Evolution of the dipole polarizability in the stable tin isotope chain
S. Bassauer, P. von Neumann-Cosel, P.-G. Reinhard, A. Tamii, S. Adachi, C. A. Bertulani, P. Y. Chan, G. Colò, A. D'Alessio, H. Fujioka, H. Fujita, Y. Fujita, G. Gey, M. Hilcker, T. H. Hoang, A. Inoue, J. Isaak, C. Iwamoto, T. Klaus, N. Kobayashi, Y. Maeda, M. Matsuda, N. Nakatsuka, S. Noji, H. J. Ong, I. Ou, N. Paar, N. Pietralla, V. Yu. Ponomarev, M. S. Reen, A. Richter, X. Roca-Maza, M. Singer, G. Steinhilber, T. Sudo, Y. Togano, M. Tsumura, Y. Watanabe, V. Werner
EEvolution of the dipole polarizability in the stable tin isotope chain
S. Bassauer a , P. von Neumann-Cosel a, ∗ , P.-G. Reinhard b , A. Tamii c , S. Adachi c , C.A. Bertulani d ,P.Y. Chan c , G. Col`o e , A. D’Alessio a , H. Fujioka f , H. Fujita c , Y. Fujita c , G. Gey c , M. Hilcker a ,T.H. Hoang c , A. Inoue c , J. Isaak a,c , C. Iwamoto g , T. Klaus a , N. Kobayashi c , Y. Maeda h , M. Matsuda i ,N. Nakatsuka a , S. Noji j , H.J. Ong k,c , I. Ou l , N. Paar m , N. Pietralla a , V.Yu. Ponomarev a , M.S. Reen n ,A. Richter a , X. Roca-Maza e , M. Singer a , G. Steinhilber a , T. Sudo c , Y. Togano o , M. Tsumura p ,Y. Watanabe q , V. Werner a a Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, D-64289 Darmstadt, Germany b Institut f¨ur Theoretische Physik II, Universit¨at Erlangen, D-91058 Erlangen, Germany c Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan d Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, Texas 75429, USA e Dipartimento di Fisica, Universit`a degli Studi di Milano and INFN, Sezione di Milano, 20133 Milano, Italy f Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan g RIKEN, Nishina Center for Accelerator-Based Science, 2-1 Hirosawa, 351-0198 Wako, Saitama, Japan h Department of Applied Physics, Miyazaki University, Miyazaki 889-2192, Japan i Department of Communications Engineering, Graduate School of Engineering, Tohoku University, Aramaki Aza Aoba,Aoba-ku, Sendai 980-8579, Japan j National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA k Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China l Okayama University, Okayama 700-8530, Japan m Department of Physics, Faculty of Science, University of Zagreb, Zagreb, Croatia n Department of Physics, Akal University, Talwandi Sabo, Bathinda Punjab-151 302, India o Department of Physics, Rikkyo University, Tokyo 171-8501, Japan p Department of Physics, Kyoto University, Kyoto 606-8502, Japan q Department of Physics, University of Tokyo, Tokyo 113-8654, Japan
Abstract
The dipole polarizability of stable even-mass tin isotopes , , , , , Sn was extracted from inelas-tic proton scattering experiments at 295 MeV under very forward angles performed at RCNP. Predictionsfrom energy density functionals cannot account for the present data and the polarizability of
Pb simul-taneously. The evolution of the polarizabilities in neighboring isotopes indicates a kink at
Sn while allmodel results show a nearly linear increase with mass number after inclusion of pairing corrections.
Keywords: , , , , , Sn(p,p (cid:48) ), E p = 295 MeV, θ lab = 0 ◦ − ◦ , relativistic Coulomb excitation,photoabsorption cross sections, dipole polarizability
1. Introduction
Determination of the nuclear Equation of State(EoS) is one of the major goals of current nuclearphysics research [1], both experimentally and theo-retically. Its knowledge is e.g. required for an under-standing of astrophysical events like core-collapsesupernovae [2] or the formation [3] or the mass and ∗ Corresponding author
Email addresses: [email protected] (S. Bassauer), [email protected] (P. von Neumann-Cosel) radius [4, 5] of neutron stars. In particular, theobservation of a neutron star merger through thedetection of gravitational waves [6] and the asso-ciated electromagnetic spectrum provides a multi-tude of new experimental information, whose inter-pretation crucially depends on the EoS of neutron-rich matter [7, 8].The largest uncertainty of the EoS of proton-neutron asymmetric matter stems from the symme-try energy term. Since the symmetry energy can-not be measured directly, experimental observablesare sought that show a close correlation with its
Preprint submitted to Physics Letters B July 21, 2020 a r X i v : . [ nu c l - e x ] J u l roperties. The two most promising identified sofar are the thickness of the neutron skin formed inheavy nuclei and the dipole polarizability, see e.g.Ref. [9]. The volume terms of nuclear energy den-sity functional (EDF) theory – presently the mostsuccessful approach to the microscopic descriptionof heavy nuclei – are directly related to nuclear bulkparameters such as the incompressibility K or thesymmetry energy J and those bulk parameters of-ten have a near one-to-one correspondence to nu-clear observables. In particular, the symmetry en-ergy J shows a strong correlation with the slopeof symmetry energy L , the neutron skin thickness( r n − r p ), and the dipole polarizability α D whichhas attracted much attention [1, 10–17]. Accord-ingly, there is renewed interest in the measurementof the electric dipole strength or the correspondingphotoabsorption cross sections in nuclei for an ex-traction of the dipole polarizability α D from inversemoments of the E1 sum rule [18] α D = (cid:126) c π (cid:90) σ abs E d E x = 8 π (cid:90) B(E1) E x d E x , (1)where E x is the excitation energy, B(E1) the re-duced electric dipole transition strength and σ abs the photoabsorption cross section.In principle, the determination of α D requiresdata at all excitation energies. However, It is wellknown from extensive studies in the past [19, 20]that most of the E1 strength is concentrated in theIsoVector Giant Dipole Resonance (IVGDR). Fur-thermore, the contribution from the high-energy re-gion above the IVGDR is diminished by the inverseenergy weighting in Eq. (1). On the other hand, therole of low-energy strength is enhanced. Heavy nu-clei show resonance-like structures of isovector E1strength below the IVGDR, typically around theneutron threshold, often called Pygmy Dipole Res-onance (PDR). The PDR is observed in nuclei withneutron excess and thought to originate from thoseoutermost neutrons that display a soft spatial cor-relation with respect to the other nucleons formingthe core of the nucleus under study. This featurepoints towards a sensitivity