Nuclear Pasta Research Articles

Phase Diagram of Nuclear Pastas in Neutron Star Crusts

Two neural networks were trained to predict, respectively, the Euler characteristic and the curvature of nuclear pastas in neutron star crust conditions generated by molecular dynamics simulations of neutron star matter with 0.1 < x < 0.5, 0.040 fm−3 < ρ < 0.085 fm−3 (0.68 × 1014 g/cm3 < ρ < 1.43 × 1014 g/cm3), and 0.2 MeV < T < 4.0 MeV, where x is proton content, the density is ρ, and the temperature is T. The predictions of the two networks were combined to determine the nuclear pasta phase that is thermodynamically stable at a given x, ρ, and T, and a three-dimensional phase diagram that extrapolated slightly the regions of existing molecular dynamics data was computed. The jungle gym and anti-jungle gym structures are prevalent at high temperature and low density, while the anti-jungle gym and anti-gnocchi structures dominate at high temperature and high density. A diversity of structures exist at low temperatures and intermediate density and proton content. The trained models used in this work are open access and available at a public repository to promote comparison to pastas obtained with other models.

Superfluid Rivers in Spinning-down Neutron Stars

We study the motion of neutron superfluid vortices in a spinning-down neutron star, assuming axisymmetry of the flow and ignoring the motion of vortices about the rotation axis. We find that the vortex array, if initially rectilinear, is soon substantially deformed as the star spins down; vortices are swept outward by the Magnus force, accumulating in regions of the inner crust where they pin, accompanied by significant bending of the vortex array. As the star spins down to below a spin rate of ∼20 Hz (twice the spin rate of the Vela pulsar), the Magnus and pinning forces gradually compress the vortex array into dense sheets that follow spherical shells. In some cases, the vortex array bends on itself and reconnects, forming one or more tori of vortex rings that contain superfluid “rivers” with significant angular momentum. Vortex sheets are likely to form near the base of the inner crust in the regime of nuclear pasta.

Do short range correlations inhibit the appearance of the nuclear pasta?

Do short range correlations inhibit the appearance of the nuclear pasta?

Correction to: Anisotropic electron transport in the nuclear pasta phase

Correction to: Anisotropic electron transport in the nuclear pasta phase

Anisotropic electron transport in the nuclear pasta phase

ABSTRACTThe presence of nuclear pasta is expected to modify the transport properties in the mantle of neutron stars. The non-spherical geometry of the pasta nuclear clusters leads to anisotropies in the collision frequencies, impacting the thermal and electrical conductivity. We derive analytical expressions for the anisotropic collision frequencies using the Boltzmann equation in the relaxation time approximation. The average parallel, perpendicular, and Hall electrical conductivities are computed in the high-temperature regime above crustal melting, considering incoherent elastic electron-pasta scattering and randomly oriented pasta structures. Numerical values are obtained at different densities and temperatures by using the Indiana University–Florida State University (IUFSU) parametrization of the non-linear Walecka model to determine the crustal structure. We find that the anisotropy of the collision frequencies grows with the length of the pasta structures and, independently of the magnetic field, the presence of rod and slab phases decreases the conductivity by more than one order of magnitude. Our numerical results indicate that, even if the pasta structures might survive above the crustal melting point, no strong anisotropies are to be expected in the conduction properties in this temperature regime, even in the presence of a very high magnetic field.

Elastic properties of nuclear pasta in a fully three-dimensional geometry

Realistic estimations on the elastic properties of neutron star matter are carried out with a large strain (ε≲0.5) in the framework of relativistic-mean-field model with Thomas-Fermi approximation, where various crystalline configurations are considered in a fully three-dimensional geometry with reflection symmetry. Our calculation confirms the validity of assuming Coulomb crystals for the droplet phase above neutron drip density, which nonetheless does not work at large densities since the elastic constants are found to be decreasing after reaching their peaks. Similarly, the analytic formulae derived in the incompressible liquid-drop model give excellent description for the rod phase at small densities, which overestimates the elastic constants at larger densities. For slabs, due to the negligence on the variations of their thicknesses, the analytic formulae from liquid-drop model agree qualitatively but not quantitatively with our numerical estimations. By fitting to the numerical results, these analytic formulae are improved by introducing dampening factors. The impacts of nuclear symmetry energy are examined adopting two parameter sets, corresponding to the slope of symmetry energy L=41.34 and 89.39 MeV. Even with the uncertainties caused by the anisotropy in polycrystallines, the elastic properties of neutron star matter obtained with L=41.34 and 89.39 MeV are distinctively different, results in detectable differences in various neutron star activities.

