Clusters in Dense Matter and the Equation of State

Clusters in Dense Matter and the Equation of State Excellence Cluster Universe, Technische Universität München GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt in collaboration with Gerd Röpke (Universität Rostock) Thomas Klähn (Uniwersytet Wroc lawski) David Blaschke (Uniwersytet Wroc lawski) Hermann Wolter (LMU München) Maria Voskresenskaya (GSI Darmstadt) 11th Symposium on Nuclei in the Cosmos Heidelberg 19-23 July 2010

The Life and Death of Stars when the era of nuclear fusion reactions ends: last phases in the life of a massive star (8M sun M star 30M sun ) core-collapse supernova neutron star or black hole 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.) Clusters and EoS - 1

The Life and Death of Stars when the era of nuclear fusion reactions ends: last phases in the life of a massive star (8M sun M star 30M sun ) core-collapse supernova neutron star or black hole essential ingredient in astrophysical model calculations: equation of state (EoS) of dense matter dynamical evolution of supernova static properties of neutron star 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.) Clusters and EoS - 1

Thermodynamical Conditions densities: 10 9 / sat 10 ( sat 2.5 10 14 g/cm 3 ) temperatures: 0 MeV k B T 25 MeV (ˆ= 2.9 10 11 K) electron fraction: 0 Y e 0.6 M. Liebendörfer, R. Käppeli, S. Scheidegger, Universität Basel Clusters and EoS - 2

Thermodynamical Conditions densities: 10 9 / sat 10 ( sat 2.5 10 14 g/cm 3 ) temperatures: 0 MeV k B T 25 MeV (ˆ= 2.9 10 11 K) electron fraction: 0 Y e 0.6 global theoretical description of matter properties is required M. Liebendörfer, R. Käppeli, S. Scheidegger, Universität Basel Clusters and EoS - 2

Equation of State of Dense Matter many EoS developed in the past: from simple parametizations to sophisticated models many investigations of detailed aspects: often restricted to particular conditions (e.g. zero temperature) only few EoS used in astrophysical models: most well known J.M. Lattimer, F.D. Swesty (Nucl. Phys. A 535 (1991) 331) H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi (Prog. Theor. Phys. 100 (1998) 1013) most difficult problem: description of strongly interacting subsystem (hadronic or quark matter) in this talk: formation of clusters in nuclear matter in standard astrophysical EoS: only nucleons, α particle and representative heavy nucleus, suppression of cluster formation with phenomenological excluded-volume mechanism consider more microscopic model, more clusters Clusters and EoS - 3

Composition of Nuclear Matter I depends strongly on density, temperature and neutron-proton asymmetry affects thermodynamical properties Clusters and EoS - 4

Composition of Nuclear Matter I depends strongly on density, temperature and neutron-proton asymmetry affects thermodynamical properties theoretical models: different points of view chemical picture: mixture of different nuclear species and nucleons in chemical equilibrium problems: properties of constituents independent of medium interaction between particles dissolution of nuclei at high densities physical picture: correlations of nucleons formation of bound states problems: treatment of three-, four-,... many-body correlations difficult choice of interaction combination of approaches? Clusters and EoS - 4

Composition of Nuclear Matter II low densities: mixture of nuclei and nucleons models with nuclei in statistical equilibrium (NSE, virial expansion, Beth-Uhlenbeck,... ) Clusters and EoS - 5

Composition of Nuclear Matter II low densities: mixture of nuclei and nucleons models with nuclei in statistical equilibrium (NSE, virial expansion, Beth-Uhlenbeck,... ) high densities (around/above nuclear saturation): homogeneous and isotropic neutron-proton matter mean-field models (Skyrme Hartree-Fock, relativistic mean-field,... ) in between at low temperatures: liquid-gas phase transition surface effects and long-range Coulomb interaction inhomogeneous matter formation of pasta phases/lattice structures Clusters and EoS - 5

Quantum Statistical Approach I nonrelativistic finite-temperature Green s function formalism starting point: nucleon number densities (τ = p, n) n τ (T, µ p, µ n ) = 2 d 3 k (2π) 3 dω 2π f τ(ω)s τ (ω) with Fermi distribution f τ (ω) and spectral function S τ (ω) depending on self-energy Σ τ expansion of spectral function beyond quasiparticle approximation generalized Beth-Uhlenbeck descripton with medium dependent self-energy shifts/binding energies generalized scattering phase shifts from in-medium T-matrix T, n p, n n µ p, µ n free energy F(T, n p, n n ) by integration ( (F/V ) nτ thermodynamically consistent derivation of EoS = µ τ ) T,nτ Clusters and EoS - 6

Quantum Statistical Approach II medium modifications single nucleon properties self-energy shift of quasiparticle energy effective mass Clusters and EoS - 7

