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.     

Cherenkov effect in the weak interactions generated by the neutrinos and new approach for estimation of neutrino mass

It is shown that if weak interactions can generate masses and polarize matter, then the Cherenkov effect induced by these interactions at v _v > c/n appears. The effect of (resonance) enhancement of neutrino oscillations in matter (v _v < c/n) and the Cherenkov (v _v > c/n) effect are competitive processes and at definite neutrino energies the effect of (resonance) enhancement of neutrino oscillations in matter will change to the Cherenkov effect. Then neutrino vacuum oscillations will regenerate and we obtain an excellent possibility of estimating neutrino masses. And knowing estimation of mass for one (electron) neutrino we can obtain masses of the rest neutrinos by using the values for mass differences for neutrinos obtained in oscillation experiments.     

Nuclear-rainbow scattering and nucleus-nucleus potentials at short distances

The elastic scattering of strongly bound nuclei at energies of 10 to 70 MeV per nucleon shows the phenomenon of “rainbow scattering.” A nuclear rainbow appears because of deflection to negative angles. This process involves a strong overlap of nuclear densities, with values of up to twice the saturation density of nuclear matter. The ¹⁶O+¹⁶O system is studied with a high precision over a wide energy range from 7 to 70 MeV per nucleon in several laboratories. Primary Airy maxima and higher order Airy structures are observed. At all energies, excellent fits are obtained with deep potentials as deduced from the double-folding model involving a nucleon-nucleon interaction weakly dependent on the density. It is shown that Pauli blocking expected at low energies is strongly reduced if the local momenta are calculated self-consistently. Systematics confirms a refractive origin of large-angle scattering, at low energies inclusive. Thus, nuclear-rainbow scattering yields unique information about the properties of cold nuclear matter at higher densities.     

Shock waves and mach cones in fast nucleus-nucleus collisions

Mach shock waves and head shock waves occur during the interpenetration of a light high energetic nucleus with a heavy one. Collisions of¹⁶O,¹²C and⁴He ions at energies between 0.25 and 2.1 GeV/N with Ag and Cl nuclei have been investigated. The theoretical concepts and the experiment are presented and interpreted. From that a velocity of first sound in nuclear matterc _s ≈0.19c, a Mach shock velocityv _s ≈0.58c and a nuclear compression constantK=300 MeV are deduced.     

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.     

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.     

Anomaly in the nuclear curvature energy

By use of semiclassical mean-field methods, we study the dependence of the curvature-energy coefficient and other surface properties of nuclear matter upon the energy-density functional. This is done both by solving the Euler-Lagrange equation with a simplified phenomenological functional and by obtaining a variational solution with a fourth-order extended Thomas-Fermi functional. The calculated curvature-energy coefficient a_c decreases with increasing value of the bulk nuclear incompressibility coefficient K for physically relevant values of K, but always remains larger than 3 MeV in either approach when the volume-energy coefficient, saturation density, surface-energy coefficient and surface diffuseness are constrained to their experimental values. The calculated values of the curvature-energy coefficient a_c are significantly larger than experimental values obtained from analyses of fission-barrier heights and ground-state masses of nuclei throughout the periodic table. Among possible resolutions of this anomaly, we suggest that relativistic effects or correlations may play a significant role in the nuclear surface, or that the leptodermous expansion may break down in regions of large curvature, such as occurs for highly deformed shapes and for light nuclei.     

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.     

Nuclear mass formula with a Yukawa-plus-exponential macroscopic model and a folded-Yukawa single-particle potential

We use a Yukawa-plus-exponential macroscopic model and a folded-Yukawa single-particle potential to systematically calculate the ground-state masses of 4023 nuclei ranging from ¹⁶O to {279}112. The method is also used to calculate the fission-barrier heights of 28 nuclei ranging from ¹⁰⁹Cd to ²⁵²Cf. We introduce several previously neglected physical effects, including a smaller nuclear radius constant, a proton form factor, an exact diffuseness correction, an A⁰ term, a chargeasymmetry term, and microscopic zero-point energies. The nuclear radius constant is determined from elastic electron scattering and microscopic calculations of nuclear density distributions, the range of the Yukawa-plus-exponential folding function is determined from heavy-ion elastic scattering, the surface-energy constant and surface-asymmetry constant are determined from the fission-barrier heights of the 28 nuclei that are considered, and the remaining constants are determined from the ground-state masses of 1323 nuclei ranging from ¹⁶O to ²⁵⁹No for which experimental values are known with experimental errors less than 1 MeV. For the final formula, the root-mean-square error in the ground-state masses is 0.835 MeV and the root-mean-square error in the fission-barrier heights is 1.331 MeV. Some of the remaining discrepancies in the groundstate masses can be understood in terms of instabilities with respect to ε₃ and ε₆ deformations.     

