Symmetry energy from the nuclear collective motion: constraints from dipole, quadrupole, monopole and spin-dipole resonances
EEPJ manuscript No. (will be inserted by the editor)
Symmetry energy from the nuclear collective motion: constraintsfrom dipole, quadrupole, monopole and spin-dipole resonances
G. Col`o , , U. Garg , , H. Sagawa , Dipartimento di Fisica, Universit`a degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy INFN, sezione di Milano, via Celoria 16, I-20133 Milano, Italy Physics Department, University of Notre Dame, Notre Dame, Indiana 46556, USA Joint Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, Indiana 46556, USA Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, Fukushima 965-8560, Japan RIKEN Nishina Center, Wako 351-0198, JapanReceived: date / Revised version: date
Abstract.
The experimental and theoretical studies of Giant Resonances, or more generally of the nuclearcollective vibrations, are a well established domain in which sophisticated techniques have been intro-duced and firm conclusions reached after an effort of several decades. From it, information on the nuclearequation of state can be extracted, albeit not far from usual nuclear densities. In this contribution, whichcomplements other contributions appearing in the current volume, we survey some of the constraints thathave been extracted recently concerning the parameters of the nuclear symmetry energy. Isovector modes,in which neutrons and protons are in opposite phase, are a natural source of information and we illustratethe values of symmetry energy around saturation deduced from isovector dipole and isovector quadrupolestates. The isotopic dependence of the isoscalar monopole energy has also been suggested to provide a con-nection to the symmetry energy: relevant theoretical arguments and experimental results are thoroughlydiscussed. Finally, we consider the case of the charge-exchange spin-dipole excitations in which the sumrule associated with the total strength gives in principle access to the neutron skin and thus, indirectly, tothe symmetry energy.
PACS.
As is testified by the variety of contributions in this vol-ume, complementarity of the sources of information is avital component of our understanding of the symmetryenergy. In our contribution, we review several attemptsto use the nuclear collective excitations as a tool to inferthe properties of the symmetry energy. We stress that inmost of these cases, although we can only access densi-ties that are relatively close to the usual nuclear density,the information can be considered as quite accurate dueboth to well established experimental techniques and tothe availability of microscopic methods that have beentested against many other observables. This is at variancewith other situations (astrophysical observations and, tosome extent, heavy-ion collisions) in which one is poten-tially able to explore a broader range of densities, but atthe expense of facing with more global and specific uncer-tainties.We start by reviewing the basic equations related tosymmetry energy. For any nuclear system the total energymust depend both on neutron and proton densities ρ n and ρ p , E = (cid:90) d r E ( ρ n ( r ) , ρ p ( r )) , (1)where E is the energy density and we have assumed lo-cality for the sake of simplicity. In the following, we willuse q as a generic label for neutrons and protons. In finitesystems the energy can actually depend not only on thespatial densities, but also on their gradients ∇ ρ q , on thekinetic energy densities τ q , as well as on other generaliseddensities like the spin-orbit densities J q ; however, in infi-nite matter, one has a simple expression in terms of thespatial densities only (cf., e.g., Ref. [1]).Instead of ρ n and ρ p , one can use the total density ρ and the local neutron-proton asymmetry, β ≡ ρ n − ρ p ρ . (2)In asymmetric matter, we can make a further simplifica-tion on E ( ρ, β ) by making a Taylor expansion in β andretaining only the quadratic term (odd powers of β are a r X i v : . [ nu c l - t h ] D ec G. Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion obviously forbidden due to isospin symmetry), E ( ρ, β ) ≈ E ( ρ, β = 0) + E sym ( ρ ) β = E ( ρ, β = 0) + ρS ( ρ ) β . (3)The first term on the r.h.s. is the energy density ofsymmetric nuclear matter E nm , while the second term de-fines the main object of all studies in this volume, namelythe symmetry energy S ( ρ ). The symmetry energy at sat-uration S ( ρ ) is denoted by different symbols in the liter-ature viz. J , a τ or a ; we shall use J in what follows. Westress, however, that Eq. (3) is not really a simplification:the coefficient of the term in β which should follow isnegligible in most models at the densities of interest forthis work (see, e.g., Ref. [2] and in particular Fig. 1 of thatwork; this conclusion has been also systematically checkedin Ref. [3] with all the models used therein). We remindthe reader that the pressure can be written in a uniformsystem as P = − ∂E∂V (cid:12)(cid:12)(cid:12)(cid:12) A = ρ ∂∂ρ E ρ (cid:12)(cid:12)(cid:12)(cid:12) A , (4)where A is the nucleon number. Thus, although it is cus-tomary to refer to the energy per particle as the “equationof state”, the relationship with the quantity that betterfits such name, that is, the pressure as a function of thedensity, is evident from the latter equation.The