Matter, nuclear

The composition of the Earth was determined both by the chemical composition of the solar nebula, from which the sun and planets formed, and by the nature of the physical processes that concentrated materials to form planets. The bulk elemental and isotopic composition of the nebula is believed, or usually assumed to be identical to that of the sun. The few exceptions to this include elements and isotopes such as lithium and deuterium that are destroyed in the bulk of the sun s interior by nuclear reactions. The composition of the sun as determined by optical spectroscopy is similar to the majority of stars in our galaxy, and accordingly the relative abundances of the elements in the sun are referred to as "cosmic abundances." Although the cosmic abundance pattern is commonly seen in other stars there are dramatic exceptions, such as stars composed of iron or solid nuclear matter, as in the case with neutron stars. The... 

A pre-supernova model of a 9Mq star is taken from Nomoto [3], which forms a 1.38 Mq O-Ne-Mg core. We link this core to a one-dimensional implicit La-grangian hydrodynamic code with Newtonian gravity. The equation of state of nuclear matter (EOS) is taken from Shen et al. [4]. We find that a very weak explosion results, where no r-processing is expected. In order to examine the possible operation of the r-process in the explosion of this model, we artificially obtain an explosion with a typical energy of 1051 ergs by application of a multiplicative factor (= 1.6) to the shock-heating term in the energy equation. 

After a 20 year break V. H. Ambartsumyan and G. S. Sahakian initiated an intensive research on compact objects during the 1960s in Armenia. In their pioneering work on compact stars they showed, that with increasing density, hyperons appear in nuclear matter and thus a neutron star at high densities consists predominantly of hyperons. Thus, as the density increases more and more heavy particles become stable. After the discovery of quarks as basic constituents of hadrons (including hyperons) the ideas of compact stars with quark cores or stars entirely composed of quark matter were presented. 

Abstract The equation of state (EOS) of nuclear matter at finite temperature and density with various proton fractions is considered, in particular the region of medium excitation energy given by the temperature range T < 30 MeV and the baryon density range ps < 1014 2 g/cm3. In this region, in addition to the mean-field effects the formation of few-body correlations, in particular light bound clusters up to the alpha-particle (1 < A < 4) has been taken into account. The calculation is based on the relativistic mean field theory with the parameter set TM1. We show results for different values for the asymmetry parameter, and (3 equilibrium is considered as a special case. 

The equation of state (EOS), the composition and the possible occurrence of phase transitions in nuclear matter are widely discussed topics not only in nuclear theory, but are also of great interest in astrophysics and cosmology. Experiments on heavy ion collisions, performed over the last decades, gave new insight into the behavior of nuclear systems in a broad range of densities and temperatures. The observed cluster abundances, their spectral distribution... 

Clusters and Condensates in the Nuclear Matter Equation of State... 

If a singularity in the medium modified few-body T matrix is obtained, it may be taken to indicate the formation of a quantum condensates. Different kinds of quantum condensates are also considered [7, 8], They become obvious if the binding energy of nuclei is investigated [9], Correlated condensates are found to give a reasonable description of near-threshold states of na nuclei [10], The contribution of condensation energy to the nuclear matter EOS would be of importance and has to be taken into account not only in mean-field approximation but also considering correlated condensates. 

The relativistic EOS of nuclear matter for supernova explosions was investigated recently [11], To include bound states such as a-particlcs, medium modifications of the few-body states have to be taken into account. Simple concepts used there such as the excluded volume should be replaced by more rigorous treatments based on a systematic many-particle approach. We will report on results including two-particle correlations into the nuclear matter EOS. New results are presented calculating the effects of three and four-particle correlations. 

In the low-density limit, the most important effect of interaction with respect to the nuclear matter EOS is the formation of bound states characterized by the proton content and the neutron content Nt. We will restrict us to only the... 

Results for the composition of nuclear matter at temperature T = 10 MeV with proton fraction V/"1, = 0.2 are shown in Fig. 1, for symmetric matter Yp0t = 0.5 in Fig. 2. The model of an ideal mixture of free nucleons and clusters applies to the low density limit. At higher baryon density, medium effects are relevant to calculate the composition shown in Figs. 1, 2, which are described in the following sections. 

Figure 1. Composition of nuclear matter with proton fraction 0.2 as function of the baryon density, T = 10 MeV.

