aa r X i v : . [ nu c l - t h ] A p r EPJ manuscript No. (will be inserted by the editor)
Pygmy resonances and symmetry energy
C. A. Bertulani
Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429, USAReceived: date / Revised version: date
Abstract.
I present a brief summary of the first three decades of studies of pygmy resonances in nucleiand their relation to the symmetry energy of nuclear matter. I discuss the first experiments and theoriesdedicated to study the electromagnetic response in halo nuclei and how a low energy peak was initiallyidentified as a candidate for the pygmy resonance. This is followed by the description of a collective statein medium heavy and heavy nuclei which was definitely identified as a pygmy resonance. The role of theslope parameter of the symmetry energy in determining the properties of neutron stars is stressed. Thetheoretical and experimental information collected on pygmy resonances, neutron skins, and the numerouscorrelations found with the slope parameter is briefly reviewed.
PACS.
To the memory of Pier Francesco Bortignon
Collective excitation modes in nuclei are well known andhave been studied for the last 70 years. Their first unam-biguous observation was reported by Baldwin and Klaiberin [1] in photo-absorption experiments. But Bothe andGentner [2] already had a first indication that they existedwhen they obtained very large cross sections, two ordersof magnitude larger than predicted theoretically, for thephoto-production of radioactivity in several targets. Thelarge cross sections were obtained with high energy pho-tons, of about 15 MeV, and now are understood as dueto a collective nuclear response to the electric dipole (E1)field of the photon. Such E1 collective states had beenpredicted theoretically by Migdal [3]. For a detailed de-scription of giant resonances, see Ref. [4]Much of what we know about the giant resonancestoday was gathered in photo-absorption processes usingmono-energetic photons, as reported, e.g., in Refs. [5,6].The giant resonances in all nuclei are located above theparticle emission threshold. In medium and heavy nucleithe large Coulomb barrier prevents charged particle decay,and the photo-absorption cross section is usually obtainedfrom a measurement of neutron yields for a given γ -rayenergy [7,8,9]. Send offprint requests to : Giant resonances are observed in basically all nuclei. Incontrast, pygmy resonances have been mainly observed inneutron-rich nuclei. Historically, the first observation of apygmy resonance occurred in 1961 with the discovery of asignificant number of unbound states identified as a bumpin γ -rays emitted following neutron capture [10]. But thefirst use of of the name pygmy resonance , or pygmy dipoleresonance (PDR), was in 1969 when its effect on the cal-culations of neutron capture cross sections was reported[11]. The description of the PDR as a collective excitationwas first introduced in Ref. [13] based on a three fluidmodel consisting of protons and neutrons fluids in thesame orbitals, and neutrons accounting for excess neu-trons interacting less strongly with the other nucleons.The model assumed that the neutron excess would oscil-late against the N = Z core. It was much later when thefirst experimental proposal to study pygmy resonances inCoulomb excitation experiments emerged in 1987 at theJPARC facility in Japan [14]. The abnormal size of the Li nucleus was first reported inRef. [15] by measuring the momentum distributions of Lifragments in nucleon removal reactions. A superposition I thank Riccardo Raabe (KU Leuven) for this information. C. A. Bertulani: Pygmy resonances and symmetry energy of two nearly gaussian shaped distributions was necessaryto reproduce the experimental data. The wider peak waslinked to the removal of neutrons from a tightly bound Li core, whereas the narrower peak was thought to arisefrom the removal of the loosely bound valence neutronsin Li. In fact, the two-neutron separation energy in Liis only about 300 keV explaining why their removal onlyslightly “shakes” the Li core, thus explaining the narrowcomponent of the momentum distribution. A narrow mo-mentum distribution implies a large spatial extent of theneutrons, which in turn is a consequence of their smallseparation energy. These conclusions were later found tobe in agreement with Coulomb breakup experiments [16]and with theory [17,18].The momentum distributions of core fragments of haloprojectiles were initially analyzed experimentally usingthe simple Serber formula [19] dσ c d q = C | ψ ( q ) | , (1)where C is a kinematical constant and ψ ( q ) is the Fouriertransform of the ground state wave function of the nu-cleus. In fact, this formalism works rather well for loosely-bound nuclei such as Li and Be [20,21,22]. For nu-cleon removal with larger separation energies, the Serberformalism is not appropriate and yields inaccurate results[22,23,24,25,26,27].
