Physical properties of hot,dense matter: The general case

The thermodynamic properties of hot, dense matter are examined in the density range 10^?5 fm^?3 ? n ? 0.35 fm^?3 and the temperature range 0 ? T ? 21 MeV, for fixed lepton fractions Y_? = 0.4, 0.3 and 0.2 and for matter in β-equilibrium with no neutrinos. Three phases of the matter are considered: the nuclei phase is assumed to consist of Wigner-Seitz cells with central nuclei surrounded by a nucleon vapor containing also α-particles; in the bubbles phase the cell contains a central spherical bubble of nucleon vapor surrounded by dense nuclear matter; the third phase is that of uniform nuclear matter. All are immersed in a sea of leptons (electrons and neutrinos) and photons. The nuclei and bubbles are described by a compressible liquid drop model which is self-consistent in the sense that all of the constituent properties — bulk, surface, Coulomb energies and other minor contributions — are calculated from the same nuclear effective hamiltonian, in this case the Skyrme 1' interaction. The temperature dependence of all of these energies is included, for bulk and surface energies by direct calculation, for the Coulomb energy by combining in a plausible way the usual electrostatic energy and the numerical results pertaining to a hot Coulomb plasma. Lattice contributions to the Coulomb energy are an essential ingredient, and lattice modifications to the nuclear translational energy are included. A term is constructed to allow also for the reduced density of excited states of light nuclei. All of these modifications incorporate necessary physical effects which modify significantly the matter properties in some regions.     

核物质液-气相变和热核稳定性

简述了研究核物质液－气相变的理论计算及研究热核稳定性的二相平衡模型．讨论了相变的临界温度及其对核物质大小、不对称度及库仑相互作用的依赖性．比较了用各种核力或模型算得的热核极限温度及实验结果． A brief introduction of the theoretical studies of liquid gas phase transition in nuclear matter and a two phase equilibrium model for studying stability of hot nuclei is presented. The critical temperature of the phase transition and its dependence on size and asymmetry of nuclear matter and Coulomb interaction are discussed. The limiting temperatures of hot nuclei calculated with various nuclear forces or models and experimental results are compared.     

Flow probe of symmetry energy in relativistic heavy-ion reactions

Various definitions of the symmetry energy are introduced for nuclei, dilute nuclear matter below saturation density and stellar matter, which is found in compact stars or core-collapse supernovae. The resulting differences are exemplified by calculations in a theoretical approach based on a generalized relativistic density functional for dense matter. It contains nucleonic clusters as explicit degrees of freedom with medium-dependent properties that are derived for light clusters from a quantum statistical approach. With such a model the dissolution of clusters at high densities can be described. The effects of the liquid-gas phase transition in nuclear matter and of cluster formation in stellar matter on the density dependence of the symmetry energy are studied for different temperatures. It is observed that correlations and the formation of inhomogeneous matter at low densities and temperatures causes an increase of the symmetry energy as compared to calculations assuming a uniform uncorrelated spatial distribution of constituent baryons and leptons.     

The determination of the Coulomb energy of a nuclear system at the scission point upon fission

The dependence of the Coulomb energy of deformed nuclei on the shape and the nuclear matter density distribution is considered. For an ellipsoidal shape of the nucleus and a homogeneous distribution of nuclear matter, the result was obtained in a simple analytical form convenient for further use. Separate attention was paid to the consideration of a Fermi-like distribution of nuclear matter using different shapes of the nucleus that reflect several different kinds of collective motions. After these considerations, the dependence of the Coulomb energy of the fission products at the scission point of binary fission reactions on configurations, shapes and nuclear matter distributions was investigated. Calculation of the dependence of the shape of the nuclei deformation parameters at the scission point from their total kinetic energies was made in detail. Finally, the influence of shell effects on the mass yield of fission products is discussed. The text was submitted by the authors in English.     

