Low-energy E1 strength in select nuclei: Possible constraints on the neutron skins and the symmetry energy
aa r X i v : . [ nu c l - t h ] J un Low-energy E strength in select nuclei: Possible constraints on the neutron skins andthe symmetry energy Tsunenori Inakura,
1, 2
Takashi Nakatsukasa,
2, 3 and Kazuhiro Yabana
3, 2 Department of Physics, Graduate School of Science,Chiba University, Yayoi-cho 1-33, Inage, Chiba 263-8522, Japan RIKEN Nishina Center, Wako, 351-0198, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan
Correlations between low-lying electric dipole ( E
1) strength and neutron skin thickness are sys-tematically investigated with the fully self-consistent random-phase approximation using the Skyrmeenergy functionals. The presence of strong correlation among these quantities is currently underdispute. We find that the strong correlation is present in properly selected nuclei, namely in spher-ical neutron-rich nuclei in the region where the neutron Fermi levels are located at orbits with loworbital angular momenta. The significant correlation between the fraction of the energy-weightedsum value and the slope of the symmetry energy is also observed. The deformation in the groundstate seems to weaken the correlation.
PACS numbers: 21.10.Pc, 21.60.Jz, 25.20.-x
The isospin-dependent part of the nuclear equation ofstate (EOS), especially the symmetry energy, is receiv-ing current attention [1, 2]. Although the symmetryenergy at the saturation density E sym ( ρ ) is relativelywell known, its values at other densities, which have astrong impact on the description of neutron stars andsteller explosions, are poorly determined at present. In-formation on the density dependence of the symmetry en-ergy might be obtained from the neutron-skin thickness∆ r np , since the skin thickness was found to be stronglycorrelated with the slope L of the symmetry energy; L = 3 ρ E ′ sym ( ρ ) [3, 4]. However, the large uncertain-ties in measured neutron-skin thickness have practicallyprohibited us from making an accurate estimate on L .The electric dipole ( E
1) response is a fundamentaltool to probe the isovector property of nuclei. The gi-ant dipole resonance (GDR), which is rather insensitiveto the structure of an individual nucleus, provides infor-mation on the magnitude of the symmetry energy nearthe saturation density ρ . In contrast, the low-energy E E sym ( ρ ) at densities away from ρ .Among many issues on the PDR, the correlation be-tween the PDR and neutron skin is one of important sub-jects currently under dispute. If the strong correlationexists, the PDR may constrain both ∆ r np and the slopeparameter L . The calculation by Piekarewicz with therandom-phase approximation (RPA) based on the rela-tivistic mean-field model predicted a linear correlation forSn isotopes [8]. Utilizing similar arguments, the neutronskin thickness and the slope parameter were estimatedfrom available data in Pb, Ni,
Sn, and so on [5, 9].However, Reinhard and Nazarewicz performed a covari-ance analysis investigating the parameter dependence forthe Skyrme functional models, which concluded that the correlation between the PDR strength and ∆ r np is veryweak [10]. Recently, they have extended their studies tothe E q [11]. Itshould be noted that these conclusions, which seemed tocontradict to each other, were given from RPA calcula-tions for specific spherical nuclei using different ways ofanalysis.Recently, we have performed a systematic RPA cal-culation on the PDR for even-even nuclei [12] using thefinite amplitude method [13–17]. The calculation is self-consistent with the Skyrme energy functional and fullytakes into account the deformation effects. We found thatthe significant enhancement of the PDR strength takesplace in regions of specific neutron numbers. The mainpurpose of the present paper is to show that the qualityof the correlation between the PDR strength and ∆ r np are also sensitive to the neutron number of the isotopes.Namely, the strong correlation exists only in particularneutron-rich nuclei. This may provide a possible sugges-tion for future measurements to constrain ∆ r np and L . Numerical calculations —
