Energy Of Nuclear Matter Research Articles

π- and ϱ-exchange three-nucleon potentials (I)

The current algebra program of the ππ exchange three-body force (TBF) is extended to include ϱ- and π-meson exchange. A qualitative discussion is given of the origin of the potentials which are constrained by the low-energy theorems of Thirring, Beg, and Kroll and Ruderman and supplemented by Δ-isobar contributions. The leading local parts of the ϱϱ and ϱπ TBF's are presented and used to estimate the binding energy of nuclear matter. The contributions are ππ TBF ∼- −4 MeV, ϱπ TBF ∼- +2 MeV, ϱϱ TBF ∼- − 1 4 MeV . The relative contributions from model-independent low-energy theorems and from the Δ-isobar to the ϱπ and ρρ potentials are discussed. In contrast to the ππ exchange TBF which is dominated by the model-independent part, the ϱπ and pp potentials have substantial contributions from an explicit Δ-isobar. The convergence of the nuclear matter estimates with respect to exchanged meson mass indicates that future calculations should not be concerned with heavier meson exchanges. The form factors used in the estimates with these potentials are collected in an appendix.

Field-theoretical model for nuclear and neutron matter. I. Equation of state.

We analyze a field-theoretical Lagrangian model for the many-body problem of baryonic matter. The Lagrangian describes the baryonic interaction through the exchange of \ensuremath{\sigma}, \ensuremath{\pi}, \ensuremath{\omega}, and \ensuremath{\rho} mesons, and contains various models studied in the literature as particular cases. The model is solved in the relativistic Hartree approximation using the technique of the covariant Wigner function. The vacuum fluctuation effects are considered and renormalized through a counterterm procedure. The equation of state is obtained in this renormalized Hartree approximation at finite temperature and in the presence of homogeneous and isotropic meson condensates. In a first particular case (the Serot model) the parameters of the model are determined in order to explain the nuclear saturation and symmetry energy of symmetric nuclear matter at nuclear density and to account for the expected low-density behavior of dense matter after the neutron drip in a phenomenological way. In this case the incompressibility of nuclear matter at the nuclear density is 460 MeV. A second determination of the parameters of the general model leads to a \ensuremath{\sigma} model with an explicit symmetry-breaking term and is coupled to the \ensuremath{\omega} and \ensuremath{\rho} mesons in a renormalizable way. The equation of state thus obtained explains accurately all the properties of nuclear matter at nuclear density. In particular, we obtain a nuclear incompressibility of 225 MeV. The analysis of the resulting energy density as a function of the meson condensates shows that the presence of homogeneous and isotropic pion condensates or abnormal states of Lee-Wick type in neutron matter at T=0 is energetically forbidden in this approximation.

Influence of mesonic and nuclear selfenergies on the binding energy of nuclear matter

π- and ϱ-meson selfenergies are applied in nuclear matter calculations. It is shown that they lead to an additional attraction of nearly 10 MeV at nuclear matter density. However, this effect is almost exactly cancelled by nuclear selfenergy corrections.

Giant isovector monopole resonances and the symmetry energy of nuclear matter

Average excitation energies for isovector monopole resonances are computed for double magic nuclei using Skyrme interactions and RPA sum rules. The dependence of the resonance energy on the value of the nuclear matter symmetry energy is shown to be very weak when the interactions reproduce the gross bulk properties of the ground states of those nuclei.

On the stationarity of the nuclear-surface energy

We show that the specific surface energy of semi-infinite nuclear matter is stationary with respect to the limiting central density only in the charge-symmetric case. The failure of the “ σ = 0” theorem in the asymmetric case is demonstrated numerically within the context of the scaling model, and formally within the context of a very general constrained energy-density method. The non-zero value of σ is shown to be related to the existence of the neutron skin.

