Pb – Pygmy Dipole Resonance, Dipole Polarizability, Neutron Skin and Symmetry Energy

Electric Dipole Response of 208Pb – Pygmy Dipole Resonance, Dipole Polarizability, Neutron Skin and Symmetry Energy –

A. Tamii1, I. Poltoratska2, P. von Neumann-Cosel2,Y.Fujita1,T.Adachi1,C.A.Bertulani3,J.Carter4, M. Dozono5,H.Fujita1, K. Hatanaka1,D.Ishikawa1,M.Itoh6,Y.Kalmykov2, T. Kawabata7, A.M. Krumbholz2,E.Litvinova8,9,H.Matsubara5, K. Nakanishi5, R. Neveling10,H.Okamura1,H.J.Ong1, B. ¨ozel-Tashenov8, V.Yu. Ponomarev2,A.Richter2,11, B. Rubio12, H. Sakaguchi1,Y.Sakemi6, Y. Sasamoto13, Y. Shimbara1,Y.Shimizu5,F.D.Smit10, T. Suzuki1,Y.Tameshige14,J.Wambach2,R.Yamada15, M. Yosoi1 and J. Zenihiro5

1Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, Japan 2Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany 3Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, Texas 75429, USA 4School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa 5RIKEN Nishina Center, Wako, Saitama 351-0198, Japan 6Cyclotron and Radioisotope Center, Tohoku University, Sendai 980-8578, Japan 7Department of Physics, Kyoto University, Kyoto 606-8502, Japan 8GSI Helmholtzzentrum für Schwerionenforschung, D-64291 Darmstadt, Germany 9Institut für Theoretische Physik, Goethe-Universität, 60438 Frankfurt am Main, Germany 10iThemba LABS, Somerset West 7129, South Africa 11ECT*, Villa Tambosi, 1-38123, Villazzano (Trento), Italy 12Instituto de Fisica Corpusular, CSIC-Universidad de Valencia, E-46071 Valencia, Spain 13Center for Nuclear Study, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan 14National Institute of Radiological Sciences, Chiba 263-8555, Japan 15Department of Physics, Kyushu University, Fukuoka 812-8581, Japan

Abstract The electric dipole (E1) response of 208Pb has been precisely determined by proton inelastic scattering measurement at very forward angles. The data are quite important for studies of low-lying dipole strength, often called pygmy dipole resonance (PDR), as well as for extracting sum-rule values of the E1 reduced transition probability B(E1). The electric dipole polarizability, which is deﬁned as an inversely energy weighted sum-rule of B(E1), is much focused because of its close correlation to the neutron skin thickness and the symmetry energy of the nuclear equation of state. The determination of the symmetry energy is important for nuclear physics as well as studies of astrophysics e.g. properties and cooling process of a neutron star, X-ray burst, superburst, supernovae, and nucleosynthesis.