of the Energy-WeightedSum Rule (EWSR) exhausted by the PDR on theneutron pressure below saturation density and thusto a correlation with the properties of the EoS.However, the structure underlying the PDR andthe resulting properties are not systematically un-derstood yet [21, 22] and a simple relation to bulkproperties is questionable [23]. Recently, inelastic proton scattering at energiesof a few hundred MeV and very forward angles in-cluding 0 ◦ has been established as a new methodto extract the complete E1 strength in heavy nu-clei from low excitation energies across the giantresonance region [24]. Under this particular kine-matics selective excitation of E1 and spin-M1 dipolemodes is observed. Their contributions to the crosssections can be separated either by a MultipoleDecomposition Analysis (MDA) of the cross sec-tions [25] or independently by the measurement ofa combination of polarization transfer observables[24, 26]. Good agreement of both methods wasdemonstrated for reference cases [14, 26, 27] indi-cating that the much simpler measurement of crosssections using an unpolarized beam and employingthe MDA thereof is sufficient for a reliable extrac-tion of the E1 strength distribution.Two different driving agents for the evolution ofthe dipole polarizability are conceivable, viz. neu-tron excess and the general trend with mass num-ber A (i.e., the size) both dependent on the sym-metry energy. The chain of proton-magic tin nu-clei is of particular interest because the underlyingstructure changes little between neutron shell clo-sures N = 50 and 82 and thus highlights the roleof neutron excess. Accordingly, a variety of modelcalculations have been performed attempting to ex-plore this connection [11, 13, 15, 28–42]. While thecorrelation of α D , J and L in EDF models is ro-bust [12], quantitative predictions differ consider-ably because the static isovector properties of theinteractions are usually poorly constrained by thedata used to determine the interaction parameters.The present letter reports on a systematic study ofthe dipole polarizability in the stable even-mass tinisotopes and provides an important benchmark forthe attempts to develop interactions with predictivepower as a function of nuclear mass and neutron ex-cess.
2. Experiment
The experiments were performed at the Re-search Center for Nuclear Physics (RCNP), OsakaUniversity, using the Grand Raiden Spectrometer[43]. The proton beam had an energy E p = 295MeV. Typical beam currents were between 2 and20 nA, depending on the spectrometer angle. Datawere taken at central spectrometer angles 0 ◦ , 2 . ◦ and 4 . ◦ . Highly enriched self-supporting targets2 Energy (MeV) d σ d Ω d E ( m b / s r / M e V ) Sn(p,p’), 0 ◦ Sn(p,p’), 2.5 ◦ Sn(p,p’), 4.5 ◦ Figure 1: Double differential cross section of the
Sn(p,p (cid:48) )reaction at E p = 295 MeV and scattering angles Θ lab = 0 ◦ ,2 . ◦ and 4 . ◦ . of , , , , , Sn with areal densities be-tween 3 and 7 mg/cm were used. The dispersion-matching technique enabled measurements with en-ergy resolutions between 30 and 40 keV (full widthat half maximum, FWHM). The experimental tech-niques and the raw data analysis are described inRef. [44] and specific to the present experiment inRef. [45]. Data taking for Sn was limited to across check of experimental cross sections obtainedin a previous experiment [14, 46].Typical spectra at the three main spectrometerangles are shown in Fig. 1 for
Sn by way ofexample. The dominance of relativistic Coulombexcitation expected for the kinematics at scatter-ing angles close to 0 ◦ [24] suggest that the promi-nent excitation centered at about 15 MeV is dueto the IVGDR. At lower excitation energies a pro-nounced structure is visible which also slowly dis-appears with increasing scattering angle. The an-gular dependence indicates a dipole character of theexcited states underlying this structure as demon-strated in the next section.
3. Multipole Decomposition Analysis
An MDA of the cross section angular distribu-tions was performed based on a least-squares fit ofthe type (cid:88) i (cid:18) d σ dΩ (Θ i , E x ) exp − d σ dΩ (Θ i , E x ) th (cid:19) ≡ min , (2) whered σ dΩ (Θ , E x ) th = (cid:88) Oλ a Oλ d σ dΩ (Θ , E x , Oλ ) DWBA + b d σ dΩ (Θ) QFS , (3)with the condition that all coefficients a Oλ and b were positive. The spectra were analyzed in200 keV bins. The angular acceptance of the spec-trometer of ± . ◦ allowed to generate five datapoints per angle and energy bin, so that in total 15data points between 0 . ◦ and 5 . ◦ were available forthe MDA. The shapes of theoretical angular distri-butions for different electric ( E ) and magnetic ( M )multipolarities Oλ ( O = E, M ) calculated with thecode DWBA07 [47] and based on transition densi-ties from the Quasiparticle Phonon Model (QPM)[48] were used as input. As demonstrated for pre-vious cases [25, 27], the low momentum transfersof the experiment permit a restriction of multi-poles in Eq. (3) to E1, M1 and one multipole rep-resenting all contributions λ > −
25 MeV), where the IVGDR contri-butions are negligible, was found to be constant andassumed to also hold in the giant resonance region.Prior to the MDA, the contributions of theisoscalar giant monopole and quadrupole reso-nances were subtracted from the spectra followingthe method described in Ref. [49], again using QPMresults of the corresponding strengths. The ex-perimental strength distributions were taken fromRef. [50].Examples of the fits of Eq. (2) are displayed inFig. 2 for the case of the
Sn(p,p (cid:48) ) data. Theupper part shows selected bins at excitation ener-gies of 8, 15 and 23 MeV indicated by the verticaldashed lines in the spectra, whose angular distribu-tions are plotted in the lower part. At the lowestexcitation energy one finds a dominance of E1 crosssections close to 0 ◦ , but M1 contributions are non-negligible. Above 4 ◦ the spectra are dominated by λ > .5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 Energy (MeV) d σ d Ω d E ( m b / s r / M e V ) S n Sn(p,p’), 0 ◦ Sn(p,p’), 2.5 ◦ Sn(p,p’), 4.5 ◦ θ (deg) − − d σ d Ω ( m b / s r ) E x = E1E3 M1Sum θ (deg) E x =
15 MeV
E1E3 QFSSum θ (deg) E x =
23 MeV
Figure 2: Results of the MDA for three different excitationenergy bins at 8, 15 and 23 MeV indicated by vertical dashedlines, respectively, shown for the example of
Sn.
