Clusterized nuclear matter in PNS crust and the E sym

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1 Clusterized nuclear matter in PNS crust and the E sym Ad. R. Raduta IFIN-HH Bucharest in collaboration with: Francesca Gulminelli (LPC-Caen, France) Francois Aymard (LPC-Caen, France) Clusterized nuclear matter in PNS crust and the ETrento, sym April 10th, / 17

2 Overview The NSE model assumptions cluster definition: r-clusters versus e-clusters drawbacks versus possible improvements treatment consistency Predictions along β-eq. path, different effective interactions T=0 finite temperature In medium effects significance of E sym in charged unhomogeneous systems (E sym ) star as a function of (E sym ) NS Conclusions Clusterized nuclear matter in PNS crust and the ETrento, sym April 10th, / 17

3 Nuclear matter during CCSN A. Fantina, PhD thesis; Watanabe et al., PRC69, wide ranges of (T, ρ, Ye ) are spanned: 109 < T < K, 105 ρ 1014 g/cm3, 0 Yp 0.8 at ρ < ρ0 : clusters + nucleons Clusterized nuclear matter in PNS crust and the ETrento, April 10th, 2014 sym 3 / 17

4 N(uclear) S(tatistical) E(quilibrium) Physical circumstances: at ρ < ρ 0 matter consists of clusters and nucleons Y: equilibrium under nuclear strong interaction species abundances depend on thermo cond. and NOT on reaction rates Y/N: equilibrium under weak strong interaction: µ n = µ p + µ e (+µ ν ), µ e = µ e (ρ e ), ρ e = ρ p Phenomenological model T=0: ebaryon =min finite T: fbaryon =min Clusterized nuclear matter in PNS crust and the ETrento, sym April 10th, / 17

5 N(uclear) S(tatistical) E(quilibrium) Modeling: no interaction among clusters and unbound nucleons: Z baryon = Z cl Z unbound Cluster gas (Fisher hypothesis): Z cl = n A A>1 ω A,µ3 = 1 2 ω n A A,µ3 n A! 2πσ 2 A V F ( 2πAm 0 βh 2 ) 3/2 exp β ( fa,ī µ 3Ī ) Unbound nucleon gas: mean-field with effective interactions ( ) Z unbound = Z (HM) 0 exp β Ŵ 0 Ĥ 0 Geometrical corrections: V F = V v unbound Coulomb screening whithin WS approx. Souza et al., Ap.J. (2009), Botvina & Mishustin, Nucl. Phys. A (2010); Hempel & Schaffner-Bielich, Nucl. Phys. A (2010); Blinnikov et al., AA (2011), AR & Gulminelli, Phys. Rev. C (2010) Clusterized nuclear matter in PNS crust and the ETrento, sym April 10th, / 17

6 N(uclear) S(tatistical) E(quilibrium) Ingredients and possible improvements: treatment consistency: clusters and gas should be described by the same effective interaction/ LDM parameters fitted on Skyrme [Danielewicz & Lee, Nucl. Phys. A (2009)] f = B + ɛ Ts; s = ln ρ(ɛ); ρ 0 (δ r ) = ρ 0 0 (1 3Lδ2 r /K sym ) how to define a cluster? as a density fluctuation? as a bound system? in-medium cluster modification? cluster-cluster interaction? Built-in Uncertainity: iso-vector properties of the EOS improved NSE: AR, Gulminelli, Aymard, Eur.Phys.J (2014) Question: How much iso-vector prop. of NM impact on star composition and EOS? Clusterized nuclear matter in PNS crust and the ETrento, sym April 10th, / 17

7 Cluster definition Papakonstantinou et al., PRC2013 Cluster=density fluctuation Cluster=localized wave func. excluded volume applies excluded volume does not apply bulk energy is unmodified binding energy shift excitation en. < B excitation en. up to min(sn,sp) ρ bulk = ρ 0 (δ r ) ρ(δ) compressible clusters Clusterized nuclear matter in PNS crust and the ETrento, sym April 10th, / 17

8 The EOS input: the isovector channel E(ρ, δ) = E 0 (ρ, δ = 0)+E sym (ρ, δ = 0)δ 2 +O(δ 4 ) E sym (ρ, 0) = 1 2 E(ρ,δ) 2! δ 2 δ=0 = E sym (ρ 0, δ = 0) + Lχ + Ksym 2! χ 2 + O(χ 3 ) NN-potential ρ 0 K L K sym (fm 3 ) (MeV) (MeV) (MeV) SLY SGI SkI LNS similar iso-scalar properties different iso-vector properties asym. matter is sensitive to iso-vector EoS Clusterized nuclear matter in PNS crust and the ETrento, sym April 10th, / 17