We perform an analysis sim-ilar to Ref. [10] to investigate the Skyrme parameter de-pendence of the RPA results for nuclei of many kinds(mostly with Z ≤ x = 0, y = 0, and z = 0 planes. Weadopt the representation of the three-dimensional adap-tive Cartesian grids [19] within a sphere of the radius R box = 15 fm. The real-space representation has an ad-vantage over other representations, such as harmonic os-cillator basis, on the treatment of the continuum states.The Skyrme functional of the SkM ∗ parameter set [20]is used unless otherwise specified. The residual interac-tion in the present calculation contains all terms of theSkyrme interaction including the residual spin-orbit, theresidual Coulomb, and the time-odd components. The S n , S P D R [ e f m ] Pb, Skin thickness [fm] r = 0.55 (1 ± t (1 ± t (1 ± t (1 ± t (1 ± W (1 ± α x ± x ± x ± x ± FIG. 1: (Color online) Correlations between the PDRstrength S PDR in Sn and the neutron skin thickness ∆ r np in Pb. The cross denotes a result obtained with the orig-inal SkM ∗ parameter set. Other symbols represent resultswith the modified parameter set as shown in the right panel.The solid line indicates a linear fit. The correlation coefficientfor these parameter set is also shown. See the text for detail. pairing correlation is neglected for simplicity, which haslittle impact on E Definition of PDR strength, PDR fraction, and corre-lation coefficient—
We define the PDR strength as S PDR ≡ Z ω c S ( E E ) dE = E n <ω c X n B ( E n ) , (1)with the PDR cutoff energy ω c . The PDR fraction f PDR is the ratio of the integrated photoabsorption cross sec-tion below ω c to the total integrated cross section. f pdr = R ω c σ abs ( E ) dE R σ abs ( E ) dE = P E n <ω c n E n B ( E n ) P n E n B ( E n ) , (2)In Eqs. (1) and (2), we fix the cutoff at ω c = 10 MeV.Many former works adopted the same definition [10, 12],because of its simplicity. In light spherical neutron-richnuclei, the value of ω c = 10 MeV can reasonably sepa-rate the PDR peaks from the GDR. However, for heaviernuclei, the separation becomes more ambiguous. It is es-pecially difficult for deformed nuclei. Later, we introduceanother definition of the PDR strength using a variable ω c , to check the validity.To quantify the correlation between two quantities, weuse the correlation coefficient r . When we have datapoints for ( x i , y i ) with i = 1 , · · · , N d , it is defined by r ≡ P N d i =1 ( x i − ¯ x )( y i − ¯ y ) qP N d i =1 ( x i − ¯ x ) qP N d j =1 ( y j − ¯ y ) , (3)where ¯ x and ¯ y are the mean values of x i and y i , respec-tively. The absolute value of r does not exceed the unity.A perfect linear correlation, y i = ax i + b , corresponds to r = ± a . In the Skin thickness [fm] S P D R [ e f m ] Ni (c)r = 0.94 Ni (a)r = 0.69 Ni (b)r = 0.76 Ni Ni Ni σ a b s fr ac ti on [ % ] (d) FIG. 2: (Color online) (a)-(c) Correlations between S PDR and∆ r np in , , Ni. See the caption of Fig. 1. Calculatedcorrelation coefficients are also shown. (d) f pdr as functionsof ∆ r np for even-even Ni isotopes, calculated with the SkM ∗ parameter set. See the text for detail. followings, the correlation with r > r <
0) is referredto as “positive” (“negative”) correlation.
Neutron skin thickness in
Pb —
First, we confirmthe result in Ref. [10]. Reference [10] reported that the S PDR for
Sn has only a weak correlation with the neu-tron skin thickness defined by ∆ r np ≡ p h r i n − p h r i p of Pb. In Fig. 1, the S PDR for
Sn is shown as a func-tion of the neutron skin thickness, ∆ r np , of Pb. Theplotted 21 points are obtained by calculating ∆ r np and S PDR with the SkM ∗ functional, and with slightly modi-fied values of 10 Skyrme parameters ( t , , , , x , , , , W ,and α ). It seems to indicate some correlation, however,the calculated points are somewhat scattered.Using these 21 sample values ( N d = 21), the corre-lation coefficient r is calculated according to Eq. (3).In the present case of Fig. 1, we obtain the coefficient r = 0 .
55. The correlations between ∆ r np in Pb and S PDR in Ni and Ni, are also weak with r = 0 . − . Pb, however, the correlation is weak. This is qualita-tively consistent with the result in Ref. [10].