4. Photoabsorption cross sections
The Coulomb excitation cross sections resultingfrom the MDA were converted to equivalent pho-toabsorption cross sections using the virtual pho-ton method [51]. The virtual photon spectrum wascalculated in an eikonal approach [52] in contrastto the previous study of
Sn, where the semiclas-sical approximation was used [14, 46]. In heavynuclei the differences between both approaches aresmall (typically less than 10%) but in lighter nucleithe semiclassical approach fails [53]. A compari-son of both approaches and the typical energy de-pendence for the present kinematic conditions areshown in Sec. 3.3 of Ref. [24] for the example of
Sn. Although the experimental spectra extendabove 20 MeV, the E1 cross sections become toosmall with respect to the quasifree background fora meaningful decomposition in the MDA. Figure 3presents a comparison of the resulting photoabsorp-tion cross sections with data available from ( γ ,xn)experiments [54–56], again for the example of Sn.The error bars include contributions from system-atic uncertainties of the cross sections determina-tion and from the MDA decomposition [45] addedquadratically. Statistical errors are negligible.One finds significant differences on the low-energy flank of the IVGDR. The Livermore databy Fultz et al. [54] show the best agreement withthe present result above 12 MeV, but undershootthe data from all other experiments for E x < γ ,n) data ofRef. [56] agree best with our results. Similar differ- Energy (MeV) σ ( m b ) Sn S n Sn(p,p’), this work
Sn( γ ,xn), [50] Sn( γ ,xn), [51] Sn( γ ,n), [52] Figure 3: Photoabsorption cross sections of
Sn deducedfrom the present experiment in comparison to previous work[54–56]. ences are found for the other isotopes studied here.A detailed account is given elsewhere [45].
5. Dipole polarizability
The present data provide photoabsorption crosssections in the energy region 6 −
20 MeV forthe determination of α D from Eq. (1). Below 6MeV, B(E1) strength distributions are available for , , , Sn from nuclear resonance fluorescenceexperiments [57, 58], but were neglected for consis-tency with the other isotopes. These contributionsare generally small ( < . −
30 MeV for , , , Sn.However, we refrain from using them, since theseresults show large variations between different iso-topes but no systematic isotopic dependence as dis-cussed in Ref. [45]. Including these data would notonly alter the absolute values but also significantlymodify the isotopic dependence. Rather we employa theory-assisted estimate of strength in the regionabove 20 MeV. To that end, we performed calcula-tions at the level of quasiparticle random phase ap-proximation (QRPA) [61] and of the more detailedQPM [24–26, 62, 63]. The QRPA and QPM crosssections used to calculate the dipole polarizability4
Energy (MeV) α D (f m ) Sn S n Figure 4: Running sum of the dipole polarizability from thepresent (p,p (cid:48) ) data for the example of
Sn. Red: Contri-bution from 6 MeV to S n . Blue: Contribution from S n to20 MeV. Orange: Contribution above 20 MeV from QPMcalculations, see text for details. in the energy region above 20 MeV were convolutedwith Lorentzians whose widths were tuned to re-produce the present IVGDR data. We have donethat for different models and parametrizations andfind consistently the same contribution of about8 % to α D . A particularly encouraging result is themost elaborate test based on a fully self-consistentcontinuum RPA calculation [64] with the Skyrmeparametrization SV-bas [16], for technical reasonsperformed for the doubly magic nucleus Sn. Itprovides a proper description of the experimentalphotoabsorption cross sections [65] without furtherfolding and finds a contribution of 6 % in the some-what larger
Sn. To account for the uncertaintiesin that extrapolation, an error of 10 % is associ-ated with the contributions taken from the modelresults.Figure 4 displays the evolution of α D as a func-tion of excitation energy (the running sum) for theexample of Sn. The error band considers statis-tical and systematical uncertainties, the latter in-cluding contributions from experiment and from theMDA (for details see Ref. [45]). The figure demon-strates that the polarizability values are dominatedby the contribution of the IVGDR (blue), but thelow-energy (red) and high-energy (orange) parts arenon-negligible. The corresponding partial and totalvalues are summarized in Table 1. The low-energycontribution up to the neutron threshold ( S n ) – i.e.the part missed in ( γ ,xn) experiments – varies from13 % ( Sn) to 8 % (
Sn) due to the decrease of
Table 1: Total dipole polarizability α D of , , , , , Sn (in fm ) determined as described inthe text. Partial values are given for the contributions from6 MeV to the neutron threshold S n , from S n to 20 MeV and >
20 MeV. − S n S n − >
20 Total
Sn 0 . . . . Sn 0 . . . . Sn 0 . . . . Sn 0 . . . . Sn 0 . . . . Sn 0 . . . . S n as a function of mass number. The high-energycontribution from the QPM calculations amountsto 9 −
10 % in all isotopes.We note that a larger value for
Sn was pub-lished in Ref. [14] based on the same type of exper-iment, which after correction for the quasideuteronpart amounted to α D = 8 . . However, thedifference to the present result is not due to the(p,p (cid:48) ) data (cross sections from the previous andpresent experiments agree within error bars), butresult from averaging in Ref. [14] with the ( γ ,xn)data of Refs. [54, 55], whose contributions to α D from the IVGDR region are larger than from thepresent work, and from the particularly large pho-toabsorption strengths of Ref. [54] in the energy re-gion 20 −
30 MeV for the case of
Sn (see Ref. [45]).The new polarizability results are now discussedin comparison to theoretical predictions from nu-clear EDFs (for a general review see Ref. [66]) basedon the non-relativistic Skyrme functional and therelativistic mean field model (RMF). As argued inalready in the introduction, dipole polarizability α D is a key observable for probing the symmetry en-ergy which, in turn, is important to determine thenuclear EoS. The present data on α D in the Sn iso-topic chain provide new insights to that discussion,in particular on the decomposition of the role ofneutron excess vs. the global mass dependence.We have scrutinized a great variety of publishedEDF parametrizations, but confine the present dis-cussion to four typical representatives: SV-bas is aSkyrme functional tuned to a large pool of groundstate properties (energy, radius, surface thickness)with additional constraints on the EoS such thatit reproduces giant resonances and α D in Pb[16]. KDE0-J33 [67] is also a Skyrme functionalwith fewer ground state properties in the fit pooland more bias on bulk properties; we take here5
12 114 116 118 120 122 124 A α D (f m ) DDMEaKDE0-J33 DDPCXSV-bas Exp.
