Cluster Correlations in Dilute Matter

Transcription

1 Cluster Correlations in Dilute Matter GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt Nuclear Astrophysics Virtual Institute NUFRA 2015 Fifth International Conference on Nuclear Fragmentation From Basic Research to Applications Kemer (Antalya), Turkey October 4 11, 2015

2 Outline Introduction Correlations and Clusters, Astrophysics and Equation of State, Nuclear Matter and Stellar Matter, Theoretical Approaches, EoS for Astrophysical Applications, Constraints, Objectives Generalized Relativistic Density Functional Features of the Model, Fit of Model Parameters, Properties of Nuclear Matter, Mass Shifts, Degeneracy Factors, Constraint from Heavy-Ion Collisions, EoS Table, Neutron Star Matter Symmetry Energy and Neutron Skins Constraints, Density Dependence and Neutron Skins, Cluster Correlations on Nuclear Surface, Experimental Study of α-clustering, Conclusions Details: S. Typel, G. Röpke, T. Klähn, D. Blaschke, H.H. Wolter, Phys. Rev. C 81 (2010) M.D. Voskresenskaya, S. Typel, Nucl. Phys. A 887 (2012) 42 G. Röpke, N.-U. Bastian, D. Blaschke, T. Klähn, S. Typel, H.H. Wolter, Nucl. Phys. A 897 (2013) 70 S. Typel, H.H. Wolter, G. Röpke, D. Blaschke, Eur. Phys. J. A 50 (2014) 17 S. Typel, Phys. Rev. C 89 (2014) M. Hempel, K. Hagel, J. Natowitz, G. Röpke, S. Typel, Phys. Rev. C 91 (2015) S. Typel, M. Oertel, T. Klähn, Phys. Part. Nuc. 46 (2015) 633 S. Typel, Phys. Part. Nuc. 46 (2015) 777 Cluster Correlations in Dilute Matter - 1

3 Introduction

4 Correlations and Clusters interacting many-body systems information on correlations in spectral functions (in general complicated structure) Cluster Correlations in Dilute Matter - 2

5 Correlations and Clusters interacting many-body systems information on correlations in spectral functions (in general complicated structure) approximation: quasiparticles with self-energies in-medium change of particle properties reduction of residual correlations Cluster Correlations in Dilute Matter - 2

6 Correlations and Clusters interacting many-body systems information on correlations in spectral functions (in general complicated structure) approximation: quasiparticles with self-energies in-medium change of particle properties reduction of residual correlations quasiparticle concept very successful in nuclear physics phenomenological mean-field models (e.g. Skyrme, Gogny, relativistic) with only nucleons as degrees of freedom treatment of pairing correlations (Bogoliubov transformation) Cluster Correlations in Dilute Matter - 2

7 Correlations and Clusters interacting many-body systems information on correlations in spectral functions (in general complicated structure) approximation: quasiparticles with self-energies in-medium change of particle properties reduction of residual correlations quasiparticle concept very successful in nuclear physics phenomenological mean-field models (e.g. Skyrme, Gogny, relativistic) with only nucleons as degrees of freedom treatment of pairing correlations (Bogoliubov transformation) correlations in dilute matter clusters as quasiparticles Cluster Correlations in Dilute Matter - 2

8 Correlations and Clusters interacting many-body systems information on correlations in spectral functions (in general complicated structure) approximation: quasiparticles with self-energies in-medium change of particle properties reduction of residual correlations quasiparticle concept very successful in nuclear physics phenomenological mean-field models (e.g. Skyrme, Gogny, relativistic) with only nucleons as degrees of freedom treatment of pairing correlations (Bogoliubov transformation) correlations in dilute matter clusters as quasiparticles exact diagonalisation of interacting many-body Hamiltonian system of independent quasiparticles = many-body states! Cluster Correlations in Dilute Matter - 2

9 Correlations and Clusters interacting many-body systems information on correlations in spectral functions (in general complicated structure) approximation: quasiparticles with self-energies in-medium change of particle properties reduction of residual correlations quasiparticle concept very successful in nuclear physics phenomenological mean-field models (e.g. Skyrme, Gogny, relativistic) with only nucleons as degrees of freedom treatment of pairing correlations (Bogoliubov transformation) correlations in dilute matter clusters as quasiparticles exact diagonalisation of interacting many-body Hamiltonian system of independent quasiparticles = many-body states! strongly interacting matter in equilibrium equation of state (EoS) thermodynamic properties and chemical composition Cluster Correlations in Dilute Matter - 2

10 Astrophysics and Equation of State essential ingredient in astrophysical model calculations: Equation(s) of State of dense matter dynamical evolution of core-collapse supernovae, neutron star mergers static properties of neutron stars conditions for nucleosynthesis energetics, chemical composition, transport properties,... X-ray: NASA/CXC/J.Hester (ASU) Optical: NASA/ESA/J.Hester & A.Loll (ASU) Infrared: NASA/JPL-Caltech/R.Gehrz (Univ. Minn.) NASA/ESA/R.Sankrit & W.Blair (Johns Hopkins Univ.) Cluster Correlations in Dilute Matter - 3

11 Astrophysics and Equation of State essential ingredient in astrophysical model calculations: Equation(s) of State of dense matter dynamical evolution of core-collapse supernovae, neutron star mergers static properties of neutron stars conditions for nucleosynthesis energetics, chemical composition, transport properties,... timescale of reactions timescale of system evolution equilibrium (thermal, chemical,... ) application of EoS reasonable X-ray: NASA/CXC/J.Hester (ASU) Optical: NASA/ESA/J.Hester & A.Loll (ASU) Infrared: NASA/JPL-Caltech/R.Gehrz (Univ. Minn.) NASA/ESA/R.Sankrit & W.Blair (Johns Hopkins Univ.) Cluster Correlations in Dilute Matter - 3

12 Astrophysics and Equation of State essential ingredient in astrophysical model calculations: Equation(s) of State of dense matter dynamical evolution of core-collapse supernovae, neutron star mergers static properties of neutron stars conditions for nucleosynthesis energetics, chemical composition, transport properties,... timescale of reactions timescale of system evolution equilibrium (thermal, chemical,... ) application of EoS reasonable simulation of core-collapse supernova Temperature, T [MeV] Baryon density, log (ρ [g/cm 3 ]) wide range of thermodynamic variables (temperature, density, isospin asymmetry) global EoS required Baryon density, n B [fm 3 ] T. Fischer, Uniwersytet Wroc lawski Y e Cluster Correlations in Dilute Matter - 3

