in Ni,
Sn and
Pb [30]. Below saturation density, α D in Pb was also found to be sensitive to both the symme-try energy S ( ρ c ) and slope parameter L ( ρ c ) at the subsaturationcross density ρ c = . − [31]. Since S ( ρ c ) is well con-strained, L ( ρ c ) can be strongly constrained from experimental α D in Pb [31]. At ρ r = ρ /
3, another linear correlation wasbuilt between α − D and S ( ρ r ) [32]. Besides, α D between two dif-ferent nuclei [33], as well as α D J between two di ff erent nuclei[30], were also shown to have good linear correlations.In recent years, the electric dipole polarizabilities α D in Pb [34], Ca [35], and stable Sn isotopes [33, 36, 37] weremeasured with high resolution via polarized proton inelasticscattering at extreme forward angles [38]. For unstable nu-cleus Ni, α D was also extracted by Coulomb excitation ininverse kinematics [39]. However, there are problems whenone uses these high-resolution dipole polarizability data to con-strain isovector properties: the constraints on L or ∆ R np is eitherwith big uncertainties due to the uncertainty of J or in model-dependent ways. One way to solve the problem and constrain L sti ffl y is to find a direct and model-independent correlation be-tween α D and L . Although the previous studies have shown thatthe model-independent linear correlation only exists between α D J and L , it was only limited to stable nuclei or nuclei near β -stability line. It is well known that exotic phenomena willpresent when approaching to nuclei far from β -stability line,such as novel shell structures [40–44], new types of excitations[23, 45, 46], and so on. For E ff erent characteristics of E β -stability line, and further a ff ect α D . So an interesting question is if the linear correlation be-tween α D J and L observed in stable nuclei still holds and newcorrelations would appear in neutron-rich nuclei.Therefore, in our study we will explore the correlations be-tween α D and nuclear isovector properties such as slope param-eter L and neutron-skin thickness ∆ R np in even-even Sn iso-topes from neutron-deficient Sn to neutron-rich
Sn. Thecalculations are performed by QRPA based on Skyrme Hartree-Fock-Bogoliubov (HFB) model, in which the spherical symme-tries are imposed. The linear correlations are evaluated by aleast-square regression analysis. Based on the newly discov-ered correlations, constraints on L and neutron-skin thicknesswill be discussed.
2. Theoretical Framework
We carry out a self-consistent HFB + QRPA calculation of E V pp ( rrr , rrr ) = V h − ρ ( rrr ) ρ i δ ( rrr − rrr ) (2)where rrr = ( rrr + rrr ) /
2, and ρ = . − is the nuclear sat-uration density, while V is adjusted by fitting neutron pairinggaps of ∼ Sn according to the five-point formula [56]. Theelectric dipole polarizability α D is given by α D = π e m − , m − = X ν (cid:12)(cid:12)(cid:12) h ψ ν | F (IV)1 µ | ψ i (cid:12)(cid:12)(cid:12) E ν (3)where ψ ν and E ν are the eigenstates and eigenvalues of QRPAequations, and ψ is the ground state. m − is the inverse energy-weighted sum rule (EWSR), which is calculated using the isovec-tor dipole operator F (IV)1 µ = NA Z X p = r p Y µ − ZA N X n = r n Y µ (4)where A , N , Z denote mass number, neutron number, protonnumber, and Y µ are the spherical harmonics. In our calcula-tions, the quasiparticle energy cuto ff E cut is set as 90 MeV andthe total angular momentum cuto ff of quasiparticle j max is setas 21 /
3. Results and Discussions α D and nuclear isovector properties First of all, we study if the previously discovered linear cor-relation between α D J and L holds in the whole tin isotopesfrom neutron-deficient ones to neutron-rich ones. So in Tab. 1,Pearson correlation coe ffi cients (or Pearson’s coe ffi cients) r be-tween α D J and L in even-even Sn isotopes from Sn to
