The TheSymmetry Energy

Transcription

1 The TheSymmetry Energy from fromlow Lowto to High High Densities Hermann Wolter Ludwig-Maximilians-Universität München (LMU) The nuclear symmetry energy from the point of view of different systems - Nuclei - Nuclear Matter - Neutron star matter - Heavy Ion Collisions with special attention to the effect of correlations Not so much a talk, But rather collection of material for a discussion and a possible publication. I use material of many collaborators, but acknowledge particularly help of Stefan Typel Mini-Symposium on "Dynamics of Correlations in Dense Hadronic Matter" Wrocław Poland, December 18-0, 01

2 E( ρ, I)/ A = E( ρ ) + E ( ρ ) I + O( I B B heavy ion collisions in the Fermi energy regime Importance of Nuclear Symmetry Energy sym B I = N N 4 + ) +... Z Z Models for different compositions of NS Isospin Transport properties, (Multi-)fragmentation (diffusion, fractionation, migration) E sym (ρ Β ) (MeV) 0 Asy-stiff Asy-soft 1 ρ Β /ρ rel. heavy 3 ion collisions Constraints 0 p, n ± on the Slope π,k of SE from Structure and low-energy HIC L 60 ± 5MeV +,0 Isotopic ratios of flow, particle production

3 Equation-of-State and Symmetry Energy Extrapolate to infinite systems E( A,Z ) A = a v a s 1 A 1 / 3 a a ( N Z ) A neglect, but indicates the density dep. of the SE a c Z( Z A 1) + δ 4 / 3 ε( ρ, ρ 3 ) ρ3 ρ B 3 = E + + ) nm( ρb ) Esym ( ρb ) O ( ρb ρb ρb pair Neglect, since infinite in charged system 4 Expansion around symmetric system a a ( A) a v a = v A aa 1 / A s aa PD, J. Lee, NPA818, 36 (009) a a A small V a s a A 1 / 3 aa Esym( ρ0 )?

4 Equation-of-State and Symmetry Energy densityasymmetry dep. of nucl.matt. EOS of symmetric nuclear matter stiff saturation point 4 E( ρb, δ ) / A = Enm( ρb ) + Esym( ρb ) δ + O( δ ) +... ρ Β /ρ 0 Fairly well fixed! Soft! soft neutron matter EOS asymmetry δ density ρ δ = ρ ρ n n p ρ + ρ as Esym( ρ0 ) Symmetry energy: neutron - symm matter, rather unknown, e.g. Skyrme-like param.,b.a.li p asy-stiff asy-soft Quadratic dependence, since nuclear interactions symmetric under proton neutron exchange T>0 in nuclear matter. E( ρb, ρ3,t = 0 ) H( µ B, µ 3,T )

5 1.5 Representations of Symmetry Energy Parametrizations:. polynomial behaviour 3. Expansion around ρ ο correlations 4 E( ρb, δ ) / A = Enm( ρb ) + Esym( ρb ) δ + O( δ ) +... / 3 ε ( ρ / ρ ) + pot E sym = 0 E sym C ( ρ / ρ ) ( ρ ) = S 0 γ L ρ ρ K 0 sym ρ0 18 E 1 pot ρ 3 F 0 sym ρ ρ 0 ρ0 ( ) ρn ρ p δ = ρ + ρ Relation between γ and L n p Lattimer, Lim, arxiv Skyrme Typel, Brown L [MeV] Correlation between S 0 and L from mass fits S 0 [MeV] ρ 0 Correlation between L and K sym?

6 The symmetry energy in density functionals 1. Skyrme-type with momentum dependence e.g. B.A. Li, MDI e.g. BGBD (Catania) similar but slightly diff paramtrization also with momentum dependence E sym ( ρ ) A 3 0 = MeV Asystiff. Relativistic density functionals RMF parametrizations (functionals) (see above) Asy-soft

7 The Nuclear Symmetry Energy in different realistic models The EOS of symmetric and pure neutron matter in different manybody approaches C. Fuchs, H.H. Wolter, EPJA 30(006)5,(WCI book) Rel, Brueckner Nonrel. Brueckner Variational Rel. Mean field Chiral perturb. stiff The symmetry energy as the difference between symmetric and neutron matter: E sym = E neutr.matt E nucl.matt SE soft SE ist also momentum dependent effective mass m* n < m* p Different proton/neutron effective masses asy-stiff m * q m m = 1 + h k U q k 1 data m* n > m* p Isovector (Lane) potential: momentum dependence U Lane (k) 1 = β (U neutr U prot ) asy-soft

