Evolution of the dipole polarizability in the stable tin isotope chain

The dipole polarizability of stable even-mass tin isotopes 112,114,116,118,120,124 was extracted from inelastic proton scattering experiments at 295 MeV under very forward angles performed at RCNP. Predictions from energy density functionals show generally larger values than those observed experimentally, but interactions capable to reproduce experimental polarizabilities over a wide mass range are still consistent within experimental and theoretical uncertainties. The evolution of the polarizabilities in neighboring isotopes exhibits a kink at $^{120}$Sn while all model results show a smooth increase with mass number.


Introduction
Determination of the nuclear Equation of State (EoS) is one of the major goals of current nuclear physics research [1], both experimentally and theoretically. Its knowledge is e.g. required for an understanding of astrophysical events like core-collapse supernovae [2] or the formation of neutron stars [3]. In particular, the observation of a neutron star merger through the detection of gravitational waves [4] and the associated electromagnetic spectrum provides a multitude of new experimental information, whose interpretation crucially depends on the EoS of neutron-rich matter [5,6].
The largest uncertainty of the EoS of protonneutron asymmetric matter stems from the symmetry energy term. Since the symmetry energy cannot be measured directly, experimental observables are sought that show a close correlation with its properties. The two most promising identified so far are the thickness of the neutron skin formed in heavy nuclei and the dipole polarizability, see e.g. Ref. [7]. In energy density functional (EDF) theory -the most successful approach to the microscopic description of heavy nuclei -both quantities show a strong correlation with the leading parameter (called J) and its derivative with respect to density (called L ) of a Taylor expansion of the symmetry energy term around saturation density [1,[8][9][10][11].
Accordingly, there is renewed interest in the measurement of the electric dipole strength or the corresponding photoabsorption cross sections in nuclei for an extraction of the dipole polarizability α D from inverse moments of the E1 sum rule [12] where E x is the excitation energy, B(E1) the reduced electric dipole transition strength and σ abs the photoabsorption cross section. In principle, the determination of α D requires data at all excitation energies. However, It is well known from extensive studies in the past [13,14] that most of the E1 strength is concentrated in the IsoVector Giant Dipole Resonance (IVGDR). Furthermore, the contribution from the high-energy region above the IVGDR is diminished by the inverse energy weighting in Eq. (1). On the other hand, the role of low-energy strength is enhanced. Heavy nuclei show resonance-like structures of isovector E1 strength below the IVGDR, typically around the neutron threshold, often called Pygmy Dipole Resonance (PDR). The PDR in observed in nuclei with neutron excess and thought to originate from those outermost neutrons that display a soft spatial correlation with respect to the other nucleons forming the core of the nucleus under study. This feature points towards a sensitivity of the Energy-Weighted Sum Rule (EWSR) exhausted by the PDR on the neutron pressure below saturation density and thus to a correlation with the properties of the EoS. However, the structure underlying the PDR and the resulting properties are not systematically understood yet [15,16] and a simple relation to bulk properties is questionable [17].
Recently, inelastic proton scattering at energies of a few hundred MeV and very forward angles including 0 • has been established as a new method to extract the complete E1 strength in heavy nuclei from low excitation energies across the giant resonance region [18]. Under this particular kinematics selective excitation of E1 and spin-M1 dipole modes is observed. Their contributions to the cross sections can be separated either by a Multipole Decomposition Analysis (MDA) of the cross sections [19] or independently by the measurement of a combination of polarization transfer observables [18,20]. Good agreement of both methods was demonstrated for reference cases [20][21][22] indicating that the much simpler measurement of cross sections using an unpolarized beam and employing the MDA thereof is sufficient for a reliable extraction of the E1 strength distribution.
The present letter reports on a systematic study of the dipole polarizability in the stable even-mass tin isotopes. The chain of proton-magic tin nuclei is of particular interest because the underlying structure changes little between neutron shell closures N = 50 and 82. The evolution of the dipole polarizability should be driven by the neutron excess and thus by the symmetry energy. Accordingly, a variety of model calculations have been performed attempting to explore this connection [9,11,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. While the correlation of α D , J and L in EDF models is robust [10], quantitative predictions differ considerably because the static isovector properties of the interactions are usually poorly constrained by the data used to determine the interaction parameters. The present results thus provide an important benchmark for the attempts to develop interactions with predictive power as a function of nuclear mass and neutron excess.

