陽子非弾性散乱による E1 応答測定と中性子スキン A. Tamii Research Center for Nuclear Physics (RCNP), Osaka University RIBF 討論会 at 北海道大学 February 21, 2013 安定核の B(E1) 分布を測る

Outline 1. Electric Dipole Response, Neutron Skin and Symmetry Energy Dipole Polarizability, Pygmy Dipole Resonance) 2. Experimental Method proton inelastic scattering at forward angles 3. Results 4. Summary

Electric Dipole Response, Neutron Skin and Symmetry Energy

SnSn SpSp Particle （ neutron ） separation energy 0 PDR? IVGDR g.s. oscillation of neutron skin against core? oscillation between neutrons and protons E1 1 - core neutron skin Low-Lying Dipole Strength Electric Dipole (E1) Response

SnSn SpSp ( ,xn) GR and Continuum (Main Strength) Discrete (Small Strength) Particle （ neutron ） separation energy (p,p’) 0 PDR IVGDR g.s. 208 Pb( ,  ) M1 strength measured by R.M. Laszewski et al, PRL61(1988)1710 Electric Dipole (E1) Response ( ,  ’) NRF

Coulomb Excitation at 0 deg. Excited State Target Nucleus Real Photon Measurements, NRF and ( ,xn) Probing EM response of the target nucleus Decay  -rays or neutrons are measured. Select low momentum transfer (q~0) kinematical condition, i.e. at zero degrees Excited State Target Nucleus Missing Mass Spectroscopy with Virtual Photon Insensitive to the decay channel. Total strengths are measured. Only the scattered protons are measured. EM Interaction is well known (model independent) q,q,

P.-G. Reinhard and W. Nazarewicz, PRC 81, (R) (2010). Self-consistent mean field calc. based on the energy density functional theory using Skrym SV-min interaction. - SV-min parameters were determined to reproduce binding energies, r.m.s. radii, pairing gap, ls-splitting, surface thickness, etc. Correlation between the Neutron Skin Thickness and Dipole Polarizability

(Electric) Dipole Polarizability External Field Restoring Force Balanced Displacement Dipole Polarizability

My “very simplified picture” of the correlation between neutron skin thickness and dipole polarizability One dimensional nucleus without surface diffuseness E1 larger skin thickness smaller restoring force E1 larger dipole polarizability With finite diffuseness the relation becomes smoother.

(Electric) Dipole Polarizability = inversely energy weighted sum-rule of B(E1)  : excitation energy

X. Roca-Maza et al., PRL106, (2011) 208 Pb Correlation between the Neutron Skin Thickness of 208 Pb and the Slope Parameter (L)

Nuclear Equation of State (EOS) K sym is discussed from studies of e.g. isoscalar giant monopole resonances (ISGMRs). Determination of L is becoming important. Symmetry energy L: Slope Parameter EOS for Energy per nucleon Saturation Density ~0.16 fm -3 (Baryonic Pressure)

Lattimer et al., Phys. Rep. 442, 109(2007) Accreting neutron star/white dwarf, X-Ray burst, Superburst Neutron Star Mass and Radius Neutron Star Internal Structure Core-Collapse Supernova Neutron Star Cooling Nucleosynthesis K. Sumiyoshi, Astrophys. J. 629, 922 (2005) Langanke and Martinez-Pinedo Determination of the Symmetry Energy in Nuclear EOS.

Steiner et al., Phys. Rep (2005) 核子当たりのエネル ギー Nucleon Density (fm -3 ) E/A (MeV) E/N (MeV) Neutron Density (fm -3 ) Nuclear Equation of State (EOS) Neutron Matter (  =1) Prediction of the neutron matter EOS is much model dependent. Neutron Matter (  =1) Symmetry Energy (and Coulomb) Nuclear Matter (  =0) Symmetry Energy 1. pn interaction is stronger than pp and nn interactions 2. occupation of higher orbits due to Pauli exclusion principle

Saturation of Density Short range interaction ~1 fm ~0.16 fm -3 Energy increase at higher-density: 1. density dependence of the tensor interaction. 2. exchange interaction 3. short range repulsive core of the NN interaction Strong attraction due to tensor interaction. Symmetric nuclear matter N=Z 密度の飽和性 SHF RMF

X. Roca-Maza et al., PRL106, (2011) 208 Pb Correlation between the Neutron Skin Thickness of 208 Pb and the Slope Parameter (L)

