Density dependence of the nuclear symmetry energy estimated from neutron skin thickness in finite nuclei

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1 Density dependence of the nuclear symmetry energy estimated from neutron skin thickness in finite nuclei X. Roca-Maza a,b X. Viñas b M. Centelles b M. Warda b,c a INFN sezione di Milano. Via Celoria 16, Milano, Italy b Departament d Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain c Katedra Fizyki Teoretycznej, Uniwersytet Marii Curie-Sklodowskiej, Poland M. Centelles, X. Roca-Maza, X. Viñas and M. Warda, Phys. Rev. Lett (2009) M. Warda, X. Viñas, X. Roca-Maza and M. Centelles, Phys. Rev. C (2009)

2 Introducing myself... Degree in Physics: University of Barcelona from 2000 to 2005 Phd thesis: University of Barcelona from 2005 to 2010 Title: Isospin asymmetry in stable and exotic nuclei. Advisors: X. Viñas and M. Centelles Defense: 6 May 2010 Post-doc: University of Barcelona from May to October Post-doc: INFN Milano from November 2010

3 Works in which I have participated

4 The symmetry energy and the outer crust in colaboration with J. Piekarewicz Impact of the symmetry energy on the outer crust of non-accreting neutron stars. Phys. Rev. C 78 (2008)

5 The symmetry energy and the neutron skin thickness of nuclei in colaboration with X. Viñas, M. Cenetelles and M. Warda Single particle shell effects in the neutron skin thickness of nuclei within mean-field models In preparation, writing... Origin of the neutron skin thickness of 208 Pb in nuclear mean-field models Accepted in Phys. Rev. C. Analysis of bulk and surface contributions in the neutron skin of nuclei. Phys. Rev. C81 (2010) Neutron skin thickness in droplet model with surface width dependence: indications of softness of the nuclear symmetry energy. Phys. Rev. C80 (2009) Nuclear symmetry energy probed by neutron skin thickness of nuclei. Phys. Rev. Lett. 102 (2009)

6 Electron scattering in colaboration with X. Viñas, M. Cenetelles and F. Salvat Parity violating electron scattering at the kinematics of the PREX experiment and the neutron skin thickness of 208 Pb. In preparation, writing... Theoretical study of elastic electron scattering along N = 16, N = 50 and N = 82 isotonic chains. In preparation, writing... Theoretical study of elastic electron scattering off stable and exotic nuclei Phys. Rev. C 78 (2008)

7 The symmetry energy and the GMR in colaboration with X. Viñas, M. Cenetelles, S.K. Patra, B.K. Sharma, P.D. Stevenson Influence of the symmetry energy on the giant monopole resonance of neutron-rich nuclei. J. Phys. G. 37 (2010)

8 DDMEδ, new mean field effective interaction in colaboration with X. Viñas, M. Cenetelles, P. Ring and P. Schuck Relativistic mean field interaction with density dependent meson-nucleon vertices based on microscopical calculations. In preparation, writing...

9 Density dependence of the nuclear symmetry energy estimated from neutron skin thickness in finite nuclei X. Roca-Maza a,b X. Viñas b M. Centelles b M. Warda b,c a INFN sezione di Milano. Via Celoria 16, Milano, Italy b Departament d Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain c Katedra Fizyki Teoretycznej, Uniwersytet Marii Curie-Sklodowskiej, Poland c INFN sezione di Milano. Via Celoria 16, Milano, Italy M. Centelles, X. Roca-Maza, X. Viñas and M. Warda, Phys. Rev. Lett (2009) M. Warda, X. Viñas, X. Roca-Maza and M. Centelles, Phys. Rev. C (2009)

10 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 m ) and very large objects as neutron stars ( R 10 4 m ) Nuclear Physics: Neutron skin thickness in finite nuclei, stable nuclei, Heavy-Ion collisions, Giant Resonances... Astrophysics: Supernova explosion, Neutron emission and cooling of protoneutron stars, Mass-Radius relations in neutron stars, Composition of the crust of neutron stars...

11 Equation of State in asymmetric matter e(ρ, δ) = e(ρ, 0) + c sym (ρ)δ 2 + O(δ 4 ) ( δ = ρ ) n ρ p ρ Around the saturation density we can write e(ρ, 0) a v K v ɛ 2 and c sym (ρ) J Lɛ+ 1 ( 2 K symɛ 2 ɛ = ρ ) 0 ρ 3ρ 0 ρ fm 3, a v 16MeV, K v 230MeV, J 32MeV However, the values of L = 3ρ c sym (ρ)/ ρ ρ0 and K sym = 9ρ 2 2 c sym (ρ)/ ρ 2 ρ0 which govern the density dependence of c sym near ρ 0 are less certain and predictions vary largely among nuclear theories.