208 18.519.019.520.020.5
Figure 5: Dipole polarizability α D in the stable Sn isotopes(left panel) and in Pb (right panel). Note the differentscale for
Pb. The experimental values (blue dots) andtheir errors (blue band) are compared with theoretical re-sults from the Skyrme functionals KDE0-J33 [17, 67] (greendiamonds) and SV-bas [16] (purple pentagons) and the RMFfunctionals DDMEa [68] (orange squares) and DD-PCX [42](red hexagons). the one sample from the KDE0 series closest tothe present results. DDMEa is a RMF functionalwith density-dependent meson coupling [68]. Fi-nally, DD-PCX is a different fit to RMF [42] tunedto reproduce α D ( Pb) [26] after correction forthe quasideuteron part [59]. In all cases, pairingis included in the calculations. We have chosenthese four cases to cover relativistic as well as non-relativistic models and within each group two EDFswhich differ sufficiently concerning other properties.A more detailed evaluation covering larger sets ofEDFs will be given in a subsequent publication.Figure 5 shows α D in the Sn chain (left) and Pb (right) comparing data with results fromthe four selected EDF parametrizations. At firstglance, all four EDFs lie reasonably close to all data.The same holds for most of the other more recent,well tuned EDFs because isovector trends of groundstate data imprint already some information on theisovector response. A closer look reveals interest-ing differences from which we may learn about nu-clear response properties. SV-bas was tuned to thevalue of α D in Pb [26] prior to the correctionfor the quasideuteron part and found to performvery well for the older, larger value of α D in Snin Ref. [14], but now lies above the values of thepresent work. DD-PCX was tuned to α D ( Pb)after correction for the quasideuteron part [59] andthus fits perfectly for
Pb while being somewhathigh for Sn. KDE0-J33 and DDMEa come closest for the Sn chain, however at the price of underesti-mating α D ( Pb).The similarity of the ordering along the Sn chainand in
Pb shows that one can shift the α D val-ues globally up and down without sacrificing toomuch of the overall quality of a functional, a fea-ture observed already in earlier studies (see e.g.Refs. [11, 16, 17]). The trend with nucleon number A looks rather rigid leading to very similar slopesalong the Sn chain and, on a wider scale, to strictrelations between Pb and the Sn isotopes. Thisis, in fact, expected from Migdal’s hydrodynamicalmodel [69]. The rigidity of the trends with A posesan intriguing problem for the given functionals: itseems that one cannot accommodate the α D data in Pb and in the Sn isotopes simultaneously. For aquantitative estimate we can look at statistical cor-relations between observables, see e.g. Refs. [10, 70].The correlations of α D within the present Sn chainare better than 97%. The correlation of of α D be-tween Pb and the Sn isotopes is with still large(about 90%), but already less restrictive. We canalso see from Fig. 5 that the A dependence of α D allows some variations as the slope of the four EDFsis not fully identical. We have also checked the largescale trend from Pb to Sn with other EDFs andfind that the density dependence of a functionalplays a role, as already indicated in previous studieswith both RMF [13, 68, 71] and Skyrme function-als [72]). This sets the direction for future improve-ment. We have to develop EDFs describing α D overa wide range of A including also data on Ca [73], Ca [53] and Ni [74] and Zr [75] into the fit, firsttrying to exploit the leeway in present functionalsand, if necessary, to consider EDFs with more elab-orated density dependencies. Such large scale fitsshould, moreover, reduce the extrapolation uncer-tainties for the symmetry energy J which are about1.4 MeV with the present error bands for α D ( Pb)[59]. The description of the IVGDR, where a sim-ilar problem in describing the trend with A exists[76], is also expected to benefit.A further aspect of interest in Fig. 5 are thetrends with A depicted in the left panel. The fourtheoretical results shows a similar nearly linear be-havior while the experimental data slightly bendover around Sn. Recall that
Sn corresponds toa subshell closure (below the neutron 1 h / shell).Indeed, calculations with the RMF parametrizationFSU040 [71, 77], which uses the filling approxima-tion rather than pairing, delivers qualitatively theexperimental trend. This indicates that the pairing6 Energy (MeV) α D ( a r b . u n i t s ) S n Sn Exp.DDMEaDDPCXKDE0-J33SV-bas
Figure 6: Normalized running sums of the dipole polarizabil-ity in
Sn of the four EDFs under consideration in compar-ison to the experimental data. The theoretical results fromdiscrete RPA were smoothed by Lorentzians to account forescape and spreading width. strength plays a role for details of the trend causedby shell effects. Thus, the role of pairing deservesfurther careful investigations.We finally address possible relations between low-lying dipole strength in the PDR regime and α D discussed in the introduction. The inverse energyweights in the integration to α D in Eq. (1) empha-sizes the low-energy region. In order to quantifyits contribution, we show in Fig. 6 a comparisonof of the normalized experimental and theoreticalrunning sums of α D for the example of Sn. Thebounds of the PDR region in the calculations arenot sharply defined, but should lie near 11 MeV.There are some variations of the predicted contri-bution but all are well below the experimental valueof about 15% in that energy region. Note that theexperimental curve is much softer at the lower andupper ends. The empirical smoothing of the RPAspectra simulating line broadening from many-bodyconfigurations is obviously insufficient in the wingsof the spectra. This, in turn, indicates that a largepart of dipole strength observed in the PDR re-gion can be interpreted as a low-energy tail of theIVGDR [46]. A clean separation of the two com-ponents requires many-body calculations beyondRPA.
