Isoscalar Density Research Articles

Impact of the stiffness of equation of state on saturation properties of asymmetric nuclear matter and Urca cooling in neutron stars

Abstract We consider nuclear equation of state (EOS) based on the semi-realistic M3Y-Paris and M3Y-Reid effective nucleon-nucleon (NN) interactions to assess the influence of the stiffness of the isospinasymmetric nuclear matter (ANM) on its isoscalar and isovector saturation properties, and on the β-stable proton fraction in neutron star (NS) matter. Two parameterizations schemes of the CDM3Y isovector density dependence of the NN interaction are adopted, one is scaled to the isoscalar density dependence and the other (IVF1) is developed independently of it. We found that the stiffer EOS yields larger binding saturation density that decreases with the isospin asymmetry (I), and slightly decreases the maximum asymmetry (Imax) reached for bound ANM. The density-slope (L) of the symmetry energy is found to correlate directly with the stiffness of ANM, meanwhile the symmetryenergy curvature (Ksym) and the quadratic isovector incompressibility coefficient (Kτ ) are found to correlate with it inversely. We found also that the stiffer EOS indicates larger neutron-rich npeμ NS matter under β-equilibrium. Only the CDM3Y-Paris EOSs of isoscalar incompressibility K0 up to 270 MeV and that of CDM3Y-Reid with K0 = 200 MeV indicate allowed direct URCA (DU) cooling process in β-stable NS matter, at densities greater than 3.5 ρ0, based on the IVF1 density dependence. The stiffer EOS proposes a narrower density range for DU process in NS. According to the considered Paris (Reid) forces, muons can emerge in cold β-stable npe matter at ρ ≥ 0.85 ρ0 (0.79 ρ0), which slightly decreases both the neutron richness at NS core and the DU threshold density.

Exploring Hyperon Structure with Electromagnetic Transverse Densities

We explore the structure of the spin-1/2 flavor-octet baryons (hyperons) through their electromagnetic transverse densities. The transverse densities describe the distribution of charge and magnetization at fixed light-front time and enable a spatial representation of the baryons as relativistic systems. At peripheral distances $$b \sim 1/M_\pi $$ the transverse densities are computed using a new method that combines chiral effective field theory ( $$\chi $$ EFT) and dispersion analysis. The peripheral isovector densities arise from two-pion exchange, which includes the $$\rho $$ resonance through elastic unitarity. The isoscalar densities are estimated from vector meson exchange ( $$\omega , \phi )$$ . We find that the “pion cloud” in the charged $$\varSigma $$ hyperons is comparable to the nucleon, while in the $$\varXi $$ it is suppressed. The $$\varLambda $$ – $$\varSigma ^0$$ transition density is pure isovector and represents a clear manifestation of peripheral two-pion dynamics.

Symmetry energy III: Isovector skins

Isoscalar density is a sum of neutron and proton densities and isovector is a normalized difference. Here, we report the experimental evidence for the displacement of the isovector and isoscalar surfaces in nuclei, by ∼0.9 fm from each other. We analyze data on quasielastic (QE) charge exchange (p,n) reactions, concurrently with proton and neutron elastic scattering data for the same target nuclei, following the concepts of the isoscalar and isovector potentials combined into Lane optical potential. The elastic data largely probe the geometry of the isoscalar potential and the (p,n) data largely probe a relation between the geometries of the isovector and isoscalar potentials. The targets include 48Ca, 90Zr, 120Sn and 208Pb and projectile incident energy values span the range of (10–50) MeV. In our fit to elastic and QE charge-exchange data, we allow the values of isoscalar and isovector radii, diffusivities and overall potential normalizations to float away from those in the popular Koning and Delaroche parametrization. We find that the best-fit isovector radii are consistently larger than isoscalar and the best-fit isovector surfaces are steeper. Upon identifying the displacement of the potential surfaces with the displacement of the surfaces for the densities in the Skyrme–Hartree–Fock calculations, and by supplementing the results with those from analyzing excitation energies to isobaric analog states in the past, we arrive at the slope and value of the symmetry energy at normal density of 70<L<101 MeV and 33.5<aaV<36.4 MeV, respectively.