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

The meson fields op, too and po are found by solving a set of equations self-consistently as shown in [11], Also expressions for the energy density, pressure and the entropy density can be found there. The empirical values of the binding energy of nuclear matter and nuclear matter density are reproduced using the above mentioned parameterization. The nuclear matter EOS can be found expressing the chemical potentials as functions of temperature, baryon density... 

The account of two-particle correlations in nuclear matter can be performed considering the two-particle Green function in ladder approximation. The solution of the corresponding Bethe-Salpeter equation taking into account mean-field and Pauli blocking terms is equivalent to the solution of the wave equation... 

Calculations of the composition (112/ns) of symmetric nuclear matter (np = nn, no Coulomb interaction) are shown in Fig. 3 [7], At low densities, the contribution of bound states becomes dominant at low temperatures. At fixed temperature, the contribution of the correlated density 112 is first increasing with increasing density according to the mass action law, but above the Mott line it is sharply decreasing, so that near nuclear matter density (ns = ntot = 0.17 fm-3) the contribution of the correlated density almost vanishes. Also, the critical temperature for the pairing transition is shown. 

Figure 3. Fraction of correlated density for symmetric nuclear matter, T = 10 MeV. Only two-particle correlations are taken into account.

The result of this calculation is also seen in Fig. 2, to be compared with the evaluation of the correlated density shown in [5], Two particle correlations are suppressed for densities higher then the Mott density of about 0.001 fm-3, but will survive to densities of the order of nuclear matter density. 

Now we can calculate the composition replacing the binding energies by the density dependent ones. Results for the composition are shown in Figs. 1,2. It is shown that in particular a-clusters are formed in symmetric nuclear matter but they are destroyed at about nuclear matter density. In the case of asymmetric matter, triton becomes abundant. 

Isospin singlet (pn) and triplet (nn,pp) pairing in nuclear matter... 

One of the most amazing phenomena in quantum many-particle systems is the formation of quantum condensates. Of particular interest are strongly coupled fermion systems where bound states arise. In the low-density limit, where even-number fermionic bound states can be considered as bosons, Bose-Einstein condensation is expected to occur at low temperatures. The solution of Eq. (6) with = 2/j, gives the onset of pairing, the solution of Eq. (7) with EinP = 4/i the onset of quartetting in (symmetric) nuclear matter. At present, condensates are investigated in systems where the cross-over from Bardeen-Cooper-Schrieffer (BCS) pairing to Bose-Einstein condensation (BEC) can be observed, see [11,12], In these papers, a two-particle state is treated in an uncorrelated medium. Some attempts have been made to include the interaction between correlated states, see [7,13]. 

A possible application for the formation of a-like condensates are selfconjugate 4n nuclei such as 8Be, 12C, 160,20Ne, 24Mg, and others. Of course, results obtained for infinite nuclear matter cannot immediately be applied to finite nuclei. However, they are of relevance, e.g., in the local density approximation. We know from the pairing case that the wave function for finite systems can more or less reflect properties of quantum condensates. 

In certain regions of the density-temperature plane, a significant fraction of nuclear matter is bound into clusters. The EOS and the region of phase instability are modified. In the case of /3 equilibrium, the proton fraction and the occurrence of inhomogeneous density distribution are influenced in an essential way. Important consequences are also expected for nonequilibrium processes. 

The inclusion of both three and four-particle correlations in nuclear matter allows not only to describe the abundances oft, h, a but also their influence on the equation of state and phase transitions. In contrast to the mean-field treatment of the superfluid phase, also higher-order correlations will arise in the quantum condensate. 

It is well known that at lower densities the properties of the EoS are primarily determined by the SE [2], The latter is defined in terms of a Taylor series expansion of the energy per particle for nuclear matter in terms of the asymmetry parameter a = (N — Z)/A (or equivalently the proton fraction x = Z/A),... 

As noted above at higher densities the EoS is sensitive to 3NF contributions. Whereas the 3NF for low densities seems now well understood its contribution to nuclear matter densities remains unsettled. In practice in calculations of the symmetry energy in the BHF approach two types of 3NF have been used in calculations in ref.[4] the microscopic 3NF based upon meson exchange by Grange et al. was used, and in ref. [15] as well in most VCS calculations the Urbana interaction. The latter has in addition to an attractive microscopic two-pion exchange part a repulsive phenomenological part constructed in such a way that the empirical saturation point for SNM is reproduced. Also in practice in the BHF approach to simplify the computational efforts the 3NF is reduced to a density dependent two-body force by averaging over the position of the third particle. 

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