12 Experimental constraints Recent reseach in heavy-ion collisions at intermediate energy is consistent with c sym (ρ) = c sym (ρ 0 ).(ρ/ρ 0 ) γ at ρ < ρ 0. Isospin difussion γ = (L = 88 ± 25 MeV). Isoscaling γ = 0.69 (L 65 MeV) Inferred from nucleon emision ratios γ = 0.5(L 55 MeV). The GDR of 208 Pb analyzed with Skyrme forces suggests a constraint c sym (0.1 fm 3 )= MeV (γ ). The study of the PDR in 68 Ni and 132 Sn predicts L=49-80 MeV. The Thomas-Fermi model of Myers and Swiatecki fitted very precisely to binding energies of 1654 nuclei predicts an EOS that yields γ = 0.51 NEUTRON SKIN THICKNESS?

13 Neutron skin thickness

14 What is experimentally know about neutron skin thickness in nuclei? The neutron skin thickness is defined as S = r 2 n 1/2 r 2 p 1/2, where r 2 n 1/2 and r 2 p 1/2 are the rms radii of the neutron and proton distributions respectively. r 2 p 1/2 is known very accurately from elastic electron scattering measurements (e.g. r ch ( 208 Pb) = ± fm [Angeli (2004)]). r 2 n 1/2 has been obtained with hadronic probes such as: (a) Proton-nucleus elastic scattering (5.522fm < r n ( 208 Pb) < fm [Clark (2003)]). (b) Inelastic scattering excitation of the giant dipole and spin-dipole resonances (r n ( 208 Pb) = 5.67 ± 0.07 fm [Krasznahorkay (1990)]). (c) Antiprotonic atoms: Data from antiprotonic X rays and radiochemical analysis of the yields after the antiproton annihilation (r n ( 208 Pb) = 5.66 ± 0.02 fm) [Trzcińska (2001)].

15 0.2 S (fm) I S = (0.9 ± 0.15)I + ( 0.03 ± 0.02) fm A. Trzcińska et al, Phys. Rev. Lett. 87, (2001) CAN S OF 26 STABLE NUCLEI, FROM 40 Ca TO 238 U, ESTIMATED USING ANTIPROTONIC ATOMS DATA HELP IN CONSTRAINING THE SLOPE AND CURVATURE OF c sym?

16 Symmetry energy and neutron skin thickness in the Liquid Drop Model Symmetry Energy where a sym (A) = J, x A = 9J 1 + x A 4Q A 1/3 E sym (A) = a sym (A)(I + x A I C ) 2 A I = (N Z)/A, I C = e 2 Z/(20JR), R = r 0 A 1/3. Neutron skin thickness S = [ 3/5 t e 2 Z/(70J) + 5 ] 2R (b2 n bp) 2 where t = 3r 0 2 J/Q 1 + x A (I I C )

17 Neutron skin thickness t = 2r 0 3J [J a sym(a)] A 1/3 (I I C )

18 Table: Value of a sym (A) and density ρ that fulfils c sym (ρ) = a sym (A) for A = 208, 116 and 40 in MF models. J and a sym are in MeV and ρ is in fm 3. A = 208 A = 116 A = 40 Model J a sym ρ a sym ρ a sym ρ NL NL-SH FSUGold TF-MS SLy SkX SkM* SIII SGII

19 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. 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 2r 0 L ( t = 1 ɛ K sym )ɛa 1/3( ) I I C 5 3 J 2L

20 Neutron skin thickness t = 3 2r 0 L ( 1 ɛ K sym )ɛa 1/3( ) I I C 5 3 J 2L

21 Fitting procedure and results We optimize 3 2r 0 L ( t = 1 ɛ K sym )ɛa 1/3( ) I I C 5 3 J 2L using c sym = 31.6( ρ ρ 0 ) γ MeV, ɛ = 1 3(1 + ca 1/3 ), ρ 0 = 0.16fm 3 and taking as experimental baseline the neutron skins measured in 26 antiprotonic atoms. We predict (b n b p ): L = 75 ± 25 MeV

22 S (fm) experiment linear average of experiment Our fit Fe Ca Ni Fe Cd 28 Ni 56 Fe Co Zr Sn Ni Sn Te Te Cd 50 Sn Te 90 Th Te Te Zr Ca Sn I = (N Ζ) / Α Bi Pb S = (0.9 ± 0.15)I + ( 0.03 ± 0.02) fm A. Trzcińska et al, Phys. Rev. Lett. 87, (2001) U

23 Influence of the surface width (b n b p ) S = [ 3/5 t e 2 Z/(70J) + 5 ] 2R (b2 n bp 2 ) b n and b p are obtained at the ETF level.