Modification of the mean field theory of super-dense matter

We further discussed the Modified Mean Field Theory proposed in [1,2]. We found that this model has a new solution ml = 2; adding ϱ-meson, we calculated the symmetric energy of nuclear matter a 4; with the new parameters we calculated the equations of state with and without the addition of ϱ-meson; we compared the two; we also compared with the results for ml = 4.3; we made a preliminary theoretical investigation of the form of the vector meson mean field using the two-body correlation function; lastly, using the structural equation of neutron star we calculated the maximum mass for the various equations of state. The results are: MMFT-2 (ml = 2): M max = 2.22M⊙ (without ϱ- meson) M max = 2.27M⊙ (with ϱ-meson) MMFT-1 (ml = 4.3): M max = 1.69M⊙ (without ϱ- meson) M max = 1.89M⊙ (with ϱ- meson)

Phase transition of the nucleon-antinucleon plasma in a relativistic mean-field theory

Studying Walecka's mean-field theory we find that one can reproduce the observed binding energy and density of nuclear matter within experimental precision in an area characterized by a line in the coupling-constant plane. A part of this line defines systems which exhibit a phase transition around ${T}_{c}\ensuremath{\sim}200$ MeV for zero baryon density. The rest corresponds to such systems where the phase transition is absent; in that case a peak appears in the specific heat around $T\ensuremath{\sim}200$ MeV. We interpret these results as indicating that the hadron phase of nuclear matter alone indicates the occurrence of an abrupt change in the bulk properties around ${\ensuremath{\rho}}_{V}\ensuremath{\sim}0$ and $T\ensuremath{\sim}200$ MeV.

The σ-model and the binding energy of nuclear matter

Using a version of the chiral invariant σ-model, it is shown that many-body forces can eliminate the discrepancy between calculated and empirical values of the binding energy and equilibrium density of nuclear matter. Simple techniques for including such many-body forces in conventional nuclear matter calculations are suggested.

A connection between the relativistic mean field approach and effective interactions

Starting first from the relativistic mean field approach of finite nuclei, the Dirac equation is reduced to a Schrödinger equation which resembles very much the Hartree-Fock-Skyrme equation. As an alternative, a low density expansion of the energy in nuclear matter is performed and both approaches are compared. The connection with effective hamiltonians is discussed, as well as the possible relation between realistic parameters and effective ones.

Nonlinear mean field theory for nuclear matter and surface properties

Nuclear matter properties are studied in a nonlinear relativistic mean field theory. We determine the parameters of the model from bulk properties of symmetric nuclear matter and a reasonable value of the effective mass. In this work, we stress the nonrelativistic limit of the theory which is essentially equivalent to a Skyrme hamiltonian, and we show that most of the results can be obtained, to a good approximation, analytically. The strength of the required parameters is determined from the binding energy and density of nuclear matter and the effective nucleon mass. For realistic values of the parameters, the nonrelativistic approximation turns out to be quite satisfactory. Using reasonable values of the parameters, we can account for other key properties of nuclei, such as the spin-orbit coupling, surface energy, and diffuseness of the nuclear surface. Also the energy dependence of the nucleon-nucleus optical model is accounted for reasonably well except near the Fermi surface. It is found, in agreement with empirical results, that the Landau parameter F 0 is quite small in normal nuclear matter. Both density dependence and momentum dependence of the NN interaction, but especially the former, are important for nuclear saturation. The required scalar and vector coupling constants agree fairly well with those obtained from analyses of NN scattering phase shifts with one-boson-exchange models. The mean field theory provides a semiquantitative justification for the weak Skyrme interaction in odd states. The strength of the required nonlinear term is roughly consistent with that derived using a new version of the chiral mean field theory in which the vector mass as well as the nucleon mass is generated by the σ-field.

