Clusters in Nuclear Matter and the Equation of State

Transcription

1 Journa of Physics: Conference Series Custers in Nucear Matter and the Equation of State o cite this artice: Stefan ype 2013 J. Phys.: Conf. Ser View the artice onine for updates and enhancements. Reated content - Nucei in Dense Matter and Equation of State Stefan ype - Custer correations in diute matter and equation of state Stefan ype - Custer formation and dissoution in a generaized reativistic density functiona approach for dense matter Stefan ype his content was downoaded from IP address on 28/11/2017 at 10:38

2 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing Custers in Nucear Matter and the Equation of State Stefan ype GSI Hemhotzzentrum für Schwerionenforschung, Panckstraße 1, D Darmstadt, Germany E-mai: Abstract. he equation of state of dense matter is an essentia ingredient in astrophysica simuations, e.g. of core-coapse supernovae and neutron stars. At densities beow nucear saturation density, nuceon-nuceon correations strongy affect the thermodynamica properties of matter. Custers appear and the chemica composition is modified. hese effects can be considered in a generaized reativistic density functiona approach. It shows the correct imits by interpoating between the viria equation of state at ow densities and quasipartice meanfied modes at high densities. Heavy-ion coisions offer the possibiity to study the effects of correations and the modification of custer properties in diute matter. 1. Introduction During most of their ifetime, stars evove steadiy by burning ight nucei in fusion reactions creating heavier nucei in the stear core and surrounding shes. Massive stars (M star 8M sun ) deveop an iron core that eventuay coapses under its own gravitationa attraction and a vioent supernova exposion is aunched [1]. he outer parts of the former star are ejected and a neutron star or back hoe is eft as a remnant. he dynamica evoution of such a core-coapse supernova and the properties of the neutron star are determined by the the equation of state (EoS) of dense stear materia [2]. It determines the thermodynamic conditions of the expanding matter that is the site of nuceosynthesis reactions. he chemica composition of the matter has an impact on the response to the copious amount of neutrinos emitted by the core. Since the timescae of the nucear reactions is much shorter than that of the supernova evoution, an equation of state of dense matter in thermodynamica and chemica equiibrium can be used in astrophysica simuations assuming oca charge neutraity. hree parameters are sufficient to characterize the thermodynamic conditions of the system: the mass density (or baryon number density n), the temperature and the eectron fraction Y e (or neutron-proton asymmetry β = 1 2Y e ). ypica ranges of their vaues in supernova simuations are 10 9 / sat 10 (with the nucear saturations density sat g/cm 3 ), 0.1 MeV k B 50 MeV and 0 Y e 0.6. Beow sat it is usuay sufficient to consider nuceons, nucei, eectrons and therma photons as the reevant degrees of freedom in stear matter. Neutrinos are not incuded. hey are treated independenty of the EoS in simuations because they are not in equiibrium with the matter. here are many EoS avaiabe in the iterature, comprising a arge variety of approaches from simpe parametrizations to very eaborate modes. Most investigations focus on particuar aspects and properties of the matter for particuar conditions, e.g. symmetric nucear matter, neutron matter, matter in β equiibrium, ow- or high-density matter et cetera. Since a goba Pubished under icence by IOP Pubishing Ltd 1

3 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing description is required covering the fu parameter space, ony a sma number of reaistic EoS, usuay in tabuar form, is used in astrophysica simuations of core-coapse supernovae, see references [3, 4, 5] for the most we-known and widey appied EoS in astrophysica simuations and references [6, 7, 8, 9, 10, 11] for some recenty deveoped or extended modes. For the description of neutron stars, a zero-temperature EoS is usuay appropriate and a much arger number of modes can be empoyed. he short-range nuceon-nuceon interaction modifies the thermodynamic properties of dense matter substantiay. In particuar, it is responsibe for two-, three-,..., many-body correations that change the chemica composition at ow densities by creating new partice species as bound states of nuceons. hese custers have to be considered expicity in the description. Standard EoS tabes consider ony those partices that are aso incorporated in astrophysica simuations, i.e. neutrons, protons, eectrons, photons, the α partice and a representative heavy nuceus. In recent EoS, the set of partices was extended to incude more ight custers ( 2 H, 3 H, 3 He, 4 He), a fu distribution of a nucei in the nucear chart or exotic partices such as hyperons, other heavy baryons, mesons (pions, kaons,...) or quarks. At high densites around and above nucear saturation density, nucear matter is expected to be uniform and mean-fied modes with neutrons and protons as soe constituents are usuay sufficient. Here, effects of the interaction are treated in a quasipartice approach. At ow densities and temperatures beow a critica vaue of approximatey 15 MeV nucear matter wi deveop inhomogeneities with a coexistence of gas and iquid phases on a macroscopic scae. It has to be emphasized that this iquid-gas first-order phase transition is a feature of the fictitious system of nucear matter where the Couomb interaction is negected and there are no eectrons to enforce charge neutraity. In reaistic stear matter with Couomb interaction and eectrons, however, there is a formation of ow- and high-density regions separated by various shapes of the interfaces ( pasta phases ) on a microscopic scae. he transition is driven by the baance of the short-range nucear and ong-range Couomb interaction. At very ow temperatures, a soid phase with crysta/attice structures deveops. It is obvious that an appropriate construction of the phase transitions is needed and a consistent interpoations between the various regions is required in a goba EoS for astrophysica appications. It is a great chaenge to cover a aspects in a singe mode and often a combination of different approaches is required. his contribution focuses on the theoretica description of dense matter at nucear saturation density and (much) beow, in particuar the formation of custers due to correations and their dissoution with increasing density. In section 2 different concepts and approaches are discussed that aow to describe dense matter with custers at ow densities and that take more and more effects of the interaction into account. A generaized reativistic density functiona (grdf) approach is presented in section 3. It incorporates the effects of correations and covers the whoe density region up to nucear saturation density with the correct ow- and high-density imits. Medium effects on the properties of ight and heavy nucei within the grdf approach are discussed in section 4. he ow-density imit of the mode is considered in section 5 and the effects on neutron matter are studied as an exampe. he behavior of ight custer abundances and the effects of two-nuceon scattering correations are presented in section 6. he effects of custer formation on the symmetry energy are investigated in section 7 and experimenta tests of the mode predictions are mentioned. Detais and further references on the materia presented in this contribution can be found in references [12, 13, 14]. In the foowing, natura units are used such that = c = k B = 1 to simpify the equations. 2. heoretica approaches Density, temperature and neutron-proton asymmetry have a strong effect on the chemica composition and thermodynamic properties of diute matter. here are essentiay two different points of view that can serve as starting points to construct theoretica modes of the system. 2