overall trend of the symmetry energy is poorlyknown, but the main quantities on which attention hasbeen focused are J ≡ S ( ρ ) ,L ≡ ρ S (cid:48) ( ρ ) ,K sym ≡ ρ S (cid:48)(cid:48) ( ρ ) . (5) L is often referred to as the “slope parameter”.It is expected, therefore, that the isovector giant res-onances may be the main source of information for thesymmetry energy. They are collective excitations in whichmost of the nucleons participate, as it is known from thefact that they exhaust a large fraction of the appropriatesum rules. Especially in heavy nuclei we can assume thatif an isovector external field displaces the protons with re-spect to the neutrons, by creating a local proton-neutronasymmetry, the restoring force in the harmonic approxi-mation can be related to δ Eδβ , (6)if the oscillations involve only variations of β . Of coursesuch simple argument should be taken with care: the nu-cleus does not have uniform values of ρ and β ; isospin isnot an exact quantum number so that isoscalar and isovec-tor oscillations are not well separated; and, finally, quan-tum effects like shell structure or pairing may also spoilthe simple classical arguments. However, we will show inwhat follows that these warnings, although manifestingthemselves in some error bar that we must attribute toour extractions of the symmetry energy, do not prevent at all deducing values for the symmetry energy around sat-uration. We shall also show that isoscalar modes like themonopole resonance can be somewhat related to the sym-metry energy if one observes this mode along an isotopicchain.The outline of the paper is the following. In Secs. 2, 3and 4 we discuss some constraints coming from differentisovector states, that is, the well-known giant dipole reso-nance, the so-called pygmy resonances, and the isovectorgiant quadrupole resonance, respectively. In Sec. 5 we pro-vide theoretical arguments why the isoscalar monopole,when measured in nuclei with neutron excess, could alsoprovide access to some key parameter associated with thesymmetry energy; the related experimental data and quan-titative conclusions are drawn in Sec. 6. In Sec. 7 we moveto the spin-dipole mode excited by charge-exchange reac-tions, namely to the spin-dipole mode of excitation, whosetotal strength is related with the neutron skin; we reviewthe experimental findings and the associated theoreticalanalysis. Finally, we provide a short summary in Sec. 7. The relationship between the symmetry energy and themost collective and well known isovector giant resonance,namely the isovector giant dipole resonance (IVGDR),can be well elucidated by some macroscopic model. Asit has been done in Ref. [3], one can start as a guidelinefrom the hydrodynamical model of giant resonances, asproposed by E. Lipparini and S. Stringari [4]. They as-sume an energy functional (1) which is simplified yet suf-ficiently realistic, solve the macroscopic equations for thedensities and currents, and extract expressions for the mo-ments m and m − associated with an external operator F ( m k ≡ (cid:82) dE S ( E ) E k where S is the strength functionassociated with F ). The expression for m is proportionalto (1 + κ ), where κ is the well-known “enhancement fac-tor” which in the case of Skyrme forces is associated withtheir velocity dependence [1]. The expression for m − , inthe case of an isovector external operator, includes inte-grals involving E sym and F . They can be evaluated in asimple way if one assumes the validity of the leptodermousexpansion. In this way, one introduces volume and surfacecoefficients of E sym that can be denoted by b vol and b surf ,respectively. By specializing F to the isovector dipole case,the following expression is obtained: E − ≡ (cid:114) m m − = (cid:118)(cid:117)(cid:117)(cid:116) h m (cid:104) r (cid:105) b vol (cid:16) b surf b vol A − (cid:17) (1 + κ ) . (7)This equation has been found to yield values of the cen-troid energy which are in rather good agreement withthose of microscopic Hartree-Fock plus Random PhaseApproximation (HF-RPA) calculations performed with mi-croscopic Skyrme interactions [3,5]. In fact, the same equa-tion has been used in a previous study [5], in order to con-strain directly the parameters of the isovector part of theSkyrme interaction. . Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion 3 Although there is not a straightforward analytic rela-tion between Eq. (7) and a similar expression that con-tains the symmetry energy, a semi-analytic relationshiphas been found in Ref. [3]. The plausibility of such a rela-tion can be briefly discussed here. The coefficient b vol canbe identified with J . If the nucleus had a sharp surface thiswould be the only quantity appearing in Eq. (7). The rel-evance of the nuclear surface, however, manifests itself inthe correction (cid:16) b surf b vol A − (cid:17) − . One could then assumethat the r.h.s. of Eq. (7) does not scale as √ b vol ≡ (cid:112) S ( ρ ),but rather as (cid:112) S (¯ ρ ) where ¯ ρ is some value of density be-low the saturation density ρ , namely it is an average den-sity that takes into account the fact that some nucleonsare localised in the inner part of the nucleus where thedensity is ρ while others are more localised at the surfacewhere the density is lower.
11 12 13 14 15 16E -1 [MeV] f ( . ) = { S ( . )( + κ ) } / [ M e V / ] Skyrme-RPA calculationsFit
Fig. 1.