Low-order variational calculations for model nuclear matter with the healing condition

Low order calculations for the energy and density of model nuclear matter are carried out which make use of correlation functions $f(r)$ obtained by means of a differential equation. This is derived by variation of the two-body energy functional subject to the well known healing condition ($\ensuremath{\rho}\ensuremath{\int}{[1\ensuremath{-}f(r)]}^{2}D(r)d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}=c, c$ a small constant). The problem of the determination of the corresponding Lagrangian multiplier has been investigated and we have concluded to determine it from the less deep minimum of the ratio $|\frac{{E}^{(3)}}{{E}^{(2)}}|$ of the third to the second term of the factorized Iwamoto-Yamada expansion of the energy. In addition, the corresponding condition for the minimum has been studied. Results for the values of the energy (which has been approximated by the first three terms of the factorized Iwamoto-Yamada expansion), the density, and several convergence quantities have been obtained with the test potentials of Iwamoto and Yamada and Ohmura, Morita, and Yamada. Compared to other results from low and high order calculations, it is seen that those of the density are similar, but those for the energy are smaller in our case, suggesting that correlation functions quite constrained have been obtained. The possibility of improvements is finally considered.NUCLEAR STRUCTURE Correlation function, binding energy, and Fermi momentum of nuclear matter. Low order calculation, healing condition.

Thermostatic properties of semi-infinite symmetric nuclear matter

The thermal evolution of surface thickness and surface free energy of semi-infinite symmetric nuclear matter have been computed using a hot Thomas-Fermi model. Performing a leptodermous expansion of the total free energy, approximated analytical formulas for these quantities have been obtained.

Multiple Δ(3, 3) excitation and the binding energy of nuclear matter

The contributions of ring diagrams to the binding energy of nuclear matter are evaluated. A technique is developed which allows the numerical calculation of third- and fourth-order rings including isobar excitations and using realistic interactions. It is found that ring diagrams involving isobar excitation are of similar importance as the corresponding conventional pure nucleonic terms. The contribution of the ring diagrams is reduced if a nucleon-nucleon interaction is used with a weak tensor force.

π-Induced correlation energy and precritical phenomena in nuclear matter

The effects of the precursor phenomena of π-condensation in nuclear matter on the correlation energy and the compressibility are studied. In the precritical situation anomalously large values are obtained.

A variational approach to dense relativistic matter using functional techniques

The zero temperature ground state of an infinite system of baryons interacting with each other through the exchange of scalar and vector mesons is studied by means of a variational principle appropriate to relativistic systems. A trial wavefunctional is constructed which represents the fluctuation of the quantum fields about their mean values. The renormalized ground-state energy is subsequently calculated at a point where the vacuum is stable. Renormalization to all orders in the strong coupling constants is thereby obtained. A simple expression for the binding energy per particle with three free parameters is found. These parameters are fixed by fitting to the observed nucleon mass and to the values of the fermi momentum and binding energy of nuclear matter. A prediction for the binding energy and equation of state of nuclear and neutron matter is obtained for densities far away from the density of normal nuclei. Finally, a comparison is made with results obtained by other authors who have used classical-perturbative methods for the same system.

Renormalization effects in a field theory of finite nuclei

The ground-state of a finite system of nucleons coupled via scalar and vector mesons is studied using functional methods. A trial wavefunctional representing the fluctuations of the quantum fields about their mean values is used as an input for a variational principle. A simple expression for the renormalized ground-state energy density of closed-shell nuclei is obtained, and all parameters are determined by fitting to the fermi momentum and binding energy of nuclear matter, and to either the surface energy or the surface thickness of heavy nuclei. Calculations for semi-infinite nuclear matter and for40Ca yield results in reasonable agreement with experiment, but show that renormalizations can be large quantitatively and depend rather strongly on the specific mechanism for renormalization.

Variational calculations of asymmetric nuclear matter

We report on variational calculations of the energy E( ρ, β) of asymmetric nuclear matter having ϱ = ϱ n + ϱ p = 0.05 to 0.35 fm −3, and β = ( ϱ n − ϱ p / g9 = 0 to 1. The nuclear h used in this work consists of a realistic two-nucleon interaction, called v 14, that fits the available nucleon-nucleon scattering data up to 425 MeV, and a phenomenological three nucleon interaction adjusted to reproduce the empirical properties of symmetric nuclear matter. The variational many-body theory of symmetric nuclear matter is extended to treat matter with neutron excess. Numerical and analytic studies of the β-dependence of various contributions to the nuclear matter energy show that at ϱ < 0.35 fm −3 the β 4 terms are very small, and that the interaction energy EI(ρ, β) defined as E( ρ, β) − T F( ρ, β), where T F is the Fermi-gas energy, is well approximated by EI 0(ϱ) + β 2EI 2(ρ). The calculated symmetry energy at equilibrium density is 30 MeV and it increases from 15 to 38 MeV as ϱ increases from 0.05 to 0.35 fm −3.