Sn ,as well as the corresponding slopes k of the regression lines areshown based on the HFB + QRPA calculations using 19 Skyrmedensity functionals. Pearson’s coe ffi cient r is a statistic thatmeasures linear correlation between two variables, which is de-fined by the covariance of two variables divided by the productof their standard deviations. A value of | r | = r = Table 1: Pearson’s coe ffi cient r between the product of dipole polarizability andsaturated symmetry energy α D J and the slope parameter of symmetry energy L in Sn isotopes, as well as the corresponding slope k of the regression line ( α D J as a function of L ), calculated by QRPA based on HFB with 19 Skyrme densityfunctionals. Nucleus Sn Sn Sn Sn Sn Sn Sn r k (fm ) 0.844 1.066 1.383 1.543 2.272 2.880 3.5412
20 0 20 40 60 80 1005.56.06.57.07.58.08.5 D ( f m ) L (MeV)r = 0.777 Sn -20 0 20 40 60 80 1007.58.08.59.09.510.0 D ( f m ) L (MeV)r = 0.798 Sn -20 0 20 40 60 80 1008.59.19.710.310.911.512.1 D ( f m ) L (MeV)r = 0.835 Sn -20 0 20 40 60 80 1009.09.610.210.811.412.012.6 D ( f m ) L (MeV)r = 0.805 Sn -20 0 20 40 60 80 1001012141618 D ( f m ) L (MeV)r = 0.912 Sn k = 0.050 -20 0 20 40 60 80 100121416182022 k = 0.065 D ( f m ) L (MeV)r = 0.976 Sn -20 0 20 40 60 80 100141720232629 k = 0.082 D ( f m ) L (MeV)r = 0.989 Sn -20 0 20 40 60 80 100151821242730 k = 0.089 D ( f m ) L (MeV)r = 0.970 Sn Figure 1: (Color online) Plots for dipole polarizability α D against slope parameter of symmetry energy L in Sn isotopes calculated by QRPA based on HFB with 19Skyrme density functionals: SIII, SIV, SV, SVI (blue up triangles); SLy230a, SLy230b, SLy4, SLy5, SLy8 (red circles); SAMi, SAMi-J30, SAMi-J31, SAMi-J32,SAMi-J33 (green diamonds); SGI, SGII, SkM, SkM*, Ska (black squares). A regression line (red solid line) is obtained by a least-square linear fit of the calculated α D as a function of L . r is Pearson’s coe ffi cient and k (fm / MeV) is the slope of the regression line. -20 0 20 40 60 80 10089101112 -20 0 20 40 60 80 10010121416-20 0 20 40 60 80 10012141618 -20 0 20 40 60 80 1001214161820
L (MeV) D ( f m ) Sn,RPA r=0.794 L (MeV) D ( f m ) Sn,RPA r=0.817 D ( f m ) L (MeV)r=0.873
Sn, RPA D ( f m ) L (MeV)r=0.867
Sn, RPA
Figure 2: (Color online) The same as Fig. 1 but for , , , Sn without thepairing correlations. totally uncorrelated. From Tab. 1, one can see the Pearson’scoe ffi cients r in the whole Sn isotopes are all above 0 .
9, show-ing strong linear correlations between α D J and L . So it furtherproofs this linear correlation is a universal one which exists notonly in stable nuclei as revealed in previous studies [29] butalso in neutron-deficient and neutron-rich nuclei. The corre-sponding slope k of the regression line shows a clear increasetrend with the increase of neutron number. The larger k valuemeans a more rapid increase of α D J as a function of L , whichgives a smaller range of L under the same uncertainty of α D J .So the slope k of the regression line is an important quantityto select good candidate nuclei as probes of nuclear isovector properties, which will be discussed in details in Sec. 3.2.Although the above correlation is universal, it cannot pro-vide a sti ff constraint on the slope parameter of symmetry en-ergy L due to the uncertainty in the symmetry energy J . For ex-ample, by adopting J = ± Roca-Maza et al. obtained L = ± (6) expt ± (8) theor ± (12) est MeV, where the uncertainty ±
12 MeV comes from the uncertainty of J [29]. So it would bebetter to find a direct correlation between α D and L . Previousstudies have shown that L and α D have a good linear corre-lation within some specific parameter family [27], however, byincluding di ff erent parameter families, this correlation becomesbad, for example, in Pb the Pearson’s coe ffi cient r was givenas r = .
62 [29] and r = .