Statistical properties and stability of hot nuclei

Results of temperature-dependent Hartree-Fock calculations for equilibrated hot nuclei are presented, extending to the highest temperatures at which the nuclei remain stable. A subtraction procedure developed earlier for isolating the properties of the nucleus from the nucleus + vapor system is applied. The temperature dependence of various quantities characterizing hot nuclei is investigated. The influence of different effective interactions in the Hartree-Fock equations is examined. Special attention is devoted to the study of the high-temperature stability limit of hot nuclei. This limit in nuclei with the Coulomb interaction artificially switched off (i.e. uncharged nuclei) is shown to correspond to the critical temperature of the liquid-gas phase transition expected on the basis of hot nuclear matter calculations. In realistic charged nuclei the Coulomb repulsion causes a nucleus to become electrostatically unstable and to fall apart at much lower temperatures than its uncharged partner. The approach to and the temperature of this Coulomb instability are very sensitive to the choice of the nuclear interaction. Studying this instability in compound nuclei with different charge-to-mass ratio provides a sensitive measure of the temperature dependence of the nuclear surface properties as well as of certain features of the nuclear equation of state.     

Simulation of transitions between "pasta" phases in dense matter

Calculations of equilibrium properties of dense matter predict that at subnuclear densities nuclei can be rodlike or slablike. To investigate whether transitions between phases with nonspherical nuclei can occur during the collapse of a star, we perform quantum molecular dynamic simulations of the compression of dense matter. We have succeeded in simulating the transitions between rodlike and slablike nuclei and between slablike nuclei and cylindrical bubbles. Our results strongly suggest that nonspherical nuclei can be formed in the inner cores of collapsing stars.     

Analysis of fissionability data at high excitation energies

A level density formula that takes into account the smoothed volume, surface and curvature dependence of the single particle level density at the Fermi surface using the results of Balian and Bloch, is shown to be compatible with the level spacings found in neutron resonance data if complemented by a simple Ansatz for shell effects (due to Ignatyuk) and pairing effects. The three parameters involved, a scaling parameter, a shell damping energy and a pairing energy shift are compatible, respectively, with known nuclear radii, microscopic level density calculations and odd-even mass fluctuations. At excitation energies on the order of the neutron binding energy no evidence for an absolute level density problem or a different behaviour of level densities (collective contributions) for deformed nuclei as opposed to spherical nuclei is found. The proposed level density formula allows to calculate a priori macroscopic ratios of level densities, e.g. at the groundstate and at the saddle point, removing this important parameter from the analysis of fissionability data. As a first application, the fissionability of a number of actinide nuclei at excitation energies a few MeV above the fission barrier is analysed.     

The process of coulomb dissociation of weakly bound relativistic nuclei and hypernuclei within the two-cluster model

Using the analogy with the problem of ionization and excitation of atoms in the propagation of relativistic charged particles through a bulk of matter, the process of Coulomb dissociation of weakly bound relativistic nuclei and hypernuclei is theoretically investigated in the framework of the two-cluster deuteron-like model. Explicit expressions for the total cross section of Coulomb disintegration of weakly bound systems are derived taking into account the corrections due to the finite size of a target nucleus. Numerical estimations for the Coulomb dissociation of relativistic hypernuclei ³H_Λ and ⁶He_Λ are performed. It is shown that, owing to a sharp dependence of the Coulomb dissociation cross section on the binding energy, experimental measurements of the cross section allow one to determine the values of binding energy for these systems. The text was submitted by the authors in English.     