A narrow peak associated with the small separation en-ergies was also observed in Coulomb dissociation experi-ments [28,29], becoming the seed for subsequent intensiveinvestigations.The Coulomb excitation cross section for a given multi-polarity πL ( π = E or M , and L = 1 , , · · · ) and excitationenergy E is given by dσdE = n πL ( E ) E σ πLγ ( E ) , (2)where n πL is the virtual photon number [30] and σ πLγ isthe photo-nuclear cross section, which can be written interms of the electromagnetic response function dB πL /dE as [30] σ πLγ ( E ) = (2 π ) ( L + 1) L [(2 L + 1)!!] (cid:18) E ¯ hc (cid:19) L − dB πL ( E ) dE . (3) Using simple Yukawa or Hulthen functions for bound statesand plane waves for the continuum it was shown in Refs. [18,31,32] that a simple expression for the response func-tion emerges for electric multipoles, namely, dB EL ( E ) dE = 2 L − π (2 L + 1)( L !) (cid:18) ¯ h µ (cid:19) L × Z e L √ S ( E − S ) L +1 / E L +2 , (4)where e L is the effective charge, S is the separation energyand µ the reduced mass [18,31,32]. For the ubiquitouselectric dipole (E1) excitation one has dB E ( E ) dE = 3 Z e L ¯ h µπ √ S ( E − S ) / E . (5)These equations are very useful as they allow simple pre-dictions of the Coulomb response in halo nuclei in termsof the separation energy S . They have been widely usedin experimental analyses [33,34] and for comparison withmore complex theoretical models [32,35,36,37,38,39,40,41,42]. The above equations predict a peak in the responsefunction at an excitation energy of E = 8 S/ E [32].A certain confusion reigned in the literature during the1990s as to whether the peak observed in the Coulombbreakup experiments was due to a low energy resonanceor just a direct transition to the continuum, as is the casebehind the derivation of Eqs. 4 and 5 [32].Despite the appeal and usefulness of the expressionsabove, it was later on realized that in order to repro-duce many of the experimental data gathered on Coulombbreakup of halo nuclei during the 1990s and 2000s it wasnecessary to incorporate higher-order interactions, or fi-nal state interactions (FSI), in the theoretical calcula-tions. The discretization of the continuum was a crucialimprovement of the theories and the so-called continuum-discretized-coupled-channels (CDCC) calculations becamea necessary theory method to reproduce experimental dataand to probe the structure of loosely bound states andresonances appearing in halo systems. The role of higher-order couplings and the nuclear contribution to the breakupand comparison to experimental data has been studied bynumerous authors [44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63]. Calculations based on effectivefield theories started to emerge only recently [64]. Three body models for halo nuclei such as Li and Hehave obtained a similar response function as described inthe previous section [65,66,67,68], yielding dB E ( E ) /dE ∝ ( E − S ) /E / . This model shifts the three-body responsepeak to larger excitation energy E than the two bodymodel. Therefore, the separation energy still roughly de-termines the peak location of the E1 response but at ahigher energy and with a larger width. It was also shownthat final state interactions can substantially change thelocation of the low energy peak [68,54]. . A. Bertulani: Pygmy resonances and symmetry energy 3 Three-body models did not deliver much more infor-mation on the origins of the low-peak response and itsidentification as a pygmy resonance. The conclusion reachedby using these models was that the observed peak wasagain due to a direct transition from the ground state ofthe nucleus, e.g., Li or He, to the continuum [65,66,67,68]. This does no mean however that the three-body mod-els were not rich in physics details and predictions, muchon the contrary. They have helped us to understand newphenomena in “Borromean” nuclei and the unravelling ofEfimov states in nuclei [66].