Slope-dependent nuclear-symmetry energy within the effective-surface approximation

The effective-surface approximation is extended taking into account derivatives of the symmetry-energy density per particle with respect to the mean particle density. The isoscalar and isovector particle densities in this extended effective-surface approximation are derived. The improved expressions of the surface symmetry energy, in particular, its surface tension coefficients in the sharp-edged proton-neutron asymmetric nuclei take into account important gradient terms of the energy density functional. For most Skyrme forces the surface symmetry-energy constants and the corresponding neutron skins and isovector stiffnesses are calculated as functions of the Swiatecki derivative of the nongradient term of the symmetry-energy density per particle with respect to the isoscalar density. Using the analytical isovector surface-energy constants in the framework of the Fermi-liquid droplet model we find energies and sum rules of the isovector giant-dipole resonance structure in a reasonable agreement with the experimental data, and they are compared with other theoretical approaches.

Quasiparticle interaction in nuclear matter with chiral three-nucleon forces

We derive the effective interaction between two quasiparticles in symmetric nuclear matter resulting from the leading-order chiral three-nucleon force. We restrict our study to the L=0,1 Landau parameters of the central quasiparticle interaction computed to first order. We find that the three-nucleon force provides substantial repulsion in the isotropic spin- and isospin-independent component F_0 of the interaction. This repulsion acts to stabilize nuclear matter against isoscalar density oscillations, a feature which is absent in calculations employing low-momentum two-nucleon interactions only. We find a rather large uncertainty for the nuclear compression modulus due to a sensitive dependence on the low-energy constant c_3. The effective nucleon mass on the Fermi surface, as well as the nuclear symmetry energy, receive only small corrections from the leading-order chiral three-body force. Both the anomalous orbital g-factor and the Landau-Migdal parameter g'_{NN} (characterizing the spin-isospin response of nuclear matter) decrease with the addition of three-nucleon correlations. In fact, the anomalous orbital g-factor remains significantly smaller than its value extracted from experimental data, whereas g'_{NN} still compares well with empirical values. The inclusion of the three-nucleon force results in relatively small p-wave (L=1) components of the central quasiparticle interaction, thus suggesting an effective interaction of short range.

Realizing vector meson dominance with transverse charge densities

The transverse charge density in a fast-moving nucleon is represented as a dispersion integral of the imaginary part of the Dirac form factor in the timelike region (spectral function). At a given transverse distance b the integration effectively extends over energies in a range sqrt{t} ~< 1/b, with exponential suppression of larger values. The transverse charge density at peripheral distances thus acts as a low-pass filter for the spectral function and allows one to select energy regions dominated by specific t-channel states, corresponding to definite exchange mechanisms in the spacelike form factor. We show that distances b ~ 0.5 - 1.5 fm in the isovector density are maximally sensitive to the rho meson region, with only a ~10% contribution from higher-mass states. Soft-pion exchange governed by chiral dynamics becomes relevant only at larger distances. In the isoscalar density higher-mass states beyond the omega are comparatively more important. The dispersion approach suggests that the positive transverse charge density in the neutron at b ~ 1 fm, found previously in a Fourier analysis of spacelike form factor data, could serve as a sensitive test of the the isoscalar strength in the ~1 GeV mass region. In terms of partonic structure, the transverse densities in the vector meson region b ~ 1 fm support an approximate mean-field picture of the motion of valence quarks in the nucleon.

Parity-violating elastic electron scattering and nuclear structure

We discuss parity-violating elastic electron scattering as a complementary tool in the race for more precise determinations of neutron densities in nuclei. Isovector and isoscalar densities and form factors in N > Z and N = Z stable nuclei are discussed taking 208Pb and 28Si as examples. Distorted wave calculations of parity-violating asymmetries are shown and are compared to the plane wave impulse approximation. The extraction of the ratio between neutron and proton monopole form factors is discussed. The isospin mixing produced by Coulomb interaction in the ground state of N = Z nuclei with Skyrme selfconsistent mean fields is also discussed.