1 Introduction

Electric dipole (E1) response of nuclei is one of the most fundamental responses of nuclei to external fields. Historically the study of the E1 response has been initiated by Migdal with the prediction of its mean excitation energy.[1] The work influenced the studies of nuclear collective excitations and giant resonances through the discovery of the isovector electric giant dipole resonance (IVGDR or shortly GDR) [2] and theoretical work on incompressible hydro-dynamical model by Goldhaber and Teller.[3] In 1960s and 70s E1 responses in the GDR region have been systematically measured by using (γ,xn) reactions with photon beams produced by positron annihilation in flight method. The data have been compared with the Thomas-Reiche-Kuhn (TRK) sum-rule, which is an energy weighted sum-rule of B(E1). The E1 response below the neutron separation energy (Sn)has been measured by nuclear resonance fluorescence (NRF). The photon beam was produced by bremsstrahlung, tagged photon, or laser backward Compton scattering method. Recently a concentration of E1 strengths around the neutron separation energy has been reported for several heavy nuclei and neutron rich unstable nuclei. The strength are called low-lying dipole strength. Several theoretical models have predicted similar E1 strength concentration around the neutron separation energy, which is called pygmy dipole resonance (PDR), and have drawn much attention in relation to the observed low-lying E1 strength. The PDR is explained as a collective dipole oscillation of neutron skin against isospin- saturated core, in contrast that the GDR is explained as a collective dipole oscillation between neutrons and protons. The PDR strength may have close relation to the neutron skin thickness of the nucleus. In addition, since the PDR is located around the neutron separation energy, its strength distribution can have significant influence on the prediction of the nucleosynthesis in stars through the equilibrium between the (γ,n)and(n, γ) reactions at finite temperature.[4] Electric dipole polarizability, or shortly dipole polarizability, is another promising quantity to study the neutron skin thickness and symmetry energy of the nuclear equation of state. The work by P.-G. Reinhard and W. Nazarewicz has shown that the neutron skin thickness and the dipole polarizability of 208Pb has very strong correlation to each other.[5] Since the dipole polarizability can be determined by an inversely energy weighted sum-rule value of B(E1), one needs to measure the complete E1 strength distribution including the neutron separation energy region up to high excitation energy. The symmetry energy is essential to predict the mass, radius, and internal structure of a neutron star, supernova explosion dynamics, evolution of neutron star binary systems, mechanism of X-ray burst and superburst, neutron star cooling, and nucleosynthesis. Thus determination of complete E1 strength distribution is quite important for nuclear-astrophysics as well as nuclear physics. In this background, we have developed an experimental technique for measuring proton inelastic scattering at zero degrees and forward angles. The method is excellent to experimentally determine the complete B(E1) strength distribution including the PDR, GDR, and dipole polarizability. In this article, we report the experimental method and recent results on the 208Pb nucleus.

2 Proton Inelastic Scattering as a New Probe of E1 Responses: Comparison to Real-Photon Measurements

We have developed proton inelastic scattering measurement as a new probe to study the E1 response of nuclei. The overview of the E1 response of heavy nuclei is shown in Fig. 1. Historically the E1 response have been studied mainly by real-photon beam experiments. Figure 2 shows the difference between the two methods. In the upper panel, real photon is used as a beam to excite the target nucleus. A decay gamma-ray or a neutron (or a few neutrons) is detected. In this method, only a partial width, corresponding to the observed decaying channel, is measured. Thus all the decay channels must be measured to determine the total width. The flux of the photon beam decreases with the energy and background events from surrounding material or shields become larger. In the gamma decay measurement of NRF, high-purity germanium detectors are usually used for accurate measurement of the gamma energy. Since the photo-peak is used to measure the gamma energy, the detection efficiency decreases rapidly as the gamma energy increases (usually less than 1%). The gamma-ray cannot be observed if each strength is too small or the level density is too high. In addition, direct decay to the ground state is assumed in the analysis of the strength. The assumption might be reasonable but for accurate determination of the strength, cascade decay contribution must be corrected relying on a statistical decay simulation. The problem is similar to the Pandemonium effect in the beta-decay analysis. For the neutron decay channel, neutron detection efficiency is quite low at several MeV region. Usually the emitted neutrons are themalized by using material and the number of the emitted neutrons is measured. The method allows detection of multiple neutrons but the energy information of the neutrons is lost. Thus the E1 strength data can be obtained for each of the beam energy windows, which is typically around 300 keV. Proton decay channel is usually neglected. For real-photon measurements, one needs several grams of isotopically enriched target, that also limits the applicability of the measurements for specific target nuclides. The E1 strength measurement by inelastic proton scattering measurement is shown in the lower panel of Fig. 2. The target nucleus is bombarded by a proton and is excited by Coulomb interaction with exchanging a virtual photon. The scattered proton is detected. The method is probing only the excitation process and is independent of the decay process. Thus the total E1 excitation strength is observed. One of the essences of the method is to measure the reaction at intermediate energy (300 MeV) and at small scattering angles close to zero degrees, where Coulomb excitation is dominant. The detection efficiency of the scattered proton is close to 100%, and is uniform over the measured excitation energy range of 5-25 MeV. The range well covers the neutron separation energy and the GDR region for most of stable nuclei. Since the proton beam flux is much higher than real gamma-beam, only several mili-grams of enriched target is required. An energy resolution of 20-30 keV is achievable. Properties of the proton inelastic scattering measurement are summarized in comparison to real-photon measurements. • Total E1 strength is measured. The measurement is independent of the decay process or neutron separation energy. No correction of cascade decay or feeding is required. • E1 strength distribution is measured with a uniform and high (∼100%) detection efficiency over the excitation energy range of 5-25 MeV, that covers neutron separation energy and GDR. p p ( , ’) NRF or (γ, xn) NRF (γ, xn) γ detector PDR GDR γ (or xn) core pn skin A * A A (or A-x)