6. Conclusions
We have extracted the dipole polarizabilityof stable even-mass Sn isotopes from relativistic Coulomb excitation using 295 MeV inelastic pro-ton scattering at very forward angles. This allowsto deduce precise data on the photoabsorption crosssection up to 20 MeV. The technique provides, inparticular, data indpendent of the neutron emis-sion threshold. The results permit detailed studiesof isotopic trends of isovector properties of nucleicarried in the IVGDR, the dipole polarizability α D ,and the low-lying dipole strength (PDR) [45].We have exemplified the potential of the new datawith a brief discussion of the case of the dipolepolarizability α D , an observable whose direct re-lation to isovector bulk properties (symmetry en-ergy) makes it particularly important for theoret-ical developments. Although practically all up-to-date EDF parametrizations provide at once roughlyacceptable values for α D , there are instructive dif-ferences in detail. The new α D values are system-atically lower than the old value for Sn whichcalls for a new fine-tuning of EDF parametriza-tions. Furthermore, comparison with α D in Pbshows that present EDFs, relativistic as well asnon-relativistic, cannot match the trend of α D from Pb to the Sn region.The trends of α D along the Sn chain raise anotherintriguing question. The development with increas-ing nucleon number A indicates a bend at Snwhen deduced from the data. This is most likely asignature of shell effects implying that α D in open-shell nuclei is not only driven by bulk properties.Surprisingly, the EDF calculations with pairing pro-duce a smooth, nearly linear increase mass numberwhile calculations neglecting pairing qualitativelyalso show a pronounced kink. The mismatch callsfor a deeper analysis of the role of nuclear pairing.The present experimental results challenge thedevelopment of EDF parametrizations aiming fora systematic reproduction the dipole polarizabilityacross the nuclear chart. Because of the strong cor-relation, such models will then provide improvedpredictions for the neutron skin thickness and pa-rameters of the symmetry energy which, in turn, areimportant for extrapolation to star matter. Com-bined with results expected from future studies ofneutron-rich unstable Sn isotopes using relativisticCoulomb breakup with the R3B setup at FAIR [78]which – in contrast to the pioneering experimentby Adrich et al. [65] – will include information onthe strength below neutron threshold, a unique setof data along an isotopic chain will be available toconstrain isovector properties of nuclei and nuclearmatter.7 cknowledgements The experiments were performed at RCNP un-der the experimental program E422. The authorsthank the accelerator group for providing excel-lent beams. We are indebted to J. Piekarewicz forenlightening discussions on the impact of shell ef-fects on the polarizability in the Sn isotope chainand to B.K. Agrawal for providing calculationsfor the KDE functional. This work was fundedby the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) – Projektnummer279384907 – SFB 1245, by JSPS KAKENHI, GrantNo. JP14740154 and by MEXT KAKENHI, GrantNo. JP25105509. G.C. and X.R.-M. acknowledgefunding from the European Unions Horizon 2020 re-search and innovation program under grant agree-ment No. 654002. C.A.B. was supported in partby U.S. DOE Grant No. DE-FG02-08ER41533 andU.S. NSF Grant No. 1415656.
References [1] X. Roca-Maza, N. Paar, Prog. Part. Nucl. Phys.101 (2018) 96.[2] H. Yasin, S. Sch¨afer, A. Arcones, A. Schwenk,Phys. Rev. Lett. 124 (2020) 092701.[3] F. ¨Ozel, P. Freire, Annu. Rev. Astron. Astroph. 54(2016) 401.[4] S. Bogdanov, S.Guillot, P.S. Ray, M.T. Wolff, D.Chakrabarty, W.C.G. Ho, M. Kerr, F.K. Lamb,A. Lommen, R.M. Ludlam, R. Milburn,S. Mon-tano, M.C. Miller, M. Baub¨ock, F. ¨Ozel, D. Psaltis,R.A. Remillard, T.E. Riley, J.F. Steiner, T.E.Strohmayer, A.L. Watts, K.S. Wood, J. Zeldes, T.Enoto, T. Okajima, J.W. Kellogg, C. Baker, C.B.Markwardt, Z.Arzoumanian, K.C. Gendreau, As-troph. J. Lett. (2019) L25.[5] S. Bogdanov, F.K. Lamb, S. Mahmoodifar, M.C.Miller, S.M. Morsink, T.E. Riley, T.E. Strohmayer,A.K. Tung, A.L. Watts, A.J. Dittmann, D.Chakrabarty, S. Guillot, Z.Arzoumanian, K.C.Gendreau, Astroph. J. Lett. (2019) L26.[6] B.P. Abbott et al., Phys. Rev. Lett. 119 (2017)161101.