4 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing In the chemica picture one considers a mixture of nuceons and nucei in chemica equiibrium. It is assumed that the properties of the partices composed of nuceons are independent of the medium, e.g. the binding energies of nucei do not change with density or temperature. In order to take into account correation effects, the interaction has to be specified between a different constituents individuay. Describing the dissoution of the nucei at high densities is a major probem. Usuay, a simpe geometric concept such as the excuded-voume mechanism is introduced. In the physica picture ony nuceons are considered as fundamenta constituents. heir mutua interaction introduces correations and eads to the formation of bound states and scattering resonances. Appropriate theoretica methods have to be deveoped to treat the two-, three-,... many-body correations inside the medium. he nuceon-nuceon interaction is the ony and essentia ingredient in this type of approach. In the foowing, a number of theoretica approaches is presented that improve the description of nucear matter considering different effects of the nucear interaction. hese modes combine both pictures to various degrees Low-density modes Nucear statistica equiibrium mode he most simpe approach to describe diute nucear matter is a nucear statistica equiibrium (NSE) mode that assumes an idea mixture of nuceons (p,n) and nucei (X) with mass numbers A = N +Z in chemica equiibrium Zp+Nn A Z X. (1) hus the nonreativistic chemica potentias µ i of the partices are reated by Zµ p +Nµ n = µ X B X (2) with the binding energies B X > 0 of the nucei X with mass m X = Zm p + Nm n B X. Mutua interactions are negected. Assuming nonreativistic kinematics and Maxwe-Botzmann statistics, the grand canonica potentia is given by the simpe expression Ω(,V,µ i ) = V g ( i µi ) λ 3 exp (3) i i with therma waveenghts λ i = 2π/(m i ) and degeneracy factors g i. he summation over the index i comprises protons, neutrons and a nucei. Excited states x of nucei are usuay taken into account by introducing temperature dependent degeneracy factors g i () = (2J gs i +1)+ x (2J x i +1)exp ( E x with ground state and excited state spins J gs i and Ji x, respectivey. From Eq. (3) a thermodynamica properties, e.g. the equations of state, such as pv = N and E = 3N/2 with N = V i n i, for a mixture of idea gases with partia densities n i are derived. In particuar, the individua partice number densities are given by n i = Ω µ i = g ( i µi ) exp. (5),V,µj i Considering the ratio n X /(n Z p nn n ) for the reaction (1), one finds the aw of mass action that is we known in chemistry. he basic NSE mode negects a interactions between the constituents and cannot describe the dissoution of nucei with increasing density. In order to incude such an effect, the excudedvoume mechanism can be empoyed that suppresses the abundance of nucei at high densities, see, e.g., reference [6]. λ 3 i ) (4) 3

5 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing Viria equation of state wo-, three-,..., many-body correations due to the interaction between the constituents can be incorporated in the description by using the viria equation of state (VEoS). In this approach the grand canonica partition function is expanded in powers of the partice fugacities z i = exp(µ i /). he expansion is ony vaid for sma expansion parameters z i n i λ 3 i 1 and thus can be appied ony for ow densities and not too ow temperatures. Introducing the dimendioness custer (viria) coefficients b i = g i, b ij,...the grand canonia potentia in the viria expansion Ω(µ i,,v) = V i b i z i λ 3 i + ij b ij z i z j λ 3/2 i λ 3/ (6) j is obtained. For independent partices without interaction, the second, third,... viria coefficient vanish and the NSE resut is recovered. he effect of correations between partices i and j is encoded in the second viria coefficient. In cassica mechanics it is given by b ij = 1 2 g ij λ 3/2 i λ 3/2 j [ d 3 r ij {exp V ] } ij(r ij ) 1 with the two-body interaction potentia V ij depending on the distance r ij and the degeneracy factor g ij of the two-body state. G.E. Beth and E. Uhenbeck derived the quantum mechanica generaization of b ij [15, 16]. he integration over phase space in cassica mechanics has to be repaced by a summation over quantum states. hen, the second viria assumes the form b ij () = 1+δ ij 2 ( ) mi +m 3/2 j mi m j (7) ( de D ij (E)exp E ) ±δ ij g i 2 5/2. (8) he quantity D ij (E) = k ( g (ij) k δ E E (ij) k ) + g (ij) π dδ (ij) de (9) can be seen as the difference of the eve densities between the correated and uncorreated twobody system. It contains contributions from bound states at energies E (ij) k < 0 and scattering states with phase shifts δ (ij) in channes k and, respectivey. he ast term in (8) is a quantum statistica correction with positive (negative) sign if i and j are identica bosons (fermions). In the two-nuceon system, the binding energy of the deuteron and the scattering phase shifts are known experimentay. Hence, b ij depends ony on measured data and the ow-density behavior of the EoS can be estabished mode-independenty [17]. It can serve as a benchmark for other approaches. With increasing density, however, the power series approximation wi fai. he dissoution of custers such as the deuteron cannot be accounted for in the VEoS Quantum statistica mode Medium effects on the properties of nucei in matter can be incuded in a quantum statistica (QS) or generaized Beth-Uhenbeck approach that was formuated using thermodynamic Green s functions [18]. he EoS of an interacting many-body system is derived from the tota nuceon number density n(µ,,v) = j de 2π A(j,E)f +(E) (10) 4