77 [28]. Here we recheck the cor-relation between the dipole polarizability α D and the slope pa-rameter L of symmetry energy for the whole tin isotopes fromneutron-deficient ones to neutron-rich ones, as shown in Fig.1, to see if the previous conclusions still hold. In stable nu-cleus Sn, for some specific Skyrme parameter family, suchas SAMi (green diamonds) or SIII-SVI (up blue triangles), onecan observe a good linear correlation, in agreement with Ref.[27]. However, when one includes more di ff erent Skyrme pa-rameter sets, the linear correlation becomes poor, and the Pear-son coe ffi cient r is around 0.8, again in agreement with the casein Pb [28, 29]. Similar situations still exist in nuclei not farfrom the stability line such as , , Sn.However, the cases become totally di ff erent in the neutron-rich nuclei. The coe ffi cients are above 0 . A ≥ α D and L in the neutron-rich Sn isotopes. After A ≥ α D and L is even better than the onebetween α D J and L . We stress here the assessments are carriedout by di ff erent Skyrme functional families. For the neutron-rich nuclei of A ≥
140 with a clear linear correlation, we fur-3
00 110 120 130 140 150 1600510152025
Mass Number A D ( f m ) SLy4Sn isotopes total PDR QRPARPAQRPARPA
Figure 3: (Color online) The dipole polarizabilities as functions of mass number A in even-even Sn isotopes calculated by QRPA (square line) and RPA (circleline) using Skyrme functional SLy4. The total dipole polarizabilities (red) andthe contributions from PDR (blue) are shown respectively. ther give the slopes k of the regression lines. It is seen that k becomes larger with the increase of neutron number, which im-plies that the more neutron rich the nucleus is, the better probeit can be served as for nuclear isovector properties, seeing de-tailed discussions in Sec. 3.2.To understand the above strong linear correlations in neutron-rich Sn isotopes, we first investigate the role of pairing corre-lations. So in Fig. 2 the correlations between α D and L in , , , Sn are studied without considering pairing e ff ects.For stable nucleus Sn, the correlation between α D and L issimilar as the case with pairing correlations, where the Pear-son’s coe ffi cient is only slightly reduced without the inclusionof pairing correlations. However, for these three neutron-richnuclei , , Sn, the linear correlations become much worse,where the Pearson’s coe ffi cients are largely reduced to the val-ues 0 . .
873 and 0 . .
9. Itshows that the pairing correlations play important roles in keep-ing strong linear correlations between α D and L in neutron-richSn isotopes.On the other hand, for neutron-rich nuclei, the PDR appearsin the low-energy part of E α D in Sn isotopes in Fig. 3, wherethe total dipole polarizabilities and contributions from PDR asfunctions of mass number A in even-even Sn isotopes calculatedby QRPA and RPA using Skyrme functional SLy4 are plotted.According to the dipole strength distributions and the transitiondensities, di ff erent energies are selected as the upper boundariesof PDR for di ff erent Skyrme functionals, which are 9 . . . . . -20 0 20 40 60 80 100 120024681012 Sn Sn Sn D ( f m ) o f P DR L (MeV) r = 0.86r = 0.91r = 0.89r = 0.94 Sn Figure 4: (Color online) Plots for dipole polarizability contributed by PDRagainst slope parameter of symmetry energy in , , , Sn isotopes calcu-lated by QRPA based on HFB with 19 Skyrme density functionals: SIII, SIV,SV, SVI (blue up triangles); SLy230a, SLy230b, SLy4, SLy5, SLy8 (red cir-cles); SAMi, SAMi-J30, SAMi-J31, SAMi-J32, SAMi-J33 (green diamonds);SGI, SGII, SkM, SkM*, Ska (black squares). A linear fit is done for each nu-cleus (red solid line) with a corresponding Pearson’s coe ffi cient r . for SIV, and 13 . Sn, the PDR appears and starts to con-tribute to the dipole polariziability α D . With the neutron num-ber increases, the contribution from PDR becomes larger andlarger, which dominates the evolution trend with mass numberof the total α D . With the pairing correlations being turned o ff ,the contribution from PDR to α D is greatly reduced, which al-most keeps a small constant with the increase of neutron num-ber. As a result, the total α D is also reduced a lot, and its in-crease trend with mass number becomes as slow as that before Sn. Before
Sn, the pairing correlations only have verysmall influences on α D . Therefore, it can be seen that the pair-ing correlations play their important roles on dipole polarizia-bilities and further the linear correlations between α D and L through PDR.
