A nuclear matter calculation with the tensor-optimized Fermi sphere method with central interaction

The propagation of a deuteron through nuclear matter is examined in terms of a simple nonlocal but separable potential for the n-p interaction. It is found that the binding energy of the deuteron in nuclear matter depends strongly on the relative orientation of the deuteron spin and center-of-mass momentum, when both the Pauli exclusion principle and the tensor force component in the neutron-proton interaction are included in the calculation. A thorough discussion of the physical mechanism involved is presented. It is shown that this effect is associated with the presence of a new type of spin dependent interaction of the T/sub p/ type, in the deuteron-nucleus optical potential. The nuclear matter calculations are applied to the realistic case of a deuteron scattered by a heavy nucleus through a simple model which is valid for deuteron incident energies in excess of 100 MeV. The T/sub p/ force thus produced is found to have a non-negligible strength over a wide range of high incident deuteron energies and to be very sensitive to high momentum components of the nucleon-nucleon interaction. The mechanism examined here is also expected to generate a T/sub p/ force and a modified T/sub R/ force at low energies. opticalmore » potential. The nuclear matter calculations are applied to the realistic case of a deuteron scattered by a heavy nucleus through a simple model which is valid for deuteron incident energies in excess« less

We report on the recent developments of a new effective field theory for nuclear matter [1, 2, 3]. We present first the nuclear matter chiral power counting that takes into account both short- and long-range inter-nucleon interactions. It also identifies non-perturbative strings of diagrams, related to the iteration of nucleon-nucleon interactions, which have to be re-summed. The methods of unitary chiral perturbation theory has been shown to be a useful tool in order to perform those resummations. Results up to next-to-leading order for the ground state energy per particle of nuclear matter, the in-medium chiral quark condensate and pion self-energy are discussed.

The effects of the $\ensuremath{\alpha}$ particles in nuclear matter at low densities are investigated within three different parametrizations of relativistic models at finite temperature. Both homogeneous and inhomogeneous matter (pasta phase) are described for neutral nuclear matter with fixed proton fractions and stellar matter subject to $\ensuremath{\beta}$ equilibrium and trapped neutrinos. In homogeneous matter, $\ensuremath{\alpha}$ particles are present only at densities below 0.02 fm${}^{\ensuremath{-}3}$ and their presence decreases with increase of the temperature and, for a fixed temperature, the $\ensuremath{\alpha}$ particle fraction decreases for smaller proton fractions. A repulsive interaction is important to mimic the dissolution of the clusters in homogeneous matter. The effect of the $\ensuremath{\alpha}$ particles on the pasta structure is very small except close to the critical temperatures and/or proton fractions, when it may still predict a pasta phase while no pasta phase would occur in the absence of light clusters. It is shown that for densities above 0.01 fm${}^{3}$ the $\ensuremath{\alpha}$-particle fraction in the pasta phase is much larger than that in homogeneous matter.

A set of density dependent nucleon-nucleon (N-N) interactions has been examined in nuclear matter calculations by varying spin-isospin contributions. Two sets of potentials have been considered. One having a density dependentσ-function type short range part followed by one term long range Gaussian part while the other having a density dependentσ-function part followed by two Gaussian terms. The strength parameters of the potential have been fitted to the saturation properties of nuclear matter, i.e. binding energy per particle of 15.5 MeV atk F=1.35 fm−1. Several sets of these two potentials have been generated by varying the strength parameterM of Majorana exchange operatorP M. It is seen thatM indirectly controls the spin, iso-spin contribution to the interaction potential and thus affects the nuclear matter properties such as compressibility and symmetry energy considerably, while variation of these quantities with the range parameterμ for givenM is moderate at lowM values while at higherM values it is quite large.

Realistic density dependent CDM3Yn versions of the M3Y interaction have been used in an extended Hartree-Fock (HF) calculation of nuclear matter (NM), with the nucleon single-particle potential determined from the total NM energy based on the Hugenholtz-van Hove theorem that gives rise naturally to a rearrangement term (RT). Using the RT of the single-nucleon potential obtained exactly at different NM densities, the density- and energy dependence of the CDM3Yn interactions was modified to account properly for both the RT and observed energy dependence of the nucleon optical potential. Based on a local density approximation, the double-folding model of the nucleus-nucleus optical potential has been extended to take into account consistently the rearrangement effect and energy dependence of the nuclear mean-field potential, using the modified CDM3Yn interactions. The extended double-folding model was applied to study the elastic $^{12}$C+$^{12}$C and $^{16}$O+$^{12}$C scattering at the refractive energies, where the Airy structure of the nuclear rainbow has been well established. The RT was found to affect significantly the real nucleus-nucleus optical potential at small internuclear distances, giving the potential strength close to that implied by the realistic optical model description of the Airy oscillation.

