The method of unitary transformations in the theory of nuclear matter

The theory of nuclear matter is investigated by means of the method of unitary transformations in the special case of point transformations. The general formula for the energy per particle as a function of the density is given in Hartree-Fock approximation being neglected the induced three and more body forces for reasonable correlation functions. This function always shows saturation due to a term proportional tok _F ⁵ in the direct part of the approximation. The physical connexion of this term with the scattering amplitudes of the potential is shown. We point out the equivalence to orderk _F ³ of the energy per particle function of the unitary method with Jastrow and separation methods in their simplest form. The saturation properties are calculated for certain classes of correlations using the realistic potential of Gammel, Christian and Thaler.     

Superfluid states in β-stable nuclear matter

Two superfluid states of nuclear matter, which are supposed to play an important role in neutron stars, are discussed: the first one due to the proton-proton ¹ S ₀ pairing in β-equilibrium nuclear matter; the second one due to the anisotropic neutron-neutron ³ PF ₂ pairing in neutron matter. Since the two phases appear at high density of nuclear matter, the three-body forces were added to the pairing interaction and the strong correlation effects in the single-paricle spectrum. The energy gaps, obtained solving the extended BCS equations, significantly deviate from the values without medium effects so as to limit the role of these two superfluid states in the interpretation of phenomena occurring in the neutron-star core.     

Variational calculations of asymmetric nuclear matter

We report on variational calculations of the energy E(ρ, β) of asymmetric nuclear matter having ? = ?_n + ?_p = 0.05 to 0.35 fm^?3, and β = (?_n ? ?_p/g9 = 0 to 1. The nuclear h used in this work consists of a realistic two-nucleon interaction, called v₁₄, 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 ? < 0.35 fm^?3 the β⁴ 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 EI₀(?) + β²EI₂(ρ). The calculated symmetry energy at equilibrium density is 30 MeV and it increases from 15 to 38 MeV as ? increases from 0.05 to 0.35 fm^?3.     

Medium polarization and pairing in asymmetric nuclear matter

The many-body theory of asymmetric nuclear matter is developed beyond the Brueckner–Hartree–Fock approximation to incorporate the medium polarization effects. The extension is performed within the Babu–Brown induced interaction theory. After deriving the particle–hole interaction in the form of Landau–Migdal parameters, the effects of the induced component on the symmetry energy are investigated along with the screening of ¹ S ₀ proton–proton and ³ PF ₂ neutron–neutron pairing, which are relevant for the neutron-star cooling. The crossover from repulsive (screening) to attractive (anti-screening) interaction going from pure neutron matter to symmetric nuclear matter is discussed.     

Resummation of in-medium ladder diagrams: s-wave effective range and p-wave interaction

An exploratory study of chiral four-nucleon interactions in nuclear and neutron matter is performed. The leading-order terms arising from pion-exchange in combination with the chiral 4??-vertex and the chiral NN3??-vertex are found to be very small. Their attractive contribution to the energy per particle stays below 0.6 MeV in magnitude for densities up to ?? = 0.4 fm^?3. We consider also the four-nucleon interaction induced by pion-exchange and twofold ??-isobar excitation of nucleons. For most of the closed four-loop diagrams the occurring integrals over four Fermi spheres can either be solved analytically or reduced to easily manageable one- or two-parameter integrals. After summing the individually large contributions from 3-ring, 2-ring and 1-ring diagrams of alternating signs, one obtains at nuclear matter saturation density ?? ₀ = 0.16 fm^?3 a moderate contribution of 2.35 MeV to the energy per particle. The curve $\bar E(\rho )$ rises rapidly with density, approximately with the third power of ??. In pure neutron matter the analogous chiral four-body interactions lead, at the same density ?? _n , to a repulsive contribution that is about half as strong. The present calculation indicates that long-range multi-nucleon forces, in particular those provided by the strongly coupled ??N??-system with its small mass-gap of 293 MeV, can still play an appreciable role for the equation of state of nuclear and neutron matter.     