13 15 17 19 210.30.40.50.6 14 16 18 20 22 240.40.50.60.7 k = 0.034 r = 0.939 R np ( f m ) D (fm ) Sn k = 0.031 r = 0.947 R np ( f m ) D (fm ) Sn Figure 5: (Color online) Plots for neutron-skin thickness ∆ R np against dipolepolarizability α D in , Sn calculated by QRPA based on HFB with 19Skyrme density functionals: SIII, SIV, SV, SVI (blue up triangles); SLy230a,SLy230b, SLy4, SLy5, SLy8 (red circles); SAMi, SAMi-J30, SAMi-J31,SAMi-J32, SAMi-J33 (green diamonds); SGI, SGII, SkM, SkM*, Ska (blacksquares). A regression line (red solid line) is obtained by a least-square linearfit of the calculated ∆ R np as a function of α D . r is Pearson’s coe ffi cient and k (fm − ) is the slope of regression line. able 2: Constraints on the slope parameter of symmetry energy L from experi-mental dipole polarizability values α exp. D [30, 34–36, 39] using linear correlationbetween α D J and L obtained by skyrme QRPA calculations using 19 Skyrmefunctionals. The Pearson’s coe ffi cient r and the slope k of the regression linefitted by α D J as a function of L are also given. J = . ± . ∆ L min denotes the uncertainty coming from the uncertainty of J . Nucleus α exp. D (fm ) r k (fm ) L (MeV) ∆ L min (MeV) Pb 19 . ± .
60 0.97 2.68 39 . ± . ± . Ni 3 . ± .
31 0.96 0.56 33 . ± . ± . Ca 2 . ± .
22 0.96 0.33 14 . ± . ± . Sn 7 . ± .
50 0.97 1.10 12 . ± . ± . Sn 7 . ± .
58 0.97 1.15 10 . ± . ± . Sn 7 . ± .
51 0.97 1.23 12 . ± . ± . Sn 7 . ± .
87 0.97 1.32 18 . ± . ± . Sn 8 . ± .
60 0.97 1.38 17 . ± . ± . Sn 7 . ± .
56 0.98 1.47 8 . ± . ± . In Fig. 4 we further study the correlation between dipolepolarizabilities α D contributed by PDR and the slope parameter L of symmetry energy in Sn,
Sn,
Sn,
Sn isotopes.It shows that polarizability α D of PDR has a good correlationwith the slope parameter L in general, which enhances the linearcorrelations between the total α D and symmetry energy slopeparameter L .Apart from the correlation between α D and L , the correla-tion between α D and another important isovector property, i.e.,neutron-skin thickness, is also investigated, and the plots forneutron-skin thickness against dipole polarizability in , Snare shown in Fig. 5. Not surprisingly, the linear correlations be-tween ∆ R np and α D in Sn and
Sn are strong with r = . r = .
947 respectively, since the neutron-skin thickness ∆ R np and L are reported to have a good linear correlation when | N − Z | is large [15]. The slopes k of regression lines, fitted by ∆ R np as a function of α D , are generally small in these neutron-rich nuclei, suggesting that α D in neutron-rich nuclei can pro-vide an e ff ective constraints on neutron-skin thickness of thecorresponding nuclei. α D as a probe of nuclear isovector properties In Sec. 3.1, the correlations between α D (or α D J ) and nu-clear isovector properties, e.g., L , ∆ R np , are investigated for thewhole tin isotopes, so in the following, we will analyse whatinformation we can obtain from these correlations, and whichnucleus could be treated as a proper probe of nuclear isovectorproperties in terms of dipole polarizabilities.Experimentally, the dipole polarizabilities of Pb [34], Ni[39], Ca [35], and stable Sn isotopes [36] were measured withhigh resolution. The correlations between α D J and L are al-ways strong for both stable nuclei and nuclei far from stabilityline from previous studies and our results in Sec. 3.1. So inTab. 2, we show the constraints on the slope parameter of sym-metry energy L from experimental dipole polarizability values α exp. D using correlation between α D J and L in these experimen-tally measured nuclei. The correlations between α D J and L are (c) D J ( f m M e V ) o f P b r = 0.950 (d) D (fm ) of Sn D J ( f m M e V ) o f S n r = 0.964r = 0.765 D ( f m ) o f P b (a) r = 0.969 (b) D ( f m ) o f S n D (fm ) of Sn Figure 6: (Color online) The dipole polarizability α D (a) in Pb and (b) in