6 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing with the spectra functions A(j,E) for a singe-partice state j ( p j,σ j,τ j ) and the therma distribution function f ± (E) = {exp[(e µ)/]±1} 1 (11) for fermions (+) or bosons ( ), respectivey. he spectra function depends on the compex vaued sef-energy Σ(j, z) that contains the information on the interaction. After expanding A(j,E) for sma imaginary parts of the sef-energy, the tota density spits into two parts n = n free +2n corr. he first contribution n free = j f +[e(j)] is the density of free quasipartices withquasipartice energies e(j). heyaresoutions ofe(j) = p 2 j /(2m j)+reσ[j,e(j)] anddepend on the momentum in genera. he correation density n corr = k g (2) k f (E cont B k )+ P,P>P Mott g (2) P de π 2sin2 δ dδ de f (E cont +E) (12) receives contributions from bound states and from scattering states. his form resembes the structure of the second viria coefficient (8) in the VEoS. In contrast to b ij, the properties of the two-body states in the QS mode depend on their c.m. momentum P with respect to the medium. Due to the action of the Paui principe that bocks states by the medium, there are no two-body bound states for P beow the Mott momentum P Mott. he continuum edge E cont is defined as the energy of the scattering states with zero reative momentum. he binding energies B k and the in-medium scattering phase shifts δ depend on the medium properties and P. hey have to be determined from the in-medium matrix in genera. he Bose-Einstein distribution functions f appears correcty for the two-nuceon states. here is an additiona 2sin 2 δ factor in the continuum contribution as compared to the integra in equation (8). It reduces the strength of the expicit scattering correations since the sef-energies contain a part of the correation effect due to the interaction. here are different ways to determine the medium-dependent shift of the nucear binding energies. For ight nucei with mass number A 4 they can be cacuated by soving the appropriate in-medium Schrödinger equation for the composite system with reaistic nuceonnuceon potentias or with nuceon sef-energies taken from phenomenoogica mean-fied modes. Resuts are discussed in subsection 4.1. For heavier nucei (A > 4) a different approach based on a Wigner-Seitz ce cacuation with an energy density functiona can be used, see subsection Intermediate/high-density modes wo major casses of theoretica approaches can be distinguished that describe nucear matter successfuy around the nucear saturation density. here are phenomenoogica approaches with effective interactions, e.g. nonreativistic Hartree-Fock modes with the Skyrme or Gogny interaction and reativistic mean-fied modes, and there are ab-initio approaches that use reaistic nuceon-nuceon interactions fitted to two- (and three-) nuceon bound and scattering states, e.g. (non-)reativistic Brueckner-Hartree-Fock cacuations. Resuts of the phenomenoogica modes depend on a sma number of parameters that are usuay determined by fits to properties of finite nucei. In this way, shortcomings of the mode and the approximations in the many-body approach can be counterbaanced. In constrast, the interaction in the ab-initio modes is given from the outset and the quaity of the resuts is determined by the sophistication of the many-body method. Unfortunatey, the atter modes have some imitations in their appication, e.g. typicay a restriction to uniform nucear matter. In order to provide an EoS for astrophysica appications that is appicabe for a wide range of temperature, density and neutron-proton asymmetry, mean-fied modes are the prevaiing choice in order to obtain quantitativey reasonabe resuts. hese modes can be formuated in 5