原子核对称能对中子星壳层结构的影响

中子星内壳层中存在原子核、中子、电子等非均匀分布的物质。在Wigner-Seitz近似下,共存相方法和自洽Thomas-Fermi近似方法是描述这种非均匀物质的有效方法。中子在非均匀物质所占的比例远远大于其他组分,因此原子核的对称能对非均匀物质的性质会产生十分重要的影响,而原子核对称能的密度依赖关系在核物质饱和密度附近有较大的不确定性。采用相对论平均场理论描述核子间相互作用,研究原子核对称能对中子星内壳层的密度范围、pasta相结构、壳核相变密度等性质的影响,探寻其中可能存在的关联。计算结果表明,原子核对称能及其密度依赖性在决定中子星内壳层非均匀物质的性质中起着重要作用,这与之前相关研究中得到的结论基本相符。Within Wigner-Seitz approximation, both the coexisting phases method and the self-consistent Thomas-Fermi approximation can be used to describe the nonuniform matter consisting of nuclei, neutrons, and electrons, which may coexist in the inner crust of neutron star. Since the neutron fraction is very large, nuclear symmetry energy may have an important impact on the properties of nonuniform matter. However, the density dependence of nuclear symmetry energy around saturation density is still rather uncertain. This paper focuses on the influence of nuclear symmetry energy on the density range of inner crust, pasta phase structure, and crust-core transition density of neutron star, where the relativistic mean field theory is adopted to describe the nucleon-nucleon interaction. It is turned out that the nuclear symmetry energy and its density dependence play an import role in determining the properties of nonuniform matter in the inner crust of neutron star, which is consistent with the former related studies.     

Physical properties of hot, dense matter: The bulk equilibrium approximation

Starting from a Skyrme effective nucleon-nucleon interaction, we calculate the properties of hot matter at densities up to a little beyond that of nuclear matter. We neglect effects associated with finite nuclear size, such as surface and Coulomb energies, but include all effects associated with the bulk properties of the neutron-proton matter over a range of proton concentrations. The region of coexistence of two distinct nuclear phases is mapped out, as a function of the average proton fraction of the two-phase mixture. We present the neutron and proton chemical potentials, and the densities and proton fractions of the two coexisting phases, as functions of density and temperature. The density dependence of the nuclear contribution to the pressure is given along isotherms for a range of proton fractions. Including electron, positron and photon contributions to the entropy, we calculate the density dependence of the pressure along adiabats, and discuss the distribution of the entropy among the components. Some information is also given about hot matter in β-equilibrium. Comparison is made with earlier work.     

Threshold anomaly in non-central forces

The behaviour of the threshold anomaly for non-central potentials, which account for collective excitations in heavy-ion collisions, is investigated. It is shown that the non-central potentials should exhibit an energy dependence at energies in the vicinity of the Coulomb barrier. This energy dependence is, however, different from that of the elastic optical potential, occurring at lower energies. It if further shown that there are corrections to the traditional collective model such that, if the transition potential is expressed as the derivative of the optical potential, the corresponding deformation length will be complex and energy-dependent. Simple model calculations are presented.     

中子星内部强子-夸克相变的有限尺度效应研究

在致密星体内部极高密度条件下,强子物质可能发生退禁闭相变成为夸克物质,即强子-夸克相变。这种相变过程对于中子星的性质有着重要影响。考虑库仑能和表面能的影响,即有限尺度效应,相变过程中的混杂相包含了被称为pasta相的几何结构。强子-夸克共存相的平衡条件是通过求总能量的最小值得到的。采用相对论平均场（RMF）模型来描述强子物质相,采用Nambu-Jona-Lasinio（NJL）模型来描述夸克物质相。有限尺度效应一定程度上增加了中子星的最大质量,增加幅度取决于强子-夸克表面张力的大小。有限尺度效应能够降低混杂相的范围,其结果介于Gibbs结构和Maxwell结构的结果之间。研究结果表明,中子星中可能包含一个混杂相的核心部分,其大小受到表面张力等参数的影响。It is generally considered that hadron matter may undergo a deconfinement phase transition becoming quark matter at very high density in massive neutron stars. This hadron-quark phase transition has important impact on neutron stars, which has received much attention. We consider finite-size effect in this phase transition process, which contains the impact of Coulomb energy and surface energy. By including this effect, the mixed phase forms the pasta structures. The equilibrium conditions for coexisting hadronic and quark phases are derived by minimizing the total energy including the surface and Coulomb contributions. We employ the relativistic mean-field(RMF) model to describe the hadronic phase, while the Nambu-Jona-Lasinio(NJL) model is used for the quark phase. We conclude that the finite-size effect will raise the stiffness of EOS, and then increase the maximum mass of neutron stars, which depend on the value of surface tension. Our results show that finite-size effects can significantly reduce the region of the mixed phase, and the results lie between those from the Gibbs and Maxwell constructions. We show that a massive star may contain a mixed phase core and its size depends on the surface tension of the hadron-quark interface.     

