Clusters in Dense Matter and Equation of State

Custers in Dense Matter and Equation of State GSI Hemhotzzentrum für Schwerionenforschung, Darmstadt Nucear Astrophysics Virtua Institute NEOS 2012 Nucear Equation of State for Compact Stars and Supernovae November 28 30, 2012 Frankfurt Institute for Advanced Studies (FIAS) Custers and EoS - 0

Outine Motivation Correations in Dense Matter Theoretica Approaches Generaized Reativistic Density Functiona Light and Heavy Nucei Low-Density Limit and Neutron Matter Light Custers and Continuum Correations Symmetry Energy Summary References: S. Type, G. Röpke, T. Kähn, D. Baschke, H.H. Woter, Phys. Rev. C 81 (2010) 015803 M. D. Voskresenskaya, S. Type, Nuc. Phys. A 887 (2012) 42 G. Röpke, N.-U. Bastian, D. Baschke, T. Kähn, S. Type, H.H. Woter, Nuc. Phys. A 897 (2013) 70 Custers and EoS - 1

The Life and Death of Stars when the fue for nucear fusion reactions is consumed: ast phases in the ife of a massive star (M star 8M sun ) core-coapse supernova (CC-SN) neutron star (NS) or back hoe (BH) X-ray: NASA/CXC/J.Hester (ASU) Optica: NASA/ESA/J.Hester & A.Lo (ASU) Infrared: NASA/JPL-Catech/R.Gehrz (Univ. Minn.) NASA/ESA/R.Sankrit & W.Bair (Johns Hopkins Univ.) Custers and EoS - 2

The Life and Death of Stars when the fue for nucear fusion reactions is consumed: ast phases in the ife of a massive star (M star 8M sun ) core-coapse supernova (CC-SN) neutron star (NS) or back hoe (BH) essentia ingredient in astrophysica mode cacuations: Equation of State (EoS) of dense matter dynamica evoution of supernova static properties of neutron star conditions for nuceosynthesis energetics, chemica composition, transport properties,... X-ray: NASA/CXC/J.Hester (ASU) Optica: NASA/ESA/J.Hester & A.Lo (ASU) Infrared: NASA/JPL-Catech/R.Gehrz (Univ. Minn.) NASA/ESA/R.Sankrit & W.Bair (Johns Hopkins Univ.) Custers and EoS - 2

Core-Coapse Supernovae Timescae of Reactions Timescae of System Evoution thermodynamic equiibrium construction of Equation of State simuation of CC-SN Baryon density, og 10 (ρ [g/cm 3 ]) 10 2 6 7 8 9 10 11 12 13 14 15 0.5 0.45 0.4 Temperature, T [MeV] 10 1 10 0 0.35 0.3 0.25 0.2 0.15 0.1 0.05 10 1 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Baryon density, n [fm 3 ] B Y e T. Fischer, GSI Darmstadt Custers and EoS - 3

Core-Coapse Supernovae Timescae of Reactions Timescae of System Evoution thermodynamic equiibrium construction of Equation of State Reevant Parameters: density: 10 9 / sat 10 with nucear saturation density sat 2.5 10 14 g/cm 3 (n sat = sat /m n 0.15 fm 3 ) temperature: 0 MeV k B T 50 MeV (ˆ= 5.8 10 11 K) eectron fraction: 0 Y e 0.6 simuation of CC-SN Temperature, T [MeV] 10 2 10 1 10 0 10 1 Baryon density, og 10 (ρ [g/cm 3 ]) 6 7 8 9 10 11 12 13 14 15 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Baryon density, n [fm 3 ] B T. Fischer, GSI Darmstadt Y e 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Custers and EoS - 3

EoS for Astrophysica Appications many EoS deveoped in the past: from simpe parametizations to sophisticated modes many investigations of detaied aspects: often restricted to particuar conditions ony few reaistic goba EoS used in astrophysica simuations chaenge: covering of fu parameter space in a singe mode combination of different approaches required Custers and EoS - 4

Correations in Dense Matter ow densities two-, three-,... many-body correations due to short-range NN interaction modification of thermodynamic properties bound states appear as new partice species change of chemica composition high densities (around/above nucear saturation density) uniform neutron-proton matter mean-fied effects dominate quasipartice picture in between at ow temperatures inhomogeneous matter nucear matter: iquid-gas phase transition (no Couomb interaction, no eectrons, no charge neutraity) stear matter: formation of attice structures/ pasta phases (charge neutraity, interpay of surface effects and ong-range Couomb interaction) Custers and EoS - 5