Neutron density Proton density Neutron rms radius Proton rms radius Density distribution of protons and neutrons in a nucleus My “simple explanation” of the correlation between the neutron skin thickness and the slope parameter (L)

Smaller S  2 at higher  Larger S  2 at lower  larger skin My “simple explanation” of the correlation between the neutron skin thickness and the slope parameter (L)

Larger S  2 at higher  Smaller S  2 at lower  smaller skin

Neutron skin thickness Density dependence of the symmetry energy My “simple explanation” of the correlation between the neutron skin thickness and the slope parameter (L) Larger S  2 at higher  Smaller S  2 at lower  smaller skin Energy minimization = equilibrium condition

Parity Violation at Jefferson Lab PREX, MOLLER, & PVDIS Experiments Thomas Jefferson National Accelerator Facility Robert Michaels Hall A 1/16

PREX at J-Lab: Z 0 of weak interaction : sees the neutrons proton neutron Electric charge 1 0 Weak charge Neutron form factor Parity Violating Asymmetry T.W. Donnelly, J. Dubach, I. Sick C.J. Horowitz NPA503, 589, /16 Model Independent Determination of Neutron Skin Thickness Neutron Skin Thickness Measurement by Electroweak Interaction C. J. Horowitz, S. J. Pollock, P. A. Souder, R. Michaels PRC 63, , 2001 measurement at q=0.475 fm -1

Neutron Skin Thickness Measurement by Electroweak Interaction PREX at J-lab S. Abrahamyan et al., PRL108, (2012)

Neutron Skin Thickness Measurement by Electroweak Interaction PREX PREX Result: S. Abrahamyan et al., PRL108, (2012) Theor. Calc.: X. Roca-Maza et al., PRL106, (2011) Alternative approach at RCNP by using electromagnectic interaction The model independent determination of  R np by PREX and future projects is quite important.

SnSn SpSp Particle （ neutron ） separation energy 0 PDR? IVGDR g.s. oscillation of neutron skin against core? oscillation between neutrons and protons E1 1 - core neutron skin Low-Lying Dipole Strength Electric Dipole (E1) Response

Missing mass measurement. Independent of the decay property of excited states, decay threshold. No feeding from upper excited states Measure of the total (not partial) B(E1). At 0 deg, E1 states are dominantly excited by Coulomb interaction. Coulomb interaction is well known and is model independent. High-resolution (20-30keV). High (~90%) and uniform detection efficiency. Single shot measurement in a broad excitation energy region of 5-25MeV. Required amount of target is small (several mili-gram) Even for fragmented tiny strengths, sum of B(E1) is measurable if it is sizable. E1/M1 decomposition by two methods: polarization transfer and angular distribution of the C.S. (MDA) Proton Inelastic Scattering at Forward Angles

Experimental Method High-Resolution (p,p’) measurement at close to zero degrees AT et al., NIM A605, 326 (2009)

High-resolution Spectrometer Grand Raiden High-resolution WS beam-line (dispersion matching) Research Center for Nuclear Physics, Osaka Univ. ~30 km west from Kyoto

Spectrometers in the 0-deg. experiment setup Intensity : 3 ~ 8 nA As a beam spot monitor in the vertical direction Transport : Dispersive mode Polarized Proton Beam at 295 MeV Focal Plane Polarimeter

Grand Raiden in the 0deg Measurement Setup L R double scattering polarized

Spin Precession in the Spectrometer  p : precession angle with respect to the beam direction  b : bending angle of the beam g: Lande’s g-factor  : gamma in special relativity

GDR PDR region Spectrum

Spectrum at Lower E x

All the E1 excited states below S n known by ( ,  ’) has been observed in (p,p’). The consistency of the extracted B(E1) is excellent. I. Poltoratska, PhD thesis B(E1) of discrete states

 S=1 (M1)  S=0 (E1) Concentration of spin-M1 strength

E1/M1 Decomposition by Spin Observables spinflip / non-spinflip separation* (model-independent) Polarization observables at 0° -1 for  S = 1, M1 excitations 3 for  S = 0, E1 excitations E1 and M1 cross sections can be decomposed T. Suzuki, PTP 103 (2000) 859 At 0° D SS = D NN

Multipole Decomposition Analysis Neglect data at  >4: (p,p´) response too complex Included E1/M1/E2 or E1/M1/E3 (little difference). Phenomenological B.G. at High Ex.