13 Nuclear Matter and Stellar Matter nuclear matter only strongly interacting particles no electromagnetic interaction Cluster Correlations in Dilute Matter - 4

14 Nuclear Matter and Stellar Matter nuclear matter only strongly interacting particles no electromagnetic interaction below nuclear saturation density liquid-gas phase transition ( non-congruent ) coexistence of low/high-density phases with different isospin asymmetries precursor of clustering pressure p [kev fm -3 ] density n [fm -3 ] T = 10 MeV asymmetry β T = 10 MeV asymmetry β pressure p [MeV fm -3 ] 10 1 symmetric nuclear matter density n [fm -3 ] temperature T [MeV] symmetric nuclear matter chemical potential µ [MeV] Cluster Correlations in Dilute Matter - 4

15 Nuclear Matter and Stellar Matter nuclear matter only strongly interacting particles no electromagnetic interaction below nuclear saturation density liquid-gas phase transition ( non-congruent ) coexistence of low/high-density phases with different isospin asymmetries precursor of clustering stellar matter hadrons and leptons strong and electromagnetic interaction specific condition: charge neutrality formation of inhomogeneous matter new particle species, pasta phases quenching of liquid-gas phase transition lattice formation at low temperatures phase transition: liquid/gas solid pressure p [kev fm -3 ] density n [fm -3 ] T = 10 MeV asymmetry β T = 10 MeV asymmetry β pressure p [MeV fm -3 ] 10 1 symmetric nuclear matter density n [fm -3 ] temperature T [MeV] symmetric nuclear matter chemical potential µ [MeV] particle number density n i [fm -3 ] grdf, spherical Wigner-Seitz cell A heavy = Z heavy = 48.2 p n e 2 H 3 H 3 He 4 He T = 5 MeV n = 0.01 fm -3 Y p = radius r [fm] Cluster Correlations in Dilute Matter - 4

16 Theoretical Approaches to the EoS hadronic ab-initio methods with realistic interactions interactions: potential models, meson-exchange, chiral forces, RG evolved (Argonne, Urbana, Tucson-Melbourne, Nijmegen, Paris, Bonn,... ) two-body NN interaction (in vacuum) well constrained by experiment, three-body forces less, large uncertainties for YN, YY, etc. Cluster Correlations in Dilute Matter - 5

17 Theoretical Approaches to the EoS hadronic ab-initio methods with realistic interactions interactions: potential models, meson-exchange, chiral forces, RG evolved (Argonne, Urbana, Tucson-Melbourne, Nijmegen, Paris, Bonn,... ) two-body NN interaction (in vacuum) well constrained by experiment, three-body forces less, large uncertainties for YN, YY, etc. many-body methods: BHF/DBHF, SCGF, CBF, VMC, GFMC, AFDMC,... Cluster Correlations in Dilute Matter - 5

18 Theoretical Approaches to the EoS hadronic ab-initio methods with realistic interactions interactions: potential models, meson-exchange, chiral forces, RG evolved (Argonne, Urbana, Tucson-Melbourne, Nijmegen, Paris, Bonn,... ) two-body NN interaction (in vacuum) well constrained by experiment, three-body forces less, large uncertainties for YN, YY, etc. many-body methods: BHF/DBHF, SCGF, CBF, VMC, GFMC, AFDMC,... QCD-based/inspired descriptions Lattice QCD, DS, (P)NJL, bag models,... Cluster Correlations in Dilute Matter - 5

19 Theoretical Approaches to the EoS hadronic ab-initio methods with realistic interactions interactions: potential models, meson-exchange, chiral forces, RG evolved (Argonne, Urbana, Tucson-Melbourne, Nijmegen, Paris, Bonn,... ) two-body NN interaction (in vacuum) well constrained by experiment, three-body forces less, large uncertainties for YN, YY, etc. many-body methods: BHF/DBHF, SCGF, CBF, VMC, GFMC, AFDMC,... QCD-based/inspired descriptions Lattice QCD, DS, (P)NJL, bag models,... effective field theories (EFT) chiral EFT, nuclear lattice EFT Cluster Correlations in Dilute Matter - 5

20 Theoretical Approaches to the EoS hadronic ab-initio methods with realistic interactions interactions: potential models, meson-exchange, chiral forces, RG evolved (Argonne, Urbana, Tucson-Melbourne, Nijmegen, Paris, Bonn,... ) two-body NN interaction (in vacuum) well constrained by experiment, three-body forces less, large uncertainties for YN, YY, etc. many-body methods: BHF/DBHF, SCGF, CBF, VMC, GFMC, AFDMC,... QCD-based/inspired descriptions Lattice QCD, DS, (P)NJL, bag models,... effective field theories (EFT) chiral EFT, nuclear lattice EFT methods not always applicable (temperatures, asymmetries, densities) many EoS for neutron matter & neutron star matter, but no global EoS for astrophysical applications available from these approaches Cluster Correlations in Dilute Matter - 5

21 Theoretical Approaches to the EoS hadronic ab-initio methods with realistic interactions interactions: potential models, meson-exchange, chiral forces, RG evolved (Argonne, Urbana, Tucson-Melbourne, Nijmegen, Paris, Bonn,... ) two-body NN interaction (in vacuum) well constrained by experiment, three-body forces less, large uncertainties for YN, YY, etc. many-body methods: BHF/DBHF, SCGF, CBF, VMC, GFMC, AFDMC,... QCD-based/inspired descriptions Lattice QCD, DS, (P)NJL, bag models,... effective field theories (EFT) chiral EFT, nuclear lattice EFT methods not always applicable (temperatures, asymmetries, densities) many EoS for neutron matter & neutron star matter, but no global EoS for astrophysical applications available from these approaches only phenomenological models for global EoS at present Cluster Correlations in Dilute Matter - 5

22 EoS for Astrophysical Applications constituents: mostly considered are nucleons, nuclei (light/heavy/representative), leptons, photons,... Cluster Correlations in Dilute Matter - 6

