Symmetry energy in bulk matter from neutron skin thickness in finite - PowerPoint PPT Presentation

Symmetry energy in bulk matter from neutron skin thickness in finite nuclei nas a , M. Centelles a , X. Roca-Maza a and M. Warda a , b X. Vi˜ a Departament d’Estructura i Constituents de la Mat` eria and Institut de Ci` encies del Cosmos, Universitat de Barcelona, Barcelona, Spain b Katedra Fizyki Teoretycznej, Uniwersytet Marii Curie-Sklodowskiej, Poland M. Centelles, X. Roca-Maza, X. Vi˜ nas and M. Warda, Phys. Rev. Lett. 102 122502 (2009) Phys. Rev. C82 054314 (2010) M. Warda, X. Vi˜ nas, X. Roca-Maza and M. Centelles, Phys. Rev. C80 024316 (2009) M. Warda, X. Vi˜ nas, X. Roca-Maza and M. Centelles, Phys. Rev. C81 054309 (2010) X. Roca-Maza, M. Centelles, X. Vi˜ nas and M. Warda, Phys. Rev. Lett. 106 252501 (2011)

Why is important the nuclear symmetry energy ? The nuclear symmetry energy is a fundamental quantity in Nuclear Physics and Astrophysics because it governs, at the same time, important properties of very small entities like the atomic nucleus ( R ∼ 10 − 15 m ) and very large objects as neutron stars ( R ∼ 10 4 m ) • Nuclear Physics: Neutron skin thickness in finite nuclei, structure of neutron rich nuclei, Heavy-Ion collisions, Giant Resonances.... • High-Energy Physics: Test of the Standard Model through atomic parity non-conservation observables. • Astrophysics: Supernova explosion, Neutron emission and cooling of protoneutron stars, Mass-Radius relations in neutron stars, Composition of the crust of neutron stars...

∆ r np = � r 2 � 1 / 2 − � r 2 � 1 / 2 n p

Equation of State in asymmetric matter � δ = ρ n − ρ p � e ( ρ, δ ) = e ( ρ, 0) + c sym ( ρ ) δ 2 + O ( δ 4 ) ρ Around the saturation density we can write � � e ( ρ, 0) ≃ a v + 1 c sym ( ρ ) ≃ J − L ǫ + 1 ǫ = ρ 0 − ρ 2 K v ǫ 2 2 K sym ǫ 2 and 3 ρ 0 ρ 0 ≈ 0 . 16 fm − 3 , a v ≈ − 16 MeV , K v ≈ 230 MeV , J ≈ 32 MeV However, the values of K sym = 9 ρ 2 ∂ 2 c sym ( ρ ) /∂ρ 2 | ρ 0 L = 3 ρ∂ c sym ( ρ ) /∂ρ | ρ 0 and which govern the density dependence of c sym near ρ 0 are less certain and predictions vary largely among nuclear theories.

Symmetry energy and neutron skin thickness in the Liquid Drop Model • Symmetry Energy x A = 9 J J 4 Q A − 1 / 3 a sym ( A ) = , 1 + x A E sym ( A ) = a sym ( A )( I + x A I C ) 2 A where I C = e 2 Z / (20 JR ) , R = r 0 A 1 / 3 I = ( N − Z ) / A , . • Neutron skin thickness � t − e 2 Z / (70 J ) + 5 � � 2 R ( b 2 n − b 2 S = 3 / 5 p ) where t = 3 r 0 J / Q ( I − I C ) = 2 r 0 3 J [ J − a sym ( A )] A 1 / 3 ( I − I C ) 2 1 + x A M. Centelles, M. Del Estal and X. Vi˜ nas, Nucl. Phys. A635 , 193 (1998)

The c sym ( ρ ) - a sym ( A ) correlation • There is a genuine relation between the symmetry energy coefficients of the EOS and of nuclei: c sym ( ρ ) equals a sym ( A ) of heavy nuclei like 208 Pb at a density ρ = 0 . 1 ± 0 . 01 fm − 3 practically independent of the mean field model used to compute them. • A similar situation occurs down to medium mass numbers, at lower densities. • We find that this density can be very well simulated by ρ ≈ ρ A = ρ 0 − ρ 0 / (1 + cA 1 / 3 ) , where c is fixed by the condition ρ 208 = 0 . 1 fm − 3 . • Using the equality c sym ( ρ ) = a sym ( A ) and the LDM , the neutron skin thickness can be finally written as: � 3 2 r 0 L 1 − ǫ K sym � � ǫ A 1 / 3 � � S = I − I C 5 3 2 L J • See also Lie-Wen Chen Phys. Rev. C83 , 044308 (2011)

S = (0 . 9 ± 0 . 15) I + ( − 0 . 03 ± 0 . 02) fm A. Trzci´ nska et al, Phys. Rev. Lett. 87 , 082501 (2001) Assuming c ( ρ ) = 31 . 6( ρ/ρ 0 ) γ with ρ 0 =0.16 fm − 3 we predict ( b n = b p ): L = 75 ± 25 MeV

Influence of the surface width ( b n � = b p ) � 3 5 2 R ( b 2 n − b 2 p ) = 0 . 31 I ( NL 3) − 0 . 15 I ( SGII ) 5 b n and b p are obtained semiclassically at ETF level M.Centelles et al. NPA 635 , 193 (1998).