Comparison between the two methods Total  S = 1  S = 0

E1 Photo-absorption Cross Section in the GDR region I. Poltoratska, PhD thesis

E1 Response in 208 Pb Quasiparticle Phonon Model 3 phonons up to 8.2 MeV 2 phonons in the GDR region V.Yu. Ponomarev Relativistic Quasiparticle Time-Blocking Approximation 2QP×1 phonon E. Litvinova et al., PRC 78 (2008) , PRC 79 (2009) This Exp. AT et al., PRL107, (2011)

Relativistic Quasiparticle Time Blocking Approximation Quasiparticle Phonon Model up to 130 MeV 20.1±0.6 fm 3 /e 2 I. Poltoratska, PhD thesis Electric Dipole Polarizability

B(E1) Distribution and Dipole Polarizability of 208 Pb AT et al., PRL107, (2011) ( ,all)

Results

Calc. P.-G. Reinhard and W. Nazarewicz, PRC81, (R) (2010) fm 20.1±0.6 fm 3 /e 2 PREX fm 0.49 proton elastic scattering fm Antiproton Atoms 0.18±0.02 fm J. Zenihiro et al., PRC82, (2010) Friedman and Gal, Phys. Rep. 452, 89 (2007) S. Abrahamyan et al., PRL108, (2012)

Correlation Between Dipole Polarizability and Neutron Skin Thickness J. Piekarewicz, W. Nazarewicz, et al., PRC85, (2012)

Correlation Between Dipole Polarizability and Neutron Skin Thickness

J. Piekarewicz, W. Nazarewicz, et al., PRC85, (2012) Correlation Between Dipole Polarizability and Neutron Skin Thickness

Neutron Skin Thickness Measurement by Electromagnetic Interaction

Neutron Skin Thickness of 208 Pb 0.168±0.022 fm (this work)

Based on the work by X. Roca-Maza et al., PRL106, (2011) DP: Dipole Polarizability L  ±15 MeV Determination of Symmetry Energy

Preliminary Gaussian weight func. L  ±18 MeV Determination of Symmetry Energy

M.B. Tsang et al., PRC86, (2012). I. Tews et al., PRC (2013) DP: Dipole Polarizability HIC: Heavy Ion Collision (Tsang) PDR: Pygmy Dipole Resonance (Carbone) IAS: Isobaric Analogue State FRDM: Finite Range Droplet Model (nuclear mass analysis) n-star: Neutron Star Observation (analysis by Steiner et al.)  EFT: Chiral Effective Field Theory

Preliminary L  45±18 MeV J=30.9±1.5 MeV M.B. Tsang et al., PRC86, (2012). Determination of Symmetry Energy DP: I. Tews et al., PRC (2013) and this work DP: Dipole Polarizability HIC: Heavy Ion Collision (Tsang) PDR: Pygmy Dipole Resonance (Carbone) IAS: Isobaric Analogue State FRDM: Finite Range Droplet Model (nuclear mass analysis) n-star: Neutron Star Observation (analysis by Steiner et al.)  EFT: Chiral Effective Field Theory

Preliminary L  45±18 MeV J=30.9±1.5 MeV M.B. Tsang et al., PRC86, (2012). I. Tews et al., PRC (2013) and this work DP: Dipole Polarizability HIC: Heavy Ion Collision (Tsang) PDR: Pygmy Dipole Resonance (Carbone) IAS: Isobaric Analogue State FRDM: Finite Range Droplet Model (nuclear mass analysis) n-star: Neutron Star Observation (analysis by Steiner et al.)  EFT: Chiral Effective Field Theory Determination of Symmetry Energy DP:

Determination of Symmetry Energy FRDM2011a FRDM2012 ? L=70±15 MeV L=53.5±? MeV P. Möller, JUSTIPEN workshop October, 2012 P. Möller et al., PRL 108, (2012)

A. Carbone PRC81, (R) (2010) Correlation between the TRK sum rule of PDR and L

Application of the PDR : constraints on the symmetry energy Exp. Data: 68 Ni : O. Wieland et al, PRL 102, (2009) 132,130 Sn: A. Klimkiewicz et al., PRC 76, (R) (2007) 208 Pb: I. Poltoratska, PvNC, AT, et al., PRC 85, (R) (2012) Theoretical dependences of pygmy EWSR on J and L are determined using relativistic energy density functionals spanning the range of J and L values. Available experimental data provide constraints on theoretical models. Similar approach but different theory  A. Carbone et al, PRC 81, (R) (2010) DD-ME Slide by N. Paar

Determination of Symmetry Energy ? 208 Pb PDR Energy-Weighted Sum DD-ME Model dependence is probably large.