Correlation associated with Eq. (8) in the case of
Pb.The empty symbols correspond to the microscopic Skyrme-RPA calculations and the line is a linear fit whose correlationcoefficient is 0.91. The complete information about the Skyrmemodels that have been employed, can be found in Ref. [3] fromwhich this figure has been adapted. The arrow indicates theexperimental value for E − from Ref. [6]. In Ref. [3] it has been found indeed that such a cor-relation between E − (calculated within HF-RPA) and (cid:112) S (¯ ρ ) exists. One has to consider the term (1 + κ ), whilethe value of (cid:104) r (cid:105) does not vary significantly when onechanges the model used to calculate the dipole. For heavynuclei like Sn or
Pb the value of ¯ ρ is around 0.1 fm − whereas this value tends to lower in, e.g., Ca. The mi-croscopic HF-RPA results for E − have been calculatedusing a broad set of Skyrme forces and a strong linearcorrelation of the type (cid:112) S (¯ ρ )(1 + κ ) = a + bE − ( RP A ) (8)has been found. Typically, the correlation coefficient isaround ≈ κ (in other word, of the total dipoleEWSR up to high enough energies) is not available. How- ever, by using the case of Pb and inserting the experi-mental value for the IVGDR in Eq. (8), it has been foundthat 23 . < S (0 . < . , (9)where the error takes into account the uncertainty of thelinear fit as well as of the experimental value of κ . Recently, much interest has been devoted to the dipolestrength below the IVGDR. While in light, neutron-richhalo nuclei this strength may be specially enhanced due tothe large transition probability of weakly bound neutronsto continuum states (“threshold effect”), this is not thecase in medium-heavy nuclei with neutron excess. Thesenuclei may be either stable or unstable but the neutronexcess gives rise only to a neutron skin and not to a halo.In such systems a possible, peculiar mode of vibration hasbeen proposed, namely the oscillation of the neutrons ofthe skin with respect to the (essentially N ≈ Z) core. Thefrequency of this mode should be lower than the IVGDRand the name “Pygmy Dipole Resonance” (PDR) has beenintroduced in the literature.Experimentally, low-lying dipole strength has been foundin several nuclei (see, e.g., Fig. 2 of Ref. [7], and the tworeview papers [8,9]). To understand the microscopic na-ture of this strength, namely to establish whether it corre-sponds to the PDR picture that we have just described, orwhether it is a non-collective state, is hard if not impossi-ble especially when measurements are not exclusive ones.Typically the PDR strength may arrive up to a few % ofthe dipole EWSR. Among the cases in which this strengthappears unambiguously, we mention the nuclei
Sn [7]and Ni [10].In Ref. [11], using those two nuclei, another type ofcorrelation between dipole properties and the symmetryenergy has been found. This correlation is between thepercentage of EWSR exhausted by the PDR and the slopeparameter L defined in Eq. (5). This correlation appearsclearly, although it is not perfect, when L is plotted againstthe fraction of EWSR calculated not only by means ofSkyrme HF-RPA, but by means of relativistic RPA ontop of Relativistic Mean Field (RMF) as well. By usingsuch correlation plot, and the aforementioned experimen-tal data, values of L have been extracted for the two nucleiunder study. The separate values of L have been extractedwith an error that takes into account experimental errorsand uncertaintes in the fits of Fig. 2. The two values over-lap and if one assumes one can perform a weighted averagefor both mean values and uncertainties, one obtains L = 64 . ± . . (10)Within the choice of our models, this implies J = 32.3 ± G. Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion
12 34 262524232221201918171615141312111098765 Ni E W S R [ % ] L [MeV]
12 34 5 67 89101112 1314 15 161718 192021222324 2526 Sn E W S R [ % ] L [MeV]
PDR NiPDR Sn Sn Ni r fit =0.91r fit =0.95 E W S R [ % ] L [MeV]
12 345 6 7 8910 1112 1314 15 161718 19 202122232425 26
L=64.8±15.7J=32.3±1.3r fit =0.90 J [ M e V ] L [MeV]
Fig. 2.
In the two upper panels the correlation between the slope parameter L and the fraction of EWSR exhausted by thePDR is displayed, for the two nuclei Ni and
Sn respectively. In the lower-left panel the results for the two nuclei are showntogether, and the experimental findings are used to deduce and allowed value for L (as it is discussed in the main text). Inthe lower-right panel a correlation plot L - J is displayed, so that a value for J is also deduced. Taken from Ref. [11] (where thedetailed correspondence bwteen numbers and models used can also be found). more than one author [12,13], the neutron skin is in factwell correlated with L (this can be understood, since if L increases the symmetry energy undergoes larger changesgoing from the inner part of the nucleus to the surface, andso the system finds it energetically more convenient to lo-calise the excess neutrons on the surface). Consequently,the values of the neutron skins ∆R extracted from theprevious value of L are ∆R (cid:0) Ni (cid:1) = 0 . ± .
015 fm; ∆R (cid:0) Sn (cid:1) = 0 . ± .
024 fm; ∆R (cid:0) Pb (cid:1) = 0 . ± .
024 fm . (11)We do not dispose of a model that can help as a guide-line to understand in detail the correlation between theEWSR of the PDR and L . In a simplified picture, if thePDR is really a mode in which the excess neutrons partic-ipate and their dynamics is decoupled from the IVGDR,then the correlation we have discussed can be intuitivelyunderstood. In fact, the percentage of EWSR of the PDRincreases with the number of neutrons belonging to the“skin” [15], and the skin increases with L as we have justdiscussed.However, this picture may not be valid in all nucleiwhere the PDR shows up. In Ref. [16], it has been claimedthat the PDR has a single-particle character: this analysis has been carried out using a specific local energy func-tional, namely SV-bas. In Ref. [17] the microscopic char-acter of the PDR has been analysed in detail in the nuclei Ni,