Sn as a function of the dipole polarizability in
Sn. The dipole polarizabil-ity α D (c) in Pb and (d) in
Sn times the symmetry energy at saturationdensity J as a function of the dipole polarizability in Sn. Calculations aredone by QRPA based on HFB with 19 Skyrme density functionals: SIII, SIV,SV, SVI (blue up triangles); SLy230a, SLy230b, SLy4, SLy5, SLy8 (red cir-cles); SAMi, SAMi-J30, SAMi-J31, SAMi-J32, SAMi-J33 (green diamonds);SGI, SGII, SkM, SkM*, Ska (black squares). r is the Pearson’s coe ffi cient. Uti-lizing the experimental values of α D in Pb [30, 34] and in
Sn [36], andassuming J = . ± . Sn is predictedto be between 14.13 and 16.25 fm . obtained by QRPA calculations using 19 Skyrme density func-tionals as done in Sec.3.1. The corresponding Pearson’s coe ffi -cients r and slopes k of regression lines fitted by α D J as a func-tion of L are also given in the table. It can be seen that the lin-ear correlations are well kept for all these nuclei with r > . J = . ± . L value. Theuncertainty of L is determined by ∆ L = (cid:0) J ∆ α D + α D ∆ J (cid:1) / k ,where ∆ α D and ∆ J are the uncertainties of α D and J , respec-tively. From Tab. 2, it can be seen that L have a remarkableuncertainties which are all larger than ±
30 MeV. In the limit-ing case ∆ α D =
0, the uncertainty of slope parameter ∆ L min comes only from the the uncertainty of J , which is also given inTab. 2. It shows the uncertainty of J contributes more than halfof the total uncertainties of L , which hinders the e ff ective con-straints on L from the correlation between α D J and L . However,with the increase of neutron number in Sn isotopes, ∆ L min hasthe tendency to become smaller. This is because the slope k ofregression line increases faster than the dipole polarizabity α D with the increase of neutron number, and hence α D / k becomessmaller. So the uncertainty caused by ∆ J would become smallif one finds a nucleus with a small α D / k value.Based on the analysis of neutron-rich Sn isotopes in Sec.3.1, a strong correlation between α D and L appears in neutron-rich nuclei (seeing Fig. 1) where the PDR gives a considerablecontribution to the inverse energy-weighted sum rule m − . Soit provides a more e ff ective way to constrain L directly fromdipole polarizability. Moreover, the slope k of regression line(in Fig. 1) becomes larger with the increase of neutron num-ber, which makes the constraints on L from this correlation in5 able 3: Predictions of the dipole polarizabilities in neutron-rich Sn isotopesfrom experimental dipole polarizabilities of Pb [30, 34] and
Sn [36, 37]using the correlations shown in Fig. 6 (c) and (d). The constrained values ofslope parameter of symmetry energy L and neutron-skin thickness of neutron-rich Sn isotopes are also given from the correlations shown in Figs. 1 and Figs.5. The Pearson’s coe ffi cients r and slopes of regression line k fitted by dipolepolarizability α D as a function of L , as well as by neutron-skin thickness ∆ R np as a function of α D , are also shown respectively.Nuclei α PD (fm ) α D as a function of L ∆ R np as a function of α D r k (fm / MeV) L (MeV) r k (fm − ) ∆ R np (fm) Sn 11 . ± .
91 0.91 0.050 18 . ± . . ± . Sn 12 . ± .
96 0.93 0.054 19 . ± . . ± . Sn 13 . ± .
99 0.94 0.057 20 . ± . . ± . Sn 13 . ± .
02 0.96 0.060 21 . ± . . ± . Sn 14 . ± .
04 0.97 0.063 21 . ± . . ± . Sn 15 . ± .
06 0.98 0.065 22 . ± . . ± . Sn 15 . ± .
09 0.98 0.068 22 . ± . . ± . Sn 16 . ± .
12 0.99 0.071 23 . ± . . ± . Sn 16 . ± .
16 0.99 0.075 23 . ± . . ± . Sn 17 . ± .
21 0.99 0.078 23 . ± . . ± . Sn 17 . ± .