As the evidence accumulated of a non-negligible strengthin the low energy spectrum of dipole excitations, also forheavier nuclei, other models for the pygmy resonancesemerged [13,69,70]. It was natural to extend the Goldhaber-Teller (GT) [71] and Steinwedel-Jensen (SJ) [72] hydro-dynamical models to explain pygmy resonances as a col-lective response to external fields in “soft” neutron-richnuclei.Using the concepts described in Ref. [73] one can showthat the radial transition density of pygmy resonances canbe described in the hydrodynamical model by the equation[68] δρ ( r ) = r π R (cid:20) Z GT α GT ddr + Z SJ α SJ KR j ( kr ) (cid:21) ρ ( r ) , (6)where Z i are the effective charges in the GT and SJ models[68], α i are admixture coefficients of the GT and SJ collec-tive vibration modes, such that α GT + α SJ = 1, K = 9 . R is the mean nuclear radius. Here, j ( kr ) is the spher-ical Bessel function of order 1, and k = 2 . /R . The tran-sition density in Eq. 6 emulates a pygmy resonance basedon a collective dipole vibration of protons and neutronsand is known as soft dipole mode .If one choses α SJ = 0, Goldhaber and Teller [71] gavea simple prescription for the resonance energy of collectivevibrations in nuclei, E R = (cid:18) S ¯ h aRm N (cid:19) / , (7)where a is the approximate size of the nuclear distributionthickness and m N is the nucleon mass. According to Gold-haber and Teller [71], S in the equation above is not theseparation energy of a nucleon but the energy needed toextract one proton from the neutron environment (or oneneutron from the proton environment). It is the part of thepotential energy due to the neutron-proton interaction inthe nuclear environment, assumed to be proportional tothe symmetry energy ∝ ( N − Z ) /A .Golhaber and Teller [71] used S = 40 MeV, a = 1 − E ≃ −
20 MeV for a medium heavy nu-cleus, consistent with the experimentally found centroidsof giant dipole resonances. For neutron rich nuclei, the ex-tension of this theory needs to include a relation of S to the symmetry energy. Such a relationship is only possiblewith a microscopic model due to the nature of the fine-structure of the pygmy resonance closely related to thecoupling of phonon states with complex configurations inthe nucleus. In the case of halo nuclei a hydrodynami-cal model is likely unfit, but it works if one assumes that S = S (here the separation energy). For halo nuclei theproduct aR is also expected to be proportional to S − ,and we obtain the proportionality E P DR ∼ βS , with β ofthe order of one . Using for example Li, with a = 1 − R = 3 fm, and S = 0 . E P DR = 1 − The random phase approximation (RPA) is a useful toolto describe the nuclear response function in terms of mi-croscopic degrees of freedom. In its simplest form, one cancalculate the response to a weak time-dependent field ofthe form V x ( r ) cos( ωt ) by solving the RPA equations inthe self-consistent method [74] δρ RP A ( r ) = Z Π RP A ( r , r ′ ) V x ( r ′ ) d r ′ , (8)where δρ RP A ( r ) is the self-consistent transition densityand Π RP A ( r , r ′ ) satisfies the implicit equation Π RP A ( r , r ′ ) = Π ( r , r ′ ) + Z d r d r × Π ( r , r ) δV x ( r ) δρ ( r ) Π RP A ( r , r ′ ) . (9)Here, ρ ( r ) and Π ( r , r ′ ) are the corresponding densitiesand response function defined in terms of occupied andunoccupied orbitals in the nucleus. Other variations of theRPA exist and are explained in details elsewhere, for ex-ample using the XY-formalism [75,76] (for a recent review,see Ref. [41]).The first application of the RPA formalism to obtainthe electromagnetic response in weakly-bound nuclei wasdone in Ref. [77]. This was followed up in Ref. [32,35]where a comparison with two-body models and nucleonclustering was done. In the RPA calculations a peak atsmall energies appears around a few MeV which was in-terpreted as a pygmy resonance, or alternatively just theeffect of a small separation energy in the nuclei and of a di-rect transition to the continuum. At least for light nuclei.But for medium heavy and heavy nuclei, a fine structure ofthe resonance was revealed leading to the belief that manynucleons and many states are involved in the transition. C. A. Bertulani: Pygmy resonances and symmetry energy