Sn and
Pb: while for certain Skyrme energyfunctionals (those characterised by a larger value of L )the PDR shows up as a well-defined peak so that it canbe truly defined as a resonance, in other cases this does nothappen. Consequently, it may be more cautious to referto the low-lying dipole strength as composed of “pygmydipole states” (PDSs). These states have also a mixedisospin character and in the isoscalar response they showup more clearly, and display more coherence of the micro-scopic particle-hole (p-h) amplitudes, than in the isovectorresponse. A similar analysis, leading to consistent conclu-sions, has been performed in Ref. [18]. A possible compo-nent of toroidal motion in the low-energy dipole strengthhad been initially claimed in the microscopic calculationsof Ref. [19] and recently revived (see, e.g., [20] and ref-erences therein). About the nature of the pygmy states,and the question whether any relation with the symmetryenergy or the neutron skin has to be expected, the readercan also consult the contribution by J. Piekarewicz to thisvolume (and references therein), as well as the works onthe PDR by T. Inakura and co-workers [21,22].In conclusion, the nature of the PDR is still stronglydebated because of its complex and mixed (isoscalar/isovector, . Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion 5 surface/volume, irrotational/toroidal) character. Certainly,more experimental and theoretical investigation should beenvisaged. For the time being, it remains intriguing howdespite all warnings the EWSR carried by the PDR couldlead to very reasonable values of J and L as it has beenshown in the first part of this Section.We do not discuss in this contribution other aspectsrelated to IV dipole observables and the symmetry en-ergy. The relevance of the total dipole polarizability [23],and the fact that L turns out to be correlated with theproduct of J times the dipole polarizability [24], are thor-oughly discussed in the contribution to this volume by J.Piekarewicz. The properties of the isovector giant quadrupole reso-nance (IVGQR) have not been determined so accuratelyfor some time, due to the lack of experimental probeshaving good selectivity. Recently, at the HI γ S facility,it has been demonstrated that the scattering of polar-ized photons can provide a direct measurement of theIVGQR properties, without the uncertainties associatedwith hadronic probe experiments [25]. This experimentalachievement has motivated the attempt to extract infor-mation about the symmetry energy.The energy of the IVGQR receives contribution fromthe unperturbed p-h configurations at 2¯ hω excitation en-ergy, plus some correlation energy related to the residualinteraction. Since the residual interaction is in the isovec-tor channel, it can be naturally linked with the symme-try energy. In Ref. [26] the quantum harmonic oscillatormodel has been applied to the IVGQR case and, with mildassumptions and taking care of the fact that the unper-turbed energy can be related to the effective mass and, inturn, to the isoscalar GQR energy, the main result is thefollowing formula: E IVGQR ≈ (cid:34) ( E ISGQR ) ε ∞ A / (cid:18) S ( ρ A ) ε F ∞ − (cid:19)(cid:35) / , (12)where ε F ∞ is is the Fermi energy for symmetric nuclearmatter at saturation density, and S ( ρ A ) is the symmetryenergy at some average nuclear density for the nucleushaving mass number A. As we have done above in theIVGDR case, we can choose this average density as 0.1fm − . A first important outcome of the previous equationis that the same value of S (0 .
1) that we have derived inEq. (9) is consistent with the experimental energies of theISGQR and IVGQR, and turns out to be further validatedby this fact.Both Skyrme and relativistic mean-field models do in-deed follow quite well the scaling predicted by Eq. (12). Inthe case of the Skyrme models, we demonstrate this factin Fig. 3. The models have been built by using the sameprotocol of the recent parameter set SAMi [27]. In some ofthem all nuclear matter parameters have been kept fixedas in the original force, but the effective mass has been
SAMi-m65SAMi-m70SAMi-m75SAMi-m80SAMi-m85 m/m* ( E x I V ) ( M e V ) SAMi-J27 r =0.990 (a) SAMi-J28SAMi-J29SAMi-J30SAMi-J31
SAMi-J27 SAMi-J29 SAMi-J30 SAMi-J31SAMi-m65SAMi-m70SAMi-m75SAMi-m80SAMi-m85 ∆ r np (fm)450500550 ( E x I V ) ( M e V ) SAMi-J28 (b) r =0.980 Fig. 3.
Sensitivity of the energy of the IVGQR in
Pb tothe effective mass (upper panel) and to the neutron skin (lowerpanel). For details about the models whose names appear inthe panels, and for an explanation of the emerging correlations,see the main text. Taken from Ref. [26]. changed: thus, SAMi-m85 means that m ∗ /m is 0.85; inothers, all nuclear matter properties have been kept fixedbut the symmetry energy at saturation has been changed:in this case, SAMi-J27 means that J =27 MeV.From the upper panel of Fig. 3 it is clear that if thesymmetry energy properties are kept fixed, the energy in-creases if the square root of the effective mass decreases.This is due to the first term of Eq. (12), as the ISGQRenergy is known to scale with (cid:112) m/m ∗ [28]. In the sameway, it is expected that the second term in Eq. (12) playsa role and produces an increase of the IVGQR energy with S (0 . J , and forincreasing values of this quantity the result of the fit is acomparatively more important decrease at the same timeof L and S (0 . J = 32 ± L = 37 ±