Symmetric nuclear matter is studied within the Brueckner-Hartree-Fock (BHF) approach and is extending to the self-consistent Green’s function (SCGF) approach. Both approximations are based on realistic nucleon-nucleon interaction; that is, CD-Bonn potential is chosen. The single-particle energy and the equation of state (EOS) are studied. The Fermi energy at the saturation point fulfills the Hugenholtz-Van Hove theorem. In comparison to the BHF approach, the binding energy is reduced and the EOS is stiffer. Both the SCGF and BHF approaches do not reproduce the correct saturation point. A simple contact interaction should be added to SCGF and BHF approaches to reproduce the empirical saturation point.

An equation of state (EoS) for symmetric nuclear matter is constructed using the density dependent M3Y effective interaction and extended for isospin asymmetric nuclear matter. Theoretically obtained values of symmetric nuclear matter incompressibility, isobaric incompressibility, symmetry energy and its slope agree well with experimentally extracted values. Folded microscopic potentials using this effective interaction, whose density dependence is determined from nuclear matter calculations, provide excellent descriptions for proton, alpha and cluster radioactivities, elastic and inelastic scattering. The nuclear deformation parameters extracted from inelastic scattering of protons agree well with other available results. The high density behavior of symmetric and asymmetric nuclear matter satisfies the constraints from the observed flow data of heavy-ion collisions. The neutron star properties studied using $\beta$-equilibrated neutron star matter obtained from this effective interaction reconcile with the recent observations of the massive compact stars.

Study of strange particles produced at (sub)threshold energies in nucleus-nucleus collisions can deliver insights into fundamental questions about the in-medium properties of hadrons in dense baryonic matter. Many theoretical calculations for the production and propagation of strangeness at SIS energies lead to various predictions concerning the existence and the magnitude of in-medium effects for strange particles in nuclear matter. FOPI measured the flow of strange particles in Ni + Ni collisions at 1.93A GeV and the results are compared with transport models which are favorable to in-medium effect.

The experimental determination of nuclear masses throughout the periodic table yields — as one of the most important consequences — a well defined value for the binding energy of Nuclear Matter (Bethe, Swiaticki). In the same time the corresponding density was determined through extended measurements of nuclear radii and nuclear charge distributions (Hofstaedter). On the other hand, a comparison of these quantities with the results from a numerical calculation based on general theoretical principles (Bethe-Brueckner theory) represents a decisive test of microscopic theories of nuclear matter and hence of nuclear structure. It is, however, well-known for many years (compare F. Coester, Phys.Rev. Cl, 1970, 769) that all calculations based on any expression for the fundamental nuclear forces, which,in turn,is in reasonable agreement with the experimental scattering phases of the 2-nucleon problem, will not yield simultaneously the right experimental values for binding and density of nuclear matter (i.e. for reasonable binding values the density is too high and vice versa). Even using the (otherwise satisfactory) description of the nuclear force based on a boson exchange model (compare as an example K. Holinde,K.Erkelenz, R. Alzetta, Nucl.Phys. A 194 1972, 161), a characteristic discrepancy remains. The main purpose of this short contribution is to give some general reasons for this disturbing fact: 1) Any explicite expression for the nuclear force, which is determined through the boson exchange in between two free nucleons, is changed in a characteristic way (by the order of magnitude of a few percents) if the nucleons in question are embedded into nuclear matter. This fact is mainly due to a typical displacement of some relevant intermediate nucleon states which occur during the exchange process. In this way the boson theory of nuclear forces is — in principle — able to lift the characteristic discrepancy (apparent already in the 3-body problem) between the forces to be used in the two-body (scattering) problem and the nuclear matter properties. In order to calculate this new effect, bosonic variables must be introduced explicitely in the entire calculation (compare D. Schutte, Nucl.Phys. A221, 1974, p. 450–460, and K. Kotthoff, Doctor Thesis, Bonn, 1974). While this fundamental enlargement of nuclear theory yields corrections of the order of a few percents to all relevant nuclear properties, it might yield large changes in the case of extremely high nuclear densities which occur in Neutron Stars where this method will lead in a natural way to the so-called boson condensation. 2) Apart from the introduction of the additional bosonic variables, it appears of fundamental importance to include also the inner degrees of freedom of the nucleons themselves. This means that the virtual excitation (through boson absorption) of the nucleons into their various higher lying resonance states must also be considered. For numerical reasons only the first state — the so-called ∆ -resonance (1236 Mev) — must be considered. This kind of ‘polarisation’ is in fact inevitable and yields a most important contribution to the middle-range attractive part of nuclear forces (compare A.M. Green and P. Haapakoski “The effect of the ∆(1236) resonance,” reprint from Research Inst.f.Theor.Phys., Helsinki) and is thus — to a large extent - responsible for nuclear binding. (In addition it replaces the “unphysical” scalar σ-boson which had so far been used in most boson-theoretical deductions of nuclear forces). The contribution from the virtual ∆ -excitation to the nuclear force is,again, different in the cases of free and of “embedded” nucleons. While the introduction of bosonic variables leads to a needed increase of nuclear binding, the effect of the ∆-resonance yields an important adjustment (decrease) of nuclear density through its characteristic strongly repulsive short-range contribution to the nucleon-nucleon interaction.(There remains, however, the still unsolved problem of a numerical evaluation of the effect due to a characteristic change of the mesonic self-energies of nucleons when embedded into nuclear matter).