Theoretica Approaches different points of view chemica picture mixture of different nucear species and nuceons in chemica equiibrium properties of constituents independent of medium interaction between partices? dissoution of nucei at high densities? n n p t p n d n p n p p α p n n p n p physica picture interaction between nuceons correations formation of bound states/resonances treatment of two-, three-,..., many-body correations? choice of interaction? n n n p n n p n n p p p n p p n p p n p n n p n p Custers and EoS - 6

Theoretica Methods Improving the description step by step: idea mixture of independent partices, no interaction Nucear Statistica Equiibrium/Law of Mass Action most simpe approach, suppression of nucei excuded voume mechanism mixture of interacting partices/correations Viria Equation of State mode-independent ow-density benchmark considering medium effects with increasing density Quantum Statistica/Generaized Beth-Uhenbeck Approach correations of quasipartices with medium-dependent properties, microscopic origin of custer dissoution/mott effect (action of Paui principe) Custers and EoS - 7

Generaized Reativistic Density Functiona I extension of phenomenoogica reativistic mean-fied approaches usua degrees of freedom: neutrons, protons, eectrons quasipartices with medium-dependent sef-energies mesons (σ, ω, ρ, δ) and photons interaction with nuceons/eectrons via minima couping idea: incude new degrees of freedom with medium-dependent properties: 1. ight nucei (deuteron, triton, heion, α-partice) 2. two-nuceon scattering correations (nn, pp, np channes) 3. heavy nucei (A > 4) interaction via minima couping to mesons/photon with scaed strengths Custers and EoS - 8

Generaized Reativistic Density Functiona II grand canonica thermodynamic potentia Ω = pv = d 3 rω g (T,µ i,σ,δ,ω 0,ρ 0,A 0, σ, δ, ω 0, ρ 0, A 0 ) with density functiona ω g depending on temperature T, chemica potentias µ i, meson and photon fieds σ,δ,ω 0,ρ 0,A 0 fied equations with additiona rearrangement contributions consistent derivation of thermodynamic quantities parameters vacuum masses of nuceons, eectrons, nucei effective resonance energies and degeneracy factors density-dependent meson-nuceon/nuceus coupings, fitted to properties of atomic nucei medium-dependent energy shifts of custers (bound and continuum states) Custers and EoS - 9

Light Nucei shift of binding energies cacuation perturbativey/ variationay with reaistic nuceon-nuceon potentias main effect: Paui principe bocking of states in the medium! 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 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 ] Custers and EoS - 10

Light Nucei shift of binding energies cacuation perturbativey/ variationay with reaistic nuceon-nuceon potentias main effect: Paui principe bocking of states in the medium! exampe: symmetric nucear matter, nucei at rest in medium in vacuum: experimenta binding energies nucei become unbound (B i < 0) with increasing density of medium dissoution of custers at high densities Mott effect 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 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 ] Custers and EoS - 10

Heavy Nucei I inhomogeneous matter at ow densities comparison with uniform matter increase in binding energy spherica Wigner-Seitz ce cacuation generaized re. density functiona extended Thomas-Fermi approximation eectrons for charge compensation heavy nuceus surrounded by gas of nuceons sef-consistent cacuation with interacting nuceons, eectrons partice number density n i [fm -3 ] 10-1 10-2 10-3 10-4 A heavy = 147.1 Z heavy = 62.3 p n e T = 5 MeV n = 0.01 fm -3 Y p = 0.4 10-5 0 5 10 15 20 radius r [fm] Custers and EoS - 11

Heavy Nucei I inhomogeneous matter at ow densities comparison with uniform matter increase in binding energy spherica Wigner-Seitz ce cacuation generaized re. density functiona extended Thomas-Fermi approximation eectrons for charge compensation heavy nuceus surrounded by gas of nuceons and ight custers sef-consistent cacuation with interacting nuceons, eectrons and ight nucei increased probabiity of finding ight custers at surface of heavy nuceus partice number density n i [fm -3 ] 10-1 10-2 10-3 10-4 A heavy = 113.6 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 = 0.4 10-5 0 5 10 15 20 radius r [fm] Custers and EoS - 11

Heavy Nucei II traditiona approach in EoS tabes: singe-nuceus approximation (SNA) (one representative heavy nuceus) no distribution of nucei Custers and EoS - 12

