Nuclear Matter Saturation Density Research Articles

The d⁎(2380) in Neutron Stars — A New Degree of Freedom?

Elucidating the appropriate microscopic degrees of freedom within neutron stars remains an open question which impacts nuclear physics, particle physics and astrophysics. The recent discovery of the first non-trivial dibaryon, the d⁎(2380), provides a new candidate for an exotic degree of freedom in the nuclear equation of state at high matter densities. In this paper a first calculation of the role of the d⁎(2380) in neutron stars is performed based on a relativistic mean field description of the nucleonic degrees of freedom supplemented by a free boson gas of d⁎(2380). The calculations indicate that the d⁎(2380) would appear at densities around three times normal nuclear matter saturation density and comprise around 20% of the matter in the centre of heavy stars with higher fractions possible in the higher densities of merger processes. The d⁎(2380) would also reduce the maximum star mass by around 15% and have significant influence on the fractional proton/neutron composition. New possibilities for neutron star cooling mechanisms arising from the d⁎(2380) are also predicted.

Equation of state for dense nucleonic matter from metamodeling. I. Foundational aspects

A metamodeling for the nucleonic equation of state (EOS), inspired from a Taylor expansion around the saturation density of symmetric nuclear matter, is proposed and parameterized in terms of the empirical parameters. The present knowledge of nuclear empirical parameters is first reviewed in order to estimate their average values and associated uncertainties, and thus defining the parameter space of the metamodeling. They are divided into isoscalar and isovector type, and ordered according to their power in the density expansion. The goodness of the metamodeling is analyzed against the predictions of the original models. In addition, since no correlation among the empirical parameters is assumed a priori, all arbitrary density dependences can be explored, which might not be accessible in existing functionals. Spurious correlations due to the assumed functional form are also removed. This meta-EOS allows direct relations between the uncertainties on the empirical parameters and the density dependence of the nuclear equation of state and its derivatives, and the mapping between the two can be done with standard Bayesian techniques. A sensitivity analysis shows that the more influential empirical parameters are the isovector parameters $L_{sym}$ and $K_{sym}$, and that laboratory constraints at super-saturation densities are essential to reduce the present uncertainties. The present metamodeling for the EOS for nuclear matter is proposed for further applications in neutron stars and supernova matter.

$$^1S_0$$ 1 S 0 Pairing in Neutron Matter

We report calculations of the superfluid pairing gap in neutron matter for the $^1S_0$ components of the Reid soft-core $V_6$ and the Argonne $V_{4}'$ two-nucleon interactions. Ground-state calculations have been carried out using the central part of the operator-basis representation of these interactions to determine optimal Jastrow-Feenberg correlations and corresponding effective pairing interactions within the correlated-basis formalism (CBF), the required matrix elements in the correlated basis being evaluated by Fermi hypernetted-chain techniques. Different implementations of the Fermi-Hypernetted Chain Euler-Lagrange method (FHNC-EL) agree at the percent level up to nuclear matter saturation density. For the assumed interactions, which are realistic within the low density range involved in $^1S_0$ neutron pairing, we did not find a dimerization instability arising from divergence of the in-medium scattering length, as was reported recently for simple square-well and Lennard-Jones potential models (Phys. Rev. A {\bf 92}, 023640 (2015)).

Mirror Charge Radii and the Neutron Equation of State.

The differences in the charge radii of mirror nuclei are shown to be proportional to the derivative of the neutron equation of state and the symmetry energy at nuclear matter saturation density. This derivative is important for constraining the neutron equation of state for use in astrophysics. The charge radii of several neutron-rich nuclei are already measured to the accuracy of about 0.005fm. Experiments at isotope-separator and radioactive-beam facilities are needed to measure the charge radii of the corresponding proton-rich mirror nuclei to a similar accuracy. It is also shown that neutron skins of nuclei with N=Z depend upon the value of the symmetry energy at a density of 0.10 nucleons/fm^{3}.

