Electric Dipole Polarizability and the Neutron Skin

Electric dipole polarizability and the neutron skin

J. Piekarewicz,1 B. K. Agrawal,2 G. Colò,3, 4 W. Nazarewicz,5, 6, 7 N. Paar,8 P.-G. Reinhard,9 X. Roca-Maza,4 and D. Vretenar8 1Department of Physics, Florida State University, Tallahassee, FL 32306, USA 2Saha Institute of Nuclear Physics, Kolkata 700064, India 3Dipartimento di Fisica, Universitàdegli Studi di Milano 4INFN, Sezione di Milano, 20133 Milano, Italy 5 Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA 6 Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 7 Institute of Theoretical Physics, University of Warsaw, ul. Ho˙za69, PL-00-681 Warsaw, Poland 8Physics Department, Faculty of Science, University of Zagreb, Zagreb, Croatia 9Institut fürTheoretische Physik II, UniversitätErlangen-Nürnberg, Staudtstrasse 7, D-91058 Erlangen, Germany

208 The recent high-resolution measurement of the electric dipole (E1) polarizability αD in Pb [Phys. Rev. Lett. 107, 062502 (2011)] provides a unique constraint on the neutron-skin thickness 208 of this nucleus. The neutron-skin thickness rskin of Pb is a quantity of critical importance for our understanding of a variety of nuclear and astrophysical phenomena. To assess the model 208 dependence of the correlation between αD and rskin, we carry out systematic calculations for Pb, 132Sn, and 48Ca based on the nuclear density functional theory (DFT) using both non-relativistic and relativistic energy density functionals (EDFs). Our analysis indicates that whereas individual models

exhibit a linear dependence between αD and rskin, this correlation is not universal when one combines predictions from a host of diﬀerent models. By averaging over these model predictions, we provide

estimates with associated systematic errors for rskin and αD for the nuclei under consideration. 48 208 We conclude that precise measurements of rskin in both Ca and Pb—combined with the recent

measurement of αD—should signiﬁcantly constrain the isovector sector of the nuclear energy density functional.

PACS numbers: 21.10.Gv, 21.60.Jz, 21.65.Cd, 21.65.Mn

The Lead Radius Experiment (PREx) [1, 2] at the Jef- momentum transfers [6], PREx was able to determine the ferson Laboratory has recently provided the first model- following values for the neutron radius and neutron-skin +0.15 +0.15 independent determination of the neutron root-mean- thickness: rn=5.78−0.17 fm and rskin=0.34−0.17 fm [3]. 208 square (rms) radius rn of Pb [3]. Parity-violating elec- tron scattering, a powerful technique used by the PREx Another observable that is a strong indicator of isovec- collaboration, is particularly sensitive to the neutron dis- tor properties is the electric dipole polarizability αD re- tribution because the neutral weak-vector boson couples lated to the response of the nucleus to an externally ap- preferentially to the neutrons in the target [4]; the cou- plied electric field. For stable medium-to-heavy nuclei pling to the proton is suppressed by the weak mixing with a moderate neutron excess, the dipole response is angle. In spite of the many challenges that it faced, largely concentrated in the giant dipole resonance (GDR) this purely electroweak measurement may be interpreted of width 2-4 MeV that exhausts almost 100% of the with as much confidence as conventional electromagnetic energy-weighted sum rule [12]. For this isovector mode scattering experiments that have been used for decades of excitation—perceived as an oscillation of neutrons to accurately map the electric charge distribution of the against protons— the symmetry energy asym acts as the nucleus. restoring force. Models with a soft symmetry energy, namely, those that change slowly with density, predict A quantity that is related to the neutron radius is the larger values for asym at the lower densities of relevance neutron-skin thickness rskin=rn−rp, namely, the differ- to the excitation of this mode [13, 14]. In this context, ence between the rms neutron and proton radii. The the inverse energy-weighted E1 sum rule m−1—a quan- importance of the neutron skin lies in its strong depen- tity directly proportional to αD—is of particular interest dence on the poorly known isovector density ρ1 =ρn−ρp. as it is highly sensitive to the density dependence of the Given that rskin is a strong indicator of isovector proper- symmetry energy. This sensitivity suggests the existence ties, the determination of rn of a heavy nucleus is a prob- of a correlation: the larger rskin, the larger αD. Indeed, lem of fundamental importance with far reaching implica- the approximate proportionality of these two quantities tions in areas as diverse as nuclear structure [5–8], atomic is expected based on both macroscopic arguments[15, 16] parity violation [9], and neutron-star structure [10, 11]. and microscopic calculations [8, 17]. Naturally, the re- By measuring the neutron form factor of 208Pb at a cently completed high-resolution (~p,~p 0) measurement of moderate momentum transfer of q ≈ 0.44 fm−1, and the distribution of E1 strength in 208Pb over a wide range through a fairly model-independent extrapolation to low- of excitation energy [18] has created considerable excite- 2 ment. Of particular relevance to our work is the precise value of the measured electric dipole polarizability '+.-,* &$ $#'( 208 3 of Pb: αD=(20.1±0.6) fm . '+.-,*)') !!)%" It is the purpose of this work to examine possible corre-

lations between the dipole polarizability and the neutron- ) skin thickness of 208Pb. Generally, to assess a correlation 3 between two observables A and B within one given model one resorts to a least-squares covariance analysis, with the correlation coeﬃcient Pb] (fm Pb] |∆A ∆B| 208