18 MeV; (13) ∆R (cid:0) Pb (cid:1) = 0 . ± .
03 fm . (14)Note that these values are lower but still compatible withthe results (10) and (11) extracted from the PDR. The incompressibility K ∞ of infinite symmetric matter isdefined by the second derivative of the energy per parti-cle E ( ρ, β = 0) /ρ with respect to the density ρ at the G. Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion saturation point, K ∞ = 9 ρ d dρ (cid:18) E ( ρ, β = 0) ρ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ , (15)where E ( ρ, β = 0) is the isoscalar part of the energy den-sity E ( ρ, β ) for nuclear matter given in Eq. (3). The nu-clear matter incompressibility K ∞ is not a directly mea-surable quantity. Instead, the energy of isoscalar giantmonopole resonance, E ISGMR , is expressed in terms ofthe finite nucleus incompressibility K A as [29,30] E ISGMR = (cid:115) ¯ h K A m < r > m , (16)where m is the nucleon mass and < r > m is the meansquare mass radius of the ground state. The finite nucleusincompressibility can be parameterized by means of a sim-ilar expansion as the liquid drop mass formula with thevolume, surface, symmetry and Coulomb terms [28]: K A = K vol + K surf A − / + K τ δ + K Coul Z A / , (17)where δ = ( N − Z ) /A . We denote by K τ the asymmetryterm of the finite nucleus incompressibility K A becausethe symbol K sym has been already used as one of theisovector nuclear matter properties in Eq. (5). The vol-ume term K vol of the finite nucleus incompressibility K A is directly related to the nuclear matter incompressibility K ∞ . The asymmetry term K τ is related to nuclear matterproperties as [28,31] K τ = K sym + 3 L − LB, (18)where B is proportional to the third derivative of the en-ergy density with respect to the density at the saturationpoint, B = 27 ρ K ∞ d E ( ρ, β = 0) dρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ . (19)In Refs. [32,33], K τ is expressed in terms of the skewnessparameter Q as K τ = K sym − L − LQK ∞ , (20)where Q is defined as the third derivative of the energyper nucleon with respect to the density at the saturationpoint, Q = 27 ρ d ( E ( ρ, β = 0) /ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ . (21)The formulas (18) and (20) look different, but they areequivalent since B and Q are related by the equation B = 9 + QK ∞ . (22) The analytic formulas for K surf and K Coul are givenby K surf = 4 πr (cid:34) σ ( ρ ) + 9 ρ d σdρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ + 2 σ ( ρ ) B (cid:35) , (23) K Coul = 35 e r (1 − B ) , (24)where r is the radius constant defined by r = (cid:18) πρ (cid:19) / . (25)In Eq. (23), σ is the surface tension in symmetric semi-infinite nuclear matter defined by σ ( ρ ) = (cid:90) ∞−∞ (cid:20) E ( ρ, β = 0) − E ( ρ , β = 0) ρ ρ (cid:21) dz. (26) K surf can be evaluated by the extended Thomas-Fermiapproximation and the scaled HF calculations of semi-infinite nuclear matter in the Skyrme Hartree-Fock (SHF)model. These evaluations show that the approximate re-lation K surf ∼ − K ∞ holds within an accuracy of a few% in the SHF model. In relativistic models, the extendedThomas-Fermi approximation gives as a result a slightlylarger surface contribution, for example, K surf ∼ − . K ∞ in the case of NL3.It is feasible to calculate the values of K τ and K Coul by using various Skyrme Hamiltonians and relativistic La-grangians. It was pointed out in Ref. [34] that there are noclear correlations between L and K ∞ , and between K sym and K ∞ . On the other hand it is remarkable that K τ ,as a linear combination of K sym , L and B , show a clearcorrelation with K ∞ having a large correlation coefficient,especially in SHF models [34]. Correlation plots between K τ , the symmetry coefficient of the nuclear incompress-ibility, and the parameters J , L and K sym characterisingthe symmetry energy, are displayed in Figs. 4, 5 and 6,respectively [35]. The plot involving J and K τ in Fig. 4shows a clear correlation with a negative slope. We cansee also a similar correlation between L and K τ in Fig.5. Both correlation coefficents are between -0.6 and -0.7.Thus, empirical information on K τ from e.g. the isotopicdependence of the ISGMR energies may give some con-straint on these two values. We will come back to thispoint at the end of Sec. 6. On the other hand, the pointsin the plot of K sym vs. K τ are rather scattered, and itlooks difficult to constrain the value of K sym from theempirical value of K τ . We should also make the remarkthat the results of RMF and Skyrme models have ratherlarge overlap and give consistent results in the correlationplots of Figs. 4 and 5.It was pointed out that any Hamiltonian which has alarger K ∞ gives a smaller K τ [31]. The variations of K τ for the Skyrme interactions are K τ = ( − ± . (27) . Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion 7 -800-700-600-500-400-300-200-100 20 25 30 35 40 45SHFRMF J (MeV) K t ( M e V ) Fig. 4.
Correlation between the asymmetry term of the finitenucleus incompressibility K τ and the volume symmetry energy J , calculated by using various Skyrme parameter sets (SHF,open circles) and relativistic Lagrangians (RMF, filled circles). -800-700-600-500-400-300-200-100-50 0 50 100 150 200 SHFRMF
L (MeV) K t ( M e V ) Fig. 5.
Correlation between the asymmetry term of the finitenucleus incompressibility K τ and the slope parameter L , cal-culated by using various Skyrme parameter sets (SHF, opencircles) and relativistic Lagrangians (RMF, filled circles). On the other hand, the values of RMF are largely neg-ative and have more variation among the seven effectiverelativistic mean field (RMF) Lagrangians, K τ = ( − ± . (28)In principle, the value of K Coul should be model-indepen-dent. Among the 13 parameter sets of Skyrme interactions,the variation of K Coul is rather small, K Coul = ( − . ± .
7) MeV (29)compared with that of K τ . The values of K Coul in RMFshow essentially the same trend, but have a larger varia-tion. -800-700-600-500-400-300-200-100-500 -400 -300 -200 -100 0 100 200 300
SHFRMF K sym (MeV) K t ( M e V ) Fig. 6.