(p,p’ ) detector p Strength (Arbitrary) p E1 virtual photon 4 S 8 S 12 16 n p * Excitation Energy (MeV) A A

Figure 1: Illustration of E1 response of heavy nuclei. Time The pygmy dipole resonance (PDR) is located around the neutron separation energy (Sn). Historically nu- Figure 2: (upper panel) Principle of the E1response clear resonance ﬂuorescence (NRF), sometimes called measurement by a real-photon beam. Decay products, (γ,γ), has been used to study the E1 response be- γ-rays for NRF and neutrons for (γ,xn), are detected. low Sn while (γ,xn)(wherex denotes the number of (lower panel) Principle of the E1 response measure- neutrons) measurements have been used to study the ment by (p, p). The measurement is insensitive to E1 response above Sn. The proton inelastic scattering the decay channel since only the scattered proton is (p, p) measurement covers both the regions. detected.

• The measurement is applicable for several mili-grams of isotopically enriched target, while real-photon measurement requires several grams of target. Thus most of stable nuclides can be used as a target. • E1 strength in the high level-density region or in the continuum can be measured. If the summed strength is sizable, fragmented tiny strengths are also measurable. • Energy resolution of 20-30 keV is achievable although gamma decay measurement has better resolution of 3-7 keV. • Since the Coulomb excitation process is well-known, absolute E1 strength can be extracted from the measured diﬀerential cross section without uncertainty in the reaction model. Thisisthesameforreal- photon measurement.

3 Experimental Method

The experiment has been performed at the cyclotron facility of the Research Center for Nuclear Physics (RCNP), Osaka university. Details of the experimental setup, conditions, and analysis procedures can be found in references.[6, 7, 8, 9] A polarized proton beam at 295 MeV has been accelerated with a beam intensity of 2- 10 nA and a polarization of 0.7. An isotopically enriched 208Pb foil with a thickness of 5.2 mg/cm2 has been used as a target. The excitation energy resolution was 30 keV. In the polarization transfer measurement, two different focal planes [10] of the Grand Raiden spectrometer [11] have been used. The first one is the standard focal plane for which the magnetic field of the third dipole magnet, dipole for spin rotation (DSR), has not been applied as shown in the left panel of Fig. 3. The second one is the focal plane of the DSR plus mode in which the DSR magnetic field was applied to bend scattered protons by 18 degrees in addition to the bending by the D1 and D2 dipoles as shown in the right panel of Fig. 3. The focal plane detectors and the zero-degree beam dump were replaced for each of the focal plane modes. In both modes the primary beam which passed through the target is transported in the Grand Raiden spectrometer, extracted at the focal plane, and stopped in a beam dump. Polarization axis of the beam was controlled by using two superconducting solenoids which are placed in the beam line between the injector and the main cyclotrons. A sideway (S) polarization mode, in which the polarization axis is perpendicular to the beam direction and in the horizontal plane has been used in combination with the standard focal plane for the measurement of the polarization transfer coefficient DSS . A longitudinally polarization mode, in which the polarization axis is in parallel to the beam direction has been D2 D2

MP MP

60° 60°

D1 D1

DSR Q1-FC DSR Q1-FC Q2 Q2 Focal Plane Detectors SX D Focal Plane Detectors SX D Q1 Q1

Scattering Scattering Chamber Q Chamber Q Focal Plane Detectors Focal Plane Detectors Focal Plane Polarimeter Halo Monitor Beam Viewer Focal Plane Polarimeter Halo Monitor Large Acceptance Dump-Q o Large Acceptance Spectrometer (LAS) 0 B eam Dump Beam Viewer Spectrometer (LAS) Polarized protons Polarized protons