Pasta Phases in Neutron Star Mantle: Extended Thomas–Fermi vs. Compressible Liquid Drop Approaches

Nuclear pasta phases in the neutron stars mantle can affect the mechanical and transport properties of superdense matter, thus playing an important role in the dynamics and evolution of neutron stars. In this paper, we compare results obtained by the Extended Thomas–Fermi (ETF) method with the compressible liquid drop model (CLDM), based on the thermodynamically consistent description of the surface properties calculated for the two-phase plane interface and the same energy-density functional (for numerical illustration, we applied the Skyrme-type functional SLy4). Our ETF calculations found that pasta phases in cylindrical form cover a significant crustal region (both normal and inverse phases, aka spaghetti and bucatini are presented). Meanwhile, within the applied CLDM framework, which includes the thermodynamically required effect of neutron adsorption on the cluster’s surface but neglects curvature corrections, only the spaghetti phase was found to be energetically favorable in the small density range prior to crust–core transition. On the other hand, the recent CLDM of Dinh Thi et al., 2021, which, on the contrary, accounts for curvature term but neglects neutron adsorption, predicts pasta phase onset in better agreement with the ETF. This fact highlights the importance of the curvature effects and allows counting on the potential validity of the CLDMs as a convenient, transparent and accurate tool for investigation of the pasta-phase properties.

From neutron skins and neutron matter to the neutron star crust

We present the first Bayesian inference of neutron star crust properties to incorporate neutron skin data, including the recent PREX measurement of the neutron skin of 208Pb, combined with recent chiral effective field theory predictions of pure neutron matter with statistical errors. Using a compressible liquid drop model with an extended Skyrme energy-density functional, we obtain the most stringent constraints to date on the transition pressure Pcc=0.33−0.07+0.07 MeV fm−3 and chemical potential μcc=12.6−1.9+1,8 MeV (which control the mass, moment of inertia and thickness of a neutron star crust), the proton fractions that bracket the pasta phases yp=0.115−0.017+0.016 and ycc=0.041−0.006+0.007, as well as the relative mass and moment of inertia ΔMp/ΔMc≈ΔIp/ΔIc=0.54−0.09+0.05 and thickness ΔRp/ΔRc=0.129−0.030+0.019 of the layers of non-spherical nuclei (nuclear pasta) in the crust.

The role of the electric Bond number in the stability of pasta phases

The deformation of nuclei in the neutrons star crust can be modelled by the system of two charged fluids separated by the phase boundary with non-zero surface tension. The stability of highly elongated structures, called pasta phases, is discussed here. In this work the cylindrical and spherical portion of charged fluids are considered. Their behavior depends on whether they are in vacuum or in charged surroundings what is modelled by Wigner-Seitz cell. The electric Bond number appears to be crucial parameter in the analysis. Variety of different unstable modes in vacuum case is strongly reduced in the case of W-S cell. It comes from the virial theorem, which bounds the value of the Bond number and reduces the role played by electric forces. Although the analysis is motivated by the nuclear pasta phases it can be applied to any system of two charged fluids.

Nuclear pasta structures at high temperatures

We investigate nuclear pasta structures at high temperatures in the framework of relativistic mean field model with Thomas-Fermi approximation. Typical pasta structures (droplet, rod, slab, tube, and bubble) are obtained, which form various crystalline configurations. The properties of those nuclear pastas are examined in a three-dimensional geometry with reflection symmetry, where the optimum lattice constants are fixed by reproducing the droplet/bubble density that minimizes the free energy adopting spherical or cylindrical approximations for Wigner-Seitz cells. It is found that different crystalline structures can evolve into each other via volume conserving deformations. For fixed densities and temperatures, the differences of the free energies per baryon of nuclear pasta in various shapes and lattice structures are typically on the order of tens of keV, suggesting the possible coexistence of those structures. As temperature increases, the thermodynamic fluctuations are expected to disrupt the long-range ordering in nuclear pasta structures. We then estimate the critical conditions for nuclear pasta to become disordered and behave like liquid, which are found to be sensitive to the densities, temperatures, proton fractions, and nuclear shapes. If we further increase temperature, eventually the nonuniform structures of nuclear pasta become unstable and are converted into uniform nuclear matter. The phase diagrams of nuclear matter are then estimated, which should be useful for understanding the evolutions of neutron stars, supernova dynamics, and binary neutron star mergers.