Since these first exploratory works, microscopic studiesbased on different RPA models have been used to studyneutron-rich nuclei along the nuclear chart. All neutron-rich nuclei seem to display a visible structure in the re-sponse to external fields at low energies which has beenattributed to the pygmy resonance. Relativistic mean fieldmodels have also identified pygmy resonances. To cite afew of these works, we list Refs. [78,79,80,81,82,83,84,85,86,87,88,89,90,91,92]. It is therefore clear that pygmy res-onances constitute a new phenomenon in nuclear physicswhich appeared in the context of neutron-rich nuclei andthe effect is also enhanced in nuclei close to the drip-linewith small binding energies.Recently a new and powerful method has been devel-oped to study not only pygmy resonances but also nuclearlarge amplitude collective motion [93,94,95,96,97,98,99].The framework relies on the time dependent superfluidlocal density approximation (TDSLDA). This is an exten-sion of the Density Functional Theory (DFT) to super-fluid nuclei and can handle the response to external time-dependent fields. All degrees of freedom are taken into ac-count without any restrictions with all symmetries imple-mented such as translation, rotation, parity, local Galileancovariance, local gauge symmetry, isospin symmetry andminimal gauge coupling to electromagnetic fields [99].As in the time-dependent Hartree-Fock-Bogoliubov the-ory, the time evolution of the nucleus is governed by thetime-dependent mean field i ¯ h ∂∂t (cid:18) U ( r , t ) V ( r , t ) (cid:19) = (cid:18) h ( r , t ) ∆ ( r , t ) ∆ ∗ ( r , t ) − h ∗ ( r , t ) (cid:19) (cid:18) U ( r , t ) V ( r , t ) (cid:19) , (10)where h ( r , t ) is the single-particle Hamiltonian and ∆ ( r , t )is the pairing field. Both are obtained self-consistentlyfrom an energy functional, i.e., a Skyrme interaction. Thetime-dependent external electromagnetic field A entersthe hamiltonian by means of the minimal gauge coupling ∇ A = ∇ − i A / ¯ hc . The energy spectrum is obtained by aFourier transformation of the time evolution of the nucleardensity.In Ref. [97] the first TDSLDA calculations were re-ported for relativistic Coulomb excitation in a collision of U +
U. The results show that a considerable amountof electromagnetic strength occurs at low energies, around E x ∼ For very forward electron scattering and small energy trans-fers, Siegert theorem [100,101] allows one to show that theCoulomb and electric form factors appearing in electronscattering differential cross sections are proportional toeach other and one obtains for electric multipole excita-tions [68] dσdΩdE γ = X L dN ( EL ) e ( E e , E γ , θ ) dΩdE γ σ ( EL ) γ ( E γ ) , (11) valid in the long-wavelength approximation, i.e., for exci-tation energies E γ ≪ ¯ hc/R , with E e being the electronenergy, and the equivalent photon number given by [68] dN ( EL ) e ( E e , E γ , θ ) dΩdE γ = 4 LL + 1 αE e (cid:20) E e E γ sin (cid:18) θ (cid:19)(cid:21) L − × cos ( θ/
2) sin − ( θ/ E e /M A c ) sin ( θ/ × "