Symmetry energy I: Semi-infinite matter

Energy for a nucleus is considered in the macroscopic limit, in terms of nucleon numbers. Further considered for a nuclear system is the Hohenberg–Kohn energy functional, in terms of proton and neutron densities. Finally, Skyrme–Hartree–Fock calculations are carried out for a half-infinite particle-stable nuclear-matter. In each case, the attention is focused on the role of neutron–proton asymmetry and on the nuclear symmetry energy. We extend the considerations on the symmetry term from an energy formula to the respective term within the Hohenberg–Kohn functional. We show, in particular, that in the limit of an analytic functional, and subject to possible Coulomb corrections, it is possible to construct isoscalar and isovector densities out of the proton and neutron densities, that retain a universal relation to each other, approximately independent of asymmetry. In the so-called local approximation, the isovector density is inversely proportional to the symmetry energy in uniform matter, at the local isoscalar density. Generalized symmetry coefficient of a nuclear system is related, in the analytic limit of the functional, to an integral of the isovector density. We test the relations, inferred from the Hohenberg–Kohn functional, in the Skyrme–Hartree–Fock calculations of half-infinite matter. Within the calculations, we obtain surface symmetry coefficients and parameters characterizing the densities, for the majority of Skyrme parameterizations proposed in the literature. The volume-to-surface symmetry-coefficient ratio, and the displacement of nuclear isovector relative to isoscalar surfaces, both strongly increase as the slope of symmetry energy, in the vicinity of normal density, increases.

Chemical and mechanical spinodals a unique liquid-gas instability

We demonstrate that the instabilities of asymmetric nuclear matter at sub-saturation densities do not present two types of instabilities as usually discussed but a unique one. The associated order parameter is everywhere dominated by the isoscalar density and so the transition is of liquid-gas type even in the so-called chemical instability region.

A unique spinodal region in asymmetric nuclear matter

Asymmetric nuclear matter at sub-saturation densities is shown to present only one type of instabilities. The associated order parameter is dominated by the isoscalar density and so the transition is of liquid-gas type. The instability goes in the direction of a restoration of the isospin symmetry leading to a fractionation phenomenon. These conclusions are model independent since they can be related to the general form of the asymmetry energy. They are illustrated using density functional approaches.

Neutron and proton transition densities from 32,34S(p,p') at Ep=318 MeV. II. Neutron densities for 34S.

Differential cross sections and analyzing powers for low-lying states of $^{34}\mathrm{S}$ were measured using 318 MeV protons. Neutron transition densities for the ${2}_{1}^{+}$ and ${2}_{2}^{+}$ states were fitted to the data using general expansions which permit evaluation of uncertainties due to statistical and normalization errors, penetrability and distortion, and incompleteness in momentum space. We find that the experimental densities agree well with the shell model for the first ${2}^{+}$ state, but that the neutron density for the second ${2}^{+}$ state is distinctly different in shape. We also find ${\mathit{M}}_{\mathit{n}}$/${\mathit{M}}_{\mathit{p}}$=+0.27(6) for the ${2}_{2}^{+}$ state, in qualitative agreement with electromagnetic results for A=34 but in marked disagreement with the shell-model prediction of ${\mathit{M}}_{\mathit{n}}$/${\mathit{M}}_{\mathit{p}}$=-0.61. A detailed analysis of earlier data for 650 MeV, originally described as qualitatively consistent with the shell model, is also presented and shows that these data are more consistent with the present results. The shape of the isoscalar density for the ${2}_{2}^{+}$ state of $^{34}\mathrm{S}$ cannot be reproduced with the sd shell, supplemented by surface-peaked core polarization, but can be reproduced if an empirical (1f${)}^{2}$ contribution is admitted. Shell-model calculations were performed for other ${2}^{+}$ and ${4}^{+}$ states of $^{34}\mathrm{S}$, but give cross sections well below the (p,p') data for all but the lowest ${2}^{+}$ state. The data for the lowest ${1}^{\mathrm{\ensuremath{-}}}$, ${3}^{\mathrm{\ensuremath{-}}}$, and ${5}^{\mathrm{\ensuremath{-}}}$ states in both $^{34}\mathrm{S}$ and $^{32}\mathrm{S}$ are very similar.