23 EoS for Astrophysical Applications constituents: mostly considered are nucleons, nuclei (light/heavy/representative), leptons, photons,... models: often combination of different approaches (Skyrme/Gogny/relativistic mean-field models, NSE, virial EoS, density functionals, classical/quantum molecular dynamics,... ) Cluster Correlations in Dilute Matter - 6

24 EoS for Astrophysical Applications constituents: mostly considered are nucleons, nuclei (light/heavy/representative), leptons, photons,... models: often combination of different approaches (Skyrme/Gogny/relativistic mean-field models, NSE, virial EoS, density functionals, classical/quantum molecular dynamics,... ) global EoS used in astrophysical simulations: H&W W. Hillebrandt, K. Nomoto, R.G. Wolff, A&A 133 (1984) 175 LS180/220/375 J.M. Lattimer, F.D. Swesty, NPA 535 (1991) 331 STOS (TM1) H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, NPA 637 (1998) 435, PTP 100 (1998) 1013 Cluster Correlations in Dilute Matter - 6

25 EoS for Astrophysical Applications constituents: mostly considered are nucleons, nuclei (light/heavy/representative), leptons, photons,... models: often combination of different approaches (Skyrme/Gogny/relativistic mean-field models, NSE, virial EoS, density functionals, classical/quantum molecular dynamics,... ) global EoS used in astrophysical simulations: H&W W. Hillebrandt, K. Nomoto, R.G. Wolff, A&A 133 (1984) 175 LS180/220/375 J.M. Lattimer, F.D. Swesty, NPA 535 (1991) 331 STOS (TM1) H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, NPA 637 (1998) 435, PTP 100 (1998) 1013 HS (TM1,TMA,FSUgold,NL3,DD2,IUFSU) M. Hempel, J. Schaffner-Bielich, NPA 837 (2010) 210 SHT (NL3) G. Shen, C.J. Horowitz, S. Teige, PRC 82 (2010) , , PRC 83 (2011) SHO (FSU1.7, FSU2.1) G. Shen, C.J. Horowitz, E. O Connor, PRC 83 (2011) SFHo/SFHx A.W. Steiner, M. Hempel, T. Fischer, ApJ 774 (2013) 17 recently many more, also with additional degrees of freedom (hyperons, quarks) Cluster Correlations in Dilute Matter - 6

26 Constraints from laboratory experiments properties of nuclei: masses, charge/diffraction radii, surface thickness, giant resonances,... characteristic nuclear matter parameters (mostly indirect): saturation density, binding energy, compressibility, symmetry energy,... heavy-ion collisions (often by comparison to numerical simulations): collective flow, particle yields (e.g. light nuclei, mesons),... Cluster Correlations in Dilute Matter - 7

27 Constraints from laboratory experiments properties of nuclei: masses, charge/diffraction radii, surface thickness, giant resonances,... characteristic nuclear matter parameters (mostly indirect): saturation density, binding energy, compressibility, symmetry energy,... heavy-ion collisions (often by comparison to numerical simulations): collective flow, particle yields (e.g. light nuclei, mesons),... from calculations ab-initio theory, effective field theory: EoS of neutron matter/symmetric nuclear matter,... virial EoS: exact low-density limit at finite temperatures Cluster Correlations in Dilute Matter - 7

28 Constraints from laboratory experiments properties of nuclei: masses, charge/diffraction radii, surface thickness, giant resonances,... characteristic nuclear matter parameters (mostly indirect): saturation density, binding energy, compressibility, symmetry energy,... heavy-ion collisions (often by comparison to numerical simulations): collective flow, particle yields (e.g. light nuclei, mesons),... from calculations ab-initio theory, effective field theory: EoS of neutron matter/symmetric nuclear matter,... virial EoS: exact low-density limit at finite temperatures from astronomical observations properties of neutron stars: masses, radii, rotation, cooling,... core-collapse supernovae: neutrino signal, nucleosynthesis,... Cluster Correlations in Dilute Matter - 7

29 Objectives development of improved EoS model with: Cluster Correlations in Dilute Matter - 8

30 Objectives development of improved EoS model with: extended set of constituent particles nuclear matter: nucleons, nuclei/clusters,..., mesons, hyperons,..., quarks stellar matter: add electrons, muons, photons Cluster Correlations in Dilute Matter - 8

31 Objectives development of improved EoS model with: extended set of constituent particles nuclear matter: nucleons, nuclei/clusters,..., mesons, hyperons,..., quarks stellar matter: add electrons, muons, photons more serious consideration of correlations nucleon-nucleon correlations: clustering, pairing Pauli principle: dissolution of composite particles in medium (Mott effect) electromagnetic correlations: essential for solidification/melting Cluster Correlations in Dilute Matter - 8

32 Objectives development of improved EoS model with: extended set of constituent particles nuclear matter: nucleons, nuclei/clusters,..., mesons, hyperons,..., quarks stellar matter: add electrons, muons, photons more serious consideration of correlations nucleon-nucleon correlations: clustering, pairing Pauli principle: dissolution of composite particles in medium (Mott effect) electromagnetic correlations: essential for solidification/melting better constrained model parameters constraints: properties of nuclei, compact stars, heavy-ion collisions Cluster Correlations in Dilute Matter - 8

33 Objectives development of improved EoS model with: extended set of constituent particles nuclear matter: nucleons, nuclei/clusters,..., mesons, hyperons,..., quarks stellar matter: add electrons, muons, photons more serious consideration of correlations nucleon-nucleon correlations: clustering, pairing Pauli principle: dissolution of composite particles in medium (Mott effect) electromagnetic correlations: essential for solidification/melting better constrained model parameters constraints: properties of nuclei, compact stars, heavy-ion collisions correct treatment of phase transitions distinguish nuclear matter and stellar matter non-congruent phase transitions, gas/liquid/solid(crystal) phases Cluster Correlations in Dilute Matter - 8