8 Comparison of symmetry energy from microscopic approaches BHF B.A. Li, Phys. Rep. 464, 113 (008) Why is symmetry energy so uncertain?? ->In-medium ρ mass, and short range tensor correlations (Xu, BA. Li, PRC81 (010) 06461); -> effective mass scaling (Dong, Kuo, Machleidt, arxiv )

9 Clusterization on nuclear matter at low density Typel, Röpke, Klähn, Blaschke, et al., PRC81, (010) Quantumstatistical model (QS) -Includes medium modification of clusters (Mott transition) -Includes correlations in the continuum (phase shifts) -needs good model for quasi-particle energies in the mean field -In principle also possible for heavier clusters Generalized Rel. Mean Field model (RMF) -Good description of higher density phase, i.e. quasiparticle energies -Includes cluster degrees of freedom with parametrized density and temperature dependent binding energies -correlation in the continuum treated effectively -Heavier clusters treated in Wigner-Seitz cell approximation (single nucleus approximation) Global approach from very low to high densities see talk by G. Röpke

10 Particle Fractions very low density: p,n Increasing density: clusters arise: deuteron first, but then α dominates Mott density: clusters melt, homogeneous p,n matter; here heavier nuclei (embedded into a gas) become important, not yet fully implemented Hempel, Schaffner- Bielich (arxiv 0911:4073): NSE, with excluded volume with procedure T=5 MeV, b=0.3 S.Typel, G. Röpke, et al., PRC 010, arxiv 0908:344 Calculation in RMF of heavy cluster in Wigner- Seitz cell in betaequilibrium

11 Systematic calculation of binding energies of nuclei at finite density in WScell approximation (at the moment for T=0) change of WS radius with increasing density for given nucleus. non-uniform electron density at low densities: stabilization of heavier nucei y electron screening, alread destabilization of light nuclei higher densities: reduction of BE, competition with uniform matter, dissolution of nuclei T>0!

12 Symmetry energy: with (solid) and without (dashed) clusters Usually: but E A not quadratic for low temperatures with clusters. Thus use:

13 T=0 Density very small etc. Tritons plus free neutrons Only tritons for n/p=/1, Most strongly bound nucleus for βthis asymmetry Only α-particles (most strongly bound symmetric nucleus Mixture of a and free neutrons linear Usually: but E A not quadratic for low temperatures with clusters. Thus use: Neutron matter has no correlations. SE changes because of correlations in symmetric matter.

14 Slides from talk by B.A.LI at NuSYM011, Smith College, Mass, USA, June 011 Some basic issues on symmetry energy at low densities neutron +proton uniform matter at density ρ and isospin asymmetry What is the isospin-dependence of the EOS of clustered matter? as density decreases δ, ρ 1 A 1 0 δ, ρ A 0 Invariance of nuclear interaction under n-p exchange, for uniform matter 4 E( ρ, δ) = E ρ, δ = 0) + ( ρ) δ + ο( δ ) 0( Esym δi, ρ 1 E i Esym ( ρ) = ( ) E ρ pure neutron matter E( ρ) symmetric nuclear matter δ δ = 0 ρ δ For clustered matter at averge density, isospin asymmetry, there is no more n δ A i A 0 n i A V = i ni δi ρ = 0 A p invariance because of the Coulmb term in the binding energy, interactions among clusters and asymmetry between proton and neutron driplines, 4 E( ρ, δ) =? E ρ ( ρ) + E ( ) E ( ) + ο( δ ) 3 0(, δ = 0) + E s 1 δ s ρ δ + s 3 ρ δ ρ

15 Isospin dependence of the EOS of clustered matter 4 E( ρ, δ ) E δ + E ρ δ + E ρ δ ρ δ + ο( δ ) 3 0 ( ρ, = 0) s1( ) s ( ) + Es3( ) Strong indication of linear dependence on δ for both n-rich and p-rich matter Quantum statistical model S. Typel, G. Ropke, T. Klahn, D. Blaschke and H.H. Wolter, PRC 81, (010) S-matrix approach J. N. De, S. K. Samaddar, PRC 78, (008) S.K. Samaddar, J.N. De, X. Vinas and M. Centelles, PRC 80, (009) E + E x E ( ρ, x ) E D = x 0 s s1 S E E s x E s x = δ = iso sp in asym m etry Uniform matter n=0.001fm -3 T=4 MeV T= MeV Clustered matter The anharmonic behavior depends on whether all mirror nuclei are included in pairs

16 Neutron star models Model for structure of NS Models for different compositions of NS A normal NS (n,p,e) or exotic NS? A question of the density dep. of the Symmetry energy Stellar matter: neutral: n,p,e,(µ) Z y = Esym ( ρ ) N Proton fraction directly related to symmetry energy:

17 Masses and Radii of Neutron Stars Mass-Radius Relation Demorest et al. (010) PSR J1614, M=1.97*-0.04

18 Testing models of the high-density symmetry energy in Neutron stars Klähn, Blaschke, Typel, Faessler, Fuchs, Gaitanos,Gregorian, Trümper, Weber, Wolter, Phys. Rev. C74 (006) Effective β-equilibrated EOS depends strongly on symmetric EOS Proton fraction and direct URCA β equilibrium and charge neutrality : direct URCA process : p n + e threshold : + ν y = y 11%, fast neutrino cooling + e N Z = y( ε sym ) Neutron star mass dep. on Symmetry Energy Fast cooling of NS: direct URCA process Cooling curves dep. On mass and process typical neutron stars PSR J1614 Demorest proton fraction x p n + e + ν Onset of direct URCA (x>1/9) + e log Temp density n [fm -3 ] log t

19 Limits on the EoS from a Bayesian analysis of NS mass-radius observations A. Steiner, J. Lattimer, E.F.Brown, arxiv Comparison to models many Skyrme models eliminated Limits on SE power law γ Synthesis of constraints..but measurement in very different density ranges! Lattimer, Lim, arxiv

20 Separation of nuclear and Coulomb contributions? Lattice structures Not discuss here Model for structure of NS Calculation in RMF of heavy cluster in Wigner-Seitz cell in beta-equilibrium

21 Exploration of the Phase Diagram of Strongly Interacting Matter by Heavy Ion Collisions Liquid-gas coexistence 1 SIS18 SIS300 Quark-hadron coexistence 0 Supernovae IIa neutron stars Z/N Isospin degree of freedom

22 Description of heavy ion collision from initial to final state: initial final thermal thermal expansion hydrodynamics transport theory Statistical models, e.g. SMM, Botvina, et al. Statistical emission in expanding system, e.g. EES, Friedmann Hydrodynamical model, e.g. Stöcker, Maruhn, et al. Transport models, e.g. BUU, QMD, AMD, etc Transport approaches essential if system is not always in equilibrium. Many observbles are determined during the evolution and not only at the end. Especially interesting questions, like the high density phase, occur when the system is still not equilibrated.

23 Investigation of the Symmetry Energy in Different Density Ranges 1. ρ<<ρ 0 : expanding fireball in Fermi-energy heavy ion collisions. cluster correlations at low density and temperature, symmetry energy finite at low density. ρ<ρ 0 : Isospin transport in Fermi energy central and peripheral collisions, (multi-)fragmentation, 3. ρ~ρ 0 : structure and low energy excitations of (asymmetric) nuclei: skin thickness, Pygmy resonances, IAS, 4. ρ>ρ 0 : Intermediate energy heavy ion collisions: light particle emission, flow, particle production, 5. ρ>>ρ 0 : Ultrarelativistic HI collisions, dependence of mixed and deconfinement phase on asymmetry? Strategy to determine Symmetry Energy in Heavy Ion Collisions: - SE is only small part of total interaction, which has ist own uncertainties - find observables, sensitive to the SE, in particular differences in p,n or isospin partner observables

24 Învestigation of the symmetry energy in heavy ion collisions Transport theory 3 4 non-relativistic: BUU f t + r p f m U p f = I Vlasov eq.; mean field isoscalar and isovector coll [ f,σ ] EOS 1 = (π ) 3 dp -body collisions dp 3 dp δ ( p [(1 f )(1 f ) f f f f (1 f )(1 f )] loss term 4 v dσ dω 1 34 isospin 1 1 dependent, 1 pp,nn,pn gain term p 4 p 3 p 4 ) 1) Relativistic equivalent available; RMF or EFT models for EOS ) Consistency between EOS and in-medium cross sections: e.g. (Dirac) Brueckner approach 3) Approximation to a much more complicated non-equilibrium quantum transport equation (Kadanoff-Baym) by neglecting finite width of particles (quasi-particle approximation) 4) Coupled neutron and poton eqs., Isovecor effects are small relative to isoscalar quantities; differences or ratios of observables to become independent of isoscalar uncertainties 5) Collision term dissipation, NO fluctuation term Boltzmann-Langevin eq.