Abnormal nuclear matter and many-body forces

Qualitative aspects of quantum corrections to the Lee-Wick abnormal nuclear matter are studied in terms of many-body forces in the normal nuclear matter implied by the σ-model Lagrangian field theory. Using a simplified model for the scalar meson self-energy in the nuclear medium and restricting to a set of graphs which in non-relativistic normal nuclear matter reduces to the well-known random phase approximation (RPA), we have found that an abnormal nuclear state can be bound or unbound depending upon whether strongly attractive multi-body forces are present or absent in the normal matter. This is in support of our previous result obtained heuristically from some general considerations of quantum corrections. A strongly bound abnormal matter with an equilibrium density of a few times the normal nuclear matter density ρ₀ can be formed if large attractive manybody forces can be accommodated in the normal nuclear matter. However if one accepts the present status of theories of nuclear matter binding energy in which no attractive many-body forces are called for, then the abnormal state can occur only at large densities (perhaps 8 to 10 times ρ₀) and is expected to be unbound by several hundred MeV per particle.     

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.     

A generator coordinate description of the breathing mode

Isoscalar and isovector breathing mode states in ¹⁶O and ²⁰⁸Pb are described in the generator coordinate method. The MDI 4-force, which yields a nuclear matter compressibility of K_N = 200 MeV in agreement with realistic forces and a symmetry energy of S_N = 32.8 MeV, was used.     

Nuclear compression moduli and density-dependent forces

Density-dependent zero-range forces of the form of the modified delta interaction (MDI) are generalized (MDI3, MDI4) in order to yield reasonable values of the compression modulus in nuclear matter (K_N = 200 MeV). This low value can be fitted by introducing two terms with different density dependence in the force. The four free parameters of MDI3 are adjusted to reproduce the nuclear matter values of the binding energy, density and compression modulus, and to fulfil the condition that the total energy of ¹⁶O in harmonic oscillator wave functions has a minimum at the oscillator length b = 1.75 fm, corresponding to the correct rms radius. MDI4 contains in addition a two-body spin-orbit interaction. The five parameters of MDI4 are fitted to the above three nuclear matter data and by requiring that Hartree-Fock (HF) calculations in ²⁰⁸Pb yield the experimental charge rms radius and reasonable values of certain single-particle spin-orbit splittings. The quality of MDI4 is checked by comparing calculated rms radii, binding energies, and elastic electron scattering cross sections with available experimental data for doubly closed shell nuclei. As a test the energy levels and the nuclear monopole polarization of muonic ²⁰⁸Pb are calculated self-consistently yielding impressive agreement with experiment.     

Properties of nuclear matter andN=Z closed-shell nuclei

A simple three-parameter density dependent effective interaction is used to study the properties of nuclear matter, neutron matter and some bulk properties such as ground state energies and rms charge radii of three double-closed shell nuclei⁴He,¹⁶O and⁴⁰Ca. The three parameters of the effective interaction are determined by requiring to fit the binding energy and density of infinite nuclear matter at saturation density as well as ground state energy of¹⁶O in the first order perturbation theory. This interaction gives correct saturation in nuclear matter with a value of 283 MeV for compressibility. The symmetry coefficienta _T atk _F=1·36 fm^–1 is 28·58 MeV. The energy per particle in neutron matter is calculated in the range of nuclear matter densities and it compares well with those ofNemeth andSprung. Groundstate energies and rms charge radii of⁴He,¹⁶O and⁴⁰Ca are calculated using oscillator eigen functions as single particle wave functions. Results for ground state energies are in good agreement with empirical values and rms charge radii are slightly better than those obtained byMoszkowski with the MDI.The authors are thankful to the Computer Centre, Utkal University, Bhubaneswar for providing computational facilities for this work.     

The effect of the δ(1236) in nuclear matter

The three main corrections to the binding energy of nuclear matter that involve the Δ (1236) are calculated with exchange effects and realistic correlation functions. It is found that their net effect at K_F = 1.36 fm^?1 is small, but that their K_F dependence has a considerable effect on the saturation density.     

Inclusion of the Δ resonance in the optical potential between two nuclei and its application to medium energy12C-12C scattering

An energy dependent complex optical potential between two nuclei is calculated from the potential energy density for two colliding nuclear matters generated by solving the Bethe-Goldstone equation in whichNΔ and ΔΔ channels are explicitly coupled to theNN channel. By adding the contributions from the third and fourth order ring diagrams and the relativistic correction to the calculated potential energy density, the saturation property of a nuclear matter is reasonably well reproduced. This is used together with the kinetic energy density to calculate the optical potential for the¹²C+¹²C system in the energy density formalism with the local density approximation. The surface correction term and the symmetry energy term in the energy density functional are determined to reproduce the observed binding energy and the rms radius of¹²C. Using this potential, the differential cross sections for elastic¹²C-¹²C scattering atE _lab=1440 and 2400 MeV are calculated and compared with recent experimental data.     