Correlation between S PDR and ∆ r np — Next, we dis-cuss the same correlation, but between the ∆ r np and S PDR in the same nucleus. In Fig. 2, we show the resultsfor Ni ( N = 40), Ni ( N = 50), and Ni ( N = 56).The scattered data points in Fig. 2 (a) suggest a rel-atively weak correlation in Ni, while the correlationbecomes moderately strong for Ni. The calculated cor-relation coefficients are r = 0 .
69 and 0.76 for , Ni,
Skin thickness [fm] Σ S ( E ) [ e f m ] Ca Ca FIG. 3: (Color online) Same as Fig. 2 but for , Ca. respectively. In contrast, a very strong linear correlationwith r = 0 .
94 for Ni is observed in Fig. 2 (c). It is ap-parent that the linear correlation is qualitatively differentamong the isotopes.The qualitative difference in S PDR among the isotopeswas previously observed in the PDR photoabsorptioncross section [12]. In Ref. [12], we systematically calcu-lated, for even-even nuclei up to Z = 40, the PDR frac-tion f PDR . Then, we found that f pdr significantly in-creases as a function of neutron number in regions wherethe neutron Fermi levels are located at the weakly-boundlow- ℓ shells, such as s , p , and d orbits. In Ni isotopes, thiscorresponds to the region with neutron number beyond50, as illustrated in Fig. 2 (d). Thus, the present result(Fig. 2 (a)-(c)) indicates that the neutron shell effect alsohas a significant impact on the linear correlation betweenthe neutron skin thickness and the PDR strength.We confirm the same neutron shell effect in other lightspherical isotopes; A O and A Ca. For Ca isotopes, thePDR strength appears beyond N = 28 [12]. Accord-ingly, the strong linear correlation can be seen for Caand Ca, in Fig. 3. The calculated correlation coef-ficients are r = 0 .
91 and 0.96 for , Ca, respectively.These nuclei have neutrons more than 28 and the neu-tron Fermi level is located at the p shell. They are pre-dicted to have the PDR peaks around E = 8 MeV with f pdr ≈ . − .
04 [12]. In contrast, nuclei with N ≤ f pdr < .
01 and the linearcorrelation in Ca ( N = 28) indicates r = 0 .
78 whichis much weaker than , Ca. For O isotopes, because ofthe neutron occupation of the 2 s orbit, O ( N = 16)provides another example to show a significant jump in f pdr from O [12]. This nucleus has the strongest linearcorrelation with r = 0 . Sn.It indicates a relatively weak correlation with r = 0 . Sn corresponds to a kink point similar to Ni in Fig. 2. Namely, the PDR fraction in Sn isotopeswill jump up beyond N = 82 [21]. The correlation coef-ficients are summarized in the second column of Table Ifor various nuclei. Deformed nuclei —
The deformation effect seems tosomewhat weaken the correlation. Figure 4 shows two
Skin thickness [fm] Σ S ( E ) [ e f m ] Cr β = 0.17r = 0.80 Zr β = 0.36r = 0.74 FIG. 4: (Color online) Same as Fig. 2 but for deformed nuclei Cr and
Zr. See text for details.
Skin thickness [fm] Σ S ( E ) [ e f m ] r = 0.94 Ni SGII r = 0.97 Ni SIII r = 0.74 Ni SGII r = 0.66 Ni SIII
FIG. 5: (Color online) Same as Fig. 2 but for , Ni withSGII and SIII interactions. deformed nuclei, Cr with the quadrupole deformationof β = 0 .
17 and
Zr with a larger deformation of β = 0 .
36. The Cr nucleus has the same number of neu-trons as Ca carrying a comparable PDR strengths to Ca [12]. Nevertheless, the correlation in Cr, r = 0 . , Ca.
Zr has an even larger deformation and a weaker cor-relation, r =0.74, although it has sizable PDR strength.The ground-state deformation is expected to produce apeak splitting both in the PDR and GDR. Due to thecomplicated characters in the E S PDR may be contaminated by thelow-energy tail of GDR strength.