Experiment
The experiments were performed at the Research Center for Nuclear Physics (RCNP), Osaka University, using the Grand Raiden Spectrometer [39]. The proton beam had an energy E p = 295 MeV. Typical beam currents were between 2 and 20 nA, depending on the spectrometer angle. Data were taken at central spectrometer angles 0 • , 2.5 • and 4.5 • . Highly enriched selfsupporting targets of 112,114,116,118,120,124 Sn with areal densities between 3 and 7 mg/cm 2 were used. The dispersion-matching technique enabled measurements with energy resolutions between 30 and 40 keV (full width at half maximum, FWHM). The experimental techniques and the raw data analysis are described in Ref. [40]. Data taking for 120 Sn was limited to a cross check of experimental cross sections obtained in a previous experiment [21,41]. Typical spectra at the three main spectrometer angles are shown in Fig. 1 for 116 Sn by way of example. The dominance of relativistic Coulomb excitation expected for the kinematics at scattering angles close to 0 • [18] suggest that the prominent excitation centered at about 15 MeV is due to the IVGDR. At lower excitation energies a pronounced structure is visible which also slowly disappears with increasing scattering angle. The angular dependence indicates a dipole character of the excited states underlying this structure as demonstrated in the next section.

Multipole Decomposition Analysis
An MDA of the cross section angular distributions was performed based on a least-squares fit of the type with the condition that all coefficients a Oλ and b were positive. The spectra were analyzed in  [43] were used as input. As demonstrated for previous cases [19,22], the low momentum transfers of the experiment permit a restriction of multipoles in Eq. (3) to E1, M1 and one multipole representing all contributions λ > 1 (E3 in the present case). Above the particle threshold the spectra contain a phenomenological background dominated by quasifree scattering (QFS). Its angular distribution was determined at the highest excitation energies measured (23 − 25 MeV), where the IVGDR contributions are expected to be negligible. Prior to the MDA, the contributions of the isoscalar giant monopole and quadrupole resonances were subtracted from the spectra following the method described in Ref. [44], again using QPM results of the corresponding strengths. The experimental strength distributions were taken from Ref. [45]. Further details of the MDA are described in Ref. [46].

Photoabsorption cross sections
The Coulomb excitation cross sections resulting from the MDA were converted to equivalent photoabsorption cross sections using the virtual photon method [47]. The virtual photon spectrum was calculated in an eikonal approach [48] in contrast to the previous study of 120 Sn, where the semiclassical approximation was used [21,41]. In heavy nuclei the differences between both approaches are small (typically less than 10%) but in lighter nuclei the semiclassical approach fails [49]. Although the experimental spectra extend above 20 MeV, the E1 cross sections become too small with respect to the quasifree background for a meaningful decomposition in the MDA. Figure 3 presents a comparison of the resulting photoabsorption cross sections with data available from (γ,xn) experiments [50][51][52], again for the example of 116 Sn. One finds significant differences on the low-energy flank of the IVGDR. The Livermore data by Fultz et al. [50] show the best agreement with the present result above 12 MeV, but un-dershoot the data from all other experiments for E x < 12 MeV. Near neutron threshold the new (γ,n) data of Ref. [52] agree best with our results. Similar differences are found for the other isotopes studied here. A detailed account is given elsewhere [46].