CAB = , (1) [ p 2 2

∆A ∆B D providing the proper statistical measure [19]. In Eq. (1) the overline means an average over the statistical sam- ple. A value of |CAB| = 1 means that the two observables are fully correlated whereas CAB = 0 implies that they are totally uncorrelated. Recently, the statistical measure CAB was used to study correlations between various r [208Pb] (fm) nuclear observables [8] in the context of the Skyrme SV- skin min model [20]. In particular, it was concluded that good isovector indicators that strongly correlate with the neu- FIG. 1: (Color online) Predictions from 48 nuclear EDFs dis- tron radius of 208Pb are its electric dipole polarizability as cussed in the text for the electric dipole polarizability and 208 well as neutron skins and radii of neutron-rich nuclei [8]. neutron-skin thickness of Pb. Constrains on the neutron- Indeed, by relying on the strong correlation between α skin thickness from PREx [3] and on the dipole polarizability D from RCNP [18] have been incorporated into the plot. and rskin (CAB=0.98) predicted by such DFT calcula- +0.025 tions, Tamii et al. deduced a value of 0.156−0.021 fm for the neutron-skin thickness of 208Pb. chosen in this work. In particular, the up-triangles However, the correlation coefficient C cannot as- AB mark predictions from a broad choice of Skyrme EDFs sess systematic errors that reflect constraints and limita- that have been widely used in the literature: SGII, tions of a given model [8]. Such systematic uncertainties SIII, SkI3, SkI4, SkM∗, SkO, SkP, SkX, SLy4, SLy6, can only emerge by comparing different models (or suffi- (see Refs. [22, 23] for the original references), Sk255 ciently flexible variants of a model) and this is precisely [24], BSk17 [25], LNS [26], and UNEDF0 and UNEDF1 what has been done in this Letter. To measure the linear [27]. In addition, we consider a collection of relativistic dependence between two observables A and B for a sam- and Skyrme EDFs that have been systematically varied ple of several models, the correlation coefficient Cmodels is AB around an optimal model without a significant deterio- now obtained by averaging over the predictions of those ration in the quality of the fit (this is particularly true models. Although the correlation coefficient Cmodels de- AB for the case of the isovector interaction which at present termined in such a way may not have a clear statistical remains poorly constrained). Those results are marked interpretation, it is nevertheless an excellent measure of in Fig. 1 as NL3/FSU [17, 28] (circles), DD-ME [29] linear dependence. (squares), and Skyrme-SV [20] (down-triangles). Note To this end, we have computed the distribution of E1 that the “stars” in the figure are meant to represent the strength using both relativistic and non-relativistic DFT predictions from the optimal models within the chain of approaches with different EDFs. In all cases, these self- systematic variations of the symmetry energy. At first consistent models have been calibrated to selected global glance a clear (positive) correlation between the dipole properties of finite nuclei and some parameters of nuclear polarizability and the neutron skin is discerned. matter. Once calibrated, these models are used without Yet on closer examination one observes a significant any further adjustment to compute the E1 strength RE1 using a consistent random-phase approximation. The scatter in the results, especially for the standard Skyrme electric dipole polarizability is then obtained from the models. In particular, by including the predictions from all the 48 EDFs considered here the correlation inverse energy weighted sum [21]: models CAB =0.77 was obtained. However, as seen in Ta- 8π Z ∞ ble I, within each set of the systematically varied mod- α = e2 ω−1R (ω) dω . (2) D E1 els an almost perfect correlation is obtained. Note that 9 0 by imposing the recent experimental constraints on rskin 208 The relation between αD and rskin for Pb is dis- and αD, several of the models—especially those with ei- played in Fig. 1 using the predictions from the 48 EDFs ther a very soft or very stiff symmetry energy—may al- 3 ready be ruled out. Thus, if we average our theoretical '+.-,* &$ $#'( results over the set of 25 EDFs (“Set-25”) whose pre- '+.-,*)') !!)