Three-Nucleon Forces and Triplet Pairing in Neutron Matter

The existence of superfluidity of the neutron component in the core of a neutron star, associated specifically with triplet $P-$wave pairing, is currently an open question that is central to interpretation of the observed cooling curves and other neutron-star observables. Ab initio theoretical calculations aimed at resolving this issue face unique challenges in the relevant high-density domain, which reaches beyond the saturation density of symmetrical nuclear matter. These issues include uncertainties in the three-nucleon (3N) interaction and in the effects of strong short-range correlations -- and more generally of in-medium modification of nucleonic self-energies and interactions. A survey of existing solutions to the gap equations in the triplet channel shows that the separate or combined impacts of 3N forces, coupled channels, and mass renormalization range from moderate to strong to devastating, thus motivating a detailed analysis of the competing effects. In the present work we track the effects of the 3N force and in-medium modifications in the representative case of the $^3P_2$ channel, based on the Argonne V18 two-nucleon (2N) interaction supplemented by 3N interactions of the Urbana IX family. Sensitivity of the results to the input interaction is clearly demonstrated, while consistency issues arise with respect to the simultaneous treatment of 3N forces and in-medium effects. We consider this pilot study as the first step towards a systematic and comprehensive exploration of coupled-channel $^3P F_2$ pairing using a broad range of 2N and 3N interactions from the current generation of refined semi-phenomenological models and models derived from chiral effective field theory.

Nuclear charge and neutron radii and nuclear matter: Trend analysis in Skyrme density-functional-theory approach

Radii of charge and neutron distributions are fundamental nuclear properties. They depend on both nuclear interaction parameters related to the equation of state of infinite nuclear matter and on quantal shell effects, which are strongly impacted by the presence of nuclear surface. In this work, by studying the dependence of charge and neutron radii, and neutron skin, on nuclear matter parameters, we assess different mechanisms that drive nuclear sizes. We apply nuclear density functional theory using a family of Skyrme functionals obtained by means of different optimization protocols targeting specific nuclear properties. By performing the Monte-Carlo sampling of reasonable functionals around the optimal parametrization, we study correlations between nuclear matter paramaters and observables characterizing charge and neutron distributions. We demonstrate the existence of the strong converse relation between the nuclear charge radii and the saturation density of symmetric nuclear matter and also between the neutron skins and the slope of the symmetry energy. For functionals optimized to experimental binding energies only, proton and neutron radii are weakly correlated due to canceling trends from different nuclear matter parameters. We show that by requiring that the nuclear functional reproduces the empirical saturation point of symmetric nuclear matter practically fixes the charge (or proton) radii, and vice versa. The neutron skin uncertainty primarily depends on the slope of the symmetry energy. Consequently, imposing a constraint on both $\rho_0$ and $L$ practically determines the nuclear size, modulo small variations due to shell effects.

Quantum simulations of nuclei and nuclear pasta with the multiresolution adaptive numerical environment for scientific simulations

Neutron star and supernova matter at densities just below the nuclear matter saturation density is expected to form a lattice of exotic shapes. These so-called nuclear pasta phases are caused by Coulomb frustration. Their elastic and transport properties are believed to play an important role for thermal and magnetic field evolution, rotation and oscillation of neutron stars. Furthermore, they can impact neutrino opacities in core-collapse supernovae. In this work, we present proof-of-principle 3D Skyrme Hartree-Fock (SHF) simulations of nuclear pasta with the Multi-resolution ADaptive Numerical Environment for Scientific Simulations (MADNESS). We perform benchmark studies of $^{16} \mathrm{O}$, $^{208} \mathrm{Pb}$ and $^{238} \mathrm{U}$ nuclear ground states and calculate binding energies via 3D SHF simulations. Results are compared with experimentally measured binding energies as well as with theoretically predicted values from an established SHF code. The nuclear pasta simulation is initialized in the so-called waffle geometry as obtained by the Indiana University Molecular Dynamics (IUMD) code. The size of the unit cell is 24\:fm with an average density of about $\rho = 0.05 \:\mathrm{fm}^{-3}$, proton fraction of $Y_p = 0.3$ and temperature of $T=0\:$MeV. Our calculations reproduce the binding energies and shapes of light and heavy nuclei with different geometries. For the pasta simulation, we find that the final geometry is very similar to the initial waffle state. In the present pasta calculations spin-orbit forces are not included but will be added in the future. Within the MADNESS framework, we can successfully perform calculations of inhomogeneous nuclear matter. By using pasta configurations from IUMD it is possible to explore different geometries and test the impact of self-consistent calculations on the latter.