Constraints on the symmetry energy using the mass-radius relation of neutron stars

The nuclear symmetry energy is intimately connected with nuclear astrophysics. This contribution focuses on the estimation of the symmetry energy from experiment and how it is related to the structure of neutron stars. The most important connection is between the radii of neutron stars and the pressure of neutron star matter in the vicinity of the nuclear saturation density n_s. This pressure is essentially controlled by the nuclear symmetry energy parameters S_v and L , the first two coefficients of a Taylor expansion of the symmetry energy around n_s. We discuss constraints on these parameters that can be found from nuclear experiments. We demonstrate that these constraints are largely model-independent by deriving them qualitatively from a simple nuclear model. We also summarize how recent theoretical studies of pure neutron matter can reinforce these constraints. To date, several different astrophysical measurements of neutron star radii have been attempted. Attention is focused on photospheric radius expansion bursts and on thermal emissions from quiescent low-mass X-ray binaries. While none of these observations can, at the present time, determine individual neutron star radii to better than 20% accuracy, the body of observations can be used with Bayesian techniques to effectively constrain them to higher precision. These techniques invert the structure equations and obtain estimates of the pressure-density relation of neutron star matter, not only near n_s, but up to the highest densities found in neutron star interiors. The estimates we derive for neutron star radii are in concordance with predictions from nuclear experiment and theory.     

Dual origin of pairing in nuclei

The pairing correlations of the nucleus ¹²⁰Sn are calculated by solving the Nambu–Gor’kov equations, including medium polarization effects resulting from the interweaving of quasiparticles, spin and density vibrations, taking into account, within the framework of nuclear field theory (NFT), processes leading to self-energy and vertex corrections and to the induced pairing interaction. From these results one can not only demonstrate the inevitability of the dual origin of pairing in nuclei, but also extract information which can be used at profit to quantitatively disentangle the contributions to the pairing gap Δ arising from the bare and from the induced pairing interaction. The first is the strong ¹ S ₀ short-range NN potential resulting from meson exchange between nucleons moving in time reversal states within an energy range of hundreds of MeV from the Fermi energy. The second results from the exchange of vibrational modes between nucleons moving within few MeV from the Fermi energy. Short- (v _p ^bare) and long-range (v _p ^ind) pairing interactions contribute essentially equally to nuclear Cooper pair stability. That is to the breaking of gauge invariance in open-shell superfluid nuclei and thus to the order parameter, namely to the ground state expectation value of the pair creation operator. In other words, to the emergent property of generalized rigidity in gauge space, and associated rotational bands and Cooper pair tunneling between members of these bands.     

Local-potential approximation for the Brueckner G matrix and problem of optimally choosing model subspace

The validity of the local-potential approximation, which was proposed previously for the singlet-pairing problem in semi-infinite nuclear matter, is investigated in the Bethe-Goldstone equation for the Brueckner G matrix. For this purpose, use is made of the method developed earlier for solving this equation for a planar slab of nuclear matter in the case of a separable form of NN interaction. The ¹ S ₀ singlet and the ³ S ₁+³ D ₁ triplet channel are considered. The complete two-particle Hilbert space is split into a model and the complementary subspace that are separated by an energy E ₀. The two-particle propagator is calculated precisely in the first subspace, and the local-potential approximation is used in the second subspace. With an eye to subsequently employing the G matrix to calculate the Landau-Migdal amplitude, the total two-particle energy is fixed at E=2μ, where μ is the chemical potential of the system under consideration. Specific numerical calculations are performed at μ=?8 MeV. The applicability of the local-potential approximation is investigated versus the cutoff energy E ₀. It is shown that, in either channel being considered, the accuracy of the local-potential approximation is rather high for E ₀≥10 MeV.     