Heavy Nucei II traditiona approach in EoS tabes: singe-nuceus approximation (SNA) (one representative heavy nuceus) no distribution of nucei extended approach: fu tabe of nucei incuded (c.f. NSE cacuations) vacuum binding energies needed medium-dependent shift of binding energies from SNA binding energy per nuceon (T=0 MeV, np symmetric matter) BE/A [MeV] 10 8 6 4 2 AME2011 n = 0.005 fm -3 0 0 50 100 150 200 250 300 A AME2011: G. Audi, W. Meng (private communication) Custers and EoS - 12

Heavy Nucei II traditiona approach in EoS tabes: singe-nuceus approximation (SNA) (one representative heavy nuceus) no distribution of nucei extended approach: fu tabe of nucei incuded (c.f. NSE cacuations) vacuum binding energies needed medium-dependent shift of binding energies from SNA binding energy per nuceon (T=0 MeV, np symmetric matter) BE/A [MeV] 10 8 6 4 2 AME2011 n = 0.010 fm -3 0 0 50 100 150 200 250 300 A AME2011: G. Audi, W. Meng (private communication) Custers and EoS - 12

Heavy Nucei II traditiona approach in EoS tabes: singe-nuceus approximation (SNA) (one representative heavy nuceus) no distribution of nucei extended approach: fu tabe of nucei incuded (c.f. NSE cacuations) vacuum binding energies needed medium-dependent shift of binding energies from SNA binding energy per nuceon (T=0 MeV, np symmetric matter) BE/A [MeV] 10 8 6 4 2 AME2011 n = 0.015 fm -3 0 0 50 100 150 200 250 300 A AME2011: G. Audi, W. Meng (private communication) Custers and EoS - 12

Heavy Nucei II traditiona approach in EoS tabes: singe-nuceus approximation (SNA) (one representative heavy nuceus) no distribution of nucei extended approach: fu tabe of nucei incuded (c.f. NSE cacuations) vacuum binding energies needed medium-dependent shift of binding energies from SNA binding energy per nuceon (T=0 MeV, np symmetric matter) BE/A [MeV] 10 8 6 4 2 AME2011 n = 0.020 fm -3 0 0 50 100 150 200 250 300 A AME2011: G. Audi, W. Meng (private communication) Custers and EoS - 12

Heavy Nucei II traditiona approach in EoS tabes: singe-nuceus approximation (SNA) (one representative heavy nuceus) no distribution of nucei extended approach: fu tabe of nucei incuded (c.f. NSE cacuations) vacuum binding energies needed medium-dependent shift of binding energies from SNA binding energy per nuceon (T=0 MeV, np symmetric matter) BE/A [MeV] 10 8 6 4 2 AME2011 n = 0.025 fm -3 0 0 50 100 150 200 250 300 A AME2011: G. Audi, W. Meng (private communication) Custers and EoS - 12

Heavy Nucei II traditiona approach in EoS tabes: singe-nuceus approximation (SNA) (one representative heavy nuceus) no distribution of nucei extended approach: fu tabe of nucei incuded (c.f. NSE cacuations) vacuum binding energies needed medium-dependent shift of binding energies from SNA binding energy per nuceon (T=0 MeV, np symmetric matter) BE/A [MeV] 10 8 6 4 2 AME2011 n = 0.030 fm -3 0 0 50 100 150 200 250 300 A AME2011: G. Audi, W. Meng (private communication) Custers and EoS - 12

Heavy Nucei II traditiona approach in EoS tabes: singe-nuceus approximation (SNA) (one representative heavy nuceus) no distribution of nucei extended approach: fu tabe of nucei incuded (c.f. NSE cacuations) vacuum binding energies needed medium-dependent shift of binding energies from SNA medium effects: reative stabiization of heavier and exotic nucei dissoution of nucei depending on density, temperature, np-asymmetry ony preiminary resuts binding energy per nuceon (T=0 MeV, np symmetric matter) BE/A [MeV] 10 8 6 4 2 AME2011 n = 0.035 fm -3 0 0 50 100 150 200 250 300 A AME2011: G. Audi, W. Meng (private communication) Custers and EoS - 12

Low-Density Limit I ony two-body correations reevant comparison of generaized reativistic density functiona with viria Equation of State (mode-independent benchmark) fugacity expansion of thermodynamic potentia Ω consistency reations with viria coefficients and zero-density meson-nuceon coupings effective resonance energies E ij (T) (i,j = n,p) representing continuum correations effective degeneracy factors g (eff) ij (T) (cf. treatment of excited states of nucei) reativistic corrections Custers and EoS - 13