34 Objectives development of improved EoS model with: extended set of constituent particles nuclear matter: nucleons, nuclei/clusters,..., mesons, hyperons,..., quarks stellar matter: add electrons, muons, photons more serious consideration of correlations nucleon-nucleon correlations: clustering, pairing Pauli principle: dissolution of composite particles in medium (Mott effect) electromagnetic correlations: essential for solidification/melting better constrained model parameters constraints: properties of nuclei, compact stars, heavy-ion collisions correct treatment of phase transitions distinguish nuclear matter and stellar matter non-congruent phase transitions, gas/liquid/solid(crystal) phases challenge: covering of full range of thermodynamic variables in a unified model Cluster Correlations in Dilute Matter - 8

35 Generalized Relativistic Density Functional

36 Features of the Model I extension of relativistic mean-field (RMF) models basic constituents: nucleons (n,p), mesons (ω, σ, ρ), photons (γ), hyperons (optional) minimal coupling of mesons/photons to nucleons density-dependent meson-nucleon couplings suggested by Dirac-Brueckner calculations of nuclear matter meson-nucleon coupling Γ i (ρ) DD vector density ρ [fm -3 ] ω σ ρ Cluster Correlations in Dilute Matter - 9

37 Features of the Model I extension of relativistic mean-field (RMF) models basic constituents: nucleons (n,p), mesons (ω, σ, ρ), photons (γ), hyperons (optional) minimal coupling of mesons/photons to nucleons density-dependent meson-nucleon couplings suggested by Dirac-Brueckner calculations of nuclear matter meson-nucleon coupling Γ i (ρ) DD vector density ρ [fm -3 ] thermodynamically consistent formulation rearrangement contributions to self-energies and thermodynamic quantities ω σ ρ Cluster Correlations in Dilute Matter - 9

38 Features of the Model I extension of relativistic mean-field (RMF) models basic constituents: nucleons (n,p), mesons (ω, σ, ρ), photons (γ), hyperons (optional) minimal coupling of mesons/photons to nucleons density-dependent meson-nucleon couplings suggested by Dirac-Brueckner calculations of nuclear matter meson-nucleon coupling Γ i (ρ) DD vector density ρ [fm -3 ] thermodynamically consistent formulation rearrangement contributions to self-energies and thermodynamic quantities specific features of relativistic description: novel saturation mechanism for nuclear matter (vector vs. scalar self-energies) spin-orbit interaction automatically included no superluminal speed of sound ω σ ρ Cluster Correlations in Dilute Matter - 9

39 Features of the Model I extension of relativistic mean-field (RMF) models basic constituents: nucleons (n,p), mesons (ω, σ, ρ), photons (γ), hyperons (optional) minimal coupling of mesons/photons to nucleons density-dependent meson-nucleon couplings suggested by Dirac-Brueckner calculations of nuclear matter meson-nucleon coupling Γ i (ρ) DD vector density ρ [fm -3 ] thermodynamically consistent formulation rearrangement contributions to self-energies and thermodynamic quantities specific features of relativistic description: novel saturation mechanism for nuclear matter (vector vs. scalar self-energies) spin-orbit interaction automatically included no superluminal speed of sound phenomenological approach with 9 parameters determined from fit to properties of finite nuclei ω σ ρ Cluster Correlations in Dilute Matter - 9

40 Fit of Model Parameters properties of nuclei: binding energies, spin-orbit splittings properties of charge form factor (charge radius, diffraction radius, surface thickness) E(model) - E(exp) [MeV] spin-orbit splitting [MeV] ν0p DD NL3 underbound overbound 16 O 24 O 40 Ca 48 Ca 56 Ni 100 Sn 132 Sn 208 Pb π0p ν1p ν0f ν0f experiment DD ν1p π0g π1d ν2p π0h 16 O 48 Ca 56 Ni 132 Sn 208 Pb π1d charge radius [fm] diffraction radius [fm] surface thickness [fm] experiment DD 16 O 24 O 40 Ca 48 Ca 56 Ni 100 Sn 132 Sn 208 Pb DD: S. Typel, Phys. Rev. C 71 (2005) NL3: G. A. Lalazissis, J. König, P. Ring, Phys. Rev. C 55 (1997) 540 Cluster Correlations in Dilute Matter - 10

41 Properties of Nuclear Matter density-dependent meson-nucleon couplings with parametrization DD2 (refit of parametrization DD with experimental nucleon masses) S. Typel et al., Phys. Rev. C 81 (2010) neutron matter EoS consistent with limits of chiral EFT(N 3 LO) calculations I. Tews et al., PRL 110 (2013) , T. Krüger et al., PRC 88 (2013) energy per nucleon E/A [MeV] symmetric nuclear matter χeft (N 3 LO) DD2 neutron matter density n [fm -3 ] Cluster Correlations in Dilute Matter - 11

42 Properties of Nuclear Matter density-dependent meson-nucleon couplings with parametrization DD2 (refit of parametrization DD with experimental nucleon masses) S. Typel et al., Phys. Rev. C 81 (2010) neutron matter EoS consistent with limits of chiral EFT(N 3 LO) calculations I. Tews et al., PRL 110 (2013) , T. Krüger et al., PRC 88 (2013) energy per nucleon E/A [MeV] symmetric nuclear matter χeft (N 3 LO) DD2 neutron matter density n [fm -3 ] E/N-m n [MeV] [10 14 g cm -3 ] Chiral EFT N 3 LO DD2 NL3 TM1 TMA SFHo SFHx FSUgold IUFSU LS180 LS220 QB139 S n [fm -3 ] T. Fischer et al., EPJA 50 (2014) 46 Cluster Correlations in Dilute Matter - 11

43 Properties of Nuclear Matter density-dependent meson-nucleon couplings with parametrization DD2 (refit of parametrization DD with experimental nucleon masses) S. Typel et al., Phys. Rev. C 81 (2010) neutron matter EoS consistent with limits of chiral EFT(N 3 LO) calculations I. Tews et al., PRL 110 (2013) , T. Krüger et al., PRC 88 (2013) energy per nucleon E/A [MeV] symmetric nuclear matter χeft (N 3 LO) DD2 neutron matter density n [fm -3 ] L (MeV) H&W LS NL3 TM1 TMA DD2 FSUgold IUFSU SFHo SFHx J (MeV) E/N-m n [MeV] [10 14 g cm -3 ] Chiral EFT N 3 LO DD2 NL3 TM1 TMA SFHo SFHx FSUgold IUFSU LS180 LS220 QB139 S n [fm -3 ] T. Fischer et al., EPJA 50 (2014) 46 Cluster Correlations in Dilute Matter - 11