Bulk properties of rotating nuclei and the validity of the liquid drop model at finite angular momenta

Out of self-consistent semi-classical calculations performed within the so-called Extended Thomas-Fermi approach for 212 nuclei at all even angular momentum values I ranging between 0 and 80 ħ and using the Skyrme SkM¹ effective force, the I-dependence of associated liquid drop model parameters has been studied. The latter have been obtained trough separate fits of the calculated values of the strong interaction as well as direct and exchange Coulomb energies. The theoretical data basis so obtained, has allowed to make a rough quantative assessment of the variation with I of the usual volume and surface energy parameters up to spin of ∼ 30–40ħ. As a result of the combined variation of the surface and Coulomb energies, it has been shown that this I-dependence results in a significant enhancement of the fission stability of very heavy nuclei, balancing thus partially the well-known instability due to centrifugal forces.     

Finite-temperature density-dependent HF calculation of nuclear matter with gogny interaction

A finite-temperature density-dependent Hartree-Fock method is formulated starting from a variational principle for the thermodynamic potential. Based on this method, we have carried out a nuclear-matter calculation using the Gogny D1 finite-range effective interaction. The equation of state and several other thermal properties of nuclear matter so obtained are found to be rather similar to those given by the Skyrme SkM¹ interaction. The critical temperature and density for the liquid-gas phase transition of nuclear matter are found to be 15.1 MeV and 0.05 fm⁻³, respectively. The effect of the finite-temperature rearrangement potential is discussed     

Depolarization in proton inclusive inelastic scattering and in the (d,p) reaction on medium-weight nuclei

The energy-averaged depolarization parameter K_y^y′ has been measured for the inelastic scattering of 18 MeV protons from ⁵⁴Fe, ⁶³Cu and ⁹²Mo at 45°, 90° and 135°, and for 14.35 MeV protons from ⁶³Cu at 45° and 135°. In all cases K_y^y′ varies from approximately unity for scattering with low energy loss to approximately zero for inelastic scattering to high excitation energies. The change from one of these values to the other occurs over a region ≈ 6 MeV wide centered at about 5 MeV excitation. A simple two-component model fits both the K_y^y′ and inelastic crosssection data. K_y^y′ has also been measured for the ⁵⁴Fe($\text{d}$, $\text{p}$)Fe reaction with 16 MeV deuterons incident. Here K_y^y′ changes from approximately the maximum possible value, $\text{2}\text{3}$, to about zero in a 6 MeV region centered at roughly 13 MeV excitation. The ($\text{d}$,$\text{p}$) data can be fitted by an extension of the model used for the proton scattering data.     

Analyses of electron and proton scattering to low-excitation isoscalar states in 20Ne, 24Mg and 28Si

The excitation of low-lying isoscalar 2⁺ and 4⁺ states in ²⁰Ne, ²⁴Mg and ²⁸Si by electron and proton scattering is studied. Large basis models of nuclear structure are used to determine both the electromagnetic and hadronic transition densities. The analyses of the longitudinal form factors obtained from electron scattering show that little or no effective charges are required with these nuclear structure models. Proton inelastic scattering to these states then is analysed to test effective forces based upon the Paris and Hamada-Johnston interactions. At intermediate energies (155 MeV) density-dependent t-matrices from both potentials were used with fits to data giving a clear preference for that based upon the Paris interaction. For lower energies only the Hamada-Johnston t-matrix is available and comparison of analyses of 24 and 49 MeV data made using this (complex) t-matrix with those in which the (real) Paris G-matrix is used as the effective force show a clear preference for the t-matrix. This is particularly the case with analyses of polarization data and suggests that the use of the G-matrix as an effective force in nuclear reaction calculations is inadequate even at low energies.     

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.     

Inverse Quasifission in <Superscript>156,160</Superscript>Gd + <Superscript>186</Superscript>W Reactions

To investigate the impact shell effects have in the formation of neutron-rich fragments in multinucleon transfer reactions, a series of experiments to explore the binary channel in ^156,160Gd + ¹⁸⁶W reactions at energies near and above the Coulomb barrier is performed at the Flerov Laboratory’s U-400 accelerator using the CORSET setup. These experiments are aimed mainly at obtaining the production cross sections of leadlike fragments in the process of inverse quasifission. The mass, energy, and angular distributions of the binary reaction products are measured at energies of 860 and 935 MeV of ¹⁶⁰Gd ions and 878MeV in the case of ¹⁵⁶Gd ions. The excitation energies of primary fragments are estimated using their measured mass–energy distributions. Enhanced yields of products with masses of 200–215 amu are observed for both reactions. At energies above the barrier for side-to-side collisions (935 MeV), the yield of lead-like fragments is an order of magnitude larger than at energies near the Coulomb barrier, due possibly to the influence of orientation effects. The enhancement observed in the yield of reaction products with masses heavier than the target mass confirms that multinucleon transfer reactions can be used to obtain new neutron-rich isotopes, and to synthesize new superheavy elements.     