Ep = 295 MeV Ep = 295 MeV Grand Raiden Grand Raiden (GR) 0 3 m (GR) 0 Bo eam Dump D Measurements SS DLL Measurements

Figure 3: The experimental setup of the Grand Raiden spectrometer in the zero degree measurement. Two setups are used for DSS measurement (left panel) and DLL measurement (right panel).

used in combination with the DSR plus mode for the measurement of DLL . Polarization transfer coefficients were measured in an angular range of 0-2.5 degrees. Differential cross sections were measured in an angular range of 0-6 degrees. The measured excitation energy range is 5-25 MeV. Further information can be found in Ref. 9 The upper panel of Fig. 4 shows the excitation energy spectrum of the 208Pb(p, p) reaction at 0.0-2.5 degrees. The GDR is clearly seen as a bump structure centered at 13 MeV with fine structures in the lower energy tail. Many discrete states were observed below the neutron separation energy (Sn=7.368 MeV). They are mostly E1 states. As is shown in Fig. 5, all the E1 states known by NRF have been observed. Electric dipole reduced transition probability, B(E1), has been extracted from the (p, p) cross section at 0.0-0.94 degrees for each of the discrete E1 transitions. Note that absolute value of B(E1) can be determined by the (p, p)datasincetheE1 state is excited dominantly by Coulomb interaction which has essentially no model uncertainty. The extracted B(E1) is excellently consistent with the data of NRF below Sn.AboveSn strength more than known has been observed by the present method.

4 Determination of the E1 Strength Distribution

In order to extract B(E1) strength in the continuum region, decomposition of E1andM1 strength is necessary. We have applied two independent methods. The first method uses spin transfer data as described below. A quantity, name total spin transfer (Σ), is defined as ◦ ◦ 3 − (2DSS (0 )+DLL (0 )) Σ(0◦)= . (1) 4 The total spin transfer takes a value of unity for excitations with spin transfer ΔS=1, i.e. spin-M1 excitation by nuclear interaction, and takes a value of zero for excitations with ΔS=0, i.e. E1 excitation by Coulomb interaction.[12] The relation holds model-independently at zero degrees provided the parity conservation. The measured total spin transfer is shown in the lower panel of Fig. 4. The GDR bump region is dominantly composed of E1 strengths. A concentration of spin M1 strengths is observed in the 7-8 MeV region as is reported in polarized real-photon studies.[13, 14] As an alternative and independent way of decomposing the E1andM1 components, multipole-decomposition analysis (MDA) has been done as described below. Angular distribution shapes for E1, M1, and higher- multipolarities were calculated by distorted wave impulse approximation (DWIA) by using the code DWBA07 [15] and Franey and Love effective interaction.[16] Several RPA amplitudes and single particle wave functions for each excitation mode were calculated by the quasi-particle phonon model (QPM).[17] For each energy bin, the best mixture of the E1, M1, and other target wave functions were determined by the least-square fit method so as to reproduce the experimental angular distribution. Two representative angular distributions and the result of the MDA are shown in Fig. 6. 80 γ γ )

-1 208 Pb(p,p') 400 γ

MeV Ep = 295 MeV -1

θ = 0.0−2.5 ) ˚ 2 40 200 fm dE (mb sr 2 Ω e /d σ -3 2 d 0 0 Δ 1 S=1 (10 400 Σ χ Θ 0 ΔS=0

B(E1) 200

-1

Total Spin Transfer Transfer Spin Total 0 5101520 5678 Excitation Energy (MeV) Excitation Energy (MeV)

Figure 4: Measured double diﬀerential cross sections Figure 5: Electric dipole reduced transition proba- (upper panel) and total spin transfer (lower panel) of bility, B(E1), of low lying states determined by the 208 the Pb(p,p’) reaction at Ep=295 MeV and at 0-2.5 (p, p) experiment (lower panel) and real-photon mea- degrees. surements (upper panel).