Three-body correlations in nuclear matter

A momentum-space method is developed for the calculation of three-body terms in the Brueckner-Bethe method for nuclear matter. The method is similar to one used earlier for central $S$-wave potentials. Here we extend it to the full nuclear force, including tensor forces, spin-orbit forces, etc. Furthermore, we show how the method can be used to investigate the possibility of long-range correlations in nuclear matter by summing the generalized ring series. The numerical accuracy obtainable with various mesh parameters and cut-offs in momentum space, and with various truncations of partial-wave expansions, is thoroughly explored. Several angle-average approximations are used, and the estimated numerical accuracy in the three-body cluster energy is 10-15%. The method is applied to a central potential ${v}_{2}$, a semirealistic potential ${v}_{6}$ (Reid), which has a tensor force, and to the Reid potential, augmented by an interaction that is consistent with empirical scattering phase shifts in two-body partial waves with $j\ensuremath{\ge}3$. In all cases the three-body contribution to the energy is correctly given in order of magnitude by ${\ensuremath{\kappa}}_{2}{D}_{2}$, where ${D}_{2}$ is the two-body contribution and ${\ensuremath{\kappa}}_{2}$ is the usual convergence parameter of the Brueckner-Bethe method. The generalized ring series is found to converge rapidly, indicating that long-range correlations are not very important for the binding energy of nuclear matter. The Reid potential is found to saturate at the right energy but at too high a density.NUCLEAR STRUCTURE Method for solving Brueckner-Bethe three-body equations in nuclear matter developed and applied to the Reid potential.

Relativistic effects in the Bethe-Brueckner theory of nuclear matter

We extend the theory of nuclear matter to include a relativistic description of nucleon motion. In particular we allow for negative energy components in the nucleon wave function. The amplitude for these components is calculated using an extended version of the one-boson-exchange model of nuclear forces. We find that the inclusion of negative energy states (pair currents) provides a strongly density dependent repulsive interaction. (If one limits oneself to a description involving positive energy states only, this interaction appears as an effective repulsive many-body force.) Our extended theory leads to major modification of the saturation properties of nuclear matter. For example, a boson-exchange force which, in a standard calculation, leads to significant overbinding of nuclear matter at much too high a saturation density yields, in our relativistic analysis, quite good agreement with the generally accepted empirical values for the binding energy and density of nuclear matter. (This potential has strong tensor coupling for the $\ensuremath{\rho}$ meson and a weak tensor force. These features are favored at this time on the basis of other theoretical considerations.) We conclude that, contrary to current thought, nuclear matter should be treated as a relativistic system.NUCLEAR STRUCTURE Relativistic effects in the saturation properties of nuclear matter.

Variational calculations of realistic models of nuclear matter

We report variational calculations of nuclear matter with a semi-realistic Reid v 12 model, and a realistic v 14 model of the two-nucleon interaction operator. The v 14 model fits the available nucleon-nucleon scattering data up to 425 MeV lab energy, and has relatively weak L 2 and ( L · S) 2 interactions in addition to the standard central, tensor and ( L · S ). The L 2 and ( L · S) 2 interactions are treated semiperturbatively; their contribution reduces the overbinding of nuclear matter. However, the equilibrium k F = 1.7 fm −1 and E 0 = −17.5 MeV obtained with the v 14 model are both higher than their empirical values k F = 1.33 fm − and E 0 = −16 MeV. We assume that the difference between the calculated and empirical E(ρ) is entirely due to three-nucleon interactions (TNI). The TNI contributions are phenomenologically added to the nuclear matter energy, and their parameters are adjusted to obtain the correct equilibrium energy, density and compressibility. The required TNI contributions appear to be of reasonable magnitude.