Symmetry energy at subnuclear densities and nuclei in neutron star crusts

We examine how the properties of inhomogeneous nuclear matter at subnuclear densities depend on the density dependence of the symmetry energy. Using a macroscopic nuclear model we calculate the size and shape of nuclei in neutron star matter at zero temperature in a way dependent on the density dependence of the symmetry energy. We find that for smaller symmetry energy at subnuclear densities, corresponding to the larger density symmetry coefficient $L$, the charge number of nuclei is smaller and the critical density at which matter with nuclei or bubbles becomes uniform is lower. The decrease in the charge number is associated with the dependence of the surface tension on the nuclear density and the density of a sea of neutrons, whereas the decrease in the critical density can be generally understood in terms of proton clustering instability in uniform matter.

Figure 1

The sets of $(L,K_0)$ (crosses) consistent with the mass and radius data for stable nuclei. The thin lines are lines of constant $y$. The labels A–I denote the sets for which we perform detailed calculations of the ground-state properties of inhomogeneous nuclear matter at subnuclear densities. For comparison, the values calculated from two mean-field models [TM1 (square) and SIII (dot)], which are known to be extreme cases [28], are plotted. The plot shows that our sets of $(L,K_0)$ effectively cover such extreme cases.

Figure 2

Energy per nucleon of nuclear matter for the nine sets of $(L,K_0)$ referred to as A–I in Fig. 1. In each panel, the solid lines are the energy at $\alpha =0,0.3,0.5,0.8,1$, and the dotted line is the saturation line.

Figure 3

The charge number of spherical nuclei as a function of $n_b$, calculated for the EOS models A–I.

Figure 4

The average proton fraction as a function of $n_b$, calculated for the EOS models A–I.

Figure 5

The density region containing bubbles and nonspherical nuclei as a function of $L$, calculated for the EOS models A–I. For comparison, the density corresponding to $u=1/8$ in the phase with spherical nuclei and the onset density, $n(Q)$, of proton clustering in uniform nuclear matter, which will be discussed in Sec. 4, are also plotted by circles and crosses, respectively.

Figure 6

The equilibrium properties of matter with spherical nuclei as a function of $L$, calculated at fixed volume fraction $u=1/8$. The baryon density $n_b$ (a), the nucleon densities in the center and boundary of the cell (b), the nuclear charge number $Z$ (c), the central and average proton fractions (d), the lattice constant $a$ (e), and the effective surface tension σ (f) are plotted.

Figure 7

The proton effective potential $v_{\min }$, its bulk part $v_0$, neutron excess α, and symmetry energy coefficient $S(n)$ as a function of nucleon density, calculated for the EOS models A–I.

Figure 8

The onset density of proton clustering in uniform nuclear matter as a function of $L$. For comparison, we plot the density corresponding to $u=1/8$ in the phase with spherical nuclei, which is taken from Fig. 6a.

Figure 9

The proton chemical potential and pressure in pure neutron matter of density 0.1 fm${}^{-3}$ as a function of $L$.