27 0.99 0.082 23 . ± . . ± . neutron-rich nuclei more sti ff . For example, an uncertainty of ± . in α D of Sn, which is about the present accuracy forexperimental measurement in dipole polarizability, could con-strain L within ±
10 MeV, while with the same uncertainty of α D in Sn, L can be constrained within ± α D in neutron-rich nuclei.In Fig. 6(a), we study the correlations of α D between Pband
Sn. Although it was found that α D between two stablenuclei, e.g., between Pb and
Sn, have a good linear cor-relation [30, 33], this correlation is no longer well kept when itis extended to α D between one stable nucleus and one neutron-rich nucleus, e.g., between Pb and
Sn, as seen in Fig. 6(a).The correlation between two neutron-rich nuclei, e.g., between
Sn and
Sn, is further checked in Fig. 6(b), and it becomesstrong again. So one fails to predict α D of neutron-rich nu-clei from α D of stable nuclei directly. Since both α D J in stablenuclei and α D in neutron-rich nuclei linearly correlate with L , α D J in stable nuclei should also linearly correlate with α D inneutron-rich nuclei. This is checked by our calculations in Fig.6, where α D J in Pb (c) and in
Sn (d) as a function of α D in Sn are plotted. Good linear correlations with r = . .
964 are found respectively, which can be used for thepredictions of α D in Sn as well as other neutron-rich nuclei.Utilizing the experimental α D values of Pb and
Sn, shownin Tab. 2, and adopting J = . ± . α ∈ [12 . , . and α D ∈ [14 . , .
29] fm are obtained for Sn. Theoverlap α D ∈ [14 . , .
25] fm is finally taken as the predictedvalue for Sn.The same process can be done for other neutron-rich nu-clei. The predicted α D from Sn to
Sn are given in Tab.3, with which the corresponding constraints on L and neutron-skin thickness ∆ R np are deduced and presented in Tab. 3 from the correlations between α D and L , as well as between ∆ R np and α D . The corresponding Pearson’s coe ffi cients r of both corre-lations are shown in the table, and it can be seen that the linearcorrelations are very well kept for all these neutron-rich nuclei.Here since the L values are constrained from the linear correla-tion between α D and L directly, the uncertainties become muchsmaller compared to those shown in Tab. 2. With the increaseof neutron number, the slope of regression line fitted by α D asa function of L becomes larger, and as a result, the uncertaintyof L also becomes smaller until Sn even with an increasinguncertainty in the predicted α PD . For the neutron-skin thickness,the slope of regression line fitted by ∆ R np as a function of α D keeps almost a constant with increasing neutron numbers, yetthe uncertainties of constrained neutron-skin thickness are be-coming larger caused by the increasing uncertainties in α PD . Dueto the lack of experimental data of α D in neutron-rich nuclei, thepresent constraints on L shown in Tab. 3 in fact don’t give newinformation compared to the L values obtained from the cor-relation between L and α D J in Pb and in
Sn. However,the direct correlation between α D and L would show its specialimportance and e ff ectiveness in constraining nuclear isovectorproperties when the experimental data of α D in neutron-rich tinisotopes are available, so the measurements of dipole polariz-ability towards neutron-rich nuclei are strongly called for.
4. Summary
The correlations between electric dipole polarizability α D (or times symmetry energy at saturation density J ) and slopeparameter of symmetry energy L are studied in Sn isotopes pre-formed by QRPA based on Skyrme HFB theory. The previouslyfound correlation between α D J and L is confirmed in the wholeSn isotopes from neutron-deficient ones to neutron-rich ones.The linear correlation between α D and L is not strong in stabletin isotopes and their surroundings, however, it becomes betterfor mass number A > A ≥
140 with the correlation coe ffi cients r > .
9, wherePDR gives a considerable contribution to α D . The enhancementof this correlation between α D and L is attributed to the pairingcorrelations, which play important roles through PDR.With the available high-resolution data of α D , the constraintson L are obtained from the correlation between α D J and L .Large uncertainties of L are found, where more than half arecontributed by the uncertainty from symmetry energy ∆ J = ± . L is theone with a small α D / k value, where k is the slope of regressionline fitted by α D J as a function of L . In stable Sn isotopes, the α D / k becomes smaller towards neutron-rich side.With the strong correlation between α D and L in neutron-rich Sn isotopes, L can be constrained directly and more sti ffl yif experimental data of α D with high resolution in these nu-clei are known. At the moment, α D in neutron-rich nuclei arepredicted using the linear correlation between α D J in a stablenucleus with experimental data and α D in a neutron-rich nu-cleus. The measurements of electric dipole polarizability to-wards neutron-rich nuclei are called for.6 cknowledgement This work is partly supported by National Natural ScienceFoundation of China under Grant No. 12075104, 11675065 and11875152, Fundamental Research Funds for the Central Uni-versities under Grant No.lzujbky-2019-11, and Strategic Prior-ity Research Program of Chinese Academy of Sciences, GrantNo. XDB34000000.