12 + (cid:18) E e E γ (cid:19) LL + 1 sin (cid:18) θ (cid:19) + tan (cid:18) θ (cid:19) . (12)As with the case of Coulomb excitation, the cross sec-tion for electron scattering in this limit, and for large elec-tron energies, are proportional to the cross sections for theelectric multipolarity EL induced by real photons.In both cases, this is a very useful relationship, as it isnearly impossible to study the interactions of real photonswith radioactive nuclei in the laboratory. The conditionsabove are also hard to establish, except for electron-ioncolliders, which have been promised in radioactive beamfacilities, but not yet fullly realized [102,103].The excitation of the electric pygmy dipole resonanceat large angles by inelastic electron scattering with exist-ing facilities has been reported in this special issue by Po-momarev et al. [43]. They demonstrate that the excitationof pygmy resonance states in (e,e’) reactions is predomi-nantly of transversal character for large scattering angles.They were also able to extract the fine structure of thepygmy states at low excitation energies.Electron scattering on halo nuclei was explored theo-retically in Ref. [104] and a comparison with fixed-targetexperiments was presented. For a given electron energy E e , the total cross section for the dissociation of halo nu-clei was shown to be σ e ( E e ) = 64 √ π e eff µc S ln (cid:18) E e S (cid:19) , (13)where µ is the reduced mass of the halo nucleus, treatedas a two-body cluster-like nucleus, and e eff is the effectivecharge involved in the transition.For normal nuclei, with S ≃ few MeV, the halo nucleuselectron-disintegration cross section is negligible. The for-mula above predicts a dependence on the inverse of theseparation energy. In an hypothetical situation, with S =100 keV, E e = 10 MeV, e eff = e , and µ = m N ∼ MeV, this equation yields a non-negligible 25 mb for thedissociation cross section. The electro-disintegration crosssection increases very slowly with the electron energy pos-ing a challenge for future experiments. In contrast, it wasshown in Ref. [104], that at low electron energies the crosssections increase much faster with E e . These conclusionsare of relevance for the designing of experiments in futureelectron-ion colliders and to resolve the energy spectrumaround the pygmy resonance. . A. Bertulani: Pygmy resonances and symmetry energy 5 Based on all evidences discussed above, we conclude thatthe existence of a PDR is verified in all theoretical meth-ods involving two-body, three-body and many-body mod-els dealing with the electromagnetic response of neutron-rich nuclei. It has also been abundantly observed in ex-periments. Next we discuss their relation to astrophysics.
The structure of neutron stars is dependent on the equa-tion of state (EOS) of infinite neutron matter with a smallproton component [105]. Since most of the experimentaldata related to EOS is obtained with nuclei with N ∼ Z ,it is important to have an accurate description of the partof the EOS depending on the fraction δ = ( N − Z ) /A ,called by symmetry energy .The energy per nucleon in nuclear matter can be ex-panded in a Taylor series around N = Z yielding ǫ ( ρ, δ ) = ǫ ( ρ,
0) + S ( ρ ) δ + · · · , (14)where here S denotes the symmetry energy. For the infi-nite matter in neutron stars one uses, δ = ( ρ n − ρ p ) /ρ ,where ρ = ρ p + ρ n is the total nuclear density. The incom-pressibility of nuclear matter is defined as K = 9 ρ ∂ ǫ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ , (15)and is perhaps the easiest quantity to relate infinite nu-clear matter to experimentally observed properties suchas monopole excitations in nuclei [106].The function S ( ρ ) can also be expanded around thesaturation density of nuclear matter, ρ ≃ .
16 fm − , S ( ρ ) = J + L3 ρ − ρ ρ + · · · , (16)where the bulk symmetry energy is given by J = S ( ρ )and the slope parameter is given by L = 3 ρ dS ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12) ρ . (17)The binding energy per nucleon at saturation is ǫ ( ρ , ≃−
16 MeV. Fits to nuclear properties such as masses andexcitation energies with microscopic models have indi-cated that J ≈
30 MeV [107]. Because the theoreticalmodels have been fitted to reproduce normal nuclei, it isnot easy to find out which value of L from these modelsis the proper one to use in the EOS of neutron stars.The EOS of homogeneous nuclear matter is a relationof the pressure and density, given by p ( ρ, δ ) = ρ dǫ ( ρ, δ ) dρ . (18) It is strongly dependent on S , because for δ = 1, i.e., neu-tron matter, and ρ ∼ ρ one obtains p = Lρ /