Momentum and density dependence of the ph interaction in continuum RPA calculations with Skyrme forces

We analyse the momentum and density dependence of the ph interaction in self-consistent RPA calculations of electromagnetic reactions with Skyrme forces. First, we calculate the V⁰⁰_ph and V⁰¹_ph spin-isospin components in nuclear matter for an SK3 interaction. At the nuclear surface, they have decreasing values with increasing q-values up to momenta q=3 fm⁻¹. As a second point, we show that the predictions of the ¹²C(e, e') charge response at 400 MeV/c remain practically unchanged when the zero-range quadratic momentum dependence of the SK3 interaction is replaced by the momentum dependence associated with a Yukawa short-ranged force.     

Nuclear matter approach to the heavy-ion optical potential: (II). Imaginary part

The heavy-ion optical potentials are constructed in a nuclear matter approach, for the ¹⁶O + ¹⁶O, ⁴⁰Ca + ¹⁶O and ⁴⁰Ca + ⁴⁰Ca elastic scattering at the incident energies per nucleon E^lab/A ? 45 MeV. The energy density formalism is employed assuming that the complex energy density of colliding heavy ions is a functional of the nucleon density ?(r), the intrinsic kinetic energy density τ⁽²⁾(r) and the average momentum of relative motion per nucleon K_r(≦ 1.5 fm^?1). The complex energy density is numerically evaluated for the two units of colliding nuclear matter with the same values of ρ, τ⁽²⁾ and K_r. The Bethe-Goldstone equation is solved for the corresponding Fermi distribution in momentum space using the Reid soft-core interaction. The “self-consistent” single-particle potential for unoccupied states which is continuous at the Fermi surface plays a crucial role to produce the imaginary part. It is found that the calculated optical potentials become more attractive and absorptive with increasing incident energy. The elastic scattering and the reaction cross sections are in fair agreement with the experimental data.     

Strong interaction physics from hadronic atoms

Hadronic atoms provide a unique laboratory for studying strong interactions and nuclear medium effects at zero kinetic energy. Previous results from analyses of strong-interaction data consisting of level shifts, widths and yields in π⁻, K⁻, p̄ and ∑⁻ atoms are reviewed. Recent results from fits to comprehensive sets of data in terms of density-dependent optical potentials that respect the low-density limit, where the interaction tends to the free hadron nucleon value, are discussed. The importance of using realistic nuclear density distributions is highlighted. The introduction of density dependence in most cases significantly improves the fit to the data and leads to some novel results. For K⁻ atoms, a substantial attraction of order 200 MeV in nuclear matter is suggested, with interesting repercussions for K̄ condensation and the evolution of strangeness in high-density stars. For p̄ atoms it is found that a reasonable p-wave strength can be accommodated in the fitted optical potential, in agreement with the energy dependence observed for some low-energy p̄N reactions. For ∑⁻ atoms, the fitted potential becomes repulsive inside the nucleus, implying that Σ hyperons generally do not bind in nuclei in agreement with recent measurements. This repulsion significantly affects calculated masses of neutron stars.     

Atomic masses above146Gd derived from a shell model analysis of high spin states

From a shell model analysis of high-spin states in neutron deficient nuclei above¹⁴⁶Gd we have derived the ground state masses of theN=82 and 83 isotones of Eu, Tb, Dy, Ho, and Er. The results can be used to calculate the energies of aligned multiparticle yrast configurations. They also link ten α-decay chains to the nuclei with known masses, providing many new absolute mass values which are compared with predictions. An examination of the two-proton separation energies atN=82 shows an 0.5 MeV break in the nuclear mass surface atZ=64.     

Equation of state of dense nuclear matter

An equation of state for cold nuclear matter for the region of densities ρ_nm−4ρ_nm, where ρ_nm is empirical nuclear-matter density, is constructed. We begin from the detailed calculation of Day and Wiringa for the two-body interactions; these give a saturation density of ∼2ρ_nm. This density is brought down to ρ_nm by the addition of relativistic corrections. Additional binding is obtained from three-body forces. A reasonable picture is obtained with the Day-Wiringa compression modulus for the two-body calculation, but the picture can be further improved by choosing this to be smaller.Our equation of state is similar to that of Friedman and Pandharipande in the region of nuclear matter denstiy ρ_nm, but, due to higher-order terms in the loop correction, is substantially softer at high density. Basically what happens is that the many-body effects saturate with increasing density, leaving only the two-body interactions.With this equation of state, prompt supernova explosions are very powerful when the compression modulus of neutron-rich matter (twice as many neutrons as protons) is ∼150 MeV, which corresponds to K_nm ∼ 190 MeV for symmetric nuclear matter.Analysis shows that hot nuclear matter formed in heavy ion collisions demands a very stiff equation of state. We understand this as arising from the strong velocity dependence in the real part of the optical model potential which follows chiefly from the Lorentz character of the interactions, the vector mean field growing with increasing density and the scalar one decreasing. This gives a substantial repulsive contribution to the energy per particle and produces a stiff effective equation of state for several hundred MeV heavy-ion collisions. With increasing degree of equilibration the magnitude of the repulsive energy decreases since equilibration decreases the effective momentum. Given the strong velocity dependence in the interaction, the hot equation of state can be reconciled with the cool one.