Low-Density Limit II VEoS: expansion of Ω in fugacities z i = exp(µ i /T) Ω = TV i b i z i λ 3 i + ij b ij z i z j λ 3/2 i λ 3/2 j + ijk z i z j z k b ijk +... λ i λ j λ k with virira coefficients b i, b ij, b ijk and therma waveengths λ i = 2π/(m i T) definition of effective resonance energies E ij (without reativistic corrections) nn channe (simiar for pp channe): 1 λ 3 n ( b nn + g ) n 2 5/2 np channe 2b np λ 3 n g d λ 3 d exp ( Bd = 1 λ 3 nn T ) = 1 λ 3 np g (nn) I (nn) g (np) I (np) with viria integras I (ij) = de π dδ (ij) de exp( ) E T Custers and EoS - 14

Low-Density Limit II VEoS: expansion of Ω in fugacities z i = exp(µ i /T) Ω = TV i b i z i λ 3 i + ij b ij z i z j λ 3/2 i λ 3/2 j + ijk z i z j z k b ijk +... λ i λ j λ k with virira coefficients b i, b ij, b ijk and therma waveengths λ i = 2π/(m i T) definition of effective resonance energies E ij (without reativistic corrections) nn channe (simiar for pp channe): 1 λ 3 n ( b nn + g ) n 2 5/2 np channe 2b np λ 3 n g d λ 3 d exp ( Bd = 1 λ 3 nn T ) = 1 λ 3 np g (nn) I (nn) = ± 1 g (nn) λ 3 0 exp nn g (np) I (np) = 1 λ 3 np ( E ) nn T 1 ( [±g (npt) 0 ]exp E ) npt T t=0 with viria integras I (ij) = de π dδ (ij) de exp( ) E T Custers and EoS - 14

Continuum Correations I effective resonance energies g (ij) de π dδ (ij) ( de exp E T ) ( = ±g (ij) 0 exp E ) ij T E nn [MeV] 30 25 20 15 10 5 S exp S exp +P exp +D exp a nn a nn +r nn unitary imit modified a nn +r nn 0 0 5 10 15 20 T [MeV] effective-range expansion for s-wave phase shifts: kcotδ (ij) 0 = 1 a ij + r ij 2 k2 anaytica resuts ow T: I (ij) 0 (T) a ij µij T/(2π) unitary imit: E ij (T) = T n2 Custers and EoS - 15

Low-Density Limit III comparison of grdf with VEoS consistency reations (without reativistic corrections) Custers and EoS - 16

Low-Density Limit III comparison of grdf with VEoS consistency reations (without reativistic corrections) nn channe (simiar for pp channe): 1 λ 3 n ( b nn + g n 2 5/2 np channe 2b np λ 3 n g d λ 3 d exp ) = 1 λ 3 nn ( Bd T ) = 1 λ 3 np g (nn) I (nn) g (np) I (np) with viria integras I (ij) = de π dδ (ij) de exp( ) E T Custers and EoS - 16

Low-Density Limit III comparison of grdf with VEoS consistency reations (without reativistic corrections) nn channe (simiar for pp channe): 1 λ 3 n ( b nn + g ) n 2 5/2 np channe 2b np λ 3 n g d λ 3 d exp ( Bd = 1 λ 3 nn T ) g (nn) I (nn) = 1 ( g (eff) λ 3 nn (T)exp E ) nn nn T = 1 λ 3 np g (np) I (np) = 1 λ 3 np t=0 g2 nc + λ 6 n2t 1 ( g (eff) npt (T)exp E ) npt T g ng p C λ 3 n λ3 p T with viria integras I (ij) = de π dδ (ij) de exp( E T) and zero-density coupings C ± = C ω C σ ±C ρ C δ C k = Γ2 k m 2 k k = ω,σ,ρ,δ definition of effective degeneracy factors g (eff) ij (T) (cf. excited states in nucei) Custers and EoS - 16

Continuum Correations II effective degeneracy factors (without reativistic corrections) g (nn) de π dδ (nn) ( de exp E T ) ( = g nn (eff) (T)exp E ) nn T gn 2 λ 3 nnc + λ 6 n 2T... g (eff) (T) 8 6 4 2 0-2 -4-6 nn( 1 S 0 ) nn( 1 S 0 ) + re. eff. np( 3 S 1 ) np( 3 S 1 )+re. eff -8 0 10 20 30 T [MeV] reativistic effects for high temperatures Custers and EoS - 17