[7] F.J. Fattoyev, J. Piekarewicz, C.J. Horowitz, Phys.Rev. Lett. , 172702 (2018).[8] A.G. Chaves, T. Hinderer, J. Phys. G: Nucl. Part.Phys. 46 (2019) 123002.[9] M. Thiel, C. Sfienti, J. Piekarewicz, C.J. Horowitz,M. Vanderhaeghen, J. Phys. G: Nucl. Part. Phys.46 (2019) 093003.[10] P.-G. Reinhard, W. Nazarewicz, Phys. Rev. C 81(2010) 051303(R).[11] J. Piekarewicz, B. K. Agrawal, G. Col`o, W.Nazarewicz, N. Paar, P.-G. Reinhard, X. Roca-Maza, and D. Vretenar, Phys, Rev. C (2012)041302. [12] X. Roca-Maza, M. Brenna, G. Col`o, M. Centelles,X. Vi˜nas, B.K. Agrawal, N. Paar, D. Vretenar, J.Piekarewicz, Phys. Rev. C 88 (2013) 024316.[13] W. Nazarewicz, P.-G. Reinhard, W. Satu(cid:32)la, D.Vretenar, Eur. Phys. J. A 50 (2014) 20.[14] T. Hashimoto, A.M. Krumbholz, P.-G. Reinhard,A. Tamii, P. von Neumann-Cosel, T. Adachi, N.Aoi, C.A. Bertulani, H. Fujita, Y. Fujita, E.Ganioˇglu, K. Hatanaka, C. Iwamoto T. Kawabata,N.T. Khai, A. Krugmann, D. Martin, H. Mat-subara, K. Miki, R. Neveling, H. Okamura, H.J.Ong, I. Poltoratska, V.Yu. Ponomarev, A. Richter,H. Sakaguchi, Y. Shimbara, Y. Shimizu, J. Simo-nis, F.D. Smit, G. S¨usoy, J.H. Thies, T. Suzuki,M. Yosoi, J. Zenihiro, Phys. Rev. C 92 (2015)031305(R).[15] J. Erler, P.-G. Reinhard, J. Phys. G 42 (2015)034026.[16] P. Kl¨upfel, P.-G. Reinhard, T.J. B¨urvenich, J.A.Maruhn, Phys. Rev. C 79 (2009) 034310.[17] C. Mondal, B.K. Agrawal, M. Centelles, G. Col`o,X. Roca-Maza, N. Paar, X. Vi˜nas, S.K. Singh, S.K.Patra, Phys. Rev. C 93 (2016) 064303.[18] O. Bohigas, N. Van Giai, D. Vautherin, Phys. Lett.B 102 (1981) 105.[19] B.L. Berman, S.C. Fultz, Rev. Mod. Phys. 47(1975) 713.[20] S.S. Dietrich, B.L. Berman, At. Data Nucl. DataTables 38 (1988) 199.[21] D. Savran, T. Aumann, A. Zilges, Prog. Part. Nucl.Phys. 70 (2013) 210.[22] A. Bracco, E. Lanza, A. Tamii, Prog. Part. Nucl.Phys. 106 (2019) 360.[23] P.-G. Reinhard, W. Nazarewicz, Phys. Rev. C 87(2013) 014324.[24] P. von Neumann-Cosel, A. Tamii, Eur. Phys. J A55 (2019) 110.[25] I. Poltoratska, P. von Neumann-Cosel, A. Tamii, T.Adachi, C.A. Bertulani, J. Carter, M. Dozono, H.Fujita, K. Fujita, Y. Fujita, K. Hatanaka, M. Itoh,T. Kawabata, Y. Kalmykov, A.M. Krumbholz, E.Litvinova, H. Matsubara, K. Nakanishi, R. Nevel-ing, H. Okamura, H.J. Ong, B. ¨Ozel-Tashenov,V.Yu. Ponomarev, A. Richter, B. Rubio, H. Sak-aguchi, Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y.Shimizu, F.D. Smit, T. Suzuki, Y. Tameshige, J.Wambach, M. Yosoi, J. Zenihiro, Phys. Rev. C 85(2012) 041304(R).[26] A. Tamii, I. Poltoratska, P. von Neumann-Cosel,Y. Fujita, T. Adachi, C.A. Bertulani, J. Carter,M. Dozono, H. Fujita, K. Fujita, K. Hatanaka,D. Ishikawa, M. Itoh, T. Kawabata, Y. Kalmykov,A.M. Krumbholz, E. Litvinova, H. Matsubara, K.Nakanishi, R. Neveling, H. Okamura, H. J. Ong,B. ¨Ozel-Tashenov, V.Yu. Ponomarev, A. Richter,B. Rubio, H. Sakaguchi, Y. Sakemi, Y. Sasamoto,Y. Shimbara, Y. Shimizu, F.D. Smit, T. Suzuki, Y.Tameshige, J. Wambach, R. Yamada, M. Yosoi, J.Zenihiro, Phys. Rev. Lett. 107 (2011) 062502.[27] D. Martin, P. von Neumann-Cosel, A. Tamii, N.Aoi, S. Bassauer, C.A. Bertulani, J. Carter, L.M.Donaldson, H. Fujita, Y. Fujita, T. Hashimoto, K.Hatanaka, T. Ito, A. Krugmann, B. Liu, Y. Maeda,K. Miki, R. Neveling, N. Pietralla, I. Poltoratska,V.Yu. Ponomarev, A. Richter, T. Shima, T. Ya- amoto, M. Zweidinger, Phys. Rev. Lett. 119(2017) 182503.[28] N. Tsoneva, H. Lenske, Ch. Stoyanov, Phys. Lett.B 586 (2004) 213.[29] D. Vretenar, T. Nikˇsi´c, N. Paar, P. Ring, Nucl.Phys. A 731 (2004) 281.[30] J. Piekarewicz, Phys. Rev. C 73 (2006) 044325.[31] S.P. Kamerdzhiev, Phys. At. Nucl. 69 (2006) 1110.[32] J. Terasaki, J. Engel, Phys. Rev. C 74 (2006)044301.[33] N. Tsoneva, H. Lenske, Phys. Rev. C 77 (2008)024321.[34] E. Litvinova, P. Ring, V. Tselyaev, Phys. Rev. C78 (2008) 014312.[35] E.G. Lanza, F. Catara, D. Gambacurta, M.V. An-dres, Ph. Chomaz, Phys. Rev. C 79 (2009) 054615.[36] E. Litvinova, P. Ring, V. Tselyaev, Phys. Rev.Lett. 105 (2010) 022502.[37] I. Daoutidis, P. Ring, Phys. Rev. C 83 (2011)044303.[38] A. Avdeenkov, S. Goriely, S. Kamerdzhiev, S. Kre-wald, Phys. Rev. C 83 (2011) 064316.[39] P. Papakonstantinou, H. Hergert, V.Yu. Pono-marev, R. Roth, Phys. Rev. C 89 (2014) 034306.[40] J. Piekarewicz, Eur. Phys. J. A 50 (2014) 25.[41] S. Ebata, T. Nakatsukasa, T. Inakura, Phys. Rev.C 90 (2014) 024303.[42] E. Y¨uksel, T. Marketin, N. Paar, Phys. Rev. C 99(2019) 034318.