Isospin asymmetry in stable and exotic nuclei

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1 Isospin asymmetry in stable and exotic nuclei Xavier Roca Maza 6 May 2010 Advisors: Xavier Viñas i Gausí and Mario Centelles i Aixalà

2 Motivation: Nuclear Chart Relative Neutron excess I (N Z )/(N + Z ) stable nuclei I exotic nuclei I 0.25

3 Motivation: Rare Ion Beam Facilities

4 Thesis works Symmetry energy Nuclear symmetry energy probed by neutron skin thickness of nuclei. Phys. Rev. Lett. 102 (2009) Neutron skin thickness in droplet model with surface width dependence: indications of softness of the nuclear symmetry energy. Phys. Rev. C 80 (2009) Impact of the symmetry energy on the outer crust of non-accreting neutron stars. Phys. Rev. C 78 (2008) Relativistic Mean Field interaction with density dependent meson-nucleon vertices:. Analysis of bulk and surface contributions in the neutron skin of nuclei. Accepted in Phys. Rev. C. Influence of the symmetry energy on the giant monopole resonance of neutron-rich nuclei. Accepted in J. Phys. G. Theoretical study of elastic electron scattering off stable and exotic nuclei Phys. Rev. C 78 (2008)

5 The Symmetry Energy and the Neutron Skin Thickness of nuclei

6 Definitions Symmetry energy Energy per particle: e(ρ, δ) The energy per particle (EoS) in asymmetric nuclear matter (infinite system) of total density ρ = ρ n + ρ p and asymmetry δ = ρn ρp ρ n+ρ p can be written as, e(ρ, δ) = e(ρ, δ = 0)+c sym (ρ)δ where, c sym (ρ) 1 2 e(ρ, δ) 2 δ 2 δ=0 e ( MeV ) neutron matter e(ρ,δ=1) Saturation c sym (ρ)~ (0.16 fm 3, 16.0 MeV) -10 e(ρ,δ=0) symmetric matter ,05 0,1 0,15 0,2 0,25 ρ ( fm 3 )

7 Symmetry energy in Nuclear Models 75 NL3, L= MeV G2, L= 107 MeV FSUGold, L= 60.4 MeV DD-ME2, L= 51.3 MeV c sym (ρ) (MeV) ,5 1 1,5 2 ρ/ρ 0

8 Symmetry energy around the saturation density Bulk parameters The symmetry energy is usually characterized in the literature by the parameters of a Taylor expansion around the saturation density ρ 0, c sym (ρ) c sym (ρ 0 ) + c sym(ρ) ρ where ɛ ρ 0 ρ 3ρ 0 J Lɛ K symɛ 2 ρ0 (ρ ρ 0 ) c sym (ρ) 2 ρ 2 ρ0 (ρ ρ 0 ) 2 With any nuclear interaction one can calculate J, L and K sym

9 The symmetry energy of a finite nucleus The semi-empirical mass formula for the binding energy It is based on the fact that the baryon density and the binding energy per nucleon are, approximately, the same for all nuclei. In its simplest form, B(N, Z ) = a v A a s A 2/3 Z a 2 (N Z ) c a 2 A 1/3 sym A The symmetry energy of a finite nucleus in the Droplet Model J a sym (A) = J 4 Q A 1/3

10 The neutron skin thickness of a nucleus Definition R np = r n 2 1/2 r p 2 1/2 Neutron skin thickness in the Droplet Model R np = t = 3 2 r J 0 Q = 2r 0 3J ( 3 t e2 Z 5 70J + 2 (b )) 2 2 n b p 5R I c 1Z 12J A 1/ J 4Q A 1/3 ( J asym (A) ) A 1/3 (I c 1Z 12J A 1/3 It is usually assumed that b n b p 1 fm )