The isospin dependence of the effective pairing interaction is discussed on the basis of the Bardeen, Cooper, and Schrieffer theory of superfluid asymmetric nuclear matter. It is shown that the energy gap, calculated within the mean field approximation in the range from symmetric nuclear matter to pure neutron matter, is not linearly dependent on the symmetry parameter owing to the nonlinear structure of the gap equation. Moreover, the construction of a zero-range effective pairing interaction compatible with the neutron and proton gaps in homogeneous matter is investigated, along with some recent proposals of isospin dependence tested on the nuclear data table.

The structure of nuclear systems derived from low momentum nucleon–nucleon potentials

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.

The parabolic approximation to the equation of state of the isospin asymmetric nuclear matter (ANM) is widely used in the literature to make predictions for the nuclear structure and the neutron star properties. Based on the realistic M3Y-Paris and M3Y-Reid nucleon-nucleon interactions, we investigate the effects of the higher-order symmetry energy on the proton fraction in neutron stars and the location of the inner edge of their crusts and their core-crust transition density and pressure, thermodynamically. Analytical expressions for different-order symmetry energy coefficients of ANM are derived using the realistic interactions mentioned above. It is found that the higher-order terms of the symmetry energy coefficients up to its eighth-order (E$_{sym8}$) contributes substantially to the proton fraction in $\beta$ stable neutron star matter at different nuclear matter densities, the core-crust transition density and pressure. Even by considering the symmetry energy coefficients up to E$_{sym8}$, we obtain a significant change of about 40$\%$ in the transition pressure value from the one based on the exact equation of state. Using equations of state based on both Paris and Reid effective interactions which provide saturation incompressibility of symmetric nuclear matter in the range of 220 MeV$\leq$K$_0$$\leq$270 MeV, we estimate the ranges 0.090 fm$^{-3}$$\leq$$\rho_t$$\leq$0.095 fm$^{-3}$ and 0.49 MeV fm$^{-3}$$\leq$P$_t$$\leq$ 0.59 MeV fm$^{-3}$ for the liquid core-solid crust transition density and pressure, respectively. The corresponding range of the proton fraction at this transition density range is found to be 0.029$\leq$x$_{p(t)}$$\leq$0.032.

We investigate ground-state properties of the superheavy nucleus with [ital N]=184 and [ital Z]=114, [sup 298]114, using conventional relativistic mean-field theory and density-dependent mean-field theory, which reproduces Dirac-Brueckner calculations in nuclear matter. Our calculations provide support for [ital N]=184 and [ital Z]=114 being magic numbers. Density-dependent mean-field theory does not predict [sup 298]114 to be stable against [beta] decay.

Several problems encountered on the way to a microscopic derivation of covariant density functionals are discussed. The starting point are semi-microscopic functionals adjusted to ab-initio calculations in nuclear matter. Since it is impossible to derive in this way the strength of tensor terms in such functionals we propose a solution of the relativistic Brueckner- Hartree-Fock equations in finite nuclei and show first results.