Universal behaviors —
The property of the linear cor-relation is very robust with respect to choice of theSkyrme energy functionals. In Fig. 5, we show the samecorrelation plot as Fig. 2 calculated with the parameterset of SkM ∗ replaced by SIII [22] and SGII [23]. All thethree Skyrme functionals yield a relatively weak corre- TABLE I: Calculated correlation coefficients r between ∆ r np and S PDR for selected nuclei. The SkM ∗ parameter set is adoptedas the central values. The values of variable ω c are also listed. Note that we cannot identify a prominent PDR peak for Ca. r ( v ) are obtained with the variable cutoff energies ω c in the fourth row. The correlation coefficients larger than 0.9 are shownin boldface. O Ne Ca Ca Ca Ni Ni Ni Cr Zr r r ( v ) ω c [ MeV ] 8.29 9.95 - 10.49 9.41 11.48 8.73 8.59 9.82 8.36 lation for Ni with r = 0 . − .
75 and a strong linearcorrelation for Ni with r > .
94. The strong correlationwith r ≈ .
95 is also confirmed for O and Ca.The slope of the straight line, obtained by linear fit,turns out to be universal too, with respect to differentSkyrme energy functionals. All these three parametersets (SkM*, SGII, and SIII) produce the similar slope, dS PDR /d (∆ r np ) = 13 −
16 e fm for Ni. We observethe linear correlation of f pdr instead of S PDR , as well,with respect to ∆ r np . However, in this case, the slopeobtained by the linear fit has a sizable dependence onfunctionals. Correlation among different energy functionals —
In-stead of slightly modifying the Skyrme parameters, wenext examine the correlation adopting many differentSkyrme functionals corresponding to a variety of values ofthe L parameter; SIII, SGII, SkM ∗ , SLy4 [25], SkT4 [26],SkI2, SkI3, SkI4, SkI5 [27], UNEDF0, and UNEDF1 [28].From these eleven different parameter sets, we estimatethe correlation coefficient r in Eq. (3) with N d = 11.Again, we have found a weak correlation with r = 0 . Ni, and a strong correlation r = 0 .
89 for Ni.We also examine the correlation between the slope pa-rameter of the symmetry energy L and the PDR fraction f PDR , in Ni and Ni. This leads to the similar coeffi-cients, r = 0 .
37 and 0 .
84 for Ni and Ni, respectively.Thus, to quantitatively constrain ∆ r np and L , the mea-surement of the PDR in the very neutron-rich Ni ismore favored than in Ni.The small correlation coefficient between L and f PDR for Ni ( r = 0 .
37) turns out to be due to the fact that thechoice of ω c = 10 MeV has different meaning for differ-ent functionals. Namely, the different energy functionalsproduce different PDR peak energies, some of which arebelow 10 MeV but some are above that. The tail of theGDR strength also depends on the choice of the energyfunctionals. Therefore, to make a more sensible analysisfor this study, we should use the variable cutoff ω c . Thiswill be discussed below. Use of variable ω c — The PDR strength (1) and PDRfraction (2) based on variable ω c are hereafter referredto as S ( v )PDR and f ( v ) pdr , respectively. The variable ω c isdetermined according to the following procedure: Thecalculated (discrete) B ( E
1) values are smeared with theLorentzian with the width of γ = 1 MeV. Plotting thissmeared E S ( E E ) as a function of energy,if we can find a distinguishable PDR peak and its en- Excitation Energy [MeV] E s t r e ng t h [ e f m ] Ni SkM* PDR GDR
FIG. 6: (Color online) Calculated E B ( E n ),vertical lines) for Ni in units of e fm and those smearedwith the width of γ = 1 MeV ( S ( E E ), solid curve) in unitsof e fm /MeV. According to the procedure described in thetext, the cutoff energy is determined as ω c = 8 .