7 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing the anguage of energy density functionas with we-known techniques for deriving and soving the reevant equations. Reativistic approaches are preferred by severa reasons. Lorentz covariance is a fundamenta principe in physics constraining the form of the possibe interactions. he spin of nuceon is naturay incuded in the reativistic description and the spin-orbit interaction appears with the correct strength in nucei. At high densities, reativistic effects are important due to the high Fermi momenta of the nuceons and a superumina speed of sound wi not occur in these modes. Beyond the kinematica effects, strong vector and scaar potentias appear in nucear matter that are responsibe for a new saturation mechanism. he mathematica formuation is quite eegant with a cose connection to modes based on quark degrees of freedom. Despite many virtues, mean-fied modes with ony nuceons as basic constituents fai at ow densities. Expicit correations are not considered and the formation and dissoution of nucei is not incorporated into the description. 3. Generaized reativistic density functiona In conventiona reativistic mean-fied (RMF) modes, neutrons and protons are the fundamenta degrees of freedom. hey are treated as quasipartices with sef-energies that depend on the medium properties and take into account the effects of the interaction. Eectrons (and myons) can be added in astrophysica appications when the EoS of charge neutra stear matter is needed and the Couomb interaction has to be incuded. he nucear interaction is modeed by an exchange of mesons (usuay ω, σ, ρ, δ) that coupe minimay to the nuceons. hey are represented by boson fieds that are generay treated as cassica fieds. Simiary, photons can be represented by the corresponding eectromagnetic potentia. he idea of the generaized reativistic density functiona (grdf) approach is to introduce new degrees of freedom in the description in order to incude nucei as constituents and to consider correations in an effective way. In reference [12] the Lagrangian density of a RMF mode was extended by introducing ight nucei ( 2 H, 3 H, 3 He and 4 He). hey are represented by corresponding spin 1, 1/2 and 0 fieds. Nucei interact with the meson and photon fieds ike the nuceons, ony the coupings are rescaed with the appropriate factors reated to their neutron and proton content. In addition, it is assumed that the binding energies of the nucei are medium dependent quantities. In the particuar approach of reference [12], this dependence enters through functions of the temperature and vector meson fieds. Nuceon-nuceon scattering correations, i.e. the nn, np and pp scattering states, were added to the mode in reference [13]. he compete positive energy continuum in a particuar channe is represented by a singe state with an effective resonance energy. his energy depends on the medium properties ike the binding energy of nucei. he degeneracy factor of the effective continuum state depends on temperature. See section 5 for detais. his corresponds to the treatment of excited states in nucei in NSE modes in subsection he couping to the meson and photon fieds is modeed in the same way as for nucei. Finay, heavy nucei with mass number A > 4 can be incorporated in a simiar way as ight nucei. However, their binding energy shifts in the medium have to be determined using a different strategy, see subsection 4.2 for detais and first resuts. In practica appications of RMF modes it is not sufficient for a quantitative description to assume minima nuceon-meson coupings with constant strength Γ im between partices i and mesons m. In order to incorporate a medium dependence of the effective nucear interaction, a modification of the mode is needed. One approach uses additiona terms in the Lagrangian density with noninear sef-interactions of the mesons. Usuay, a poynomia dependence on various combinations of the meson fieds is introduced. However, stabiity probems can occur for certain parametrizations and an extrapoation of the mode to the high-density region has to be considered with care. In a second approach, motivated by resuts of Dirac-Brueckner 6

8 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing cacuations of nucear matter [19], the coupings Γ ik are assumed to depend on the medium density. he functiona form of the dependence can be chosen fexiby with a we controed asymptotic behavior. he grdf approach uses density dependent meson-nuceon coupings. he number of free parameters in phenomenoogica RMF modes for nuceonic matter is rather sma. hey are determined in fits to properties of nucei and nucear matter parameters that can be extracted from experiments. Vaues for the partice masses are usuay cose to experimenta numbers with the exception of the σ meson that is not we constraint from measurements. he meson-nuceon coupings are specified at a certain reference density ref and a simpe functiona form of their density dependence with few parameters is introduced. When nucei are introduced in the grdf approach, additiona parameters need to be specified that determine the medium dependence of the binding energies. Besides constraints from nucear physics, a comparison of theoretica predictions with astronomica observations can be used to test EoS modes, see, e.g., reference [20]. he Lagrangian formuation of the RMF modes can be converted to a density functiona form that serves as a starting point to derive a reevant equations and thermodynamica quantities. In the present appication, the grand canonica thermodynamica potentia Ω = d 3 r ω g (,µ i,σ,δ,ω 0,ρ 0, σ, δ, ω 0, ρ 0, A) (13) with a potentia density ω g is very convenient. he temperature, the chemica potentias µ i of a partices i, the meson and photon fieds σ, δ, ω 0, ρ 0, A and their derivatives are the independent variabes in this approach. he grdf incudes nuceons, bound states of nucei and two-nuceon scattering states in the baryonic sector. heir masses are given by m i = N i m n +Z i m p (1 δ in )(1 δ ip )B (vac) i. (14) with the vacuum binding energy B (vac) i 0 of the composite systems (B (vac) i scattering states). he energy of a baryon or epton has the reativistic form with scaar potentias and vector potentias = 0 for the E i = k 2 +(m i S i ) 2 +V i (15) S i = Γ iσ σ +Γ iδ δ (1 δ in )(1 δ ip ) B i (16) V i = Γ iω ω 0 +Γ iρ ρ 0 +Γ iγ A 0 +(δ in +δ ip )V r (17) that depend on the meson fieds σ, δ, ω 0, ρ 0 and the photon fied A. he scaar sef-energy (16) contains the medium dependent binding energy shift B i 0. he rearrangment contribution V r = Γ ω ω 0n ω +Γ ρ ρ 0n ρ Γ σ σn σ Γ δ δn δ (18) appears in the vector sef-energy. his term is due to the density dependence of the coupings Γ ik = g ik Γ k ( ) with scaing factors g iω = g iσ = N i + Z i, g iρ = g iδ = N i Z i, g iγ = Z i and = n n +n p. Negecting antipartice and boson condensate contributions, the potentia density in equation (13) can be written as ω g = d 3 [ ( k g i (2π) 3 n 1±exp E )] i µ i (19) i 1 ( m 2 2 ω ω0 2 +m2 ρ ρ2 0 m2 σ σ2 m 2 δ δ2 + ω 0 ω 0 + ρ 0 ρ 0 σ σ δ δ + A 0 A ) 0 + ( Γ ωω 0 n ω +Γ ρρ 0 n ρ Γ σσn σ Γ δ δn δ) (nn +n p ) 7