Nucleon effective E-mass in neutron-rich matter from the Migdal–Luttinger jump

The well-known Migdal-Luttinger theorem states that the jump of the single-nucleon momentum distribution at the Fermi surface is equal to the inverse of the nucleon effective E-mass. Recent experiments studying short-range correlations (SRC) in nuclei using electron-nucleus scatterings at the Jefferson National Laboratory (JLAB) together with model calculations constrained significantly the Migdal-Luttinger jump at saturation density of nuclear matter. We show that the corresponding nucleon effective E-mass is consequently constrained to $M_0^{\ast,\rm{E}}/M\approx2.22\pm0.35$ in symmetric nuclear matter (SNM) and the E-mass of neutrons is smaller than that of protons in neutron-rich matter. Moreover, the average depletion of the nucleon Fermi sea increases (decreases) approximately linearly with the isospin asymmetry $\delta$ according to $\kappa_{\rm{p}/\rm{n}}\approx 0.21\pm0.06 \pm (0.19\pm0.08)\delta$ for protons (neutrons). These results will help improve our knowledge about the space-time non-locality of the single-nucleon potential in neutron-rich nucleonic matter useful in both nuclear physics and astrophysics.

Observational constraints on neutron star masses and radii

Precise and reliable measurements of the masses and radii of neutron stars with a variety of masses would provide valuable guidance for improving models of the properties of cold matter with densities above the saturation density of nuclear matter. Several different approaches for measuring the masses and radii of neutron stars have been tried or proposed, including analyzing the X-ray fluxes and spectra of the emission from neutron stars in quiescent low-mass X-ray binary systems and thermonuclear burst sources; fitting the energy-dependent X-ray waveforms of rotation-powered millisecond pulsars, burst oscillations with millisecond periods, and accretion-powered millisecond pulsars; and modeling the gravitational radiation waveforms of coalescing double neutron star and neutron star -- black hole binary systems. We describe the strengths and weaknesses of these approaches, most of which currently have substantial systematic errors, and discuss the prospects for decreasing the systematic errors in each method.

Symmetry energy of cold nucleonic matter within a relativistic mean field model encapsulating effects of high-momentum nucleons induced by short-range correlations

Significant progress has been made recently in constraining the isospin-dependent parameters characterizing the SRC (short-range correlation)-modified single-nucleon momentum distribution in neutron-rich nucleonic matter using both experimental data and microscopic model calculations. Using the constrained single-nucleon momentum distribution in a nonlinear relativistic mean field (RMF) model, we study the equation of state (EOS) of asymmetric nucleonic matter (ANM), especially the density dependence of nuclear symmetry energy $E_{\rm{sym}}(\rho)$. Firstly, as a test of the model, the average nucleon kinetic energy extracted recently from electron-nucleus scattering experiments using a neutron-proton dominance model is well reproduced by the RMF model incorporating effects of the SRC-induced high momentum nucleons, while it is significantly under predicted by the RMF model using a step function for the single-nucleon momentum distribution as in free Fermi gas (FFG) models. Secondly, the kinetic symmetry energy of quasi-nucleons is found to be $E^{\rm{kin}}_{\rm{sym}}(\rho_0)=-16.94\pm13.66\,\rm{MeV}$ which is dramatically different from the prediction of $E^{\rm{kin}}_{\rm{sym}}(\rho_0)\approx 12.5$ MeV by FFG models at nuclear matter saturation density $\rho_0=0.16\,\rm{fm}^{-3}$. Thirdly, comparing the RMF calculations with and without the high momentum nucleons using two sets of model parameters both reproducing identically all empirically constraints on the EOS of symmetric nuclear matter (SNM) and the symmetry energy of ANM at $\rho_0$, the SRC-modified single-nucleon momentum distribution is found to make the $E_{\rm{sym}}(\rho)$ more concave around $\rho_0$ by softening it significantly at both sub-saturation and supra-saturation densities, leading to an isospin-dependent incompressibility of ANM in better agreement with existing experimental data.

Modelling hybrid stars in quark-hadron approaches

The density in the core of neutron stars can reach values of about 5 to 10 times nuclear matter saturation density. It is, therefore, a natural assumption that hadrons may have dissolved into quarks under such conditions, forming a hybrid star. This star will have an outer region of hadronic matter and a core of quark matter or even a mixed state of hadrons and quarks. In order to investigate such phases, we discuss different model approaches that can be used in the study of compact stars as well as being applicable to a wider range of temperatures and densities. One major model ingredient, the role of quark interactions in the stability of massive hybrid stars is discussed. In this context, possible conflicts with lattice QCD simulations are investigated.