44 Properties of Nuclear Matter density-dependent meson-nucleon couplings with parametrization DD2 (refit of parametrization DD with experimental nucleon masses) S. Typel et al., Phys. Rev. C 81 (2010) neutron matter EoS consistent with limits of chiral EFT(N 3 LO) calculations I. Tews et al., PRL 110 (2013) , T. Krüger et al., PRC 88 (2013) DD2: very reasonable nuclear matter parameters n sat = fm 3 a V = MeV K = MeV J = MeV L = MeV L (MeV) H&W LS NL3 TM1 TMA DD2 FSUgold IUFSU SFHo SFHx J (MeV) E/N-m n [MeV] energy per nucleon E/A [MeV] symmetric nuclear matter density n [fm -3 ] [10 14 g cm -3 ] Chiral EFT N 3 LO DD2 NL3 TM1 TMA SFHo SFHx FSUgold IUFSU LS180 LS220 QB139 S 0.7 χeft (N 3 LO) DD2 neutron matter n [fm -3 ] T. Fischer et al., EPJA 50 (2014) 46 Cluster Correlations in Dilute Matter - 11

45 Features of the Model II stellar matter strong and electromagnetic interaction hadrons and leptons specific condition: charge neutrality Cluster Correlations in Dilute Matter - 12

46 Features of the Model II stellar matter strong and electromagnetic interaction hadrons and leptons specific condition: charge neutrality additional hadronic degrees of freedom: light nuclei ( 2 H, 3 H, 3 He, 4 He) heavy nuclei (A > 4), full table proton number Z DZ modified DZ original AME neutron number N Cluster Correlations in Dilute Matter - 12

47 Features of the Model II stellar matter strong and electromagnetic interaction hadrons and leptons specific condition: charge neutrality additional hadronic degrees of freedom: light nuclei ( 2 H, 3 H, 3 He, 4 He) heavy nuclei (A > 4), full table experimental binding energies: AME 2012 (M. Wang et al., Chinese Phys. 36 (2012) 1603) extension: DZ10 predictions (J. Duflo, A.P. Zuker, Phys. Rev. C 52 (1995) R23) shell effects included, not only average heavy nucleus proton number Z DZ modified DZ original AME neutron number N Cluster Correlations in Dilute Matter - 12

48 Features of the Model II stellar matter strong and electromagnetic interaction hadrons and leptons specific condition: charge neutrality additional hadronic degrees of freedom: light nuclei ( 2 H, 3 H, 3 He, 4 He) heavy nuclei (A > 4), full table experimental binding energies: AME 2012 (M. Wang et al., Chinese Phys. 36 (2012) 1603) extension: DZ10 predictions proton number Z DZ modified DZ original AME neutron number N (J. Duflo, A.P. Zuker, Phys. Rev. C 52 (1995) R23) shell effects included, not only average heavy nucleus considered as quasi-particles with scalar and vector potentials additional medium modifications of composite particles (mass shifts, internal excitations) dissolution of nuclei, Mott effect NN scattering correlations included correct low-density limit, virial EoS Details: S. Typel et al., Phys. Rev. C 81 (2010) , Eur. Phys. J. A 50 (2014) 17, M.D. Voskresenskaya and S. Typel, Nucl. Phys. A 887 (2012) 42, M. Hempel et al., Phys. Rev. C 91 (2015) Cluster Correlations in Dilute Matter - 12

49 Mass Shifts concept applies to composite particles: clusters light and heavy nuclei nucleon-nucleon continuum correlations Cluster Correlations in Dilute Matter - 13

50 Mass Shifts concept applies to composite particles: clusters light and heavy nuclei nucleon-nucleon continuum correlations two major contributions to mass shifts m i = E (strong) i + E (Coul) i Cluster Correlations in Dilute Matter - 13

51 Mass Shifts concept applies to composite particles: clusters light and heavy nuclei nucleon-nucleon continuum correlations two major contributions to mass shifts m i = E (strong) i + E (Coul) i strong shift E (strong) i effects of strong interaction Pauli exclusion principle: blocking of states in the medium reduction of binding energies dissolution at high densities: Mott effect (no excluded-volume mechanism) binding energy B d [MeV] binding energy B h [MeV] T = 0 MeV T = 5 MeV T = 10 MeV T = 15 MeV T = 20 MeV density n [fm -3 ] H density n [fm -3 ] binding energy B t [MeV] binding energy B α [MeV] density n [fm -3 ] 30 3 He 25 4 He H density n [fm -3 ] Cluster Correlations in Dilute Matter - 13

52 (strong) Ei strong shift effects of strong interaction Pauli exclusion principle: blocking of states in the medium reduction of binding energies dissolution at high densities: Mott effect (no excluded-volume mechanism) (Coul) electromagnetic shift Ei only in stellar matter electron screening of Coulomb field increase of binding energies Cluster Correlations in Dilute Matter - 13 T = 0 MeV T = 5 MeV T = 10 MeV T = 15 MeV T = 20 MeV H He H density n [fm ] binding energy Bt [MeV] 3 binding energy Bα [MeV] (Coul) + Ei binding energy Bh [MeV] (strong) mi = Ei density n [fm ] He density n [fm ] density n [fm ] 10 8 BE/A [MeV] concept applies to composite particles: clusters light and heavy nuclei nucleon-nucleon continuum correlations two major contributions to mass shifts binding energy Bd [MeV] Mass Shifts 6 AME n = fm -3 n = fm -3 n = fm -3 2 n = fm T = 0 MeV -3 n = fm -3 n = fm -3 n = fm A

53 Degeneracy Factors of Nuclei g i (T) = g (gs) i + Emax 0 dε i(ε)exp( ε/t) contribution of ground-state g (gs) i = 2J (gs) i + 1 (experimental values if available) contributions of excited states with density of states i(ε) widely used: Fermi gas model with backshift i, level density parameter a i problem: divergence for ε i 0 Cluster Correlations in Dilute Matter - 14