Dipole polarizability
The present data provide photoabsorption cross sections in the energy region 6 − 20 MeV for the determination of α D from Eq. (1). Below 6 MeV, B(E1) strength distributions are available for 112,116,120,124 Sn from nuclear resonance fluorescence experiments [53,54], but were neglected for consistency with the other isotopes. These contributions are generally small (< 0.5 % of the total dipole polarizability). In Ref. [55] it was argued that the contributions of the quasideuteron mechanism [56], which dominates the photoabsorption for excitation energies above 30 MeV in the present case, should be excluded from the integration of Eq. (1). Such a nonresonant process is not included in the model calculations. Data are available from Ref. [50] in the excitation region 20 − 30 MeV for 116,118,120,124 Sn. However, we refrain from using them, since these results show large variations between different isotopes but no systematic isotopic dependence. Rather we employ a theoryassisted estimate of strength in the region above 20 MeV. To that end, we performed calculations at the level of quasiparticle random phase approximation (QRPA) [57] and of the more detailed QPM [18-20, 58, 59]. The QRPA and QPM cross sections used to calculate the dipole polarizability in the energy region above 20 MeV were convoluted with Lorentzians whose widths were tuned to reproduce the present IVGDR data. We have done that for different models and parametrizations and find consistently the same contribution of about 8 % to α D . A particularly encouraging result is the most elaborate test based on a fully self-consistent continuum RPA calculation [60] with the Skyrme parametrization SV-bas [61], for technical reasons performed for the doubly magic nucleus 132 Sn. It provides a proper description of the experimental photoabsorption cross sections [62] without further folding and finds a contribution of 6 % in the somewhat larger 132 Sn. To account for the uncertainties in that extrapolation, an error of 10 % is associated with the contributions taken from the model results.  Figure 4 displays the evolution of α D as a function of excitation energy (the running sum) for the example of 116 Sn. The error band considers statistical and systematical uncertainties, the latter including contributions from experiment and from the MDA (for details see Ref. [46]). The figure demonstrates that the polarizability values are dominated by the contribution of the IVGDR (blue), but the low-energy (red) and high-energy (orange) parts are non-negligible. The corresponding partial and total values are summarized in Table 1. The low-energy contribution up to the neutron threshold (S n ) -i.e. the part missed in (γ,xn) experiments -varies from 13 % ( 112 Sn) to 8 % ( 124 Sn) due to the decrease of S n as a function of mass number. The high-energy contribution from the QPM calculations amounts to 9 − 10 % in all isotopes.  (56) We note that a larger value for 120 Sn was published in Ref. [21] based on the same type of experiment, which after correction for the quasideuteron part amounted to α D = 8.59(37) fm 3 . However, the difference to the present result is not due to the (p,p ) data (cross sections from the previous and present experiments agree within error bars), but result from averaging in Ref. [21] with the (γ,xn) data of Refs. [50,51], whose contributions to α D from the IVGDR region are larger than from the present work, and from the particularly large photoabsorption strengths of Ref. [50] in the energy region 20 − 30 MeV for the case of 120 Sn.
The new polarizability results are now discussed in comparison to theoretical predictions from nuclear EDFs (for a general review see Ref. [63]) based on the non-relativistic Skyrme functional and the relativistic mean field model (RMF). A much discussed key entry to self-consistent models are nuclear bulk parameters such as, e.g., the incompressibility K or the symmetry energy J [1]. On the other hand, those bulk parameters often have a near one-to-one correspondence to nuclear observables. A pronounced correlation exists between the symmetry energy parameters J and L, the neutron skin thickness (r n − r p ), and the dipole polarizability α D , which has attracted much attention [1,9,11,21,37,61,64]. The case is important because an understanding of the symmetry energy is crucial for the description of nuclear matter in stars [1,65,66]. The present new data on the dipole polarizability in the Sn isotopic chain provide novel insights to that discussion. We have scrutinized a great variety of published EDF parametrizations, but confine the present discussion to a few typical representatives. A more detailed evaluation will be given in a subsequent publication. Figure 5 shows α D in the Sn chain (left), 208 Pb (middle), and the slope of values in Sn (right), comparing data with results from four selected EDF parametrizations (obtained from QRPA and/or linear response to dipole perturbation). At first glance all four EDFs lie close to all data. The same holds for most of the more recent, well tuned EDFs because isovector trends of ground state data imprint already some information on the isovector response. A closer look reveals interesting differences from which we may learn more about nuclear response properties. SVbas [61] was tuned to the value of α D in 208 Pb [20] prior to the correction for the quasideuteron part and found to perform very well for the older, larger value of α D in 120 Sn in Ref. [21], but lies above the values of the present work. DD-PCX [38] was tuned to α D ( 208 Pb) after correction for the quasideuteron part [55]. KDE0-J33 [64,67] and DDMEa [68] come closest for the Sn chain, however at the price of underestimating α D ( 208 Pb). The similarity of the ordering along the Sn chain and in 208 Pb shows that one can shift the α D values globally up and down without sacrificing too much of the overall quality of a functional, a feature observed already in earlier studies (see e.g. Refs. [9,61,64]). However, the trend with nucleon number A is much more rigid leading to very similar slopes along the Sn chain and, on a wider scale, to strict relations between 208 Pb and the Sn isotopes. This is, in fact, already expected from Migdal's hydrodynamical model [69]. The rigidity of the trends with A poses an intriguing problem for the given functionals: one cannot accommodate α D data simultaneously in 208 Pb and Sn. However, a closer look at the slopes in Sn and in the relation from 208 Pb to Sn reveals that there are some differences which could possibly be exploited for improvements. First explorations, particularly on the large scale trend from 208 Pb to Sn indicate that the density dependence of a functional plays a role (seen from systematic variations in RMF [11,68,70] as well as Skyrme functionals [71]). Data on α D over a wide range of A (this addresses also the information on 40 [74], is also expected to benefit. Considering the slopes depicted in the right panel of Fig. 5, the four theoretical results agree with the data when averaged over the chain. However, the detailed trend differs significantly. The data show a kink at 120 Sn while all models discussed here produce a straight trend. Recall that 120 Sn corresponds to a subshell closure (below the 1h 11/2 shell). Indeed, calculations with the RMF parametrization FSU040 [70,75], which uses the filling approximation rather than pairing, delivers qualitatively the same pattern, namely a pronounced kink. This indicates that the pairing strength plays a role for details of the trend along isotopic chains and shell effects have an impact on α D in the Sn isotopes deserving further careful investigations.