9 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing where the upper (ower) signs appies to fermions (bosons). he source densities in equations (18) and (19) are determined by n k = i g ikn i for k = ω,ρ and n k = i g ikn s i for k = σ,δ, respectivey, with the individua vector densities n i = g i d 3 [ k (2π) 3 exp ( Ei µ i ) ±1] 1 (20) and scaar densities n s i = g i d 3 k m i S i (2π) 3 [exp k 2 +(m i S i ) 2 ( Ei µ i ) ±1] 1. (21) he reativistic chemica potentias of the baryonic partices are given by µ i = N i µ n +Z i µ p = µ i +m i (22) and the degeneracy factors g i can depend on the temperature. Note that n i and n s i have to be considered as functions of the independent variabes of Ω. A fied equations can be derived in the usua way from ω g. hey incude rearrangement contributions that ensure the thermodynamica consistency of the mode. E.g. the temperature dependence of the binding energy shifts and degeneracy factors generates contributions to the entropy, see reference [13] for detais. he set of fied equations has to be soved sef-consistenty in order to determine the two independent chemica potentias µ n and µ p for given tota nuceon density n and asymmetry β = 1 2Y p with tota proton fraction Y p. 4. Medium effects on nucear binding energies he essentia feature of the grdf approach is the medium dependence of the nucear binding energies causing the suppression of the custer abundances with increasing density. he main origin of this shift is the action of the Paui principe. Nuceons of the background medium occupy ow-momentum states that are no onger avaiabe for the formation of custers. An increase of the temperature reduces the bocking effect due to the increased diffuseness of the Fermi distribution Light nucei For ight custers (A 4), the medium dependent shifts of the binding energies in the grdf approach are adopted from the resuts of the QS mode. hey are parametrized as a function of temperature and an effective density that is reconstructed from the vector meson fieds, see reference [12] for detais. he density dependence of the binding energies for constant temperature are depicted in figure 1 of reference [12]. At zero density of the medium, the binding energy of a custer is given by the experimenta vaue in vacuum. With increasing density, the binding energy becomes smaer crossing the zero ine at a certain point depending on. his indicates that the custer becomes unbound. Different extrapoations to high densities were expored in reference [13]. More recent parametrizations of the energy shifts, incuding the momentum dependence, can be found in reference [21] Heavy nucei he binding energy shifts of heavy custers are cacuated on the basis of the reativistic density functiona. hus no additiona mode parameters are required. For this purpose, the formation of heavy custers is modeed by spherica Wigner-Seitz ce cacuations with an inhomogeneous density distribution of nuceons and eectrons under the condition of a vanishing tota charge. An 8

10 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing 10-1 (a) 10-1 (b) partice number density n i [fm -3 ] p n e partice number density n i [fm -3 ] p n e 2 H 3 H 3 He 4 He radius r [fm] radius r [fm] Figure 1. Density distribution of partices in a spherica Wigner-Seitz ce from a grdf cacuation in an extended homas-fermi approximation without (a) and with (b) ight custers for temperature = 5 MeV, tota nuceon density n = 0.01 fm 3 and proton fraction Y p = 0.4. Figure adapted from reference [14]. extended homas-fermi approximation is used to to sove the fied equations sef-consistenty with a partices interacting. No particuar shape of the density distributions is assumed as, e.g. in the EoS cacuations in references [3, 4]. he mode goes beyond a simpe oca density approximation because the density functiona takes into account the finite range of the interactions. However, she effects are negected. For given temperature = 5 MeV, tota density n = 0.01 fm 3 and tota proton fraction Y p = 0.4, the density distribution of partices inside the Wigner-Seitz ce is depicted in figure 1. Nuceons form a heavy custer in the center of the ce that is surrounded by a gas of nuceons. Eectrons are amost uniformy distributed inside the ce since they form a highy degenerate Fermi gas. hey effectivey screen the Couomb potentia of the protons resuting in a vanishing potentia on the surface of the ce consistent with the tota charge neutraity. When ight custers are considered in the cacuation in addition to nuceons and eectrons, an interesting effect is observed, see the right pane of figure 1. Due to the high density inside the heavy nuceus, ight custers can ony appear in the in the surrounding ow-density nuceon gas. here is an enhancement of the ight custer abundancies at the radius of the heavy custer. his coud be an indication of strong few-nuceon correations on the nucear surface. he binding energy of a nuceus at a certain density of the medium can be determined by comparing the energies of the Wigner-Seiz ce cacuation with uniform and nonuniform density distributions. In the imit of zero tota density, the radius of the ce approaches infinity and the resut with the inhomogeneous partice distributions corresponds to that of a nuceus surrounded by a coud of eectrons, a neutra atom. Since the extended homas-fermi approximation does not take into account she effects, it is more reasonabe to extract ony reative shifts of the binding energies from the Wigner-Seitz ce cacuations than to use the absoute binding energies inside the medium. Correspondingy, the vacuum binding energies from experiment or mass tabes have to be added to these shifts. Experimenta binding energies per nuceon [22] as a function of the mass number A are depicted in in figure 2 by back points. he maximum near A = 60 and she effects cose to douby magic nucei are ceary visibe. When the binding energy shifts in the medium of a 9

