Nuclear pasta and symmetry energy in the relativistic point-coupling model

Nonuniform structure of low-density nuclear matter, known as nuclear pasta, is expected to appear not only in the inner crust of neutron stars but also in core-collapse supernova explosions and neutron-star mergers. We perform fully three-dimensional calculations of inhomogeneous nuclear matter and neutron-star matter in the low-density region using the Thomas-Fermi approximation. The nuclear interaction is described in the relativistic mean-field approach with the point-coupling interaction, where the meson exchange in each channel is replaced by the contact interaction between nucleons. We investigate the influence of nuclear symmetry energy and its density dependence on pasta structures by introducing a coupling term between the isoscalar-vector and isovector-vector interactions. It is found that the properties of pasta phases in the neutron-rich matter are strongly dependent on the symmetry energy and its slope. In addition to typical shapes like droplets, rods, slabs, tubes, and bubbles, some intermediate pasta structures are also observed in cold stellar matter with a relatively large proton fraction. We find that nonspherical shapes are unlikely to be formed in neutron-star crusts, because the proton fraction obtained in $\beta$ equilibrium is somewhat small. The inner crust properties may lead to a visible difference in the neutron-star radius.

Figure 1

Binding energy per nucleon of ${}^{208}\mathrm{Pb}$ vs the symmetry energy slope $L$ with different choices of ${\rho }_{\mathrm{fix}}$ based on the PC-PK1 parametrization.

Figure 2

Symmetry energy $E_{\mathrm{sym}}$ as a function of the baryon density ${\rho }_{\mathrm{B}}$ for the generated models with different $L$. The symmetry energy is fixed at a density of $0.12{\mathrm{fm}}^{-3}$.

Figure 3

Proton density distributions in pasta phases for $Y_p=0.5$ obtained using the RMF-PC model with $L=40\mathrm{MeV}$.

Figure 4

Energy per nucleon $E$ for different configurations observed around the transition from droplets to rods.

Figure 5

Proton density distributions in typical pasta phases at a fixed proton fraction of $Y_p=0.3$. The results obtained with $L=40\mathrm{MeV}$ (left panels) are compared to those with $L=113\mathrm{MeV}$ (right panels).

Figure 6

Proton density distributions in typical pasta phases at a fixed proton fraction of $Y_p=0.05$. The results obtained with $L=40\mathrm{MeV}$ (left panels) are compared to those with $L=113\mathrm{MeV}$ (right panels).

Figure 7

Density distributions of protons, neutrons, and electrons along a line passing through the center of the droplets. The results obtained with $L=40\mathrm{MeV}$ (solid lines) are compared to those with $L=113\mathrm{MeV}$ (dashed lines).

Figure 8

Energy per nucleon $E$ as a function of the baryon density ${\rho }_B$ for $Y_p=0.5$, 0.3, 0.1, and 0.05 using the models with $L=40\mathrm{MeV}$ (left panels) and $L=113\mathrm{MeV}$ (right panels). For comparison, the results of uniform matter are also displayed by dashed lines.

Figure 9

Proton fractions as a function of the baryon density in neutron-star matter with nonuniform and uniform distributions.

Figure 10

Density distributions of protons, neutrons, and electrons in $\beta$ equilibrium along a line passing through the center of the droplets. The results obtained with $L=40\mathrm{MeV}$ are compared to those of $L=113\mathrm{MeV}$.

Figure 11

Energy per nucleon $E$ as a function of the baryon density ${\rho }_B$ using the models with $L=40\mathrm{MeV}$ (a) and $L=113\mathrm{MeV}$ (b). For comparison, the results of uniform matter are also displayed by dashed lines.

Figure 12

Crust-core transition density ${\rho }_{B,t}$ (a) and proton fraction at the transition point $Y_{p,t}$ (b) as a function of the symmetry energy slope $L$ using the generated models based on the PC-PK1 parametrization.

Figure 13

Mass-radius relations of neutron stars by using different combinations of the core and crust segments. The results using the inner crust EOS with $L=40$ and 113 MeV are shown by thick and thin lines, respectively. The colored horizontal bars indicate the mass measurements of PSR J1614–2230 [100, 101, 102], PSR $\mathrm{J0348}+0432$ [103], and PSR $\mathrm{J0740}+6620$ [104], while the mass constraint from GW190814 [67] is shown in pink color. The constraints from NICER [69] and GW170817 [59] are also indicated.

Figure 14

Dimensionless tidal deformability $\mathrm{\Lambda }$ as a function of the neutron-star mass $M$. The vertical line represents the constraints on ${\mathrm{\Lambda }}_{1.4}$ from the analysis of GW170817 [59].