The decompositions of spin-ﬂip (ΔS=1) and spin-non-ﬂip (ΔS=0) cross sections are compared in Fig. 7 between the polarization analysis and the MDA. The agreement between the two independent methods is satisfactory within the experimental uncertainty.

80 208 208 Ex = 7.26-7.37 MeV (a) Pb(p,p´) (b) Pb(p,p´) E = 295 MeV 40 E = 295 MeV 1 p p Θ = 0° - 2.5° Θ = 0° - 2.5° 40 20

-1 10 Total E1 M1 0 0 d d / (mb/ sr) Multipole Decomposition Multipole Decomposition E2 Polarization Observables ΔS=1 10 Polarization Observables ΔS=1 0 24640 lab (deg) 5

1 Ex = 7.37-7.41 MeV 20 0 dE (mb / sr MeV) (mb dE dE (mb / sr MeV) (mb dE Ω

0 Ω -5 / d -1 σ ΔS=0 / d ΔS=0 10 2 σ d 120 2 40

Total d E1 M1 80

d d / (mb/ sr) 20 10-2 40 0246 (deg) lab 0 0 56789 10 12 14 16 18 20 Figure 6: Angular distribution Excitation Energy (MeV) Excitation Energy (MeV) of the cross sections for two excitation energy bins. E2com- Figure 7: Comparison between two independent decomposition methods, mul- ponent is negligible in the lower tipole decomposition and polarization analysis, for lower (higher) excitation panel. energy region in the left (right) panel.

The photo-absorption cross section in the GDR region has been extracted by MDA as shown in Fig. 9 by employing semi-classical description of Coulomb excitation.[18] The result is consistent with the data of (γ,xn) measurement [19] and total photo-nuclear cross sections by using tagged-photon method.[20] The overall B(E1) distribution determined by the (p, p) is shown in Fig. 9. The bump centered at ∼13 MeV corresponds to the GDR, and the strength concentration at around 7-9 MeV corresponds to the PDR. Now we have obtained an experimental technique to accurately determine the full B(E1) strength distribution of stable nuclei from low (∼5MeV)tohigh(∼20 MeV) excitation energies which fully cover the PDR and GDR.

208Pb(γ,xn) 208 γ 600 Pb( ,all) 208Pb(p,p')

400 (mb)

abs σ 200

0 8 101214161820 Excitation Energy (MeV)

Figure 8: Comparison of photo-absorption cross sections determined by (p, p)(circle),(γ,xn) [19] (solid line) and tagged gamma-absorption [20] (square) measurements in the GDR region of 208Pb.

B(E1)↑ dB(E1)↑/dEx ) x 2 0.4 GDR

fm PDR 2

0.2 /0.2 MeV) 2 fm dB(E1)↑/dE 2 B(E1)↑(e 0.0 (e 5 101520 Excitation Energy (MeV)

Figure 9: The measured B(E1) distribution of 208Pb. The bump centered at ∼13 MeV corresponds to the giant dipole resonance, and the strength concentration at around 7-9 MeV corresponds to the low-lying dipole strength or often called pygmy dipole resonance.