11 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing 10 8 BE/A [MeV] AME2011 n = fm -3 n = fm -3 n = fm -3 n = fm -3 n = fm -3 n = fm -3 n = fm A Figure 2. Binding energy per nuceon of nucei with mass number A in vacuum (AME2011 [22]) and in the medium at temperature = 0 MeV for various densities n. Figure from reference [14]. certain density are taken into account, a cear change in the A dependence is observed. At ow medium densities, the screening of the Couomb fied by the eectrons causes a stronger binding of the nucei. he effect is stronger for nucei with arge charge numbers. In contrast, at higher medium densities, the binding energies of the nucei reduce substantiay because there is ess increase of binding energy when a custer is formed from a uniform nuceon distribution. Light custers dissove at ower medium densities than heavier custers. he sope of the distribution and the position of the most strongy bound nucei change with the medium density, too. raditiona EoS tabes for astrophysica appications, see, e.g. Refs. [3, 4], consider ony one representative nuceus, the so-caed singe-nuceus approximation (SNA). Ony more recent NSE-type modes [6, 7] cover the the fu distribution of nucei. hese modes usuay take into account a correction of the binding energy due to the Couomb screening effect but the reduction of the nucear binding energies at higher medium densities is generay not considered. A significant change in the abundance distribution can be expected from the grdf cacuations and it remains to be seen how this effect modifies the resuts of astrophysica simuations. 5. Low-density imit At ow densities and finite temperatures, the VEoS represents the correct description of the matter properties because ony two-body correations are reevant. hus, the EoS in the grdf approach shoud reproduce the mode-independent ow-density resuts of the VEoS. Consistency reations can be derived by comparing the fugacity expansions of the grand thermodynamica potentia Ω in both modes. hey can be used to determine the effective resonance energies E ij and degeneracy factors of the nuceon-nuceon scattering states g (eff) ij that both depend on. One finds that there are aready corrections to the first order coefficient b i in equation (13) of the VEoS due to the reativistic kinematics. Introducing the viria integras for continuum correations of partices i and j I (ij) de = π dδ (ij) ( de exp E ), (23) 10

12 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing interna energy per nuceon [MeV] idea gas reativistic Fermi gas reativistic viria EoS standard RMF grdf density n [fm -3 ] Figure 3. Interna energy per nuceon for neutron matter at temperature = 10MeVasafunctionofthetota neutron density n in different approximations. Figure adapted from reference [14]. the contribution from different scattering channes can be coected in a singe exponentia term [ g (ij) I (ij) = ±g (ij) 0 exp E ] ij() (24) that defines E ij (). he sign on the right-hand side of equation (24) is determined uniquey by the sign of the eft-hand side. he effective resonance energies usuay increase smoothy with temperature as depicted in figure 2 of reference [13]. At ow temperatures, ony the s-wave channe contributes substantiay to the viria integra and the effective-range expansion of the phase shift can be used to obtain anaytica resuts for the viria integras. Negecting reativistic correction, the consistency reations 1 λ 3 nn for the nn channe and 1 λ 3 np g (nn) I (nn) = 1 [ λ 3 g nn (eff) ()exp E nn() nn g (np) I (np) = 1 λ 3 np 1 t=0 ] g2 n C + λ 6 n 2 [ g (eff) npt ()exp E ] npt() g ng p C λ 3 n λ3 p for the np channe are found from a comparison of the second viria coefficients in the VEoS and grdf approaches. he sum of the two possibe isospin channes t = 0, 1 is expicity indicated in equation (26). he occurrence of a term that depends on the meson coupings Γ k at zero density through the coefficients C ± = C ω C σ ±C ρ C δ with C k = [Γ k (0)] 2 /m 2 k is the important feature in these reations. Since the effective resonance energies E ij () are aready defined by reation (24), equations (25) and (26) serve as the definition of the effective degeneracy factors g nn (eff) () and g (eff) npt (). Again, a smooth dependence on the temperature is found as depicted in figure of reference [13]. In the zero-temperature imit, the consistency reations simpify substantiay and two conditions are found that connect the differences C ω C σ and C ρ C δ with the scattering engths of the reevant four nuceon-nuceon s-wave scattering channes. Conventiona RMF parametrizations vioate these reations, see references [13, 14]. (25) (26) 11