References [1] B.-A. Li, L.-W. Chen, and C. M. Ko, Phys. Rep. , 113 (2008).[2] M. Oertel, M. Hempel, T. Klahn, and S. Typel, Rev. Mod. Phys. ,015007 (2017).[3] M. Dutra, O. Lourenco, J. S. Martins, A. Delfino, J. stone, and P. Steven-son, Phys. Rev. C , 035201 (2012).[4] M.Dutra, O.Lourenco, S.S.Avancini, A.Delfino, D.P.Menezes,C.Providencia, S.Typel, and J.R.Stone, Phys. Rev. C , 055203(2014).[5] J. M. Lattimer and M. Prakash, Astrophys. J , 426 (2001).[6] Z. W. Liu, Z. Qian, R. Y. Xing, J. R. Niu, and B. Y. Sun, Phys. Rev. C ,025801 (2018).[7] M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W. G. Lynch,and A. Steiner, Phys. Rev. Lett. , 122701 (2009).[8] J. M. Lattimer and M. Prakash, Phys. Rep. , 127 (2016).[9] X. Roca-Maza and N. Paar, Prog. Part. Nucl. Phys. , 96 (2018).[10] W.D.Myers and W.J.Swiatecki, Nucl. Phys. A , 267 (1980).[11] M.Warda, X.Vinas, X.Roca-Maza, and M. Centelles, Phys. Rev. C ,024316 (2009).[12] B. A. Brown, Phys. Rev. Lett. , 5296 (2000).[13] L.-W. Chen, C. M. Ko, and B.-A. Li, Phys. Rev. C , 064309 (2005).[14] N. Wang and T. Li, Phys. Rev. C , 011301(R) (2013).[15] B. A. Brown, Phys. Rev. Lett. , 122502 (2017).[16] J. Yang and J. Piekarewicz, Phys. Rev. C , 014314 (2018).[17] G. Colo, L. Cao, N. V. Giai, and L. Capelli, Comp. Phys. Comm. ,142 (2013).[18] J. Terasaki, J. Engel, M. Bender, and J. Dobaczewski, Phys. Rev. C ,034310 (2005).[19] E.Khan and N. V. Giai, Phys. Lett. B , 253 (2000).[20] G.Giambrone, S. Scheit, F. Barranco, P. Bortignon, G.Colo, D.Sarchi, andE.Vigezzi, Nucl. Phys. A , 3 (2003).[21] M. Martini, S. Peru, and M. Dupuis, Phys. Rev. C , 034309 (2011).[22] N.Paar, P.Ring, T.Niksic, and D.Vretenar, Phys. Rev. C , 034312(2003).[23] N. Paar, D. Vretenar, E. Khan, and G. Colo, Rep. Prog. Phys. , 691(2007).[24] P. Ring, Z. yu Ma, N. V. Giai, D. Vretenar, A. Wandelt, and L. gang Cao,Nucl. Phys. A , 249 (2001).[25] T.Niksic, D.Vretenar, and P.Ring, Phys. Rev. C , 064302 (2002).[26] P. Klupfel, P.-G. Reinhard, T. Burvenich, and J. Maruhn, Phys. Rev. C ,034310 (2009).[27] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C , 051303(R) (2010).[28] J. Piekarewicz, B. K. Agrawal, G. Colo, W. Nazarewicz, N. Paar, P.-G.Reinhard, X. Roca-Maza, and D. Vretenar, Phys. Rev. C , 041302(R)(2012).[29] X. Roca-Maza, M. Brenna, G. Colo, M. Centelles, X. Vinas, B. Agrawal,N. Paar, D. Vretenar, and J. Piekarewicz, Phys. Rev. C , 024316 (2013).[30] X. Roca-Maza, X. Vinas, M. Centelles, B. Agrawal, G. Colo, N. Paar,J. Piekarewicz, and D. Vretenar, Phys. Rev. C , 064304 (2015).[31] Z. Zhang and L.-W. Chen, Phys. Rev. C , 064317 (2014).[32] Z. Zhang and L.-W. Chen, Phys. Rev. C , 031301 (2015).[33] T. Hashimoto, A. M. Krumbholz, P.