Lattimer et al., arXiv v1(2012) Constant term of the Symmetry Energy Slope Parameter of the Symmetry Energy Determination of Symmetry Energy Determination of the dipole polarizability of 208 Pb strongly constrains the symmetry energy. Calc. for 208 Pb dipole polarizability by J. Piekarewicz “The concordance of experimental, theoretical and observational analyses suggests that neutron star radii, in the mass range 1 M -2 M, lie in the narrow window 11 km < R < 12 km.” See also, M.B. Tsang et al., PRC86, (2012). Lattimer et al., arXiv v1(2012)

0-5.3 deg 90 Zr(p,p’) PDR region deg deg C. Iwamoto, et al., PRL108, (2012).  E=~20 keV

A. Krugmann et al., analysis in progress

Summary We have accurately determined the B(E1) distribution and dipole polarizability of 208 Pb by using proton inelastic scattering at forward angles. 208 Pb … suitable for deducing constraints on model parameters. Coulomb excitation … reaction mechanism is well-known dipole polarizability … sum rule to reduce model dependence The measured dipole polarizability has a strong correlation with the neutron skin thickness and the symmetry energy parameters. The data give a strong constraints on theoretical models.

Summary Measured nuclei: DP, PDR, spin-M1 (data analysis in progress) 96 Mo, Dirk Martin 48 Ca, Jonny Birkhan 90 Zr, C. Iwamoto (PDR-region, published in PRL108, (2012)) 120 Sn, A.M. Krumbholtz, T. Hashimoto 154 Sm, A. Krugmann 88 Sr, 92 Mo, 70 Zn: Measurements in plan Stable Zr Isotopes Stable Sn Isotopes ⇔ comparison with unstable nuclei (GSI data)

Thank You

Correlation Between Dipole Polarizability and Neutron Skin Thickness The dipole polarizability of 208 Pb has been precisely measured. If the neutron skin thickness of 208 Pb can been precisely determined, the model parameters can be best constrained by the two numbers. Model-independent determination of the neutron skin thickness, i.e. PREX, is very important, but the present accuracy is insufficient. As an alternative way, the neutron skin thickness can be extracted from the dipole polarizability with help of theoretical models since the two numbers are well-correlated.

E1/M1 Decomposition by Spin Observables spinflip / non-spinflip separation* (model-independent) Polarization observables at 0° -1 for  S = 1, M1 excitations 3 for  S = 0, E1 excitations E1 and M1 cross sections can be decomposed T. Suzuki, PTP 103 (2000) 859 At 0° D SS = D NN

Preliminary  R np ( 208 Pb)=0.168±0.024 fm Gaussian weight func. Determination of Neutron Skin Thickness

Preliminary L  45±18 MeV S 0 =30.9±1.5 MeV M.B. Tsang et al., PRC86, (2012). J.M. Lattimer et al., arXiv: v1 Determination of Symmetry Energy

Lattimer et al., arXiv v1(2012) Constant term of the Symmetry Energy Slope Parameter of the Symmetry Energy Determination of Symmetry Energy Determination of the dipole polarizability of 208 Pb strongly constrains the symmetry energy. Calc. for 208 Pb dipole polarizability by J. Piekarewicz “The concordance of experimental, theoretical and observational analyses suggests that neutron star radii, in the mass range 1 M -2 M, lie in the narrow window 11 km < R < 12 km.”

E1 Response in 208 Pb Quasiparticle Phonon Model 3 phonons up to 8.2 MeV 2 phonons in the GDR region V.Yu. Ponomarev Relativistic Quasiparticle Time-Blocking Approximation 2QP×1 phonon E. Litvinova et al., PRC 78 (2008) , PRC 79 (2009) This Exp.

Determination of the Symmetry Energy in Nuclear EOS. is important for nuclear physics, - property of neutron matter - property of neutron rich nuclei - calculation of heavy ion-collision process as well as for - nuclear astrophysics related to neutron star

Momentum bite and back ground condition were different between the two measurements, but...

After making background subtraction, agreement of the two measurements is excellent!