3. Therefore,the slope parameter L is crucial to describe the EOS ofneutron matter. Experimental and theoretical analyses ofvarious kinds of nuclear structure and nuclear reactionsshow that L is not well known, varying with the range 0and 150 MeV [108,109]. The explosion mechanism of core-collapse supernovae is also dependent on the EOS of nu-clear matter and its characteristics out of neutron/protonsymmetry [105,110,111,112,113,114,115]. The role of the pressure in obtaining the structure of neu-tron stars is determined by the solutions of the equationsof hydrostatic equilibrium. They are supposed to be gov-erned by the Tolman-Oppenheimer-Volkoff (TOV) equa-tions, an extension of Newton’s laws including special rel-ativity and general relativity corrections. The TOV equa-tions are a set of coupled differential equations of the form(here we use c = 1) dpdr = − G ρ ( r ) m ( r ) r (cid:20) p ( r ) ρ ( r ) (cid:21) (cid:20) πr p ( r ) m ( r ) (cid:21) × (cid:20) − Gm ( r ) r (cid:21) − (19) dm ( r ) dr = 4 πr ρ ( r ) , (20)where m ( r ) is the enclosed mass profile of the star up toradius r and G is the gravitational constant. The solutionsof the TOV equations for a given EOS are able to predictthe mass and radii of neutron stars. As reported in Ref.[116], there is a large variation among observational dataand theories for masses of neutron stars.In order to solve the TOV equations one needs to knowthe relationship of pressure and energy density (EOS), p ( ρ ) and as we discussed above, this relates very closely tothe slope parameter L . And how much do we know aboutit? This is only part of the puzzle, but the one which mightneed most improvements. There exist various theoretical models to describe the bulkproperties of nuclei, such as their masses and nucleondensity distributions. A widely used model relies on theHartree-Fock-Bogoliubov (HFB) theory using Skyrme in-teractions [75]. The Skyrme forces are contact interactionsembodying coordinate, spin and isospin dependence. Theenergy density functional ǫ [ ρ ] is a straightforward byprod-uct of the model. As mentioned above, this density depen-dence is exactly what one needs to deduce the mass andradii of neutron stars using the TOV equations [110]. C. A. Bertulani: Pygmy resonances and symmetry energy
There are hundreds of Skyrme interactions that havebeen devised and that are able to describe successfully alimited number of nuclear properties. Some of them haverecently been fitted by means of a computational descrip-tion of global masses and other nuclear properties [117].This new era of intense developments in computationalpower has allowed for a better constraint of the interac-tions appropriate to fit global properties of nuclei. How-ever, when extrapolated to densities below and above thesaturation density, the equations of state ǫ ( ρ ) tend to di-verge [118]. Therefore, there is a strong interest in theliterature to pinpoint those Skyrme interactions that bet-ter describe neutron matter properties. A glimpse of thisdifficulty is shown in Table 1 where we show some of thepredictions of the Skyrme models for the input needed toneuron star properties. It is evident that, whereas K and J tend to agree among many of the predictions, the slopeparameter L is the least constrained. Table 1.
Properties of nuclear matter at the saturation densityas predicted by some Skyrme models. All numbers are in unitsof MeV. The parameters for the Skyrme forces were taken from[119,120,121,122,123,124,125,126]. For more details see Ref.[127] K J L K J L
SIII 355. 28.2 9.91 SLY5 230. 32.0 48.2SKP 201. 30.0 19.7 SKXS20 202. 35.5 67.1SKX 271. 31.1 33.2 SKO 223. 31.9 79.1HFB9 231. 30.0 39.9 SKI5 255. 36.6 129.