13 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing 10 0 partice fraction X i density n [fm -3 ] p n 2 H 3 H 3 He 4 He Figure 4. Partice fractions X i = n i /n in nucear matter with temperature = 10 MeV and tota proton fraction Y p = 0.4 as a function of the tota nuceon density n in the grdf approach without heavy custers. Cacuation without (dashed ines) and with (fu ines) nuceon-nuceon scattering correations. Heavy custers with mass number A > 4 are not considered in this cacuation. Figure adapted from reference [14]. In pure neutron matter, various effects on the ow-density EoS are most easiy compared. here is no bound state and two-nuceon correations appear ony in the nn scattering channe. In figure 3 the interna energy per neutron E/N is depicted as a function of the tota neutron density in different approximations for = 10 MeV. For a nonreativistic idea gas, it is just E/N = 3/2 = 15 MeV independent of n. he interna energy per nuceon at zero density ist shifted to higher vaues by reativistic effects. With quantum statistica corrections of a Fermi gas, an increase of E/N with the density is observed. he exact VEoS exhibits a different sope due to the considered correations but a standard RMF cacuation does not reproduce this behavior. In constrast, the grdf mode with the constribution of the effective nn scattering correation perfecty reproduces the VEoS at ow densities. 6. Light custers and continuum correations At densities beow nucear saturation density, the composition of matter changes substantiay when the thermodynamica variabes, n and Y p are varied. he evoution of the partice mass number fractions X i = A i n i /n is shown in figure 4 as a function of the tota nuceon density n for = 10 MeV and Y p = 0.4 for the grdf mode using the parametrization DD2 [12]. A cacuation without nuceon-nuceon continuum correations (dashed ines) is compared to a cacuation with these correations (fu ines). Protons and neutrons dominate the composition of matter at very ow densities with a tiny fraction of deuterons. he abundance of heavier custers is negigibe at these densities. With increasing density, custer contributions to the composition become more important. First, three-body ( 3 H, 3 He) and then four-body ( 4 He) correations appear and the number of free nuceons is reduced. he tota custer fraction reaches a maximum at approximatey 1/10 of the nucear saturation density. Increasing the density further causes a reduction of the custer abundancies and finay they disappear. hus the grdf mode can describe the dissoution of custers in matter, i.e. the Mott effect, by assuming a medium dependence of the custer properties, more precisey the binding energies. In figure 4 a substantia effect of two-nuceon scattering correations on the custer fractions can be observed. hey ead to a reduction of the number of deuteron-ike correations and a redistribution of the remaining partice fractions. he partice fractions in the grdf mode correspond to those those of quasipartices and not to those of the origina constituents. he 12

14 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing scaed symmetry energy E sym (n)/e sym (n sat ) without custers with custers scaed density n/n sat Figure 5. Scaed symmetry energy of nucear matter at zero temperature as function of the scaed density in the grdf mode with parametrization DD2 without (bue dashed ine) and with (red fu ine) custer formation taken into account. Figure adapted from reference [14]. finite sef-energy of the quasipartices accounts for some part of the correation strength due to the interaction and the size of expicit correations is reduced. 7. Symmetry energy hedensitydependenceofthenucearmatter symmetryenergye sym (n)iscurrentyinvestigated with much experimenta and theoretica effort, see references [23, 24] for constraints and the importance in nucear physics and in astrophysica appications. Usuay, E sym (n) is defined as the second derivative of the energy per nuceon E(n,β)/A in nucear matter with respect to the neutron-proton asymmetry β, thus representing the curvature of the energy per nuceon in the direction of isospin asymmetry. In many cases, a quadratic approximation is sufficienty precise and the symmetry energy can be obtained by the finite-difference formua E sym (n) = 1 [ E 2 A (n,1) 2E A (n,0)+ E ] A (n, 1) (27) that compares symmetric nucear matter (β = 0) with pure neutron (proton) matter (β = ±1). At finite temperatures, one has to distingish the interna symmetry energy E sym and the free symmetry energy F sym. he density dependence of E sym at ow densities is strongy modified by the appearance of custers, see figure 5 for matter at zero temperature. E sym (n) approaches zero in the imit of vanishing density n in conventiona mean-fied modes of uniform matter without custer degrees of freedom. When the formation of custers is taken into account, the system becomes more bound, in particuary for symmetric nucear matter and at ow temperatures. In contrast, the energy per nuceon of neutron matter is much ess affected by correations. As a consequence, the symmetry energies rises as compared to that in mean-fied modes without correations. For vanishing temperature it even approaches a finite vaue in the imit of zero density. At higher temperatures, the effects wi be ess pronounced but possiby sti detectabe in experiments. It has to be emphasized that the symmetry energy (27), extracted from the tota energies per nuceon, contains aso Couomb contributions if custers are taken into account. his is in contrast to the Bethe-Weizsäcker mass formua for nucei where Couomb and symmetry energy contributions are considered separatey. In principe, the eectromagnetic contribution to the tota energy can be extracted unambiguousy from the mode cacuations and the density dependence of the pure nucear symmetry energy can be determined. he anaysis of heavy-ion coisions can hep to observe the correation effects on the symmetry energy by determining the thermodynamica properties of the expanding system as compete 13