Links between heavy ion and astrophysics

Heavy-ion experiments provide important data to test astrophysical models. The high-density equation of state can be probed in HI collisions and applied to the hot protoneutron star formed in core collapse supernovae. The parity radius experiment (PREX) aims to accurately measure the neutron radius of ²⁰⁸Pb with parity-violating electron scattering. This determines the pressure of neutron-rich matter and the density dependence of the symmetry energy. Competition between nuclear attraction and Coulomb repulsion can form exotic shapes called nuclear pasta in neutron star crusts and supernovae. This competition can be probed with multifragmentation HI reactions. We use large-scale semiclassical simulations to study nonuniform neutron-rich matter in supernovae. We find that the Coulomb interactions in astrophysical systems suppress density fluctuations. As a result, there is no first-order liquid-vapor phase transition. Finally, the virial expansion for low-density matter shows that the nuclear vapor phase is complex with significant concentrations of alpha particles and other light nuclei in addition to free nucleons.     

Influence of hadronic hard core radius on detonations and deflagrations in quark matter

The parameters for deflagration and detonations transitions between baryon-rich quark matter and hadronic matter are calculated at high densities and low temperatures. The equation of state of hadronic matter takes into account the hard core repulsion character of nuclear forces and the bag pressure is included in the quark phase. The transition is considered first directly from the quark phase to the hadronic phase. It is found that the velocity of the front separating the quark and nuclear phases as well as the energy flux are substantially larger than in the case where no hard core repulsion is taken into account. Most of the calculated quantities are only weakly dependent on the temperature in the region considered. The transitions from the quark phase to the mixed phase and from the mixed phase to the hadronic phase are also considered. No realistic solutions are found for the case where the temperature remains zero throughout the transition. At small temperatures and high densities solutions are obtained where the transition involves only minimal superheating. The energy flux for such a transition is small.     

Analysis of fission excitation functions and the determination of shell effects at the saddle point

A method is proposed to deduce the shell correction energy corresponding to the fission transition state shape of nuclei in the mass region around 200, from an analysis of the first chance fission values of the ratio of fission to neutron widths, (Γ _f /Γ _n )₁. The method is applied to the typical case of the fissioning nucleus²¹²Po, formed by alpha bombardment of²⁰⁸Pb. For the calculation of the neutron width, the level densities of the daughter nucleus after neutron emission were obtained from a numerical calculation starting from shell model single particle energy level scheme. It is shown that with the use of standard Fermi gas expression for the level densities of the fission transition state nucleus in the calculation of the fission width, an apparent energy dependence of the fission barrier height is required to fit the experimental data. This energy dependence, which arises from the excitation energy dependence of shell effects on level densities, can be used to deduce the shell correction energy at the fission transition state point. It is found that in the case of²¹²Po, the energy of the actual transition state point is higher than the energy of the liquid drop model (LDM) saddle point by (3 ± 1) MeV, implying significant positive shell correction energy at the fission transition state. Further, the liquid drop model value of level density parametera is found to be a few per cent smaller for the saddle point shape as compared to its spherical shape.     

Excitonic insulator: dependence of energy gap width on carrier density

On the basis of the Ashcroft empty core model potential, the equation for the constant of Coulomb interaction in the theory of excitonic insulator is modified. It is shown that in this case the dependence of the energy gap width on the charge carrier density obeys the Mott criterion in the limit of low densities. The conformity of the theory with some experimental data concerning metal–insulator phase transitions in doped semiconductors and transition metal compounds is discussed.     