%" dictions fall within the RCNP value of αD, we obtain rskin=(0.168±0.022) fm, a value that is fairly close to the one obtained in Ref. [18]. Moreover, if we further limit (a) the models to those that are consistent with the PREx constraint of rskin greater than 0.17 fm, then the number of EDFs is reduced to 10 (“Set-10”) and the theoretical prediction becomes rskin=(0.189±0.017) fm. Clearly, in order to provide a meaningful constraint on current theoretical models, an accuracy of at least 0.03 fm on Sn] (fm) 132 the experimental value of the neutron radius is required. [ Based on the central value of rn=5.78 fm reported by the skin PREx collaboration [3], one concludes that a follow-up models 0.5% measurement would be very helpful. CAB =0.997 In anticipation of the follow-up PREx experiments it is pertinent to ask weather parity-violating experiments in other nuclei may be warranted [30]. Using either lighter (b) nuclei or nuclei with a larger neutron excess will increase PREx the parity-violating asymmetry. To assess the merit of 0.24 additional measurements of neutron radii, we have computed data-to-data relations between the neutron-skin 208 thickness of Pb and the neutron-skin thickness of two 0.2 doubly-magic neutron-rich nuclei: stable 48Ca and un- Ca] (fm) r 132 stable Sn. We are well aware that parity-violating ex- 48 periments on radioactive nuclei are unlikely to happen in [ the foreseeable future. Yet parity-violating experiments skin 0.16 on stable targets may serve to calibrate experiments with r hadronic probes that, while still challenging, could even- Cmodels =0.852 tually be used to extract the neutron radius of short lived 0.12 AB systems such as 132Sn. 0.12 0.16 0.20 0.24 0.28 In Fig. 2a we display model predictions for the neutron- 132 208 skin thickness of Sn as a function of the corresponding rskin [ Pb] (fm) 208 rskin in Pb. The displayed correlation is both strong and fairly model independent. Indeed, the correlation FIG. 2: (Color online) Predictions from the 48 nuclear EDFs coefficient obtained by considering all 48 EDFs is—as in used in the text for the neutron-skin thickness of 208Pb and the case of the systematically varied forces listed in Ta- 132Sn (a) and 48Ca (b). Constrains on the neutron-skin thick- ble I—very close to unity. This suggests that new exper- ness from PREx [3] have been incorporated into the plot. 132 imental information on rskin in Sn will not provide additional constraints on the theoretical models used in the present work, provided that an accurate measurement of both 48Ca and 208Pb—and incorporating the recent mea- 208 208 the neutron-skin thickness of Pb may be carried out. surement of αD in Pb—one should be able to signifi- Averaging our results, a theoretical estimate for rskin in cantly constrain the isovector sector of the nuclear EDF. 132 48 Sn of (0.232±0.022) fm is obtained with Set-25 and of The theoretical model-averaged estimate for rskin in Ca (0.254±0.016) fm with Set-10. In addition, we predict a is (0.176±0.018) fm for Set-25 and (0.189±0.017) fm for 3 3 value of (10.081±0.150) fm for αD using Set-25. Set-10. Moreover, a prediction of (2.306±0.089) fm for 48 The situation for the case of the neutron-skin thick- αD in Ca is obtained with Set-25. ness in 48Ca shown in Fig. 2b is different. Whereas In summary, we have examined the correlation be- the correlation coefficient among the three systematically tween the electric dipole polarizability and neutron-skin varied models (NL3/FSU, DD-ME, and Skyrme-SV) re- thickness of 208Pb using a large ensemble of 48 reason- mains close to unity (see Table I) there is a significant able nuclear energy density functionals. Physical argu- spread in the predictions of all 48 models that is driven ments based on a macroscopic analysis suggest that these primarily by the traditional Skyrme forces. This sug- two isovector observables should be correlated, although 208 gests that an accurate measurement of rskin in Pb this correlation may display some systematic model de- 48 is not sufficient to significantly constrain rskin in Ca. pendence. In fact, we have found that as accurately- Conversely, by measuring the neutron-skin thickness of calibrated models are systematically varied around their 4

208 132 48 αD[ Pb] rskin[ Sn] rskin[ Ca] model model model Model CAB Slope Intercept CAB Slope Intercept CAB Slope Intercept Skyrme 0.9959 29.0847 15.5290 0.9992 1.0568 0.0555 0.9768 0.5989 0.0798 DD-ME 0.9939 31.9907 14.5206 1.0000 1.0575 0.0500 0.9997 0.5272 0.0849 NL3/FSU 0.9941 29.8864 13.9692 0.9999 1.0429 0.0547 0.9868 0.5028 0.0897

TABLE I: Least-square correlation coeﬃcient, slope (in fm2), and intercept (in fm3) between various observables and the neutron-skin thickness of 208Pb for the various systematically-varied models considered in the text. The dipole polarizability is expressed in units of fm3 while the neutron skins in units of fm. Slope and intercept are obtained by ﬁtting a straight line through the data.

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