Isospin quartic term in the kinetic energy of neutron-rich nucleonic matter

The energy of a free gas of neutrons and protons is well known to be approximately isospin parabolic with a negligibly small quartic term of only $0.45$ MeV at the saturation density of nuclear matter $\rho_0=0.16/\rm{fm}^3$. Using an isospin-dependent single-nucleon momentum distribution including a high (low) momentum tail (depletion) with its shape parameters constrained by recent high-energy electron scattering and medium-energy nuclear photodisintegration experiments as well as the state-of-the-art calculations of the deuteron wave function and the equation of state of pure neutron matter near the unitary limit within several modern microscopic many-body theories, we show for the first time that the kinetic energy of interacting nucleons in neutron-rich nucleonic matter has a significant quartic term of $7.18\pm2.52\,\rm{MeV}$. Such a large quartic term has significant ramifications in determining the equation of state of neutron-rich nucleonic matter using both terrestrial and astrophysical observables.

Correlations between the nuclear matter symmetry energy, its slope, and curvature

By using point-coupling versions of finite range nuclear relativistic mean field models containing cubic and quartic self interactionsin the scalar field σ, a nonrelativistic limit is achieved. This approach allows for an analytical expression for the symmetry energy (J) as a function of its slope (L) in a unified form, namely, L = 3J + f(m*, ρo, Bo, Ko), where the quantities m*, po, Bo and Ko are bulk parameters at the nuclear matter saturation density ρo. This result establishes a linear correlation between L and J which is reinforced by exact relativistic calculations we have performed. An analogous analytical correlation can also be found for J, L and the symmetry energy curvature (Ksym). Based on these results, we propose a graphic constraint in L × J plane which finite range models should satisfy.

Neutron–proton effective mass splitting in neutron-rich matter and its impacts on nuclear reactions

The neutron–proton effective mass splitting in neutron-rich nucleonic matter reflects the spacetime nonlocality of the isovector nuclear interaction. It affects the neutron/proton ratio during the earlier evolution of the Universe, cooling of proto-neutron stars, structure of rare isotopes and dynamics of heavy-ion collisions. While there is still no consensus on whether the neutron–proton effective mass splitting is negative, zero or positive and how it depends on the density as well as the isospin-asymmetry of the medium, significant progress has been made in recent years in addressing these issues. There are different kinds of nucleon effective masses. In this mini-review, we focus on the total effective masses often used in the non-relativistic description of nuclear dynamics. We first recall the connections among the neutron–proton effective mass splitting, the momentum dependence of the isovector potential and the density dependence of the symmetry energy. We then make a few observations about the progress in calculating the neutron–proton effective mass splitting using various nuclear many-body theories and its effects on the isospin-dependence of in-medium nucleon–nucleon cross-sections. Perhaps, our most reliable knowledge so far about the neutron–proton effective mass splitting at saturation density of nuclear matter comes from optical model analyses of huge sets of nucleon–nucleus scattering data accumulated over the last five decades. The momentum dependence of the symmetry potential from these analyses provide a useful boundary condition at saturation density for calibrating nuclear many-body calculations. Several observables in heavy-ion collisions have been identified as sensitive probes of the neutron–proton effective mass splitting in dense neutron-rich matter based on transport model simulations. We review these observables and comment on the latest experimental findings.

The symmetry energy, neutron skin thickness and isovector dipole response of neutron-rich nuclei

The isotopic evolution of the relationship between the symmetry energy at saturation density of nuclear matter (J ), neutron skin thickness (ΔR ) and relevant observables related to isovector dipole excitations in neutron rich 116−136 Sn isotopes has been investigated in the framework of relativistic nuclear energy density functional theory. The description employs a family of effective interactions with density dependent meson-nucleon couplings (DDME) spanning the range of values J = 30 − 38 MeV.

Early appearance ofΔisobars in neutron stars

We discuss the formation of $\ensuremath{\Delta}$ isobars in neutron star matter. We show that their threshold density strictly correlates with the density derivative of the symmetry energy of nuclear matter: the $L$ parameter. By restricting $L$ to the range of values indicated by recent experimental and theoretical analysis, i.e., $40\phantom{\rule{4.pt}{0ex}}\text{MeV}\ensuremath{\lesssim}L\ensuremath{\lesssim}62\phantom{\rule{4.pt}{0ex}}\text{MeV}$, we find that $\ensuremath{\Delta}$ isobars appear at a density of the order of 2 to 3 times the nuclear matter saturation density, i.e., the same range as for the appearance of hyperons. The range of values of the couplings of the $\ensuremath{\Delta}\mathrm{s}$ with the mesons is restricted by the analysis of the data obtained from photoabsorption, electron and pion scattering on nuclei. If the potential of the $\ensuremath{\Delta}$ in nuclear matter is close to the one indicated by the experimental data then the equation of state becomes soft enough that a ``\ensuremath{\Delta} puzzle'' exists, similar to the ``hyperon puzzle'' widely discussed in the literature. Possible solutions to this puzzle are also discussed.