25 isospin dependent, pp,nn,pn 3 4 non-relativistic: BUU f t + r p f m U p f = I Vlasov eq.; mean field coll isoscalar and isovector [ f,σ ] EOS 1 = (π ) 3 dp -body collisions dp 3 dp δ ( p [(1 f )(1 f ) f f f f (1 f )(1 f )] loss term 4 v dσ dω gain term p 4 p 3 p 4 ) Should an EoS including the effects of correlations and light clusters be used in a transport description? Light clusters can be treated a independent degrees of freedom in the tranport theory (in a similar way as we use it in a generalized RMF theory. One then obtains coupled transport equations. The are coupled vis collisions terms which create a cluster: Medium corrections in the formation of light charged particles in heavy ion reactions C. Kuhrts (U Rostock), M. Beyer (U Rostock), P. Danielewicz (NCSL, MSU), G. Roepke (U Rostock) Journal-ref: Phys.Rev. C63 (001) If not, not so clear: clusters are formed in a transport desription automatically. But they are not very realistic. Usually only formed at the end at low density. But here also different transport approaches diagree, e.g. BUU and QMD approaches

26 Dynamical Stochastic vs. Statistical Fragmentation Dynamic (BNV) Dynamic (BNV) Statistical (SMM) Statistical (SMM) Primary fragments Statistical analysis used in many multifragmentation analyses (SMM =statistical multifragmentation model) Secondary fragm. Dynamical analysis used in simulations of heavy ion collisions (BNV=Boltzmann- Nordheim-Vlasov~BUU

27 A statistical analysis to determine Symmetry Energy at low desnities S. Kowalski, J. Natowitz, et al.,prc (007) 64 Zn+( 9 Mo, 197 Au) at 35 AMeV Central collisions, reconstruction of fireball Determination of thermodyn. conditions as fct of v surf =v emission -v coul ~time of emission with specified conditions of density and temperature: temperature: isotope temperatures, double ratios H-He densities ρ p, ρ n, from yield ratios and bound clusters Isoscaling analysis (B.Tsang, et al., ) R µ n µ n Y ( N, Z ) T T 1 = e e e Y 1 ( N, Z ) N Z α N + β Z Isoscaling coefficients α and β Symmetry free energy α = 4Fsym Z1 Z (( A ) ( A ) 1 T ) E = F + T sym sym S ( NSE )

28 Scheme of Kowalski Interpretation 64Zn+(9Mo,197Au) at 35 AMeV, i.e. two systems for isoscaling analysis 3-source fit and complete reconstruction of participant, i.e. N,Z of source Participant emits light clusters (p,n,d,t,3he,α) and cools Earlier emitted particles have higher temperature and higher initial velocity =v surf,,, which is measured Interpret results by thermodynamics as a function of v surf, i.e. each shell is a piece of equilibrated dilute matter, of which T, ρ,

29 Results of thermodynamical analysis of heavy ion data Isoscaling coefficients Albergo temperatures A,Z of source for two reactions Extracted ρ-t relation for emitting sourse v surf decreases, time increases

30 Comparision of low-density symmetry energy to experiment: J. Natowitz, G. Röpke, S. Typel, HHW, PRL 104, 0501 (010) complete density range low densities E sym Single nucleus approx. (Wigner-Seitz), RMF Quantum Statistical model, T=1 MeV) Parametrization of nuclear symmetry energy of different stiffness (B.A. Li) Successfully reproduce the experimentally deduced symmery energy at low density. Symmetry energy is finite at very low density!

31 Dynamical Interpretations of Heavy Ion Collisions peripheral Isospin migration Coulomb barrier to Fermi energies Isospin fractionation, multifragm central Relativistic energies deepinelastic N/Z of PLF residue = isospin diffusion N/Z of neck fragment and velocity correlations pre-equil. dipole N/Z ratio of IMF s pre-equil. light particles Flow: directed and elliptic Production of particles, π,k Not go into any detail, but rather collect status of constraints from these investigations

32 Comparison to contraints: B.Tsang, Y. Zhang, et al.,prl 008 For densities below saturation E sym ( ρ ) = S 0 L ρ ρ 0 0 O 3 + ρ ρ + ρ0 ρ0 Current state of knowledge: Generally consistent with each other, but still rather uncertain. More work necessary However, take into account correlation between S 0 and L

33 Centelles, et al., PRL 10 (009): L 60 ± 5MeV

34 Present constraints on the symmetry energy at high density π+/π- ratio, Feng, et al. Au+Au, elliptic flow, FOPI Fermi Energy HIC, various observabl es, compilati on MSU π+/π- ratio B.A. Li, et al. Moving towards a determination of the symmetry energy in HIC but at higher density non-consistent results of simulations for pion observables.

35 Short summary: Definitions of the symmetry energy The symmetry energy in different systems - nuclei - nuclear matter - matter in neutron stars and supernovae, - matter in heavy ion collisions Presents contrants on the symmetry energy Thank you for attention and apologies for unfinished nature of talk because of various adverse circumstances