Low Density Neutron Star Matter with Quantum Molecular Dynamics: The Role of Isovector Interactions

The effect of isospin-dependent nuclear forces on the inner crust of neutron stars is modeled within the framework of Quantum Molecular Dynamics (QMD). To successfully control the density dependence of the symmetry energy of neutron-star matter below nuclear saturation density, a mixed vector-isovector potential is introduced. This approach is inspired by the baryon density and isospin density-dependent repulsive Skyrme force of asymmetric nuclear matter. In isospin-asymmetric nuclear matter, the system shows nucleation, as nucleons are arranged into shapes resembling nuclear pasta. The dependence of clusterization in the system on the isospin properties is also explored by calculating two-point correlation functions. We show that, as compared to previous results that did not involve such mixed interaction terms, the energy symmetry slope L is successfully controlled by varying the corresponding coupling strength. Nevertheless, the effect of changing the slope of the nuclear symmetry energy L on the crust-core transition density does not seem significant. To the knowledge of the authors, this is the first implementation of such a coupling in a QMD model for isospin asymmetric matter, which is relevant to the inner crust of neutron and proto-neutron stars.

Ensembles of unified crust and core equations of state in a nuclear-multimessenger astrophysics environment

We present an ensemble of unified neutron star crust and core equations of state, constructed using an extended Skyrme energy density functional through the crust and outer core, and appended by two piecewise polytropes at higher densities. The equations of state are parameterized by the first three coefficients in the density expansion of the symmetry energy J, L and \(K_{\mathrm{sym}}\), the moment of inertia of a 1.338 M\(_{\odot }\) star \(I_{1.338}\) and the maximum neutron star mass \(M_{\mathrm{max}}\). We construct an ensemble with uniform priors on all five parameters, and then apply data filters to the ensemble to explore the effect of combining neutron skin data from PREX with astrophysical measurements of radii and tidal deformabilities from NICER and LIGO/VIRGO. Neutron skins are calculated directly using the EDFs. We demonstrate that both the nuclear data and astrophysical data play a role in constraining crust properties such as the mass, thickness and moment of inertia of the crust and the nuclear pasta layers therein, and that astrophysical data better constrains \(K_{\mathrm{sym}}\) than PREX data.

Glassy quantum nuclear pasta in neutron star crusts

Nuclear pasta is an exotic form of nuclear matter that occurs in the crust of neutron stars below saturation density. Various exotic nuclear structures with cylindrical, planar, and more complicated geometries evolve from the Coulomb lattice of nuclei immersed in a fluid of neutrons. Such structures in the crust should not only affect how a neutron star cools and rotates but also the height of the ``mountains'' that the crust can sustain---potentially detectable as persistent sources of gravitational waves. By performing a large set of quantum calculations the authors show that the energy landscape of nuclear pasta consists of multiple local minima with very similar energies. Hence, at the characteristic temperatures nuclear pasta becomes a self-organized glassy amorphous solid similar to soft-matter systems found on Earth.

Cooking pasta with Lie groups

We extend the (gauged) Skyrme model to the case in which the global isospin group (which usually is taken to be SU(N)) is a generic compact connected Lie group G. We analyze the corresponding field equations in (3+1) dimensions from a group theory point of view. Several solutions can be constructed analytically and are determined by the embeddings of three dimensional simple Lie groups into G, in a generic irreducible representation. These solutions represent the so-called nuclear pasta state configurations of nuclear matter at low energy. We employ the Dynkin explicit classification of all three dimensional Lie subgroups of exceptional Lie group to classify all such solutions in the case G is an exceptional simple Lie group, and give all ingredients to construct them explicitly. As an example, we construct the explicit solutions for G=G2. We then extend our ansatz to include the minimal coupling of the Skyrme field to a U(1) gauge field. We extend the definition of the topological charge to this case and then concentrate our attention to the electromagnetic case. After imposing a “free force condition” on the gauge field, the complete set of coupled field equations corresponding to the gauged Skyrme model minimally coupled to an Abelian gauge field is reduced to just one linear ODE keeping alive the topological charge. We discuss the cases in which such ODE belongs to the (Whittaker-)Hill and Mathieu types.

Pairing effects in nuclear pasta phase within the relativistic Thomas–Fermi formalism

Pairing effects in non-uniform nuclear matter, surrounded by electrons, are studied in the protoneutron star early stage and in other conditions. The so-called nuclear pasta phases at subsaturation densities are solved in a Wigner–Seitz cell, within the Thomas–Fermi approximation. The solution of this problem is important for the understanding of the physics of a newly born neutron star after a supernova explosion. It is shown that the pasta phase is more stable than uniform nuclear matter on some conditions and the pairing force relevance is studied in the determination of these stable phases.