[43] M. Fujiwara, H. Akimune, I. Daito, H. Fujimura,Y. Fujita, K. Hatanaka, H. Ikegami, I. Katayama,K. Nagayama, N. Matsuoka, S. Morinobu, T. Noro,M. Yoshimura, H. Sakaguchi, Y. Sakemi, A. Tamii,and M. Yosoi, Nucl. Instrum. Methods A 422(1999) 488.[44] A. Tamii, Y. Fujita, H. Matsubara, T. Adachi,J. Carter, M. Dozono, H. Fujita, K. Fujita, H.Hashimoto, K. Hatanaka, T. Itahashi, M. Itoh, T.Kawabata, K. Nakanishi, S. Ninomiya, A. Perez-Cerdan, L. Popescu, B. Rubio, T. Saito, H. Sak-aguchi, Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y.Shimizu, F. Smit, Y. Tameshige, M. Yosoi, J. Zeni-hiro, Nucl. Instrum. Methods A 605 (2009) 326.[45] S. Bassauer, P. von Neumann-Cosel, P.-G. Rein-hard, A. Tamii, S. Adachi, C.A. Bertulani,P.Y. Chan, A. D’Alessio, H. Fujioka, H. Fujita,Y. Fujita, G. Gey, M. Hilcker, T.H. Hoang, A. In-oue, J. Isaak, C. Iwamoto, T. Klaus, N. Kobayashi,Y. Maeda, M. Matsuda, N. Nakatsuka, S. Noji,H.J. Ong, I. Ou, N. Pietralla, V.Yu. Ponomarev,M.S. Reen, A. Richter, M. Singer, G. Steinhilber,T. Sudo, Y. Togano, M. Tsumura, Y. Watan-abe, V. Werner, submitted to Phys. Rev. C;arXiv:2007.06010.[46] A.M. Krumbholz, P. von Neumann-Cosel, T.Hashimoto, A. Tamii, T. Adachi, C.A. Bertulani,H. Fujita, Y. Fujita, E. Ganioˇglu, K. Hatanaka,C. Iwamoto, T. Kawabata, N.T. Khai, A. Krug-mann, D. Martin, H. Matsubara, R. Neveling, H.Okamura, H.J. Ong, I. Poltoratska, V.Yu. Pono-marev, A. Richter, H. Sakaguchi, Y. Shimbara, Y.Shimizu, J. Simonis, F.D. Smit, G. S¨usoy, J.H.Thies, T. Suzuki, M. Yosoi, J. Zenihiro, Phys. Lett.B 744 (2015) 7.[47] J. Raynal, program DWBA07, NEA Data Service NEA1209/08.[48] V. G. Soloviev, Theory of Atomic Nuclei: Quasi-particles and Phonons (Institute of Physics, Bris-tol, 1992).[49] L.M. Donaldson, C.A. Bertulani, J. Carter, V.O.Nesterenko, P. von Neumann-Cosel, R. Neveling,V.Yu. Ponomarev, P.-G. Reinhard, I.T. Usman,P. Adsley, J.W. Br¨ummer, E.Z. Buthelezi, G.R.J.Cooper, R.W. Fearick, S.V. F¨ortsch, H. Fujita,Y. Fujita, M. Jingo, W. Kleinig, C.O. Kureba, J.Kvasil, M. Latif, K.C.W. Li, J.P. Mira, F. Nemu-lodi, P. Papka, L. Pellegri, N. Pietralla, A. Richter,E. Sideras-Haddad, F.D. Smit, G.F. Steyn, J.A.Swartz, A. Tamii, Phys. Lett. B 776 (2018) 133.[50] T. Li, U. Garg, Y. Liu, R. Marks, B.K. Nayak, P.V.Madhusudhana Rao, M. Fujiwara, H. Hashimoto,K. Nakanishi, S. Okumura, M. Yosoi, M. Ichikawa,M. Itoh, R. Matsuo, T. Terazono, M. Uchida, Y.Iwao, T. Kawabata, T. Murakami, H. Sakaguchi,S. Terashima, Y. Yasuda, J. Zenihiro, H. Akimune,K. Kawase, M.N. Harakeh, Phys. Rev. C 81 (2010)034309.[51] C.A. Bertulani, G. Baur, Phys. Rep. 163 (1988)299.[52] C.A. Bertulani, A.M. Nathan, Nucl. Phys. A 554(1993) 158.[53] J. Birkhan, M. Miorelli, S. Bacca, S. Bassauer,C.A. Bertulani, G. Hagen, H. Matsubara, P.von Neumann-Cosel, T. Papenbrock, N. Pietralla,V.Yu. Ponomarev, A. Richter, A. Schwenk, A.Tamii, Phys. Rev. Lett. 118 (2017) 252501.[54] S.C. Fultz, B.L. Berman, J.T. Caldwell, R.L.Bramblett, M.A. Kelly, Phys. Rev. 186 (1969)1255.[55] A. Leprˆetre, H. Beil, R. Berg`ere, P. Carlos, A. D.Miniac, A. Veyssi`ere, K. Kernbach, Nucl. Phys. A219 (1974) 39.[56] H. Utsunomiya, S. Goriely, M. Kamata, T. Kondo,O. Itoh, H. Akimune, T. Yamagata, H. Toyokawa,Y. W. Lui, S. Hilaire, A.J. Koning, Phys. Rev. C80 (2009) 055806.[57] B. ¨Ozel-Tashenov, J. Enders, H. Lenske, A.M.Krumbholz, E. Litvinova, P. von Neumann-Cosel,I. Poltoratska, A. Richter, G. Rusev, D. Savran, N.Tsoneva, Phys. Rev. C 90 (2014) 024304.[58] K. Govaert, F. Bauwens, J. Bryssinck, D. DeFrenne, E. Jacobs, W. Mondelaers, L. Govor,V.Yu. Ponomarev, Phys. Rev. C 57 (1998) 2229.[59] X. Roca-Maza, X. Vi˜nas, M. Centelles, B.K.Agrawal, G. Col`o, N. Paar, J. Piekarewicz, D.Vretenar, Phys. Rev. C 92 (2015) 064304.[60] K.P. Schelhaas, J.M. Henneberg, M. Sanzone-Arenh¨ovel, N. Wieloch-Laufenberg, U. Zurm¨uhl,B. Ziegler, M. Schumacher, F. Wolf, Nucl. Phys.A 489 (1988) 189.[61] P. Ring and P. Schuck, The Nuclear Many-BodyProblem, Springer, New York 1980.[62] N. Ryezayeva, T. Hartmann, Y. Kalmykov, H.Lenske, P. von Neumann-Cosel, V.Yu. Ponomarev,A. Richter, A. Shevchenko, S. Volz, J. Wambach,Phys. Rev. Lett. 89 (2002) 272502.[63] I. Poltoratska, R.W. Fearick, A.M. Krumbholz, E.Litvinova, H. Matsubara, P. von Neumann-Cosel,V.Yu. Ponomarev, A. Richter, A. Tamii, Phys.Rev. C 89 (2014) 054322.