PREX at J-Lab: Z 0 of weak interaction : sees the neutrons proton neutron Electric charge 1 0 Weak charge Neutron form factor Parity Violating Asymmetry T.W. Donnelly, J. Dubach, I. Sick C.J. Horowitz NPA503, 589, /16 Model Independent Determination of Neutron Skin Thickness Neutron Skin Thickness Measurement by Electroweak Interaction C. J. Horowitz, S. J. Pollock, P. A. Souder, R. Michaels PRC 63, , 2001

Neutron Skin Thickness Measurement by Electroweak Interaction PREX at J-lab S. Abrahamyan et al., PRL108, (2012)

Neutron Skin Thickness Measurement by Electroweak Interaction PREX PREX Result: S. Abrahamyan et al., PRL108, (2012) Theor. Calc.: X. Roca-Maza et al., PRL106, (2011) Other approaches at RCNP using electromagnectic interaction The model independent determination of  R np by PREX and future projects is quite important.

79 Collaborators RCNP, Osaka University A. Tamii, H. Matsubara, K. Hatanaka, H. Sakaguchi Y. Tameshige, M. Yosoi and J. Zenihiro Dep. of Phys., Osaka University Y. Fujita Dep. of Phys., Kyoto University T. Kawabata CNS, Univ. of Tokyo K. Nakanishi, Y. Shimizu and Y. Sasamoto CYRIC, Tohoku University M. Itoh and Y. Sakemi Dep. of Phys., Kyushu University M. Dozono Dep. of Phys., Niigata University Y. Shimbara IKP, TU-Darmstadt P. von Neumann-Cosel, A.M. Krumbholtz, Y. Kalmykov, I. Poltoratska, V.Yu. Ponomarev, A. Richter and J. Wambach KVI, Univ. of Groningen T. Adachi and L.A. Popescu IFIC-CSIC, Univ. of Valencia B. Rubio and A.B. Perez-Cerdan Sch. of Science Univ. of Witwatersrand J. Carter and H. Fujita iThemba LABS F.D. Smit Texas A&M Commerce C.A. Bertulani NSCL, MSU E. Litvinova RCNP-E282 and Theory Supports

スキルム相互作用（ Skyrm Interaction) 主に重い原子核の平均場近似での計算に使われる有効相互作用で、 原子核の巨視的性質（質量、飽和密度等）をうまく再現するよう にパラメータを現象論的に決めたもの。 平均場計算に使いやすい様にパラメータ化されている。 δ 型の相互作用で密度依存性があり、 2 体力、 3 体力を含む。 極めて多くの種類がある。 理論計算の前提条件にあう相互作用を選んだり、モデル依存性を 議論したりしやすい。 平均場近似での有効相互作用の例

( 時間依存） Hartree Fock （波動関数を根底におく）とは異なるアプ ローチ ( エネルギー ) 密度汎関数法 ( Energy) Density Functional Method (Theory) 原子核の全ての物理量を、エネルギー密度分布関数の関数として 記述しようとする理論的試み。 波動関数でなく、密度（通常は波動関数の絶対値の 2 乗）を根底 に置く、量子多体系計算。 量子化学の分野で発達した。 原子核の平均場近似計算に取り入れられ威力を発揮しつつある。 （計算コストを大幅に下げる） 1998 Walter Kohn ノーベル賞

82 杉本・村岡「原子核構造 学」 原子核の電荷分布 電子弾性散乱による測定。

Saturation of Density Short range interaction ~1 fm ~0.16 fm -3 Energy increase at higher-density: 1. density dependence of the tensor interaction. 2. exchange interaction 3. short range repulsive core of the NN interaction Strong attraction due to tensor interaction. Nucleon-nucleon interaction Symmetric nuclear matter N=Z 密度の飽和性

SHF RMF Nuclear Matter Incompressibility

Giant Resonance (GMR) by M. Itoh

Giant Monopole Resonance: GMR T. Li et al., PRC99, (2007)  120 Sn  120 Sn * E in E out E recoil ExEx

Giant Monopole Resonance: GMR

diffuseness Diffuseness of the nuclear surface is created 1. mainly by the tunneling effect of the bound nucleons in a nuclear potential 2. and by zero-energy oscillation of the nuclear shape vibration may also contribute to the diffuseness. tunneling

Constraining the symmetry energy from dipole polarizability J=(32.6±1.4) MeV Theoretical constraints on the symmetry energy at saturation density (J) and slope of the symmetry energy (L) from dipole polarizability (α D ) using relativistic nuclear energy density functionals Exp. data from polarized proton inelastic scattering, α D =18.9(13)fm 3 /e 2 A. Tamii et al., PRL. 107, (2011) L=(50.9±12.6) MeV DD-ME Slide by N. Paar

Determination of Symmetry Energy ? 208 Pb PDR Energy-Weighted Sum DD-ME 208 Pb DP DD-ME