Correlation between the asymmetry term of the finitenucleus incompressibility K τ and the second derivative of thesymmetry energy K sym , calculated by using various Skyrmeparameter sets (SHF, open circles) and relativistic Lagrangians(RMF, filled circles). The study of the isoscalar giant monopole resonance (IS-GMR) provides a direct experimental connection to nu-clear incompressibility in finite nuclear systems. The cen-troid energy of ISGMR, E ISGMR , can be related to thenuclear incompressibility of finite nuclear matter, K A , asgiven by Eq. (16). The ISGMR strength distribution canbe determined experimentally via inelastic scattering ofisoscalar probes. The most commonly-used, and effective,probe for such investigations has been the α -particle (the He nucleus). In these investigations, inelastic scatteringmeasurements are performed off a particular target at veryforward angles, including 0 ◦ .The importance of making measurements at such ex-treme forward angles, including 0 ◦ , is twofold: the crosssection for the ISGMR peaks at 0 ◦ , and the L =0 angu-lar distribution is most distinct at the very forward an-gles. These measurements are, however, extremely diffi-cult since the primary beam passes very close to the scat-tered particles at these angles and one requires a combi-nation of a high-quality, halo-free beam, and an appropri-ate magnetic spectrometer. The high-resolution magneticspectrometer Grand Raiden, at the Research Center forNuclear Physics (RCNP) at Osaka University, Japan [36]is a most suitable such instrument; similar measurementsare being carried out at the Texas A & M University cy-clotron facility as well [37]. An unmatched asset of GrandRaiden is that its optical properties allow for collectionof inelastic scattering spectra practically free of all instru-mental background that had been a bane of such measure-ments in the past.The inelastic scattering spectra are analyzed using mul-tipole decomposition analysis (MDA) [38,39] to extractthe ISGMR strength distributions, the centroid of which G. Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion can give the compressibility, K A of the nucleus underinvestigation. Examples of such “background-free”’ spec-tra, as well as the details of the experimental techniquesand analysis procedures for these measurements have beenprovided in several recent reports from the RCNP work(see, for example, Refs. [40,41,33]).To go from K A to K ∞ , one builds a class of energyfunctionals, E ( ρ ) [cf. Eq. (1)], with different parametersthat allow calculations for nuclear matter and finite nu-clei in the same theoretical framework. The parameter-set for a given class of energy functionals is characterizedby a specific value of K ∞ . The ISGMR strength distri-butions are obtained for different energy functionals in aself-consistent RPA calculation. The K ∞ associated withthe interaction that best reproduces the ISGMR centroidenergies is, then, considered the correct value [28].Following this procedure, both relativistic and non-relativistic calculations give K ∞ =240 ±
20 MeV [42,43,44,45,46]. These accurately calibrated relativistic and non-relativistic models reproduce very well the ISGMR cen-troid energies in the “standard” nuclei, Zr and
Pb.However, it has been established in recent measurementson the Sn and Cd isotopes [41,33] that this value of K ∞ significantly overestimates E ISGMR for these “open shell”nuclei. In other words, it would appear that the Sn andCd nuclei are “softer”, considering the E ISGMR from justthese nuclei would yield an appreciably lower value for K ∞ . Pairing correlations have been suggested as a reasonfor this softening; yet, the results are not conclusive [47,48,49].As noted in the previous Section, K A may be param-eterized as: K A ≈ K vol (1 + cA − / ) + K τ (cid:18) N − ZA (cid:19) + K Coul Z A / . (30)Here, c ≈ − K Coul is essentially a model-independentterm (in the sense that the deviations from one theoret-ical model to another are quite small) [31]; and K τ isthe asymmetry term. Although closely related, the finite-nucleus asymmetry term K τ should not be confused withthe corresponding term in infinite nuclear matter–a quan-tity also denoted by K τ at times, but which should actu-ally be written as K ∞ τ (we have introduced this quantityin Eq. (18) above, and showed that it should not be con-fused with K sym either; in fact, the asymmetry coefficientof the finite nucleus incompressibility does not take con-tribution merely from the second derivative of the symme-try energy). K ∞ τ should never be regarded as the A → ∞ limit of the finite-nucleus asymmetry K τ . Yet the fact that K τ is both experimentally accessible and strongly corre-lated with K ∞ τ is vital in placing stringent constraints onthe density dependence of the symmetry energy. It is thestrong sensitivity of K ∞ τ to the density dependence of thesymmetry energy that makes this investigation of criticalimportance in constraining the EOS of neutron-rich mat-ter. This asymmetry term, K τ , can be obtained by investi-gating the ISGMR over a series of isotopes for which theneutron-proton asymmetry, ( N − Z ) /A , changes by an ap-preciable amount. Coming back to Eq. (30), for a seriesof isotopes, the difference K A − K Coul Z A − / may beapproximated to have a quadratic relationship with theasymmetry parameter ((N - Z)/A)), of the type y = A+ Bx , with K τ being the coefficient, B, of the quadraticterm. (N-Z)/A K - K Z A ( M e V ) Sn Cd A C ou l - / Fig. 7.
The difference K A − K Coul Z A − / in the Sn andCd isotopes plotted as a function of the asymmetry parame-ter, ( N − Z ) /A . The data are from Refs. [41,33]. The valuesof K A have been derived using the customary moment ratio (cid:112) m /m − for the energy of ISGMR, and a value of 5.2 ± K Coul (see previous Section). The solidlines correspond to K τ = - 550 MeV. Such an investigation was carried out by Li et al. overthe even- A − Sn isotopes [40,41] and by Patel et al. over the even- A , − Cd isotopes [33]. The Sn iso-topes yielded a value of K τ = − ±
100 MeV, the Cdisotopes resulted in K τ = − ±
75 MeV. Not only are thetwo values thus obtained in excellent agreement with eachother, but also are consistent with values indirectly ob-tained from several other measurements: K τ = − ± K τ = − +120 − MeV obtained from constraints placedby neutron-skin data from anti-protonic atoms across themass table [51]; and, K τ = − ±
50 MeV obtained fromtheoretical calculations using different Skyrme interactionsand relativistic mean-field (RMF) Lagrangians [31]. InFig. 7, we show the data for the Sn and Cd isotopes fromRefs. [41,33] along with quadratic fits with a commonvalue of K τ = −
550 MeV.From the correlation plots in Figs. 4 and 5, one mayextract the symmetry energy coefficients J and L fromthe empirical value K τ = −
550 MeV. We must take intoaccount the error on this latter quantity ( ±
100 MeV), aswell as the uncertainties on the linear fits. In this way, J is found to lie in the range 27.7-35.6 MeV. On the other . Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion 9 Fig. 8.