54 Degeneracy Factors of Nuclei g i (T) = g (gs) i + Emax 0 dε i(ε)exp( ε/t) contribution of ground-state g (gs) i = 2J (gs) i + 1 (experimental values if available) contributions of excited states with density of states i(ε) widely used: Fermi gas model with backshift i, level density parameter a i problem: divergence for ε i 0 here: improved model with explicit separation of ground and excited states (M.K. Grossjean, H. Feldmeier, Nucl. Phys. A 444 (1985) 113) i(ε) = π 24 a i a (n) i a (p) i ( exp β i ε+ a ) i β i (β i ε 3 ) 1/2 ( 1 exp [ ( β iεexp a ) i β i a i β i )] 1/2 a 2 i β i 2 = a i ε [ ( )] 1 exp a i β i a i = a (n) i +a (p) i, no divergence for ε 0 Cluster Correlations in Dilute Matter - 14

55 Degeneracy Factors of Nuclei g i (T) = g (gs) i + Emax 0 dε i(ε)exp( ε/t) contribution of ground-state g (gs) i = 2J (gs) i + 1 (experimental values if available) contributions of excited states with density of states i(ε) widely used: Fermi gas model with backshift i, level density parameter a i problem: divergence for ε i 0 here: improved model with explicit separation of ground and excited states (M.K. Grossjean, H. Feldmeier, Nucl. Phys. A 444 (1985) 113) i(ε) = π 24 a i = a (n) i a i a (n) i a (p) i ( exp β i ε+ a ) i β i (β i ε 3 ) 1/2 ( 1 exp [ ( β iεexp +a (p) i, no divergence for ε 0 a ) i β i a i β i )] 1/2 a 2 i β i 2 = a i ε [ ( )] 1 exp a i β i further modifications: pairing correction, high-temperature cut-off (at flashing temperature ) dissociation of nuclei Cluster Correlations in Dilute Matter - 14

56 Constraint from Heavy-Ion Collisions emission of light nuclei in heavy-ion collisions at Fermi energies determination of density and temperature S. Kowalski et al. PRC 75 (2007) J. Natowitz et al. PRL 104 (2010) R. Wada et al. PRC 85 (2012) Cluster Correlations in Dilute Matter - 15

57 Constraint from Heavy-Ion Collisions emission of light nuclei in heavy-ion collisions at Fermi energies determination of density and temperature S. Kowalski et al. PRC 75 (2007) J. Natowitz et al. PRL 104 (2010) R. Wada et al. PRC 85 (2012) thermodynamic conditions as in neutrinosphere of cc supernovae ( femtonovae ) Cluster Correlations in Dilute Matter - 15

58 Constraint from Heavy-Ion Collisions emission of light nuclei in heavy-ion collisions at Fermi energies chemical equilibrium constants of α particles from M. Hempel et al., PRC 91 (2015) determination of density and temperature S. Kowalski et al. PRC 75 (2007) J. Natowitz et al. PRL 104 (2010) R. Wada et al. PRC 85 (2012) thermodynamic conditions as in neutrinosphere of cc supernovae ( femtonovae ) particle yields chemical equilibrium constants K c [i] = n i /(n Z i p n N i n ) L. Qin et al., PRL 108 (2012) K c [ ] (fm 9 ) Exp. (Qin et al. 2012) ideal gas HS(DD2), no CS, A 4 SFHo, no CS, A 4 LS220, HIC mod., cor. B STOS, HIC mod. SHT(NL3) SHO(FSU2.1) grdf QS T (MeV) Cluster Correlations in Dilute Matter - 15

59 Constraint from Heavy-Ion Collisions emission of light nuclei in heavy-ion collisions at Fermi energies chemical equilibrium constants of α particles from M. Hempel et al., PRC 91 (2015) determination of density and temperature S. Kowalski et al. PRC 75 (2007) J. Natowitz et al. PRL 104 (2010) R. Wada et al. PRC 85 (2012) thermodynamic conditions as in neutrinosphere of cc supernovae ( femtonovae ) particle yields chemical equilibrium constants K c [i] = n i /(n Z i p n N i n ) L. Qin et al., PRL 108 (2012) mixture of ideal gases/nse description not sufficient medium effects/correlations important K c [ ] (fm 9 ) Exp. (Qin et al. 2012) ideal gas HS(DD2), no CS, A 4 SFHo, no CS, A 4 LS220, HIC mod., cor. B STOS, HIC mod. SHT(NL3) SHO(FSU2.1) grdf QS T (MeV) Cluster Correlations in Dilute Matter - 15

60 Equation of State Table range of variables temperature: 0.1 MeV T 100 MeV 76 mesh points baryon density: fm 3 n b 1 fm mesh points hadronic charge fraction: 0.01 Y q mesh points in total mesh points Cluster Correlations in Dilute Matter - 16

61 Equation of State Table range of variables temperature: 0.1 MeV T 100 MeV 76 mesh points baryon density: fm 3 n b 1 fm mesh points hadronic charge fraction: 0.01 Y q mesh points in total mesh points information on thermodynamic properties (pressure, entropy, energies, chemical potentials) chemical composition (nucleons, leptons, light and heavy clusters) microscopic quantities (mean-field potentials) Cluster Correlations in Dilute Matter - 16

62 Equation of State Table range of variables temperature: 0.1 MeV T 100 MeV 76 mesh points baryon density: fm 3 n b 1 fm mesh points hadronic charge fraction: 0.01 Y q mesh points in total mesh points information on thermodynamic properties (pressure, entropy, energies, chemical potentials) chemical composition (nucleons, leptons, light and heavy clusters) microscopic quantities (mean-field potentials) availability data tables will be released on CompOSE website ( Cluster Correlations in Dilute Matter - 16

63 Neutron Star Matter I conditions: charge neutrality and β equilibrium dependence on baryon density for constant temperature hadronic charge fraction hadronic charge fraction Y q T = MeV β equilibrium baryon density n b [fm -3 ] pressure pressure p [MeV fm -3 ] β equilibrium baryon density n b [fm -3 ] T = MeV Cluster Correlations in Dilute Matter - 17

64 Neutron Star Matter I conditions: charge neutrality and β equilibrium dependence on baryon density for constant temperature hadronic charge fraction hadronic charge fraction Y q T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV baryon density n b [fm -3 ] β equilibrium pressure pressure p [MeV fm -3 ] β equilibrium baryon density n b [fm -3 ] T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV Cluster Correlations in Dilute Matter - 17