Conclusions
We have extracted the dipole polarizability of stable even-mass Sn isotopes from relativistic Coulomb excitation using 295 MeV inelastic proton scattering at very forward angles. This allows to deduce precise data on the photoabsorption cross section up to 20 MeV. The technique provides, in particular, high resolution data below particle emission threshold. The results permit detailed studies of isotopic trends of crucial isovector properties of nuclei carried in the IVGDR, the dipole polarizability α D , and the low-lying dipole strength (PDR).
We have exemplified the great potential of the new data for further development of nuclear energy density functionals with a brief discussion of the case of the dipole polarizability α D , an observable whose direct relation to isovector bulk properties (symmetry energy) makes it particularly important for theoretical developments. Although practically all up-to-date EDF parametrizations provide at once roughly acceptable values for α D , there are instructive differences in detail. The new α D values are systematically lower than the old value for 120 Sn which calls for a new fine-tuning of EDF parametrizations. Furthermore, comparison with α D in 208 Pb shows that present EDFs, relativistic as well as non-relativistic, cannot match the trend of α D from 208 Pb to the Sn region. This poses a challenge to further development of EDFs. Experimentally, a better constraint of the high-energy contribution above the IVGDR would be important, which can be expected from next-generation photoabsorption experiments at ELI-NP [76].
The trends of α D along the Sn chain raise another intriguing question. The development of the slope with increasing nucleon number A shows a kink at 120 Sn when deduced from the data. This is most likely a signature of shell effects implying that α D in open-shell nuclei is not only driven by bulk properties. Surprisingly, the EDF calculations with pairing produce an extremely flat trend for the slope while calculations neglecting pairing qualitatively also show a pronounced kink. The mismatch calls for a deeper analysis indeed of the role of nuclear pairing.
The present experimental results challenge the development of EDF parametrizations capable of systematically reproducing the dipole polarizability across the nuclear chart. Because of the strong correlation, such models will then provide improved predictions for the neutron skin thickness and parameters of the symmetry energy which, in turn, are important for extrapolation to star matter. Combined with results expected from future studies of neutron-rich unstable Sn isotopes using relativistic Coulomb breakup with the R3B setup at FAIR [77] which -in contrast to the pioneering experiment by Adrich et al. [62] -will include information on the strength below neutron threshold, a unique set of data along an isotopic chain will be available to constrain isovector properties of nuclei and nuclear matter.