24 Surface contribution to the neutron skin thickness R (b2 n bp) 2 = σ sw I = (0.3 J Q + c)i

25 Fit and results c = 0.07 c = EXP 0.9 I R np (fm) I With ρ 0 = 0.16 fm 3 and 28 J 35 MeV, and 0.05 c 0.07 fm J Q =

26 Neutron skin thickness R np (fm) in 208 Pb NL3Λ ν1 NL3Λ ν2 TM1 NL3 NL-SH FSUGold NL3Λ ν3 NL1 SkM* SkX SLy4 SkP T6 SGII D1S SIII 0.1 SVI J / Q NL3Λ ν1 NL3Λ ν2 FSUGold NL3Λ ν3 NL3 TM1 SkM* SkX SLy4 SkP T6 SGII D1S SIII SVI NL1 NL-SH L (MeV) NL3Λ ν1 SVI TM1 NL3 NL-SH NL3Λ ν2 FSUGold SLy4 NL3Λ ν3 SkX SkM* SGII SkP T6 D1S SIII NL J / Q L (MeV) L = MeV

27 Constraints on the slope of the symmetry energy

28 From the very small to the very big: the symmetry energy and the outer crust of a neutron star

29 Introduction R (Km) ρ (gr/cm 3 ) v scape /c g/g Earth (surface) P (dyn/cm 2 ) Orientative properties of a typical neutron star of mass M = M Sun.

30 Formalism Total energy per nucleon e(a, Z, ρ = ρ n + ρ p ) = e N (A, Z) + e lat (A, Z, ρ) + e el (ρ) The different contributions e N (A, Z) = M(A,Z) A Z e lat (A, Z, ρ) = C 2 lat A 4/3 p F where C lat = and p F = (3π 2 ρ) 1/3 = p Fel (A/Z) 1/3 (N el = Z) e el (ρ) = m4 el 8π 2 ρ ( xf y F (x 2 F + y 2 F ) ln(x F + y F ) ) where x F p Fel and y F ɛ F el m el = 1 + xf 2

31 Composition of the outer crust Composition FSUGold (a) Protons Neutrons Ni Sr Kr N=50 Se Sn N=82 Cd Pd Ru Mo Zr Sr Kr Composition NL3 (b) N=82 N=32 Fe Sr Kr N= Se ρ(10 11 g/cm 3 ) Ge Zn Ni Mo Zr SrKr

32 The stiffer the symmetry energy the more exotic the composition of the outer crust and the larger the neutron skin of medium and heavy elements R NL3 np ( 208 Pb) = 0.28 fm and R FSUGold ( 208 Pb) = 0.20 fm np

33 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. 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, (R) (2008)).

34 Thank you for your attention

35 Extra material

36 Some technical details The surface stiffness coeficient Q and the surface widths b n and b p are obtained from self-consistent calculations of the neutron and proton density profiles in asymmetric semi-infinite nuclear matter. To this end one has to minimize the total energy per unit area with the constraint of conservation of the number of protons and neutrons with respect to arbitrary variations of the densities. E const S = [ ε(z) µn ρ n (z) µ p ρ p (z) ] dz, where ε(z) is the nuclear energy density functional. In the non-relativistic framework the densities ρ n and ρ p obey the coupled local Euler-Lagrange equations: δε(z) δρ n µ n = 0, δε(z) δρ p µ p = 0. The relative neutron excess δ = (ρ n ρ p )/(ρ n + ρ p ) is a function of the z-coordinate. When z, the densities ρ n and ρ p approach the values of asymmetric uniform nuclear matter in equilibrium with a bulk neutron excess δ 0.

37 From the calculated density profiles one computes: z oq = zρ q(z)dz ρ q(z)dz, From the relation b 2 q = (z z 0q) 2 ρ q(z)dz. ρ q(z)dz t = z 0n z 0p = 3r 0 2 J Q δ 0, one can evaluate Q from the slope of t at δ 0 = 0. The distance t and the surface widths b n and b p in finite nuclei with neutron excess I = (N Z)/A are obtained using δ 0 given by: I + 3 c 1 Z 2 8 Q δ 0 = A 5/ J 4 Q A 1/3

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