Values of K ∞ and K τ calculated from the parametersets of various interactions as labeled [31]. The vertical andhorizontal lines indicate the experimental ranges of K ∞ andK τ , as determined from the GMR work. hand, the correlation between K τ and L is weaker and wecannot get a meaningful constraint on L .The “experimental” values thus obtained from the IS-GMR for K ∞ and K τ taken together can provide a meansof selecting the most appropriate of the interactions usedin EOS calculations. In Fig. 8, we plot the K ∞ and K τ for a number of interactions used in nuclear structure andEOS calculations. It would appear, indeed, that a vast ma-jority of the interactions fail to meet the criterion estab-lished by these measurements. A caveat to this statement,though: the K τ obtained in these measurements is only an“average” value, and the data cannot disentangle the vol-ume symmetry from higher-order effects like the surfacesymmetry. Thus, this average value has been identifiedwith the volume symmetry only, and compared with thevolume symmetry coefficient provided by the models. Itis possible, then, to execute similar fits including higher-order terms and obtain very different values for K τ [52];however, the “appropriateness” of the values of the extraterms thus obtained remains unclear. As was mentioned in Section 3, the neutron skin gives animportant information about the constraints on the sym-metry energy. It is known that the model-independent nonenergy-weighted sum rule of charge exchange spin-dipole(SD) excitations is directly related to the neutron skinthickness [53]. Recently, SD excitations were studied in Zr by the charge-exchange reactions Zr(p,n) Nb [54]and Zr(n,p) Y [55], and the model-independent sumrule for the SD excitations were extracted in Ref. [56] byusing multipole decomposition analysis (MDA) [57]. Thecharge exchange reactions ( He, t ) on Sn isotopes were also studied to extract the neutron skin thickness [58]. How-ever, one needs the counter experiment ( t, He) or (n,p)on Sn isotopes in order to extract the model-independentsum rule value from experimental data. This counter ex-periment is missing in the case of Sn isotopes.The operators for λ − pole SD transitions are definedas ˆ S λ ± = (cid:88) i t i ± r i [ σ ⊗ Y l =1 (ˆ r i )] λ =0 , , , (31)with the isospin operators being denoted as t ± = t x ± it y . The model-independent sum rule for the λ − pole SDoperator ˆ S λ ± can be obtained as S λ − − S λ + = (cid:88) i ∈ all | (cid:104) i | ˆ S λ − | (cid:105) | − (cid:88) i ∈ all | (cid:104) i | ˆ S λ + | (cid:105) | = (cid:104) | [ ˆ S λ − , ˆ S λ + ] | (cid:105) = (2 λ + 1)4 π ( N (cid:104) r (cid:105) n − Z (cid:104) r (cid:105) p ) . (32)The sum rule for the spin-dipole operator (31) then be-comes S − − S + = (cid:88) λ ( S λ − − S λ + ) = 94 π ( N (cid:104) r (cid:105) n − Z (cid:104) r (cid:105) p ) . (33)It should be noted that the sum rule (33) is directly re-lated to the difference between the mean square radiusof neutrons and protons with the weight of neutron andproton numbers.Let us now discuss the integrated SD strength. Theintegrated SD strength m ( E x ) = (cid:88) λ π = 0 − , − , − (cid:90) E x dB ( λ π ) dE (cid:48) dE (cid:48) (34)is plotted as a function of the excitation energy E x in Fig.9 for the operators ˆ S λ − and ˆ S λ + in Eq. (31). The value S − is obtained by integrating up to E x = 50 MeV from theground state of the daughter nucleus Nb ( E x = 57 MeVfrom the ground state of the parent nucleus Zr), whilethe corresponding value S + is evaluated up to E x = 26MeV from the ground state of Y ( E x = 27.5 MeV fromthe ground state of Zr). This difference between the twomaximum energies of the integrals stems from the isospindifference between the ground states of the daughter nu-clei, i.e., T=4 in Nb and T=6 in Y. That is, the 23.6MeV difference originates from the difference in excitationenergy between the T=6 Gamow-Teller states in the (p,n)and (n,p) channels [56]. For both the S − and S + strength,the calculated results overshoot the experimental data inthe energy range E x = 20-40 MeV. These results suggesta quenching of 30-40% of the calculated strength aroundthe peak region. However, the integrated cross sections upto E x = 56 MeV in Fig. 9 approach the calculated valuesfor both the t − and t + channels.The ∆S = S − − S + value is shown as a function of E x in the lower panel of Fig. 9. We note that the ∆S value sat-urates both in the calculated and the experimental values Table 1.
Sum rule values of charge exchange SD excitations in A=90 nuclei obtained by the HF+RPA calculations [59] (S − for Nb and S + for Y). The SD strength is integrated up to E x = 50 MeV for S − and E x = 26 MeV for S + , respectively.The experimental data are taken from Ref. [56]. The SD sum rules are given in units of fm . See the text for details.SIII SGII SkI3 SLy4 λ π S − S + ∆S S − S + ∆S S − S + ∆S S − S + ∆S − − − S − = 271 ± S + = 124 ± ∆S = 147 ± Fig. 9.
Integrated charge exchange SD strength (34) excitedby the operators ˆ S − and ˆ S + in Eq. (31) on Zr. The calculatedresults are obtained by the HF+RPA model using the Skyrmeinteractions SIII, SGII, SLy4 and SkI3 [59]. The upper panelshows the S − and S + strength, while the lower panel shows the S − − S + strength. All strengths for the three multipoles λ π =0 − ,1 − and 2 − are summed up in the results. The experimentaldata are taken from Ref. [56]. No quenching factor is introducedin the calculation of the integrated strength. above E x = 40 MeV, while the empirical values S − and S + themselves increase gradually above E x = 40 MeV. This isthe crucial feature for extracting the model-independentsum rule ∆S = S − − S + from the experimental data. Theempirical values S − , S + and ∆S obtained from these anal-yses are shown in Table 1. The indicated uncertainties of S − , S + and ∆S contain not only the statistical error of thedata, but also errors due to the various input of the DWIAcalculations used in the MDA, such as the optical modelparameters and the single-particle potentials [55]. Thereis an additional uncertainty in the estimation of the SDunit cross section, namely, the overall normalization factor[56], which should be studied further experimentally. Table 2.