9 Predictions at β equilibrium: composition (Y p ) cl, (Y p ) total decrease monotonically (Y p ) total is universal lim ρ 0 N free /N tot = 0 lim ρ ρ0 N free /N tot = 1 deviations because of finite T crust-core transition EoS-dependence ρ ρ 0 : A cl increases with ρ ρ ρ 0 : A cl 0 strong EoS-dependence AR, Gulminelli, Aymard, Eur.Phys.J (2014) Clusterized nuclear matter in PNS crust and the ETrento, sym April 10th, / 17

10 Composition at β-equilibrium at T=0 Gulminelli & AR, in preparation Clusterized nuclear matter in PNS crust and thetrento, E sym April 10th, / 17

11 Predictions at β equilibrium: energetics lim ρ 0 (E/A) free 0 lim ρ ρ0 (E/A) free EOS-dependent lim ρ 0 (E/A) cl 8.5 lim ρ ρ0 (E/A) cl EOS-dependent the effective dens. average dens. moderate EoS sensitivity uncertainities due to nm EOS are washed out AR, Gulminelli, Aymard, Eur.Phys.J (2014) Clusterized nuclear matter in PNS crust and thetrento, E sym April 10th, / 17

12 Bulk in-medium corrections clusters are treated as hard sphere E WS = E A,I (ρ e ) + ɛ(ρ g, δ g ) ( V WS ) A ρ 0 (A,I ) clusters can be defined by the en. dens. functional E WS = V WS ɛ [{ρ i (r), τ i (r)}] d 3 r = E A,I (ρ e ) + δe bulk + δe surf + ɛ(ρ gn, ρ gp )V WS δe bulk = ɛ(ρ g, δ g ) A ρ 0 (A,I ) Clusterized nuclear matter in PNS crust and thetrento, E sym April 10th, / 17

13 E sym in unhomogeneous matter Test of the parabolic approximation (E/A) tot = e cl (< A cl >, < δ cl >, ρ g, δ g )x cl + e g (ρ g, δ g )(1 x cl ) δ = x cl < δ cl > +(1 x cl )δ g charge inv. is broken (Coulomb effect) e sym (1) = 1 2 e 2 δ 2 δ0 (ρ) e (1) sym e(ρ, δ = 1) e(ρ, δ = 0) higher order correlations Clusterized nuclear matter in PNS crust and thetrento, E sym April 10th, / 17

14 E sym (T, ρ) e sym (1) = 1 2 e 2 δ 2 δ=δ0 (ρ) symbols=cluster; lines=mixture Mixture: lim xcl 0 e sym (1) esym g 0 (e.g. T = 1 MeV, ρ < fm 3 ) lim xcl 1 e sym (1) esym cl (e.g. T = 1 MeV, ρ > 10 4 fm 3 ) x cl x g, e sym (1) >> e sym (ρ 0 ) (e.g. T = 1 MeV, < ρ < 10 4 fm 3 ) Clusterized component: comparison with multiframentation data; Yennello, Trautmann talks AR, Gulminelli, Aymard, Eur.Phys.J (2014) Clusterized nuclear matter in PNS crust and thetrento, E sym April 10th, / 17

15 E sym meaning e sym (1) = 1 2 e 2 δ 2 δ=δ0 (ρ) e sym (2) = e(ρ, δ = 1) e(ρ, δ = 0) e (1) sym e (2) sym Clusterized nuclear matter in PNS crust and thetrento, E sym April 10th, / 17

16 (e sym ) mix versus (e sym ) NM T=0.5 MeV homogeneous matter clusters+unbound nucleons (E sym ) mix inherits the (ρ 0, 0)-uncertainity of NM at all densities Clusterized nuclear matter in PNS crust and thetrento, E sym April 10th, / 17

17 Conclusions an improved NSE where clusterized and unbound components are described by the same energy functional is proposed PNS composition and energetics is investigated along β-eq. path for different effective interactions at T=0 and finite T because clusterization, the explored effective density differs from the average density the isovector properties dependence of stellar matter is reduced with respect to the one of homogeneous matter meaning of (E sym ) mix Clusterized nuclear matter in PNS crust and thetrento, E sym April 10th, / 17