11 R np in 208 Pb Symmetry energy 0,3 Hartee-Fock R np in 208 Pb (fm) 0,2 NL3, G1 TM1, NL-SV HS NL-SH, PK1 G2 S271, NL3Λ v1 Z271, PKDD NL3Λ v2 FSUGold DD-ME1,D 3 C NL3Λ v3,tw DD-ME2 SkI2 Gs Rs GSkII SkSM* FKVW GSkI, SIV SkM*, D280, SSk SLy5 SkX, SLy4, SLy7 T6 SkP, D250, D300 D1S, SGII SIII, D1, D260 0, L (MeV) Covariant Covariant DD, PC Skyrme Gogny 0,4 0,8 1,2 J / Q NL3Λ v2 FSUGold NL3Λ v3 NL3 HS TM1 NL-SH NL3Λ v1 SkI2 Gs Rs SkM* SkX, SLy4 T6 SkP SGII SIII (a) (b) (c) SkI J a sym (A) (MeV)

12 c sym (ρ) versus a sym (A) Universal relation in mean-field models c sym (0.1fm 3 ) a sym (A = 208) A = 208 A = 116 A = 40 Model J a sym ρ a sym ρ a sym ρ NL NL-SH FSUGold TF SLy SkX SkM* SIII SGII c sym (ρ A ) = a sym (A) ρ A ρ 0 ρ 0 /( A 1/3 ) for 40 A 238

13 The neutron skin thickness and c sym (ρ) summetry properties of the EoS and the R np c sym (ρ A ) = a sym (A) c sym (ρ) = J Lɛ K symɛ 2 R np and the EoS parameters t = 3r ( 0 2J L 1 K ) ( sym 2L ɛ ɛa 1/3 I c ) 1Z 12J A 1/3 J is well determined by the experiment ( 31.6 MeV) as compared with L and K sym.

14 Experiment: Antiprotonic atoms R np (fm) 0,3 0,2 0,1 0-0,1 experiment linear average of experiment Droplet Model Fe Ca Ni Fe Cd 28 Ni 56 Fe Co Zr Sn Sn 52 Te 48 Cd 50 Sn Te 90 Th Ni Te Te Te Zr Ca Sn 0 0,1 0,2 I = (N Ζ) / Α Bi Pb ( ) γ Empirical indications at ρ < ρ 0 : c sym = J ρ ρ0 We find L = 75 ± 25 MeV U

15 Surface width contribution to the neutron skin thickness Our prediction for L points towards a relatively soft symmetry energy Mean Field models predict different surface widths for the proton and neutron density profiles and, therefore, a surface contribution to the neutron skin thickness of nuclei may be important.

16 Droplet model with surface width correction 0,3 R np (fm) 0,2 0,1 0 0,1 NL3 J/Q = 1.18 NL-SH J/Q = 1.05 SkM* J/Q = 0.77 SIII J/Q = 0.44 empty symbols: LDM with b n b p solid symbols: ETF DM inspired ansatz R np = ( 3 5 ) t e2 Z 70J + ( 0.3 J Q + p) I sw R np (fm) 0 0 0,05 0,1 0,15 0,2 I

17 Density content of the symmetry energy c = 0.07 c = EXP 0.9 I R np (fm) I With ρ 0 = 0.16 fm 3, 28 < J < 35 MeV (suggested by MF models) and 0.05 < p < 0.07 fm we found 30 < L < 80 MeV

18 Comparison with other L predictions this work Binding energies (Danielewicz) Method n-p emission ratios isoscaling isospin diffusion Binding energies Thomas-Fermi PDR GDR L (MeV) The compatible range for L range from 45 to 75 MeV

19 The Symmetry Energy and the Outer Crust of a Neutron Star

20 Introduction Symmetry energy 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.

21 Formalism Symmetry energy 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 where x F p Fel ( xf y F (x 2 F + y 2 F ) ln(x F + y F ) ) and y F ɛ F el m el = 1 + xf 2

22 Results: 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

23 R np and the outer crust R NL3 np ( 208 Pb) = 0.28 fm and R FSUGold ( 208 Pb) = 0.20 fm np The larger the neutron skin of 208 Pb, the more exotic the composition of the outer crust

24 Relativistic Mean Field interaction with Density Dependent Meson-Nucleon Vertices:

25 Relativistic Mean Field Models Standard Relativistic Mean Field Models Interaction: σ, ω and ρ mesons (and the γ) Usually fitted to finite nuclei properties (binding energies, charge radii, etc.) and to some properties of the EoS at saturation (E/A, ρ 0, K 0, etc.) The coupling constants do not depend on the density. DDMEδ Interaction: σ, ω, ρ and δ mesons (and the γ) Fitted to a microscopic EoS obtained from ab initio calculations based on a bare nucleon-nucleon potential and, in addition, to finite nuclei properties (binding energies and charge radii). The coupling constants depend on the density.