65 Neutron Star Matter I conditions: charge neutrality and β equilibrium dependence on baryon density or baryon chemical potential for constant temperature hadronic charge fraction hadronic charge fraction Y q T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV baryon density n b [fm -3 ] β equilibrium pressure no phase transition pressure p [MeV fm -3 ] β equilibrium T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV baryon chemical potential µ b -m n [MeV] Cluster Correlations in Dilute Matter - 17

66 Neutron Star Matter II conditions: charge neutrality and β equilibrium dependence on baryon density for constant temperature mass fraction of heavy nuclei 1.1 average mass number 140 mass fraction of heavy nuclei X A T = MeV β equilibrium average mass number of heavy nuclei <A> T = MeV β equilibrium baryon density n b [fm -3 ] baryon density n b [fm -3 ] Cluster Correlations in Dilute Matter - 18

67 Neutron Star Matter II conditions: charge neutrality and β equilibrium dependence on baryon density for constant temperature mass fraction of heavy nuclei 1.1 average mass number 140 mass fraction of heavy nuclei X A T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV β equilibrium average mass number of heavy nuclei <A> T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV β equilibrium baryon density n b [fm -3 ] baryon density n b [fm -3 ] Cluster Correlations in Dilute Matter - 18

68 Neutron Star Matter III conditions: charge neutrality and β equilibrium dependence on baryon density for constant temperature average neutron number 100 average proton number 60 average neutron number of heavy nuclei <N> T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV β equilibrium average proton number of heavy nuclei <Z> T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV T = MeV β equilibrium baryon density n b [fm -3 ] baryon density n b [fm -3 ] Cluster Correlations in Dilute Matter - 19

69 Symmetry Energy and Neutron Skins

70 Constraints density dependence of symmetry energy E s (n) in nuclear matter E A (n,β) = E 0(n) + E s (n)β n = n n + n p β = (n n n p )/n Cluster Correlations in Dilute Matter - 20

71 Constraints density dependence of symmetry energy E s (n) in nuclear matter E A (n,β) = E 0(n) + E s (n)β n = n n + n p β = (n n n p )/n symmetry energy at saturation J = E s (n sat ) slope coefficient L = 3n d dn E s n=nsat Cluster Correlations in Dilute Matter - 20

72 Constraints density dependence of symmetry energy E s (n) in nuclear matter E A (n,β) = E 0(n) + E s (n)β n = n n + n p β = (n n n p )/n symmetry energy at saturation J = E s (n sat ) slope coefficient L = 3n d many efforts to determine J = S v and L experimentally dn E s n=nsat χ Lagrangian and Q. Montecarlo Neutron Matter Neutron-Star Neutron Star Data Observations p & α scattering Neutron Skin charge ex. Antiprotonic Atoms Nuclear Model Fit Heavy Ion Collisions Giant Resonances N A scattering Charge Ex. Reactions Energy Levels Parity Violating e-scattering IVGQR Neutron Skin PDR Neutron Skin Mass Mass n p Emission Ratio Isospin Diffusion GDR PDR N Optical Potential Empirical Dipole Polarizability Parity Violating Asymmetry L (MeV) Hebeler et al. PRL105 (2010) and Gandolfi et al. PRC85 (2012) (R) Steiner et al. Astrophys. J. 722 (2010) 33 Lie-Wen Chen et al. PRC 82 (2010) Centelles et al. PRL 102 (2009) Warda et al. PRC 80 (2009) Möller et al. PRL 108 (2012) Danielewicz NPA 727 (2003) 233 Agrawal et al. PRL109 (2012) Famiano et al. PRL 97 (2006) Tsang et al. PRL 103 (2009) Roca-Maza et al. PRC 87 (2013) Roca-Maza et al. PRC (2013), in press Trippa et al. PRC 77 (2008) (R) Klimkiewicz et al. PRC 76 (2007) (R) Carbone et al. PRC 81 (2010) (R) Xu et al. PRC 82 (2010) PREX Collab. PRL (2012) (X. Viñas et al., EPJA 50 (2014) 27) (J.M. Lattimer, A.W. Steiner, EPJA 50 (2014) 40) Cluster Correlations in Dilute Matter - 20

73 Density Dependence and Neutron Skins correlation: neutron skin thickness r np = S = rn 2 1/2 rp 2 1/2 derivative of neutron matter EoS B.A. Brown, PRL 85 (2000) 5296 S. Typel and B. A. Brown, PRC 64 (2001) Cluster Correlations in Dilute Matter - 21

74 Density Dependence and Neutron Skins correlation: neutron skin thickness rnp = S = r 2 n 1/2 r 2 p 1/2 derivative of neutron matter EoS B.A. Brown, PRL 85 (2000) 5296 S. Typel and B. A. Brown, PRC 64 (2001) symmetry energy slope parameter L NL1 NL2 NL3* 0.3 NL3 PK1 NL-SV2 G2 Sk-T4 NL3.s25 PK1.s FSUGold Sk-Rs Ska DD-PC1 DD-ME1 DD-ME2 0.2 rnp (fm) SkM* HFB-17 SkP 0.15 HFB-8 MSk7 v Linear Fit, r = Skyrme, Gogny RMF TM1 SkI5 G1 NL-RA1 PC-F1 NL-SH PC-PK1 SkX Sk-T6 SkI2 RHF-PKA1 SV Sk-Gs RHF-PKO3 208 Pb SkMP SkSM* SIV MSL0 MSkA BCP SLy5 SLy4 D1N D1S SGII L (MeV) (X. Viñas et al., Eur. Phys. J. A50 (2014) 27) Cluster Correlations in Dilute Matter - 21