Nuclear “pasta” formation

The formation of complex nonuniform phases of nuclear matter, known as nuclear pasta, is studied with molecular dynamics simulations containing 51200 nucleons. A phenomenological nuclear interaction is used that reproduces the saturation binding energy and density of nuclear matter. Systems are prepared at an initial density of 0.10fm$^{-3}$ and then the density is decreased by expanding the simulation volume at different rates to densities of 0.01 fm$^{-3}$ or less. An originally uniform system of nuclear matter is observed to form spherical bubbles ("swiss cheese"), hollow tubes, flat plates ("lasagna"), thin rods ("spaghetti") and, finally, nearly spherical nuclei with decreasing density. We explicitly observe nucleation mechanisms, with decreasing density, for these different pasta phase transitions. Topological quantities known as Minkowski functionals are obtained to characterize the pasta shapes. Different pasta shapes are observed depending on the expansion rate. This indicates non equilibrium effects. We use this to determine the best ways to obtain lower energy states of the pasta system from MD simulations and to place constrains on the equilibration time of the system.

Probing Nuclear Symmetry Energy and its Imprints on Properties of Nuclei, Nuclear Reactions, Neutron Stars and Gravitational Waves

Significant progress has been made in recent years in constraining nuclear symmetry energy at and below the saturation density of nuclear matter using data from both terrestrial nuclear experiments and astrophysical observations. However, many interesting questions remain to be studied especially at supra-saturation densities. In this lecture note, after a brief summary of the currently available constraints on nuclear symmetry energy near the saturation density we first discuss the relationship between the symmetry energy and the isopin and momentum dependence of the single-nucleon potential in isospin-asymmetric nuclear medium. We then discuss several open issues regarding effects of the tensor force induced neutron-proton short-range correlation (SRC) on nuclear symmetry energy. Finally, as an example of the impacts of nuclear symmetry energy on properties of neutron stars and gravitational waves, we illustrate effects of the high-density symmetry energy on the tidal polarizability of neutron stars in coalescing binaries.

Chiral Fermi liquid approach to neutron matter

We present a microscopic calculation of the complete quasiparticle interaction, including central as well as noncentral components, in neutron matter from high-precision two- and three-body forces derived within the framework of chiral effective-field theory. The contributions from two-nucleon forces are computed in many-body perturbation theory to first and second order (without any simplifying approximations). In addition we include the leading-order one-loop diagrams from the next-to next-to leading order (N${}^{2}$LO) chiral three-nucleon force, which contribute to all Fermi liquid parameters except those associated with the center-of-mass tensor interaction. The relative-momentum dependence of the quasiparticle interaction is expanded in Legendre polynomials up to $L=2$. Second-order Pauli blocking and medium polarization effects act coherently in specific channels; namely, for the Landau parameters ${f}_{1}$, ${h}_{0}$, and ${g}_{0}$, which results in a dramatic increase in the quasiparticle effective mass as well as a decrease in both the effective tensor force and the neutron matter spin susceptibility. For densities greater than about half the nuclear matter saturation density ${\ensuremath{\rho}}_{0}$, the contributions to the Fermi liquid parameters from the leading-order chiral three-nucleon force scale in all cases approximately linearly with the nucleon density. The largest effect of the three-nucleon force is to generate a strongly repulsive effective interaction in the isotropic spin-independent channel. We show that the leading-order chiral three-nucleon force leads to an increase in the spin susceptibility of neutron matter, but we observe no evidence for a ferromagnetic spin instability in the vicinity of the saturation density ${\ensuremath{\rho}}_{0}$. This work sets the foundation for future studies of neutron matter response to weak and electromagnetic probes with applications to neutron star structure and evolution.

Soft nuclear equation-of-state from heavy-ion data and implications for compact stars

Measurements of kaon production at subthreshold energies in heavy-ion collisions point to a soft nuclear equation-of-state for densities up to 2-3 times nuclear matter saturation density. We apply these results to study the implications on compact star properties, especially in the context of the recent measurement of the two solar mass pulsar PSR J1614-2230. The implications are two-fold: Firstly, the heavy-ion results constrain nuclear matter at densities relevant to light neutron stars. Hence, a radius measurement could provide information about the density dependence of the symmetry energy which is a crucial quantity in nuclear physics. Secondly, the information on the nucleon potential obtained from the analysis of the heavy-ion data can be combined with restrictions from causality on the nuclear equation-of-state. From this we can derive a limit for the highest allowed compact star mass of three solar masses.