59 MeV. ergy E peak , ω c is defined as the energy corresponding tothe minimum value of S ( E E ) at E > E peak . In Fig.6, as an example, the determination of ω c is shown for Ni. Since the determination of the variable ω c requiresa noticeable PDR peak structure, it is difficult to define S ( v )PDR for most of stable isotopes.The values of ω c varies from nucleus to nucleus withina range of 10 ± ω c may also change when we slightly modify theSkyrme parameters. Although the correlation is slightlyenhanced by replacing S PDR by S ( v )PDR in most cases, theyare approximately similar, r ( v ) ≈ r . In Table I, thereare a few exceptions; Ni ( r = 0 . → r ( v ) = 0 . Ni ( r = 0 . → r ( v ) = 0 . Zr( r = 0 . → r ( v ) = 0 . ω c .On the other hand, isotopes indicating r > . ω c = 10 MeV show r ( v ) ≈ ω c as well. InNi isotopes, although the value of r ( v ) are slightly differ-ent from r , it is confirmed that the linear correlation issignificantly stronger in Ni than in Ni.For eleven different parameter sets, the correlation be-tween S ( v )PDR and ∆ r np for , Ni is shown in the upperpart of Fig. 7. A strong positive correlation ( r ( v ) > . S ( v )PDR and ∆ r np can be seenin Ni. In contrast, it is significantly weaker for Ni( r ( v ) = 0 . L [MeV] f P D R [ % ] Ni (c) r = 0.80 Ni (d) r = 0.87 Skin thickness [fm] Σ S ( E ) [ e f m ] Ni (a)r = 0.48 Ni (b)r = 0.91 SIIISGIISkM*SLy4SkT4SkI2SkI3SkI4SkI5UNEDF0UNEDF1
FIG. 7: Correlations between S ( v )PDR and ∆ r np (top panels),and between f ( v ) pdr and L (bottom) for Ni (left) and Ni(right), among eleven different Skyrme functionals. tion between f ( v )PDR and the slope parameter L of the sym-metry energy. Again, the correlation is stronger for Niwith r ( v ) = 0 .
87 than Ni with r ( v ) = 0 .
80. The correla-tion between ∆ r np and L has similar trend, r ( v ) = 0 . Ni and r ( v ) = 0 .
84 for Ni. Basic features ofthe correlation with the variable ω c are consistent withthose obtained with ω c fixed at 10 MeV. Thus, the PDRstrength in Ni with many excess neutrons can providea better constraint on L and the neutron skin, comparedto Ni.
Summary —
We have studied the correlation of thePDR and the neutron skin thickness, for nuclei with Z ≤
40 and
Sn. We have found that the strong lin-ear correlation is seen only in particular nuclei. The PDRstrength has a very strong linear correlation with the neu-tron skin thickness in spherical neutron-rich nuclei with14 < N ≤
16, 28 < N ≤
34, and 50 < N ≤
56. Inthese regions, the neutron Fermi levels are located at theloosely-bound low- ℓ shells and the PDR strengths signifi-cantly increase as the neutron number. Nuclei outside ofthese regions have weaker correlations. This linear corre-lation is robust with respect to the choice of the energyfunctional parameter set. This suggests that the experi-mental observation of PDR in properly selected neutron-rich nuclei could be a possible probe of the neutron skinthickness ∆ r np and a constraint on the slope parameter L of the symmetry energy. The linear correlation seemsto be weakened by the deformation due to the peak split-ting of the PDR and the GDR. The present result mayprovide a solution for the controversial issue on the corre-lation between the PDR and the neutron skin, for whichdifferent conclusions were reported previously [5, 6, 8, 10]. Acknowledgments
This work is partly supported by HPCI System Re-search Projects (Project ID: hp120192 and hp120287),by Collaborative Interdisciplinary Program (Project ID:13a-33, 12a-20 and 11a-21) at University of Tsukuba, andby JSPS KAKENHI Grant numbers 21340073, 24105006,25287065, and 25287066. The numerical calculationswere partially performed on the RIKEN Integrated Clus-ter of Clusters (RICC) as well. [1] A. W. Steiner et al., Phys. Rep. , 325 (2005).[2] . B. A. Li et al., Phys. Rep. , 113 (2008).[3] B. A. Brown, Phys. Rev. Lett. , 5296 (2000).[4] R. J. Furnstahl, Nucl. Phys. A , 85 (2002).[5] A. Klimkiewicz et al., Phys. Rev. C 76 , 051603(R)(2007).[6] S. Volz et al., Nucl. Phys.
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