75 Density Dependence and Neutron Skins correlation: neutron skin thickness r np = S = r 2 n 1/2 r 2 p 1/2 derivative of neutron matter EoS B.A. Brown, PRL 85 (2000) 5296 S. Typel and B. A. Brown, PRC 64 (2001) symmetry energy slope parameter L determine L from experimental measurement of r np parity violation in electron scattering PREX@Jefferson Lab, C.J. Horowitz et al., PRC 63 (2001) , PRC 85 (2012) coherent pion photoproduction MAMI@Mainz, C. Tarbert et al., PRL 112 (2014) r np (fm) Linear Fit, r = Skyrme, Gogny RMF SkX Sk-T6 HFB-17 SkM* FSUGold DD-ME1 DD-ME2 SkMP SkSM* SIV MSL0 MSkA Ska DD-PC1 Sk-Rs PK1.s24 G2 Sk-T4 NL3.s25 RHF-PKO3 SkI2 RHF-PKA1 SV Sk-Gs NL3* NL3 PK1 NL-SV2 TM1 SkI5 G1 NL-RA1 PC-F1 NL-SH PC-PK1 NL2 NL HFB-8 MSk7 v090 SkP SGII D1N SLy5 SLy4 BCP 208 Pb D1S L (MeV) (X. Viñas et al., Eur. Phys. J. A50 (2014) 27) Cluster Correlations in Dilute Matter - 21

76 Density Dependence and Neutron Skins correlation: neutron skin thickness r np = S = r 2 n 1/2 r 2 p 1/2 derivative of neutron matter EoS B.A. Brown, PRL 85 (2000) 5296 S. Typel and B. A. Brown, PRC 64 (2001) symmetry energy slope parameter L determine L from experimental measurement of r np parity violation in electron scattering PREX@Jefferson Lab, C.J. Horowitz et al., PRC 63 (2001) , PRC 85 (2012) coherent pion photoproduction MAMI@Mainz, C. Tarbert et al., PRL 112 (2014) correlation based on mean-field models, low densities at nuclear surface effects of correlations? r np (fm) HFB-8 MSk7 v090 SkP D1S Linear Fit, r = Skyrme, Gogny RMF SkX Sk-T6 HFB-17 SGII D1N SkM* FSUGold DD-ME1 DD-ME2 SLy5 SLy4 SkMP SkSM* SIV MSL0 MSkA BCP Ska DD-PC1 Sk-Rs PK1.s24 G2 Sk-T4 NL3.s25 SkI2 RHF-PKA1 SV Sk-Gs L (MeV) RHF-PKO3 (X. Viñas et al., Eur. Phys. J. A50 (2014) 27) NL3* NL3 PK1 NL-SV2 TM1 SkI5 G1 NL-RA1 PC-F1 NL-SH PC-PK1 NL2 NL1 208 Pb Cluster Correlations in Dilute Matter - 21

77 Cluster Correlations on Nuclear Surface finite temperature grdf calculations in spherical Wigner-Seitz cell, extended Thomas-Fermi approximation with nucleons, electrons and light clusters enhanced probability to find clusters at surface of heavy nuclei particle number density n i [fm -3 ] A heavy = Z heavy = 48.2 p n e 2 H 3 H 3 He 4 He T = 5 MeV n = 0.01 fm -3 Y p = radius r [fm] Cluster Correlations in Dilute Matter - 22

78 Cluster Correlations on Nuclear Surface finite temperature grdf calculations in spherical Wigner-Seitz cell, extended Thomas-Fermi approximation with nucleons, electrons and light clusters enhanced probability to find clusters at surface of heavy nuclei extension of grdf model to zero temperature: only α-particles relevant (density distribution from ground-state wave function in WKB approximation) particle number density n i [fm -3 ] A heavy = Z heavy = 48.2 p n e 2 H 3 H 3 He 4 He T = 5 MeV n = 0.01 fm -3 Y p = radius r [fm] Cluster Correlations in Dilute Matter - 22

79 Cluster Correlations on Nuclear Surface finite temperature grdf calculations in spherical Wigner-Seitz cell, extended Thomas-Fermi approximation with nucleons, electrons and light clusters enhanced probability to find clusters at surface of heavy nuclei extension of grdf model to zero temperature: only α-particles relevant (density distribution from ground-state wave function in WKB approximation) variation with neutron excess of nuclei study chain of Sn nuclei neutron and proton rms radii r n and r p neutron skin thickness r skin = r n r p neutron (proton) rms radius r n (r p ) [fm] neutron skin thickness r skin [fm] neutrons (without α correlations) protons (without α correlations) mass number A without α correlations mass number A Cluster Correlations in Dilute Matter - 22

80 Cluster Correlations on Nuclear Surface finite temperature grdf calculations in spherical Wigner-Seitz cell, extended Thomas-Fermi approximation with nucleons, electrons and light clusters enhanced probability to find clusters at surface of heavy nuclei extension of grdf model to zero temperature: only α-particles relevant (density distribution from ground-state wave function in WKB approximation) variation with neutron excess of nuclei study chain of Sn nuclei neutron and proton rms radii r n and r p neutron skin thickness r skin = r n r p systematic reduction of neutron skin experimental test? neutron (proton) rms radius r n (r p ) [fm] neutron skin thickness r skin [fm] neutrons (without α correlations) neutrons (with α correlations) protons (without α correlations) protons (with α correlations) mass number A without α correlations with α correlations S. Typel, PRC 89 (2014) mass number A Cluster Correlations in Dilute Matter - 22

81 Experimental Study of α-clustering at Nuclear Surface I clean probe for α-particles in nuclei: quasi-free (p,pα) knockout reactions Cluster Correlations in Dilute Matter - 23

82 Experimental Study of α-clustering at Nuclear Surface I clean probe for α-particles in nuclei: quasi-free (p,pα) knockout reactions experiment at RCNP Cyclotron Facility, Osaka, Japan collaboration with T. Aumann (TU Darmstadt), T. Uesaka (RIKEN) et al. proposal accepted in PAC evaluation March 2014, experiment in June 2015 Cluster Correlations in Dilute Matter - 23

83 Experimental Study of α-clustering at Nuclear Surface I clean probe for α-particles in nuclei: quasi-free (p,pα) knockout reactions experiment at RCNP Cyclotron Facility, Osaka, Japan collaboration with T. Aumann (TU Darmstadt), T. Uesaka (RIKEN) et al. proposal accepted in PAC evaluation March 2014, experiment in June MeV proton beam, 100 na current targets: nat Li (test), 112 Sn proton detection: Grand Raiden α detection: LAS spectrometer settings optimized to quasi-free kinematics p Sn α Cluster Correlations in Dilute Matter - 23