5 Pygmy Dipole Resonance

The developed experimental technique is suitable for studying PDRs in stable nuclei. Since the PDR is described as a dipole oscillation between the neutron skin and the isospin-saturated core, it is natural to expect that the information on the neutron skin thickness can be extracted from it. Actually several theoretical models predict good correlation between the energy weighted sum-rule of B(E1) in the PDR region and the neutron skin thickness.[21, 22] On the other hand poor relation is also predicted in other theories.[5] The situation is still not clear. Probably, the degree of the correlation depends on the nuclear structure, valence single-particle orbits, deformation, neutron binding energy and the N/Z ratio [23]. There is also much discussion on the property of PDRs, such as collectivity, isospin structure,[24] and relation to the soft-dipole resonance in light nuclei (see a review article of Ref. 25). Thus it is quite important to make a systematic study of PDRs over many kinds of representative nuclei. We have already started measurements on several nuclei: 120Sn as one of semi-magic heavy nuclei,[26] N=50 isotones (88Sr, 90Zr, 92Mo),[27, 28] 154Sm and 144Sm for studying the deformation effect,[29] 70Zn as an isotone of 68Ni and 130Te as an isobar of 130Sn. Systematic measurements on tin isotopes would also be important since there exist PDR data of unstable isotopes of 130,132Sn [30] and we may be able to see the PDR systematics in a long isotope-chain. For deformed nuclei, it is well-known that the GDR splits into two bumps which naively corresponds to the dipole oscillation in parallel and perpendicular direction to the deformation axis. It is quite interesting to see whether similar splitting also happens for PDRs. One of the issues that is making the discussion on PDRs difficult is that the way of integrating the PDR strength is not well-defined. Most of the theoretical and experimental works are defining the PDR strength as asumoftheB(E1) strength from low to some specific energy, usually 10 MeV. The strength of the pure PDR (neutron skin oscillation mode) must be, however, spreading into many E1 states with mixing to GDR, isoscalar giant dipole resonance (ISGDR), and low-lying single-particle single-hole states. Therefore ultimately one needs to study the transition density distribution of each excited states. One of such experimental trials, angular distribution of the 208Pb(p, p) reaction has been analyzed with MDA by including both the PDR and GDR type transition densities. The result look favoring the PDR type transition density for the strengths around the PDR region of 7.0-8.2 MeV, and favoring the GDR type transition density above 8.2 MeV. This kind of detailed study of the transition density of, if possible, individual excited state would be important in near future.

6 Dipole Polarizability and Neutron Skin Thickness

Since the full B(E1) strength distribution is measured, sum-rule values can be deduced. The energy weighted sum rule of B(E1), called Thomas-Reiche-Kuhn (TRK) sum rule, is one of the well-known sum rules. Here we introduce another sum rule, called electric dipole polarizability or shortly dipole polarizability (αD), which is deﬁned as an inversely energy weighted sum rule of B(E1). ¯hc σabs 8π S(ω) αD = dω = dω, (2) 2π2 ω2 9 ω where ω stands for the excitation energy, σabs for the photo-absorption cross section, and S(ω)=dB(E1)/dω for the E1 reduced transition probability per unit excitation energy. The dipole polarizability sum-rule up to 20 MeV has been determined as 18.9±1.3 fm3 from the (p, p) data. By taking the average of the independent data in the GDR region and by adding the value in the 20-130 MeV region measured by gamma absorption, the dipole polarizability up to 130 MeV has been determined as 20.1±0.6 fm3.

23 ) 3 SGII SkO NL3 ) 3 22 22 SkM* SkI3 Sk255 21 UNEDF0 21 SLy4 DD-ME2 SV-min 20 20.1s0.6 SIII SkP SkI4 20 FSU 20.1±0.6 Bsk17 SLy6 UNEDF1 19 SkX 19

18 0.156+0.025-0.021 0.168±0.024 Dipole Polarizability (fm

Dipole Polarizability (fm 18 17 0.10 0.15 0.20 0.25 0.30 0.12 0.14 0.16 0.18 0.20 0.22 Neutron Skin Thickness (fm) Neutron Skin Thickness (fm)

Figure 10: Correlation between the dipole polarizabil- Figure 11: Correlation between the dipole polariz- ity and the neutron skin thickness of 208Pb calculated ability and the neutron skin thickness of 208Pb cal- by a self-consistent mean ﬁeld mode using the SVmin culated with various parameter sets. The predictions Skyrme interaction in the framework of energy-density are based on the work by J. Piekarewicz et al.[31, 32] functional method.[5] The neutron skin thickness has The neutron skin thickness has been extracted as been extracted as 0.156+0.025-0.021 fm from the mea- 0.168±0.024 fm by taking weighted average of the the- sured dipole polarizability of 20.1±0.6 fm3.[7] oretical predictions.