Low-Density Limit IV zero temperature imit of consistency reations C ω C σ = π [ ann ( 1 S 0 )+a pp ( 1 S 0 )+a np ( 1 S 0 )+3a np ( 3 S 1 ) ] 2m C ρ C δ = π [ ann ( 1 S 0 )+a pp ( 1 S 0 ) a np ( 1 S 0 ) 3a np ( 3 S 1 ) ] 2m with scattering engths a ij and assuming m = m n = m p comparison of experiment with RMF parametrizations exp. DD2 [1] DD-MEδ [2] (ω,σ,ρ) (ω,σ,ρ,δ) C ω C σ [fm 2 ] 14.15 5.39 4.90 C ρ C δ [fm 2 ] 9.61 2.48 2.55 [1] S. Type et a., Phys. Rev. C 81 (2010) 015803, [2] X. Roca-Maza et a., Phys. Rev. C 84 (2011) 054309 conventiona mean-fied modes don t reproduce effect of correations at very-ow densities Custers and EoS - 18

Neutron Matter at Low Densities I comparison: different effects nonreativistic idea gas interna energy per nuceon E/A (idea gas: E/A = 3T/2) interna energy per nuceon [MeV] 15.4 15.2 15.0 14.8 14.6 T = 10 MeV idea gas 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 density n [fm -3 ] Custers and EoS - 19

Neutron Matter at Low Densities I comparison: different effects nonreativistic idea gas re. kinematics + statistics reativistic Fermi gas interna energy per nuceon E/A (idea gas: E/A = 3T/2) 15.4 T = 10 MeV interna energy per nuceon [MeV] 15.2 15.0 14.8 14.6 idea gas reativistic Fermi gas 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 density n [fm -3 ] Custers and EoS - 19

Neutron Matter at Low Densities I comparison: different effects nonreativistic idea gas re. kinematics + statistics reativistic Fermi gas interna energy per nuceon E/A (idea gas: E/A = 3T/2) 15.4 T = 10 MeV two-body correations viria EoS with reativistic correction (not incuded in standard viria EoS) mean-fied effects standard RMF mode with density dependent coupings interna energy per nuceon [MeV] 15.2 15.0 14.8 14.6 idea gas reativistic Fermi gas reativistic viria EoS standard RMF 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 density n [fm -3 ] Custers and EoS - 19

Neutron Matter at Low Densities I comparison: different effects nonreativistic idea gas re. kinematics + statistics reativistic Fermi gas interna energy per nuceon E/A (idea gas: E/A = 3T/2) 15.4 T = 10 MeV two-body correations viria EoS with reativistic correction (not incuded in standard viria EoS) mean-fied effects standard RMF mode with density dependent coupings two-body correations generaized reativistic density functiona (grdf) with contributions from nn scattering interna energy per nuceon [MeV] 15.2 15.0 14.8 14.6 idea gas reativistic Fermi gas reativistic viria EoS standard RMF grdf 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 density n [fm -3 ] Custers and EoS - 19

Neutron Matter at Low Densities II comparison: p/n in different modes (idea gas: p/n = T) 4.1 T = 4 MeV 10.1 T = 10 MeV 4.0 10.0 p/n [MeV] 3.9 3.8 3.7 Viria RMF grmf STOS SH LS 220 p/n [MeV] 9.9 9.8 9.7 Viria RMF grmf STOS SH LS 220 3.6 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 n [fm -3 ] 9.6 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 n [fm -3 ] STOS: H. Shen et a., Nuc. Phys. A 637 (1998) 435 (TM1) SH: G. Shen et a., Phys. Rev. C 83 (2011) 065808 (FSUGod) LS 220: J.M. Lattimer et a., Nuc. Phys. A 535 (1991) 331 (K = 220 MeV) Custers and EoS - 20

Light Custers and Continuum Correations partice fractions X i = A i n i n b n b = i A in i generaized reativistic density functiona 10 0 ow densities: two-body correations most important T = 10 MeV high densities: dissoution of custers Mott effect partice fraction X i 10-1 10-2 Y p = 0.4 p n 2 H 3 H 3 He 4 He 10-3 10-5 10-4 10-3 10-2 10-1 (without heavy custers) density n [fm -3 ] Custers and EoS - 21