Analytic multi-Baryonic solutions in the SU(N)-Skyrme model at finite density

We construct explicit analytic solutions of the SU(N)-Skyrme model (for generic N) suitable to describe different phases of nuclear pasta at finite volume in (3 + 1) dimensions. The first type are crystals of Baryonic tubes (nuclear spaghetti) while the second type are smooth Baryonic layers (nuclear lasagna). Both, the ansatz for the spaghetti and the ansatz for the lasagna phases, reduce the complete set of Skyrme field equations to just one integrable equation for the profile within sectors of arbitrary high topological charge. We compute explicitly the total energy of both configurations in terms of the flavor number, the density and the Baryonic charge. Remarkably, our analytic results allow to compare explicitly the physical properties of nuclear spaghetti and lasagna phases. Our construction shows explicitly that, at lower densities, configurations with N = 2 light flavors are favored while, at higher densities, configurations with N = 3 are favored. Our construction also proves that in the high density regime (but still well within the range of validity of the Skyrme model) the lasagna configurations are favored while at low density the spaghetti configurations are favored. Moreover, the integrability property of the present configurations is not spoiled by the inclusion of the subleading corrections to the Skyrme model arising in the ’t Hooft expansion. Finally, we briefly discuss the large N limit of our configurations.

The effect of the energy functional on the pasta-phase properties of catalysed neutron stars

Nuclear pasta, that is an inhomogeneous distribution of nuclear matter characterised by non-spherical clustered structures, is expected to occur in a narrow spatial region at the bottom of the inner crust of neutron stars, but the width of the pasta layer is strongly model dependent. In the framework of a compressible liquid-drop model, we use Bayesian inference to analyse the constraints on the sub-saturation energy functional and surface tension imposed by both ab-initio chiral perturbation theory calculations and experimental measurements of nuclear masses. The posterior models are used to obtain general predictions for the crust-pasta and pasta-core transition with controlled uncertainties. A correlation study allows us to extract the most influential parameters for the calculation of the pasta phases. The important role of high-order empirical parameters and of the surface tension is underlined.

Fluctuations in the nuclear pasta phase

Baryonic matter close to the saturation density is very likely to present complex inhomogeneous structures collectively known under the name of pasta phase. At finite temperature, the different geometric structures are expected to coexist, with potential consequences on the neutron star crust conductivity and neutrino transport in supernova matter. In the framework of a statistical multi-component approach, we calculate the composition of matter in the pasta phase considering density, proton fraction, and geometry fluctuations. Using a realistic energy functional from relativistic mean field theory and a temperature and isospin dependent surface tension fitted from Thomas-Fermi calculations, we show that different geometries can coexist in a large fraction of the pasta phase, down to temperatures of the order of the crystallization temperature of the neutron star crust. Quantitative estimates of the charge fluctuations are given.

Nuclear pasta structures and symmetry energy

In the framework of the relativistic mean field model with Thomas-Fermi approximation, we study the structures of low density nuclear matter in a three-dimensional geometry with reflection symmetry. The numerical accuracy and efficiency are improved by expanding the mean fields according to fast cosine transformation and considering only one octant of the unit cell. The effect of finite cell size is treated carefully by searching for the optimum cell size. Typical pasta structures (droplet, rod, slab, tube, and bubble) arranged in various crystalline configurations are obtained for both fixed proton fractions and $\beta$-equilibration. It is found that the properties of droplets/bubbles are similar in body-centered cubic (BCC) and face-centered cubic (FCC) lattices, where the FCC lattice generally becomes more stable than BCC lattice as density increases. For the rod/tube phases, the honeycomb lattice is always more stable than the simple one. By introducing an $\omega$-$\rho$ cross coupling term, we further examine the pasta structures with a smaller slope of symmetry energy $L = 41.34$ MeV, which predicts larger onset densities for core-crust transition and non-spherical nuclei. Such a variation due to the reduction of $L$ is expected to have impacts on various properties in neutron stars, supernova dynamics, and binary neutron star mergers.

Thermal fluctuations in nuclear pasta

Despite their astrophysical relevance, nuclear pasta phases are relatively unstudied at high temperatures. We present molecular dynamics simulations of symmetric nuclear matter with several topologies of `lasagna' at a range of temperatures to study the pasta-uniform transition. Using the Minkowski functionals we quantify trends in the occupied volume, surface area, mean breadth, and Euler characteristic. The amplitude of surface displacements of the pasta increase with temperature which produce short lived topological defects such as holes and filaments near melting, resulting in power laws for increasing surface curvature with temperature. We calculate the static structure factor and report the shear viscosity and thermal conductivity of pasta, finding that the shear viscosity is minimized at the melting temperature. These results may have implications for the thermoelastic properties of nuclear pasta and finite temperature corrections to the equation of state at pasta densities.