-G. Reinhard, A. Tamii, P. vonNeumann-Cosel, T. Adachi, N. Aoi, C. A. Bertulani, H. Fujita, Y. Fujita,et al., Phys. Rev. C , 031305 (2015).[34] A. Tamii, I. Poltoratska, P. von Neumann-Cosel, Y. Fujita, T. Adachi,C. A. Bertulani, J. Carter, M. Dozono, H. Fujita, K. Fujita, et al., Phys.Rev. Lett. , 062502 (2011).[35] J. Birkhan, M. Miorelli, S. Bacca, S. Bassauer, C. A. Bertulani, G. Hagen,H. Matsubara, P. von Neumann-Cosel, T. Papenbrock, N. Pietralla, et al.,Phys. Rev. Lett. , 252501 (2017). [36] S. Bassauer, P. von Neumann-Cosel, P.-G. Reinhard, A. Tamii, S. Adachi,C. A. Bertulani, P. Y. Chan, A. D. Alessio, H. Fujioka, H. Fujita, et al.,Phys. Rev. C , 034327 (2020).[37] S. Bassauer, P. von Neumann-Cosel, P.-G. Reinhard, A. Tamii, S. Adachi,C. A. Bertulani, P. Y. Chan, A. D. Alessio, H. Fujioka, H. Fujita, et al.,Phys. Lett. B , 135804 (2020).[38] P. von Neumann-Cosel and A. Tamii, Eur. Phys. J. A , 110 (2019).[39] D. M. Rossi, P. Adrich, F. Aksouh, H. Alvarez-Pol, T. Aumann, J. Benlli-ure, M. Bohmer, K. Boretzky, E. Casarejos, M. Chartier, et al., Phys. Rev.Lett. , 242503 (2013).[40] F. Wienholtz, D. Beck, K. Blaum, C. Borgmann, M. Breitenfeldt, R. B.Cakirli, S.George, F. Herfurth, J. D.Holt, M. Kowalska, et al., Nature ,346 (2013).[41] J. Liu, Y. F. Niu, and W. H. Long, Phys. Lett. B , 135524 (2020).[42] J. J. Li, W. H. Long, J. Margueron, and N. V. Giai, Phys. Lett. B , 192(2019).[43] Z. Z. Li, S. Y. Chang, Q. Zhao, W. H. Long, and Y. F. Niu, ChinesePhysics C , 074107 (2019).[44] M. Grasso, Phys. Rev. C , 034316 (2014).[45] D. Savran, T.Aumann, and A.Zilges, Prog. Part. Nucl. Phys. , 210(2013).[46] T. Aumann, Eur. Phys. J. A , 234 (2019).[47] M. Beiner, H. Flocard, N. V. Giai, and P. Quentin, Nucl. Phys. A , 29(1976).[48] N. V. Giai and H. Sagawa, Phys. Lett. B , 379 (1981).[49] H. Krivine, J. Treiner, and O. Bohigas, Nucl. Phys. A , 155 (1980).[50] J. Bartel, P. Quentin, M. Brack, C. Guet, and H.-B. Hakansson, Nucl.Phys. A , 79 (1982).[51] H. Kohler, Nucl. Phys. A , 301 (1976).[52] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schae ff er, Nucl.Phys. A , 710 (1997).[53] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schae ff er, Nucl.Phys. A , 231 (1998).[54] X. Roca-Maza, G. Colo, and H. Sagawa, Phys. Rev. C , 031306 (2012).[55] X. Roca-Maza, M. Brenna, B. K. Agrawal, P. F. Bortignon, G. Colo, L.-G.Cao, N. Paar, and D. Vretenar, Phys. Rev. C , 034301 (2013).[56] M. Bender, K. Rutz, P.-G. Reinhard, and J. Maruhn, Eur. Phys. J. A , 59(2000).[57] J.Piekarewicz, Phys. Rev. C , 044325 (2006).[58] A. Carbone, G. Colo, A. Bracco, L.-G. Cao, P. F. Bortignon, F. Camera,and O. Wieland, Phys. Rev. C , 041301(R) (2010).[59] D.Vretenar, Y.F.Niu, N.Paar, and J.Meng, Phys. Rev. C , 044317(2012)., 044317(2012).