Since the dipole polarizability corresponds to the static dipole polarization of a nucleus in a uniform external electric field, it is sensitive to the difference of the density distribution between neutrons and protons, as well as the density dependence of the symmetry energy (described in Sec. 7) which is the restoring force of the polarization. P.-G. Reinhard and W. Nazarewicz have reported a very strong correlation between the dipole polarizability and the neutron skin thickness of 208Pb by a self-consistent mean field calculation with the energy- density functional method using the SVmin Skyrme interaction.[5] The correlation is plotted in the left panel of Fig. 10. If we use the calculation, the measured dipole polarizability can be converted to the neutron skin thickness of 208Pb as 0.156+0.025-0.021 fm.[7] Soon after the work, model dependence of the correlation has been studied for various sets of model parameters.[31] The result is shown in the right panel of Fig. 11, but here we only pick up one parameter set for each of the interaction families of NL3, FSU, DD-ME, and Skyrme-SV. The neutron skin thickness of 208Pb is determined as 0.168±0.24 fm by taking average of theoretical predictions by applying a Gaussian weight function of the deviation of the predicted dipole polarizability from the experimental data of 20.1±0.6 fm3. PREX antiprotonic atom p elastic scatt. (650 MeV) p elastic scatt. (295 MeV) dipole polarizability

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 Neutron Skin Thickness of 208Pb (fm)

Figure 12: The neutron skin thickness of 208Pb measured by parity-violating electron scattering (PREX),[34, 35] analysis of the systematic of the X-ray detection data of anti-protonic atoms,[36] proton elastic scattering at 650 MeV [37] and 295 MeV,[33] and the dipole polarizability (this work).[7, 31, 32]

The accuracy of the determined neutron skin thickness is remarkably good among many experimental works on it.[33] The result is compared with other experimental works in Fig. 12. Each experimental data has its own model uncertainty, which is often diﬃcult to estimate. In our case highly accurate data could be extracted by measuring the B(E1) distribution with an electro-magnetic probe for which the reaction mechanism is accurately known, and by applying a well-deﬁned sum rule value which can reduce the model uncertainty. The procedure to evaluate the model uncertainty by using various sets of model interactions is also relatively well-established.

As an ideal model-independent determination of the neutron skin thickness of 208Pb, parity-violating asymmetry of electron elastic scattering has been measured at Jeﬀerson Laboratory.[34] They have used weak interaction, which is also known accurately, to probe the from factor of the neutron density distribution to extract the neutron skin thickness. Although the measurement is quite important and promising, the present result, 0.302±0.175(exp)±0.026(model)±0.005(strange),[35] has large statistical uncertainty. An improved measurement for getting better statistics is scheduled. Combination of the dipole polarizability data and possible accurate measurement of the neutron skin thickness can best constrain model parameters.

7 Density Dependence of the Symmetry Energy

One of the reasons why the neutron skin thickness of 208Pb is drawing much attention is that the thickness has almost linear relation to the slope parameter of the symmetry energy of the nuclear equation of state. The nuclearequationofstatecanbeexpressedbyanequationofenergyperparticle(E/A) written as a function of nucleon density (ρ) and asymmetry parameter (δ).[38]

E/A(ρ, δ)=E/A(ρ, 0) + S(ρ)δ2 + ··· (3) L Ksym 2 S(ρ)=S0 + (ρ − ρ0)+ 2 (ρ − ρ0) + ··· (4) 3ρ0 18ρ0 ρ ≡ ρn + ρp (5) ρn − ρp δ ≡ , (6) ρn + ρp