Quantum Statistical Approach II medium modifications single nucleon properties self-energy shift of quasiparticle energy effective mass cluster properties shift of quasiparticle energy from nucleon self-energies Pauli blocking medium dependent binding energies (calculation with effective nucleon-nucleon potential) quasi-particles binding energy B d [MeV] binding energy B h [MeV] 3 2 1 0 T = 0 MeV T = 5 MeV T = 10 MeV T = 15 MeV T = 20 MeV -1 0.00 0.01 0.02 10 8 6 4 2 0 symmetric nuclear matter 2 H density n [fm -3 ] -2 0.00 0.01 0.02 density n [fm -3 ] binding energy B t [MeV] binding energy B α [MeV] 10 8 6 4 2 0-2 0.00 0.01 0.02 30 15 10 5 0 3 H density n [fm -3 ] 3 He 25 4 He 20-5 0.00 0.01 0.02 density n [fm -3 ] parametrization used in generalized RMF model Clusters and EoS - 7

Generalized Relativistic Mean-Field (RMF) Model extended relativistic Lagrangian density of Walecka type with nucleons (ψ p, ψ n ), deuterons (ϕ µ d ), tritons (ψ t), helions (ψ h ), α-particles (ϕ α ), mesons (σ, ω µ, ρ µ ), electrons (ψ e ) and photons (A µ ) as degrees of freedom only minimal (linear) meson-nucleon couplings density-dependent meson-nucleon couplings Γ i parameters: nucleon/meson masses, coupling strengths/density dependence in total 10 free parameters (highly correlated) constrained from fit to properties of finite nuclei medium-dependent cluster binding energies Clusters and EoS - 8

EoS with Light Clusters - Generalized RMF Model consider 2-, 3-, and 4-body correlations in the medium presently only bound states (deuterons, tritons, helions, and alphas) scattering contributions neglected so far Mott effect: clusters dissolve at high densities correct limits at low and high densities particle fraction X i 10 0 10-1 10-2 p n 2 H 3 H 3 He 4 He T = 6 MeV Y p = 0.4 no heavy clusters/phase transition included here 10-3 10-6 10-5 10-4 10-3 10-2 10-1 10 0 medium dependence of couplings and binding energies rearrangement contributions in self-energies and source densities essential for thermodynamical consistency density n [fm -3 ] Clusters and EoS - 9

EoS with Light Clusters - Cluster Fractions symmetric nuclear matter generalized RMF model vs. NSE (thin lines) helion fraction X h deuteron fraction X d 10 0 10 0 10-1 10-2 10-3 10-4 10-5 10-5 10-4 10-3 10-2 10-1 10 0 10-1 10-2 10-3 10-4 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] density n [fm -3 ] 10 0 10 0 10-1 10-2 10-3 10-4 2 H 3 H triton fraction X t 3 He 4 He α-particle fraction X α 10-1 10-2 10-3 10-4 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] Clusters and EoS - 10

EoS with Light Clusters - Pressure/Density symmetric nuclear matter lim n 0 (p/n) = T (ideal gas) 40 generalized RMF vs. NSE 40 QS approach vs. NSE pressure/density p/n [MeV] 30 20 10 0 pressure/density p/n [MeV] 30 20 10 0 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV -10 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] -10 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] Clusters and EoS - 11

Phase Transition - Pressure and Chemical Potential symmetric nuclear matter (Maxwell construction sufficient) RMF model without (dashed lines) and with (solid lines) clusters 10 1 940 pressure p [MeV fm -3 ] 10 0 10-1 10-2 10-3 10-4 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] chemical potential µ [MeV] 930 920 910 900 890 880 870 860 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV Clusters and EoS - 12

Heavy Clusters liquid-gas phase transition: separation of low-/high-density phases, no surface or Coulomb effects Clusters and EoS - 13

Heavy Clusters liquid-gas phase transition: separation of low-/high-density phases, no surface or Coulomb effects first step in improvement: spherical Wigner-Seitz cell calculation generalized RMF model Thomas-Fermi approximation electrons for charge compensation heavy nucleus surrounded by gas of nucleons and light clusters particle number density n i [fm -3 ] 10-1 10-2 10-3 10-4 A heavy = 169.9 Z heavy = 71.3 R cell = 16.70 fm p n e without light clusters T = 5 MeV n = 0.01 fm -3 Y p = 0.4 10-5 0 5 10 15 20 radius r [fm] Clusters and EoS - 13