64] V. Tselyaev, N. Lyutorovich, J. Speth, P.-G. Rein-hard, Phys. Rev. C 97 (2017) 044308.[65] P. Adrich, A. Klimkiewicz, M. Fallot, K. Boretzky,T. Aumann, D. Cortina-Gil, U. Datta Pramanik,Th.W. Elze, H. Emling, H. Geissel, M. Hellstr¨om,K.L. Jones, J.V. Kratz, R. Kulessa, Y. Leifels,C. Nociforo, R. Palit, H. Simon, G. Surowka,K. S¨ummerer, W. Walus and the LAND-FRS Col-laboration, Phys. Rev. Lett. 95 (2005) 132501.[66] M. Bender, P.-H. Heenen, P.-G. Reinhard, Rev.Mod. Phys. 75 (2003) 121.[67] B.K. Agrawal, S. Shlomo, V.K. Au, Phys. Rev. C72 (2005) 014310.[68] D. Vretenar, T. Nikˇsi´c, P. Ring, Phys. Rev. C 68(2003) 024310.[69] A B. Migdal, J. Exp. Theor. Phys. USSR 15 (1945)81.[70] J. Dobaczewski, W. Nazarewicz, P.-G. Reinhard,J. Phys. G (2014) 074001.[71] J. Piekarewicz, Phys, Rev. C 83 (2011) 034319.[72] J. Erler, P. Kl¨upfel, P.-G. Reinhard, Phys. Rev. C82 (2010) 044307.[73] R.W. Fearick, et al., to be published.[74] D.M. Rossi, P. Adrich, F. Aksouh, H. Alvarez-Pol,T. Aumann, J. Benlliure, M. B¨ohmer, K. Boret-zky, E. Casarejos, M. Chartier, A. Chatillon,D. Cortina-Gil, U. Datta Pramanik, H. Emling,O. Ershova, B. Fernandez-Dominguez, H. Geissel,M. Gorska, M. Heil, H.T. Johansson, A. Jung-hans, A. Kelic-Heil, O. Kiselev, A. Klimkiewicz,J.V. Kratz, R. Kr¨ucken, N. Kurz, M. Labiche,T. Le Bleis, R. Lemmon, Y. A. Litvinov, K. Ma-hata, P. Maierbeck, A. Movsesyan, T. Nilsson,C. Nociforo, R. Palit, S. Paschalis, R. Plag,R. Reifarth, D. Savran, H. Scheit, H. Simon,K. S¨ummerer, A. Wagner, W. Walu´s, H. Weick,M. Winkler, Phys. Rev. Lett. 111 (2013) 242503.[75] T. Klaus, et al., to be published.[76] J. Erler, P. Kl¨upfel, P.-G. Reinhard, J. Phys. G 37(2010) 064001.[77] B.G. Todd-Rutel, J. Piekarewicz, Phys. Rev. Lett.95 (2005) 122501.[78] T. Aumann, private commmunication.(2014) 074001.[71] J. Piekarewicz, Phys, Rev. C 83 (2011) 034319.[72] J. Erler, P. Kl¨upfel, P.-G. Reinhard, Phys. Rev. C82 (2010) 044307.[73] R.W. Fearick, et al., to be published.[74] D.M. Rossi, P. Adrich, F. Aksouh, H. Alvarez-Pol,T. Aumann, J. Benlliure, M. B¨ohmer, K. Boret-zky, E. Casarejos, M. Chartier, A. Chatillon,D. Cortina-Gil, U. Datta Pramanik, H. Emling,O. Ershova, B. Fernandez-Dominguez, H. Geissel,M. Gorska, M. Heil, H.T. Johansson, A. Jung-hans, A. Kelic-Heil, O. Kiselev, A. Klimkiewicz,J.V. Kratz, R. Kr¨ucken, N. Kurz, M. Labiche,T. Le Bleis, R. Lemmon, Y. A. Litvinov, K. Ma-hata, P. Maierbeck, A. Movsesyan, T. Nilsson,C. Nociforo, R. Palit, S. Paschalis, R. Plag,R. Reifarth, D. Savran, H. Scheit, H. Simon,K. S¨ummerer, A. Wagner, W. Walu´s, H. Weick,M. Winkler, Phys. Rev. Lett. 111 (2013) 242503.[75] T. Klaus, et al., to be published.[76] J. Erler, P. Kl¨upfel, P.-G. Reinhard, J. Phys. G 37(2010) 064001.[77] B.G. Todd-Rutel, J. Piekarewicz, Phys. Rev. Lett.95 (2005) 122501.[78] T. Aumann, private commmunication.