Neutron and proton transition densities from 32,34S(p,p') at Ep=318 MeV. I. Isoscalar densities for 32S.

Differential cross sections and analyzing powers for low-lying states of $^{34}\mathrm{S}$ were measured using 318 MeV protons. Neutron transition densities for the ${2}_{1}^{+}$ and ${2}_{2}^{+}$ states were fitted to the data using general expansions which permit evaluation of uncertainties due to statistical and normalization errors, penetrability and distortion, and incompleteness in momentum space. We find that the experimental densities agree well with the shell model for the first ${2}^{+}$ state, but that the neutron density for the second ${2}^{+}$ state is distinctly different in shape. We also find ${\mathit{M}}_{\mathit{n}}$/${\mathit{M}}_{\mathit{p}}$=+0.27(6) for the ${2}_{2}^{+}$ state, in qualitative agreement with electromagnetic results for A=34 but in marked disagreement with the shell-model prediction of ${\mathit{M}}_{\mathit{n}}$/${\mathit{M}}_{\mathit{p}}$=-0.61. A detailed analysis of earlier data for 650 MeV, originally described as qualitatively consistent with the shell model, is also presented and shows that these data are more consistent with the present results. The shape of the isoscalar density for the ${2}_{2}^{+}$ state of $^{34}\mathrm{S}$ cannot be reproduced with the sd shell, supplemented by surface-peaked core polarization, but can be reproduced if an empirical (1f${)}^{2}$ contribution is admitted. Shell-model calculations were performed for other ${2}^{+}$ and ${4}^{+}$ states of $^{34}\mathrm{S}$, but give cross sections well below the (p,p') data for all but the lowest ${2}^{+}$ state. The data for the lowest ${1}^{\mathrm{\ensuremath{-}}}$, ${3}^{\mathrm{\ensuremath{-}}}$, and ${5}^{\mathrm{\ensuremath{-}}}$ states in both $^{34}\mathrm{S}$ and $^{32}\mathrm{S}$ are very similar.

Landau zero sound in finite systems

Abstract Stationary solutions of the Landau-Vlasov equation for the semiclassical distribution function of Fermi particles in the nuclear volume are found for isoscalar density excitations. It is shown that the solutions derived are superpositions of Landau zero-sound plane waves and do not satisfy the boundary conditions of quasi-particle mirror reflection at the nucleus' edge.

Transition densities in 30Si studied by electron scattering and coupled-channel calculations of 650 MeV proton scattering.

Proton and neutron transition densities in $^{30}\mathrm{Si}$ are examined by a combination of intermediate energy (e,e') and (p,p') reactions and mirror electromagnetic transition rates. This analysis is performed for the ${2}_{1}^{+}$ and ${2}_{2}^{+}$ states at 2.234 and 3.499 MeV in $^{30}\mathrm{Si}$. Electron scattering data are presented for these states. Shell-model calculations for the proton and neutron transition matrix elements and proton transition densities are compared with the electromagnetic results. The proton transition densities are reasonably predicted for the ${2}_{1}^{+}$ state but are not adequately predicted for the ${2}_{2}^{+}$ state. A microscopic coupled-channel calculation of 650 MeV (p,p') is used to test the shell-model isoscalar transition densities. Given the uncertainties present in the reaction calculation and interaction, the isoscalar density for the ${2}_{1}^{+}$ state is found to be adequate, but the density for the ${2}_{2}^{+}$ state is less accurate. The coupled-channel effect is shown to be important for the ${2}_{2}^{+}$ state. This dependence increases with energy but should be taken into account for an accurate description of (p,p') reactions at all energies.