Light Custers and Continuum Correations partice fractions X i = A i n i n b n b = i A in i ow densities: two-body correations most important high densities: dissoution of custers Mott effect effect of NN continuum correations dashed ines: without continuum soid ines: with continuum reduction of deuteron fraction, redistribution of other partices correct imits with generaized reativistic density functiona generaized reativistic density functiona partice fraction X i 10 0 10-1 10-2 10-3 T = 10 MeV Y p = 0.4 10-5 10-4 10-3 10-2 10-1 (without heavy custers) density n [fm -3 ] p n 2 H 3 H 3 He 4 He Custers and EoS - 21

Neutron Matter - Transition from Low to High Densities medium effects: different parametrization of energy shifts (inear, quadratic, poe) correct imits at ow and high densities comparison VEoS - grmf: reduction of expicit correations at ow densities p/n [MeV] 8 7 6 5 4 3 2 1 Viria RMF grmf-inear grmf-quad grmf-poe T= 4 MeV 0 0.0001 0.001 0.01 0.1 n [fm -3 ] X nn 10 0 10-1 10-2 10-3 T = 4 MeV Viria grmf-inear grmf-quad grmf-poe 0.0001 0.001 0.01 0.1 n [fm -3 ] Custers and EoS - 22

Symmetry Energy I nucear matter energy per nuceon E(n,β) with tota density n = n n +n p and asymmetry β = (n n n p )/n symmetry energy E sym (n) = 1 2 [E(n,1) 2E(n,0) + E(n, 1)] mean-fied modes without custers ow-density behavior not correct scaed symmetry energy E sym (n)/e sym (n sat ) 1.2 1.0 0.8 0.6 0.4 0.2 T = 0 MeV without custers 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 scaed density n/n sat Custers and EoS - 23

Symmetry Energy I nucear matter energy per nuceon E(n,β) with tota density n = n n +n p and asymmetry β = (n n n p )/n symmetry energy E sym (n) = 1 2 [E(n,1) 2E(n,0) + E(n, 1)] mean-fied modes without custers ow-density behavior not correct grdf with (heavy) custers increase of E sym at ow densities due to formation of custers finite symmetry energy in the imit n 0 effect smaer with increasing temperature experimenta test in heavy-ion coisions scaed symmetry energy E sym (n)/e sym (n sat ) 1.2 1.0 0.8 0.6 0.4 0.2 T = 0 MeV without custers with custers 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 scaed density n/n sat Custers and EoS - 23

Symmetry Energy II finite temperature experimenta determination of symmetry energy heavy-ion coisions 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 partices eave the source) (S. Kowaski et a., Phys. Rev. C 75 (2007) 014601) symmetry energies in RMF cacuation without custers are too sma very good agreement with QS/gBU cacuation with ight custers (J. B. Natowitz et a., Phys. Rev. Lett. 104 (2010) 202501) free symmetry energy F sym [MeV] 20 15 10 5 (a) experiment RMF without custers QS with custers 0 0 1 2 3 4 5 V surf [cm/ns] interna symmetry energy E sym [MeV] 20 15 10 5 (b) experiment RMF without custers QS with custers 0 0 1 2 3 4 5 V surf [cm/ns] Custers and EoS - 24

Summary correations in nucear matter change of composition formation and dissoution of custers modification of thermodynamic properties e.g. change of symmetry energy Custers and EoS - 25

Summary correations in nucear matter change of composition formation and dissoution of custers modification of thermodynamic properties e.g. change of symmetry energy various theoretica approaches with custers nucear statistica equiibrium viria Equation of State quantum statistica/generaized Beth-Uhenbeck approach generaized reativistic density functiona (grdf) important features of grdf approach ight & heavy nucei and two-nuceon continuum correations incuded quasipartices with medium-dependent properties dissoution of custers Mott effect correct imits at ow and high densities we constrained parameters preparation of EoS tabes for astrophysica appications Custers and EoS - 25

Thanks to my coaborators Gerd Röpke (Universität Rostock) Nies-Uwe Bastian (Universität Rostock) David Baschke (Uniwersytet Wroc awski) Thomas Kähn (Uniwersytet Wroc awski) Hermann Woter (Ludwig Maximiians-Universität München) Maria Voskresenskaya (GSI Darmstadt) for support from Exceence Custer Universe, Technische Universität München Hemhotz Association (HGF) Hemhotz Internationa Center (HIC) for FAIR Nucear Astrophysics Virtua Institute (VH-VI-417) to the organizers of the NEOS 2012 workshop for the invitation and organization to you, the audience for your attention and patience Custers and EoS - 26