Surface contribution to the neutron skin thickness � 3 5 p ) = σ sw I = (0 . 3 J 2 R ( b 2 n − b 2 Q + c ) I 5 c = 0 . 07 fm and c = − 0 . 05

Fit and results J Q = 0 . 6 − 0 . 9 L = 31 − 81 MeV

Neutron skin thickness

Constraints on the slope of the symmetry energy

What can we learn from parity-violating electron scattering ? • See C.J. Horowitz et al. Phys. Rev. C63 , 025501 (2001); Shufang Ban et al arXiv:1010.3246 [nucl-th] • A LR is the parity-violating asymmetry d σ + d Ω − d σ − d Ω • A LR ≡ d σ + d Ω + d σ − d Ω • V ± ( r ) = V Coulomb ( r ) ± V weak ( r ) (1 − 4 sin 2 θ W ) Z ρ p ( r ) − N ρ n ( r ) • V weak ( r ) = G F ˆ ˜ 2 3 / 2 G F q 2 h 4 sin 2 θ W + F n ( q ) − F p ( q ) i • A PWBA = √ LR 4 πα 2 F p ( q ) • PREX experiment E ∼ 1.05 GeV and θ ∼ 5 ◦

From parity-violating electron scattering with E =1.06 GeV and θ = 5 ◦

From parity-violating electron scattering

From parity-violating electron scattering with E =1 GeV and θ = 5 ◦

Dipole polarizability and symmetry energy The hydrodynamical model of Lipparini and Stringari (Phys. Rep. 175 , 103 (1989) (see also W.Satula et al, Phys. Rev. C74 , 011301 (2009) ) suggest a relation between the dipole polarizability and the bulk and surface contributions to the symmetry energy: s α hydro = A < r 2 > „ 1 + 5 « � 2 A (1 + κ ) J − a sym A − 1 / 3 ; E − 1 = 24 J 3 4 m α ( D ) J

Dipole polarizability and symmetry energy

Summary and Conclusions • We have described a generic relation between the symmetry energy in finite nuclei and in nuclear matter at subsaturation. • We take advantage of this relation to explore constraints on c sym ( ρ ) from neutron skins measured in antiprotonic atoms. These constraints points towards a soft symmetry energy. • We discuss the L values constrained by neutron skins in comparison with most recent observations from reactions and giant resonances. • We learn that in spite of present error bars in the data of antiprotonic atoms, the size of the final uncertainties in L is comparable to the other analyses.

• We have investigated parity-violating electron scattering in nuclear models constrained by available experimental data to extract the neutron radius and skin of 208 Pb without specific assumptions on the shape of the nucleon densities. • We have demonstrated a linear correlation, universal in mean field framework, between A pv and ∆ r np that has very small scatter. • It is predicted that a 1% measurement of A pv would allow to constrain the slope L of the symmetry energy to near a novel 10 MeV level. • We have found a simple parametrization of the parity-violating asymmetry for electron scattering in terms of the parameters C n − C p and a n − a p of the equivalent 2pF distributions.

From parity-violating electron scattering

• The generic relation between the symmetry The generic relation between the symmetry energy in finite nuclei and in nuclear matter at subsaturation plausibly encompasses other prime correlations of nuclear observables with the density content of the symmetry energy as e.g. the constrains of c sym (0 . 1) from the GDR of 208 Pb (L. Trippa et al. Phys. Rev. C77 , 061304(R) (2008)). • The properties of c sym ( ρ ) derived from terrestrial nuclei also have intimate connections to astrophysics. As an example, we can estimate the transition density ρ t between the crust and the core of a neutron star as ρ t /ρ 0 ∼ 2 / 3 + (2 / 3) γ K sym / 2 K v (J. M. Lattimer, M. Prakash, Phys. Rep. 442 , 109 (2007)). . The constraints from neutron skins hereby yield ρ t ∼ 0 . 095 ± 0 . 01 fm − 3 . This value would not support the direct URCA process of cooling of a neutron star that requires a higher ρ t . Our prediction is in consonance with ρ t ∼ 0 . 096 fm − 3 of the microscopic EOS of Friedman and Pandharipande as well as with ρ t ∼ 0 . 09 fm − 3 predicted by a recent ρ t ∼ 0 . 09 fm − 3 predicted by a recent analysis of pygmy dipole resonances in nuclei.

Neutron skin thickness de ( ρ, δ = 1) L − K + K sym dc sym ( ρ ) L − K sym = ǫ = ǫ d ρ 3 ρ 0 3 ρ 0 d ρ 3 ρ 0 3 ρ 0

What is experimentally know about neutron skin thickness in nuclei ? • The neutron skin thickness is defined as S= R n − R p , where R n and R p are the rms of the neutron and proton distributions respectively. • R p is known very accurately from elastic electron scattering measurements • R n has been obtained with hadronic probes such as: a) Proton-nucleus elastic scattering b) Inelastic scattering excitation of the giant dipole and spin-dipole resonances c) Antiprotonic atoms: Data from antiprotonic X rays and radiochemical analysis of the yields after the antiproton annihilation

S = (0 . 9 ± 0 . 15) I + ( − 0 . 03 ± 0 . 02) fm A. Trzci´ nska et al, Phys. Rev. Lett. 87 , 082501 (2001) CAN THE NEUTRON SKIN THICKNESS of 26 STABLE NUCLEI, FROM 40 Ca TO 238 U, ESTIMATED USING ANTIPROTONIC ATOMS DATA CONSTRAINT THE SLOPE AND CURVATURE OF c sym ?