−3 where ρn and ρp are neutron and proton densities, respectively, and ρ0(∼ 0.16 fm ) is the saturation density of the symmetric (δ = 0) nuclear matter. S(ρ) is called symmetry energy and represents asymmetry dependence of the energy. The first order density dependence of S(ρ) is parameterized by slope parameter L.Theslope parameter is proportional to the baryonic pressure in a neutron star and is approximately proportional to the fourth of the neutron star radius. Thus the determination of the slope parameter is quite important to calculate the properties of a neutron star, e.g. radius, internal structure, and cooling process as well as the evolution of neutron star binaries, X-ray burst, superburst, supernova explosion dynamics, and nucleosynthesis. Due to strong correlation between the neutron skin thickness of 208Pb and the slope parameter, the slope parameter can also be constrained by the dipole polarizability data. The correlation is shown in Fig. 13. By using a procedure similar to the case of the neutron skin thickness, the slope parameter can be determined as L=45±18 MeV. Similarly the constant term of the symmetry energy has been determined as S0=30.9±1.5 MeV. The numbers are preliminary. The result is plotted in Fig. 14 as well as the constraints from other studies: analysis heavy ion collision (HIC), pygmy dipole resonance (PDR), isobaric analog states (IAS), nuclear mass formula with finite range droplet model (FRSM), estimation from neutron star observations (n-star), and prediction by a chiral effective field theory (χEFT).[39, 40] Although each result has its own complicated model dependence, the results of various works look consistent to each other and the allowed region is now becoming limited. According to the report by J.M. Lattimer and Y. Lim, the neutron star radius might be determined in 11-12 km for relatively large mass range of 1-2 solar mass.[41]. However much discussion is still on going. The slope parameter determined by the dipole polarizability belongs to a group of relatively soft nuclear equation of state. On the other hand, recent observation of a neutron star of two solar mass [42] favors harder equation of state at larger density. Nuclear models need to satisfy both the requirements.[43, 44]

23 ) 3 SGII SkO NL3 100 IAS 22 HIC SkM* SkI3 Sk255 UNEDF0 21 SV-min DD-ME2 SIII SkP SLy4 FRDM

SkI4 PDR

20 UNEDF1 (MeV) 208 FSU Pb L Bsk17 20.1±0.6 SLy6 50 DP SkX DD-ME1(a4=32) 19 χEFT n-star 45±18

Dipole Polarizability (fm 18 0 50 100 150 28 30 32 34 L (MeV) S0 (MeV)

Figure 13: Correlation between the dipole polarizabil- Figure 14: Constraints on the slope parameter (L)and 208 L ity of Pb and the slope parameter ( )ofthesym- the constant term (S0) of the symmetry energy of the metry energy of the nuclear equation of state.[31, 32] nuclear equation of state. The plot is based on the The value of L has been extracted as 45±18 MeV by work by B.M. Tsang et al.,[38] I. Tews et al. [38] and taking weighted average of the theoretical predictions. this work. See the text for details.

For making further constraints on model parameters, dipole polarizability data for other nuclei are also important. One of good candidates is 48Ca, for which theoretical predictions of the neutron skin thickness are scattered and are relatively uncorrelated with that of 208Pb. Proton inelastic scattering data on 48Ca is already taken.[45] Analysis of the dipole polarizability is in progress. Note that parity-violating electron scattering measurement on 48Ca is also planned at Jeﬀerson Laboratory to extract its neutron skin thickness. In addition dipole polarizability data on various target nuclei, listed in Sec. 6, will be able to be extracted from the same measurement.

8 Summary

Thecompleteelectricdipole(E1) strength distribution in 208Pb has been determined up to 20 MeV by high- resolution proton inelastic scattering measurement at very forward angles. Multipole components in the differential cross section data have been decomposed by two independent methods: spin transfer analysis and multipole decomposition analysis. The full E1 strength distribution has been firstly obtained over a broad excitation energy range covering the pygmy dipole resonance and the giant dipole resonance. The developed experimental technique is suitable to study the systematics of pygmy dipole resonance and electric dipole polarizability of stable nuclei. The electric dipole polarizability of 208Pb up to 130 MeV was determined as 20.1±0.6 fm3 by combining the present data and other available data. The precise data constrain the neutron skin thickness of 208Pb as 0.168±0.024 and the slope parameter of the symmetry energy as L=45±18 MeV with a help of theoretical model calculations. The accuracy is quite good and would make a strong constraint on model parameters and the prediction of neutron star properties, supernova explosion dynamics, and many other interesting astrophysical predictions. Beyond these results we note that the experimental data can also be applied for extracting the spin-M1 strength distribution,[46] fine-structure and level density of the E1strengthbyusing statistical methods,[47] and for studying the radiative gamma strength and the Axel-Brink hypothesis.[48] Acknowledgements

We are indebted to the RCNP cyclotron accelerator staﬀ and operators for providing us an excellent beam. This work was supported by JSPS (Grant No. 07454051 and 14740154) and DFG(Contracts No. SFB 634 and No. 446 JAP 113/267/0-2).

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