Heavy Clusters liquid-gas phase transition: separation of low-/high-density phases, no surface or Coulomb effects first step in improvement: spherical Wigner-Seitz cell calculation generalized RMF model Thomas-Fermi approximation electrons for charge compensation heavy nucleus surrounded by gas of nucleons and light clusters first self-consistent calculation with interacting nucleons, light clusters and electrons particle number density n i [fm -3 ] 10-1 10-2 10-3 10-4 A heavy = 159.4 Z heavy = 66.2 R cell = 17.46 fm p n e 2 H 3 H 3 He 4 He with light clusters T = 5 MeV n = 0.01 fm -3 Y p = 0.4 10-5 0 5 10 15 20 radius r [fm] Clusters and EoS - 13

Symmetry Energy I general definition for zero temperature: E s (n) = 1 2 E 2 β 2 (n, β) β = n n n p A n n +n p β=0 nuclear matter parameters J = E s (n sat ) L = 3n d dn E s n=nsat correlation: neutron skin thickness slope of neutron matter EoS ( L) B. A. Brown, Phys. Rev. Lett. 85 (2000) 5296, S. Typel, B. A. Brown, Phys. Rev. C 64 (2001) 027302 symmetry energy E s [MeV] 100 80 60 40 20 T = 0 MeV without clusters NL1 NL3 NL-SH TM1 TW99 DD-ME1 DD D 3 C 0 0 0.1 0.2 0.3 0.4 0.5 density ρ [fm -3 ] Clusters and EoS - 14

Symmetry Energy I general definition for zero temperature: E s (n) = 1 2 E 2 β 2 (n, β) β = n n n p A n n +n p β=0 nuclear matter parameters J = E s (n sat ) L = 3n d dn E s n=nsat correlation: neutron skin thickness slope of neutron matter EoS ( L) B. A. Brown, Phys. Rev. Lett. 85 (2000) 5296, S. Typel, B. A. Brown, Phys. Rev. C 64 (2001) 027302 symmetry energy E s [MeV] 100 80 60 40 20 T = 0 MeV without clusters NL1 NL3 NL-SH TM1 TW99 DD-ME1 DD D 3 C with clusters and at finite temperatures: use finite differences [ E sym (n) = 1 E 2 A (n, 1) 2E A (n, 0) + E A (n, 1)] 0 0 0.1 0.2 0.3 0.4 0.5 density ρ [fm -3 ] effects of cluster formation? experimental observation? Clusters and EoS - 14

Symmetry Energy II temperature T = 0 MeV mean-field models without clusters e.g. model with momentum-dependent interaction (MDI), parameter x controls density dependence of E sym (B. A. Li et al., Phys. Rep. 464 (2008) 113) low-density behaviour not correct internal symmetry energy E sym (n)/e sym (n 0 ) 1.5 1.0 0.5 T = 0 MeV MDI, x = 1 MDI, x = 0 MDI, x = -1 0.0 0.5 1.0 1.5 density n/n 0 Clusters and EoS - 15

Symmetry Energy II temperature T = 0 MeV mean-field models without clusters e.g. model with momentum-dependent interaction (MDI), parameter x controls density dependence of E sym (B. A. Li et al., Phys. Rep. 464 (2008) 113) low-density behaviour not correct RMF model with (heavy) clusters increase of E sym at low densities due to formation of clusters finite symmetry energy in the limit n 0 internal symmetry energy E sym (n)/e sym (n 0 ) 1.5 1.0 0.5 T = 0 MeV RMF with clusters MDI, x = 1 MDI, x = 0 MDI, x = -1 0.0 0.5 1.0 1.5 density n/n 0 Clusters and EoS - 15

Symmetry Energy III finite temperature experimental determination of symmetry energy heavy-ion collisions of 64 Zn on 92 Mo and 197 Au at 35 A MeV temperature, density, free symmetry energy derived as functions of parameter v surf (measures time when particles leave the source) (S. Kowalski et al., Phys. Rev. C 75 (2007) 014601) symmetry energies in RMF calculation without clusters are too small very good agreement with QS calculation with light clusters free symmetry energy F sym [MeV] 20 15 10 5 (a) experiment RMF without clusters QS with clusters internal symmetry energy E sym [MeV] 20 15 10 5 (b) experiment RMF without clusters QS with clusters 0 0 1 2 3 4 5 V surf [cm/ns] 0 0 1 2 3 4 5 V surf [cm/ns] Clusters and EoS - 16

Summary and Outlook theoretical models of EoS with clusters quantum statistical approach (QS) generalized relativistic mean-field model (grmf) both thermodynamically consistent correct limits at low and high densities difference in details nuclear matter at low densities formation of clusters with medium dependent properties modification of thermodynamical properties/symmetry energies change of phase transition boundaries for details see Phys. Rev. C 81, 015803 (2010) and Phys. Rev. Lett. 104, 202501 (2010) Clusters and EoS - 17