Nuclear matter approach to the heavy-ion optical potential

An energy-dependent local potential for heavy-ion (HI) scattering is derived from Reid's softcore interaction using the Brueckner theory. The Bethe-Goldstone equation in momentum space is first solved with the outgoing boundary condition for two colliding systems of nuclear matter with the relative momentum K_r per nucleon. The Fermi distribution is assumed to consist of two spheres without double counting of their intersection separated by the relative momentum K_r. The angle-averaged Pauli projection function is given in the form of a one-dimensional integral. Secondly the optical potential for HI scattering is evaluated using the energy-density formalism. The energy density is calculated for two limiting cases: (i) the sudden approximation in which the spatial distribution of the two HI is described by an antisymmetrized cluster wave function, and (ii) the adiabatic limit represented by an antisymmetrized two-centre wave function. The complex HI potential is defined in terms of the energy density from nuclear matter so that both volume elements in the finite and the infinite systems have the same nucleon and kinetic energy density. This method is applied to the ¹⁶O + ¹⁶O, ⁴⁰Ca + ¹⁶O, and ⁴⁰Ca + ⁴⁰Ca potentials. The calculated results are compared with phenomenological potentials. Though in principle our approach can generate an imaginary part for the HI potential, the magnitude is too small. Reasons and possible improvements of this point are discussed.     

The spin-isospin decomposition of the nuclear symmetry energy from low to high density

The energy per particle BA in nuclear matter is calculated up to high baryon density in the whole isospin asymmetry range from symmetric matter to pure neutron matter.The results,obtained in the framework of the Brueckner-Hartree-Fock approximation with two-and three-body forces,confirm the well-known parabolic dependence on the asymmetry parameterβ=(N?Z)/A(β^2 law)that is valid in a wide density range.To investigate the extent to which this behavior can be traced back to the properties of the underlying interaction,aside from the mean field approximation,the spin-isospin decomposition of BA is performed.Theoretical indications suggest that theβ^2 law could be violated at higher densities as a consequence of the three-body forces.This raises the problem that the symmetry energy,calculated according to theβ^2 law as a difference between BA in pure neutron matter and symmetric nuclear matter,cannot be applied to neutron stars.One should return to the proper definition of the nuclear symmetry energy as a response of the nuclear system to small isospin imbalance from the Z=N nuclei and pure neutron matter.     

Variational calculations of realistic models of nuclear matter

We report variational calculations of nuclear matter with a semi-realistic Reid v₁₂ model, and a realistic v₁₄ model of the two-nucleon interaction operator. The v₁₄ model fits the available nucleon-nucleon scattering data up to 425 MeV lab energy, and has relatively weak L² and $\text{(}\text{L}\text{·}\text{S}\text{)}{}^2$ interactions in addition to the standard central, tensor and ($\text{L}\text{·}\text{S}$). The L² and $\text{(}\text{L}\text{·}\text{S}\text{)}{}^2$ interactions are treated semiperturbatively; their contribution reduces the overbinding of nuclear matter. However, the equilibrium k_F = 1.7 fm^?1 and E₀ = ?17.5 MeV obtained with the v₁₄ model are both higher than their empirical values k_F = 1.33 fm^? and E₀ = ?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.     

Finite nuclei to nuclear matter: a leptodermous approach

The liquid drop model (LDM) expansions of energy and incompressibility of finite nuclei are studied in an analytical model using Skyrme-like effective interactions to examine, whether such expansions provide an unambiguous way to go from finite nuclei to nuclear matter, and thereby can yield the saturation properties of the latter, from nuclear masses. We show that the energy expansion is not unique in the sense that, its coefficients do not necessarily correspond to the ground state of nuclear matter and hence, the mass formulas based on it are not equipped to yield saturation properties. The defect is attributed to its use of liquid drop without any reference to particles as its basis, which is classical in nature. It does not possess an essential property of an interacting many-fermion system namely, the single particle property, in particular the Fermi state. It is shown that, the defect is repaired in the infinite nuclear matter model by the use of generalized Hugenholtz–Van Hove theorem of many-body theory. So this model uses infinite nuclear matter with well defined quantum mechanical attributes for its basis. The resulting expansion has the coefficients which are at the ground state of nuclear matter. Thus a well defined path from finite nuclei to nuclear matter is found out. Then using this model, the saturation density 0.1620 fm⁻³ and binding energy per nucleon of nuclear matter 16.108 MeV are determined from the masses of all known nuclei. The corresponding radius constant r₀ equal to 1.138 fm thus determined, agrees quite well with that obtained from electron scattering data, leading to the resolution of the so-called ‘r₀-paradox’. Finally a well defined and stable value of 288±20 MeV for the incompressibility of nuclear matter K_∞ is extracted from the same set of masses and a nuclear equation of state is thus obtained.