Microscopic theories based on energy density functionalswith Skyrme forces or relativistic models suggest that thenuclear dipole polarizability α D defined as α D = ¯ h π e Z σ E γ ( E ) E dE = 8 π Z dB E ( E ) dE dEE (21)is an additional quantity able to constrain the symmetryenergy [128,129]. These are easily extracted from Coulombexcitation experiments. The reason is simple: the virtualphoton numbers entering Eq. 2 have a n E ∼ ln(1 /E ) de-pendence with energy, which together with the 1 /E termfavors the low energy part of the spectrum where pygmyresonances are located. Hence, a measurement of Coulombdissociation (or electron scattering) is nearly proportionalto the dipole polarizability.Experiments exploring Coulomb excitation, and polar-ized proton scattering off neutron-rich nuclei to extract thedipole polarizability and its relation to the slope param-eter have been reported in a several publications in Refs.[129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148], just to cite a few of them. To date, there is a large variation in the measurements of α D in the order of 20-30%. This is still too large to constrainmost of the energy functionals stemming from Skyrme andrelativistic models. Developments on nuclear reaction the-ory are also necessary to obtain the desired accuracy inthe experimental analyses [149].A correlation between α D , the strength of pygmy res-onances, and the neutron skin in nuclei, ∆r np = (cid:10) r n (cid:11) / − (cid:10) r p (cid:11) / , (22)was also shown to exist in Refs. [90,150]. Several correla-tions have been found among neutron skin, dipole polar-izability, and the slope parameter [118,150,151,152,153].The neutron excess in a nucleus builds up a neutron pres-sure that is larger than the pressure due to the protons.The neutron pressure also contributes to the energy pernucleon which is a function of the nuclear density andits nucleon asymmetry. Therefore, neutron skins in nucleiare expected to be naturally correlated to the symmetryenergy.Experiments dedicated to the measurement of neutronskins have also been the subject of large experimental in-terest, ranging from electron scattering (with the parityviolation part of the interaction), anti-proton annihila-tion, and heavy ion collisions [150,154,155,129,148]. Anelectron-ion collider directly using the electromagnetic in-teraction as a probe would be an ideal tool, but are stillfar from being fully realized with neutron-rich nuclei [102,103]. It is possible that the neutron skin can be accessed infragmentation reactions using inverse kinematics, as pro-posed in Ref. [155]. These are rather easy experimentsusing present radioactive beam facilities. The proposal isbased on the measurement of neutron-changing cross sec-tions by the detection of all fragments with at least oneneutron removed [155]. Another proposal suggests thata subtle but well-known phenomenon, the Maris effect,could be used to access information on the neutron skin[156]. These could be achieved with polarized proton tar-gets in quasi-free (p,2p) reactions. The physics cases discussed in the previous sections bothfor Coulomb excitation by heavy ions and also by electronscattering are useful to assess information on radiativecapture reactions in stars, such as (n, γ ) and (p, γ ) [157,158,159,160,161,162]. So far, the pygmy resonances havebeen mainly probed using electromagnetic excitation (see,e.g., Eq. 2) from which one can extract the photodissocia-tion cross section 3. Using the detailed balance theorem, itwas proposed in Ref. [163] that radiative capture reactionsof relevance for astrophysics could be obtained in electro-magnetic dissociation experiments. This was proved to bea very useful tool in numerous experiments, e.g., Refs.[164,165,166,167,168,169] and became a state of the arttool in nuclear accelerator facilities.The impact of the pygmy resonances in nuclear as-trophysics is also imprinted in the energy balance of the . A. Bertulani: Pygmy resonances and symmetry energy 7 reactions in the stellar medium. It has been shown, andup to now little explored, that the energy balance in thenuclear reaction networks may change the nuclear abun-dances appreciably if one includes pygmy resonances [157]. Experiments with nuclei on earth are very limited by beamintensities, detection efficiencies, etc. Guidance by theoryis crucial for experimental success, but this symbiosis alsohas limitations. Besides, nuclei are not prototypes of neu-tron stars. Perhaps the closest examples of neutron starson earth are neutron rich nuclei. But neutron stars arebound by gravity and not by the strong interaction.Theorists are limited by computational power and bythe lack of knowledge of crucial parts of the physics in-gredients necessary for their goals such as an accurate de-scription of the interface of quarks, gluons and nucleondegrees of freedom. Based on the information collected inrelativistic nuclear collisions, there is increasing evidencethat a phrase transition exists between the quark-gluonphase and the nucleons in nuclei even at low tempera-tures.If there is a phase transition in the core of neutronstars due to a large density, then the symmetry energy ob-tained with nucleonic degrees of freedom is not enough tomodel the structure of neutron stars. There is a long wayfor experimentalists to improve their devices and theoriststo refine their models of nuclear matter. Accumulated ex-perience obtained in the last decades, tells us that everyso often an idea such as the relation of spectra of pygmyresonances and of neutron skins to the symmetry energyhelp us to pave the way.
I am honored to have met Pier Francesco. I wish to thankhim for his dedication to nuclear physics and the educationof a new generation of nuclear scientists. Maybe I do thispersonally in another universe.This work was supported in part by the U.S. DOEgrant DE- FG02-08ER41533 and the U.S. National Sci-ence Foundation Grant No. 1415656.
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