26 Lagrangian Symmetry energy L = L N + L M + L int L N = Ψ (iγ µ µ M) Ψ L M = i=σ,δ ( µ Φ i µ Φ i m 2 i Φ2 i j=ω,ρ,γ ( ) 1 2 F µν (j) F (j)µν mj 2 A(j) µ A (j)µ L int = ΨΓ σ ( Ψ, Ψ)ΨΦ σ + ΨΓ δ ( Ψ, Ψ)τ ΨΦ δ ΨΓ ω ( Ψ, Ψ)γ µ ΨA (ω)µ ΨΓ ρ ( Ψ, Ψ)γ µ τψa (ρ)µ e Ψ ˆQγ µ ΨA (γ)µ )

27 Density dependence of the meson-nucleon vertices σ and ω Γ i (ρ) = g i (ρ 0 )f i (x = ρ ρ 0 ) f (x) = a 1+b(x+d)2 1+c(x+d) 2 (Same density dependence for σ and ω) ρ and δ Γ ρ (ρ) = g ρ (ρ 0 )e aρ(x 1) Γ δ (ρ) = g ρ (ρ 0 ) [ e a δ(x 1) + b δ (x 1) ] M * p M* n ( MeV ) E.N.E. van Dalen et al. Eur. Phys. J. A 31, (2007) DDMEδ 0 0,05 0,1 0,15 0,2 0,25 ρ ( fm 3 )

28 Results: EoS Symmetry energy 30 EoS from Baldo et al. NPA 736 (2004) 241 DDMEδ 20 e ( MeV ) ,05 0,1 0,15 0,2 0,25 ρ ( fm 3 )

29 Results: Binding energies B theo - B exp (MeV) stronger binding energy BCP1 DD-ME2 DDMEδ rms BCP1 = 1.77 MeV rms DD-ME2 = 2.07 MeV rms DDMEδ = 2.94 MeV rms NL3 = 3.58 MeV weaker binding energy A

30 Results: Charge Radii theo exp R ch - Rch (fm) 0,1 ρ sat. under-estimated 0,05 0-0,05 ρ sat. over-estimated BCP1 DD-ME2 DDMEδ rms BCP1 = 0.03 fm rms DD-ME2 = 0.02 fm rms = 0.03 fm rms NL3 = 0.02 fm A

31 Results: Neutron Skins 0,2 linear average of experiment 0.90(15)I (2) PRL87 (2001) DD-ME2 L=51 MeV DDMEδ L=49 MeV R np (fm) 0,1 0-0,1 0 0,05 0,1 0,15 0,2 I

32 Energy density functional Achievements of DDMEδ Reproduce the EoS of ab initio calculations up to 2ρ 0. Reproduce the binding energies and charge radii of spherical nuclei with the same level of accuracy than other RMF models. But this is a first step... Perspectives Reduce the free parameters to 4 (g σ, g ω, g ρ and m σ ) Achieve the same level of accuracy than standard RMF models in describing finite nuclei properties. Reproduce the EoS of ab initio calculations at higher densities.

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34 c sym (ρ A ) = a sym (A) L = 75 ± 25 MeV (without surface contributions to R np ) L = 55 ± 25 MeV (with surface contributions to R np ) Compatible range for predictions coming form different observables point towards a soft symetry energy L = 60 ± 15 MeV Mean Field models and the analyzed experiment on antiprotonic atoms indicate the importance of the of the surface correction to the neutron skin thickness of nuclei. 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. With the RMF model DDMEδ it is possible to reproduce the microscopic EoS obtained from ab initio calculations based on a bare nucleon-nucleon potential and to describe at the same time basic properties of finite nuclei. Symmetry energy