Properties of nuclear pastas

In this review we study the nuclear pastas as they are expected to be formed in neutron star crusts. We start with a study of the pastas formed in nuclear matter (composed of protons and neutrons), we follow with the role of the electron gas on the formation of pastas, and we then investigate the pastas in neutron star matter (nuclear matter embedded in an electron gas).Nuclear matter (NM) at intermediate temperatures (1 MeV ≲ T ≲ 15 MeV), at saturation and sub-saturation densities, and with proton content ranging from 30% to 50% was found to have liquid, gaseous and liquid–gas mixed phases. The isospin-dependent phase diagram was obtained along with the critical points, and the symmetry energy was calculated and compared to experimental data and other theories. At low temperatures (T ≲ 1 MeV) NM produces crystal-like structures around saturation densities, and pasta-like structures at sub-saturation densities. Properties of the pasta structures were studied with cluster-recognition algorithms, caloric curve, the radial distribution function, the Lindemann coefficient, Kolmogorov statistics, Minkowski functionals; the symmetry energy of the pasta showed a connection with its morphology.Neutron star matter (NSM) is nuclear matter embedded in an electron gas. The electron gas is included in the calculation by the inclusion of an screened Coulomb potential. To connect the NM pastas with those in neutron star matter (NSM), the role the strength and screening length of the Coulomb interaction have on the formation of the pastas in NM was investigated. Pasta was found to exist even without the presence of the electron gas, but the effect of the Coulomb interaction is to form more defined pasta structures, among other effects. Likewise, it was determined that there is a minimal screening length for the developed structures to be independent of the cell size.Neutron star matter was found to have similar phases as NM, phase transitions, symmetry energy, structure function and thermal conductivity. Like in NM, pasta forms at around T ≈ 1.5 MeV, and liquid-to-solid phase changes were detected at T ≈ 0.5 MeV. The structure function and the symmetry energy were also found to depend on the pasta structures.     

Core collapse supernovae in the QCD phase diagram

We compare two classes of hybrid equations of state with a hadron-to-quark matter phase transition in their application to core collapse supernova simulations. The first one uses the quark bag model and describes the transition to three-flavor quark matter at low critical densities. The second one employs a Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model with parameters describing a phase transition to two-flavor quark matter at higher critical densities. These models possess a distinctly different temperature dependence of their transition densities which turns out to be crucial for the possible appearance of quark matter in supernova cores. During the early post-bounce accretion phase quark matter is found only if the phase transition takes place at sufficiently low densities as in the study based on the bag model. The increase critical density with increasing temperature, as obtained for our PNJL parametrization, prevents the formation of quark matter. The further evolution of the core collapse supernova as obtained applying the quark bag model leads to a structural reconfiguration of the central protoneutron star where, in addition to a massive pure quark matter core, a strong hydrodynamic shock wave forms and a second neutrino burst is released during the shock propagation across the neutrinospheres. We discuss the severe constraints in the freedom of choice of quark matter models and their parametrization due to the recently observed 2M _?? pulsar and their implications for further studies of core collapse supernovae in the QCD phase diagram.     

Field-theoretical treatment of the liquid-vapor phase transition in nuclear systems

The liquid-vapor phase transition in hot nuclear matter is investigated in a field-theoretical approach employing euclidean-space (imaginary time) path-integral techniques. This approach allows us to study the nucleation due to both quantum and thermodynamic fluctuations. The bubbles of the new phase appear as instanton solutions of the euclidean-space field equations. The critical bubble sizes and associated transition probabilities are calculated. We examine the temperature and density values for which a phase transition may develop in hot nuclear matter produced in the course of a heavy-ion reaction.     

Effects of Density-Dependent Quark Mass on Phase Diagram of Color-Flavor-Locked Quark Matter

Considering the density dependence of quark mass, we investigate the phase transition between the (unpaired) strange quark matter and the color-flavor-locked matter, which are supposed to be two candidates for the ground state of strongly interacting matter. We find that if the current mass of strange quark ms is small, the strange quark matter remains stable unless the baryon density is very high. If ms is large, the phase transition from the strange quark matter to the color-flavor-locked matter in particular to its gapless phase is found to be different from the results predicted by previous works. A complicated phase diagram of three-flavor quark matter is presented, in which the color-flavor-locked phase region is suppressed for moderate densities.