Proton, neutron and charge radii of Zr. The chargeradius is obtained by folding the proton density with the protonfinite size. The sum rule values ∆S = S − − S + of spin-dipoleexcitations are calculated by Eq. (33) with the HF neutronand proton mean square radii. The experimental value of thecharge radius is taken from Ref. [60], while the experimentaldata for r n − r p are taken from [61,56]. The radii are given inunits of fm, while the SD sum rules are given in units of fm .SIII SGII SkI3 SLy4 exp r p r c ) r c ± r n ± r n − r p ± ± ∆S From ∆S , the neutron radius of Zr is extracted tobe √ < r > n = 4.26 ± (cid:112) < r > p = 4.19 fm is used. The proton radius is ob-tained from the charge radius in Table 2 by subtractingthe proton finite size correction. The experimental uncer-tainty in the neutron skin thickness obtained by protonscattering is rather large: δ np = r n − r p = 0 . ± .
07 fm.This is mainly due to the difficulty to extract the neutronradius from the analysis of the proton scattering [61]. Thesum rule analysis of the SD strength determines the neu-tron radius with 1% accuracy, which is almost the same asthat expected for the parity violation electron scatteringexperiment. The obtained value r n − r p = 0 . ± .
04 fmcan be used to disentangle the neutron matter EOS by . Col`o, U. Garg, H. Sagawa: Symmetry energy from the nuclear collective motion 11 using the strong linear correlation between the two quan-tities [12,13,14].Very recently, Wakasa performed MDA of ( p, n ) re-action cross sections on
Pb observed at RCNP, OsakaUniversity and extracted spin-dipole strength in
Bi [62].He found 100 % of the calculated sum rule strength for 1 − states, and about 70 % of the predicted strength for 0 − and 2 − states [59]. It would be of paramount importanceto perform the counter experiment Pb( n, p ) and extractempirically the model independent sum rule S − − S + fromthe two charge-exchange experiments, in order to obtainthe neutron skin value in Pb.
In this paper, we have focused on the main constraintson the symmetry energy that are provided by the experi-mental and theoretical studies of nuclear collective vibra-tions. We have not been fully exhaustive on this subject,in keeping with the fact that other contributions in thepresent volume deal with the issues we have not discussed.Thus, our discussion has concerned the isovector dipoleand isovector quadupole states, as well as the isotopic de-pendence of the isoscalar monpole energies.It is quite natural to think of the residual proton-neutron force sustaining the isovector collective motion asbeing related with the symmetry energy. However, moreeffort is needed to make this statement more quantitative.As for the standard GDR, it has been suggested that itsenergy is correlated with the value of the symmetry energyat some sub-saturation density around 0.1 fm − , S (0 .
1) -if medium-heavy nuclei are considered. It is remarkablethat the isovector GQR can be shown to lead to a con-sistent value of S (0 . J and L that can be deduced is nicely consistent with other kindsof (completely independent) analysis that are presentedin this volume: Eqs. (9), (13) and (14) substantiate thesestatements.We have also discussed the role played by the pygmystates, or resonances. Empirically, a correlation of theirfraction of EWSR with the slope parameter L has beenfound, and reasonable values of L (10) and of the neutronskin (11) have been extracted. It is puzzling, though, thatthe PDR does not display in all the considered models aclear character related to the pure skin mode. This is oneof the issues deserving further investigation.All these observables do not seem capable of constrain-ing the parameter K τ , associated with the second deriva-tive of the symmetry energy. However, a completely dif-ferent observable namely the dependence of the isoscalarmonopole energy along an isotopic chain, can provide sucha constrain. We have discussed the theoretical argumentsbehind that, and the measurements in the Sn and Cd iso-topic chains that led to K τ around −
550 MeV (with asignificant error bar still). We have also illustrated thecorrelations emerging from our theoretical study between K τ and the other parameters associated with the symme-try energy or, more generally, with the equation of state. A further source of information on the symmetry en-ergy is the charge-exchange spin-dipole resonance. In fact,the sum rule obtained from the difference between the to-tal strength in the t − channel and the total strength in the t + channel is proportional to the difference N (cid:104) r n (cid:105)− Z (cid:104) r p (cid:105) .Experiments aimed at extracting the neutron skin havebeen first performed in the Sn isotopes. More recently,in Zr, it has been possible to extract quantitatively thevalues of the total strengths and of the skin. It would behighly desirable to consider the case of
Pb as well, inkeeping with the fact that much effort is devoted to thestudy of this nucleus by using also parity-violating asym-metry measurements.In conclusion, the study of giant resonances has beenshown to provide some robust conclusions about the sym-metry energy and its density dependence around nuclearmatter saturation density. It is not completely evident howto improve on these first conclusions. Exploring nucleiwith larger proton-neutron asymmetry (unstable nuclei)is of paramount importance as the results may either con-firm the present findings or lead to some surprise. At thesame time, further theoretical work is probably needed inorder to assess which correlations with the EoS parame-ters are genuine, and which are somehow an artefact of aspecific ansatz built in the energy functional.
Acknowledgements
This work has been supported in part by the U.S. NationalScience Foundation (Grant Nos. PHY07-58100 and PHY-1068192).
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