15 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing as possibe. here are strong experimenta efforts to extract the symmetry energy and to study the properties of fragments in diute matter. Recent resuts indicate an increase of the symmetry energy as predicted by modes with correations and custer formation, see references [25, 26, 27, 28, 29] for detais. 8. Concusions he properties of interacting many-body systems, in particuar nucear matter, are strongy affected by correations. he chemica composition can change by forming custers, i.e. manybody correations in bound and scattering states, and their dissoution at high medium densities. his aspect has to be taken into account in theoretica modes for the equation of state because thermodynamica properties of dense matter are modified by these changes. Correation effects wi aso have consequences in appications such as astrophysica simuations of core-coapse supernovae or neutron star modes. It is a great chaenge to cover the reevant range in the thermodynamica variabes temperature, density and isospin asymmetry in a singe theoretica mode. Nucear matter can be described by a number of theoretica approaches that consider correations on various eves of sophistication using different concepts. At densities near nucear saturation, properties of dense matter are quite successfuy described by mean-fied modes where nuceons are considered as quasipartices. heir sef-energies depend on the medium and incorporate the effect of correations, however expicit correations are not taken into account. At ow densities, severa approaches to construct on EoS were devised in the past. Simpe nucear statistica equiibrium modes are based on an idea mixture of nuceons and nucei without taking into account the interaction between these constituents. A suppression of custers can be incorporated into the mode using a geometrica excuded voume mechanism. wobody correations are expicity incuded in the viria equation of state that serves as a mode independent benchmark at ow densities and not too ow temperatures since the resuts depend ony on experimenta data. At high densities this approach fais because it is based on a series expansion in sma partice fugacities and medium effects on the custer properties are negected. In a quantum statistica/generaized Beth-Uhenbeck approach, these effects can be incuded on a microscopic eve eading to the dissoution of custers when their binding energies vanish. A generaized reativistic density functiona approach provides an interpoation between the correct ow-density and high-density imits. his mode considers nuceons, two-nuceon continuum states and nucei as degrees of freedom. hese constituents are treated as quasipartices with medium-dependent sef-energies. he nucear interaction is described by an exchange of mesons with density dependent coupings to the nuceons, either free or bound in custers. he mode parameters are we determined by fitting to properties of finite nucei and nucear matter. Custers, i.e. composite partices in bound or scattering states, change their properties inside the nucear medium. In particuar, a reduction of the binding energies eads to their dissoution with increasing density and the Mott effect is observed. he description of ight custers has been deveoped aready in some detai. For heavier custers, the dependence of their binding energies has sti to be extracted from microscopic cacuations in the whoe range of thermodynamica variabes. he formation and dissoution of custers at ow densities causes an increase of the nucear symmetry energy at ow densities. his feature can be studied experimentay in heavy-ion coisions and is reevant in astrophysica simuations. Providing extensive EoS tabes is one important aim for the appication of the grdf approach in the future. Acknowedgments he author acknowedges the fruitfu coaboration on various topics presented in this contribution with Gerd Röpke (Universität Rostock), David Baschke (Uniwersytet Wroc awski), 14

16 11th Internationa Conference on Nuceus-Nuceus Coisions (NN2012) Journa of Physics: Conference Series 420 (2013) IOP Pubishing homas Kähn (Uniwersytet Wroc awski), Hermann Woter (Ludwig Maximiians-Universtität München) and Maria Voskresenskaya (GSI Darmstadt). He thanks the organizers of the NN2012 conference for their support. In addition, this work was supported in part by the DFG custer of exceence Origin and Structure of the Universe, by CompStar, a Research Networking Program of the European Science Foundation (ESF), by the Hemhotz Internationa Center for FAIR within the framework of the LOEWE program aunched by the state of Hesse via the echnica University Darmstadt and by the Hemhotz Association (HGF) through the Nucear Astrophysics Virtua Institute (VH-VI-417). References [1] Janka H-h, Langanke K, Marek A, Martinez-Pinedo G and Mueer B 2007 Phys. Rep [2] Gendenning N K 2000 Compact Stars: Nucear Physics, Partice Physics, and Genera Reativity (New York: Springer) [3] Lattimer J M and Swesty F D 1991 Nuc. Phys. A [4] Shen H, oki H, Oyamatsu K and Sumiyoshi K 1998 Nuc. Phys. A [5] Shen H, oki H, Oyamatsu K and Sumiyoshi K 1998 Prog. heor. Phys [6] Hempe M and Schaffner-Bieich J 2010 Nuc. Phys. A [7] Botvina A S and Mishustin I N 2010 Nuc. Phys. A [8] Shen G, Horowitz C J and eige S 2010 Phys. Rev. C [9] Shen H, oki H, Oyamatsu K and Sumiyoshi K 2011 Ap. J. Supp. Series 197, 20 [10] Shen G, Horowitz C J and eige S 2010 Phys. Rev. C [11] Shen G, Horowitz C J and O Connor E 2011 Phys. Rev. C [12] ype S, Röpke G, Kähn, Baschke D, and Woter H H 2010 Phys. Rev. C [13] Voskresenskaya M D and ype S 2012 Nuc. Phys. A [14] ype S 2012 Custers in Nucear Matter and the Equation of State for Astrophysica Appications Proc. Int. Workshop XII Hadron Physics (Apri 22-27, 2012, Bento Gonçaves, RS, Brazi) ed M V Machado (Mevie, NY: American Institute of Physics) to be pubished [15] Beth G E and Uhenbeck E 1936 Physica [16] Beth G E and Uhenbeck E 1937 Physica [17] Horowitz C J and Schwenk A 2006 Nuc. Phys. A [18] Schmidt M, Röpke G and Schuz H 1990 Ann. Phys. (N.Y.) [19] ype S and Woter H H 1999 Nuc. Phys. A [20] Kähn, et a Phys. Rev. C [21] Röpke G 2011 Nuc. Phys. A [22] Audi G and Meng W 2011 private communication [23] sang M B et a Constraints on the symmetry energy and neutron skins from experiments and theory Preprint arxiv: [24] Lattimer J M and Lim Y 2012 Constraining the Symmetry Parameters of the Nucear Interaction Preprint arxiv: [25] Kowaski A et a Phys. Rev. C [26] Natowitz J B et a Phys. Rev. Lett [27] Hage K et a Phys. Rev. Lett [28] Qin L et a Phys. Rev. Lett [29] Wada R et a Phys. Rev. C