Self-consistent band calculation of the slab phase in the neutron-star crust

Fully self-consistent band calculation has been performed for slab phase in neutron-star inner crust using the Barcelona-Catania-Paris-Madrid energy density functional. Optimized slab structure is calculated at given baryon density either with fixed proton ratio or with the beta-equilibrium condition. Numerical results indicate neutron band gaps in order of keV to tens of keV, and the mobility of dripped neutrons are enhanced by the Bragg scattering, which leads to the macroscopic effective mass, ${\overline{m}}_z^{*}/m_n=0.65-0.75$ near the bottom of the inner crust in neutron stars. However, there is an ambiguity for definition of free neutrons, especially very near the critical density between the slab and the uniform phases, which may effectively lead to somewhat larger values of effective mass. We also compare the results of the band calculation with those of the Thomas-Fermi (TF) approximation. The TF approximation becomes invalid at low density and at high proton ratio.

Figure 1

(a) Schematic picture of the slab phase. The slabs are parallel to the $x\text{−}y$ plane. (b) Schematic illustration of potential for neutrons $U_n\left(z\right)$ in the $z\text{−}x$ plane.

Figure 2

Energy per nucleon $E_B/A$ as a function of lattice constant $a$, for the case of proton fraction $Y_p=0.05$ and baryon density $n_B=0.04{\mathrm{fm}}^{-1}$. The blue solid line shows the one obtained with the band calculation adopting 30 points of the Bloch wave number, while the blue dashed line indicates that of periodic boundary condition. Those obtained with different $N_k$ are shown by solid lines with different colors.

Figure 3

Calculated nuclear energy per nucleon $E_B/A$ at beta equilibrium as a function of baryon density $n_B$. The results of TF approximation are shown by blue diamonds (${\lambda }_W=1/36$) and green triangles (${\lambda }_W=1/144$), while those of the band calculation are by red circles. The solid line represents $E_B/A$ for the uniform matter.

Figure 4

Calculated proton ratio $Y_p$ for slab phase as a function of baryon density $n_B$. See the caption of Fig. 3.

Figure 5

Calculated neutron and proton density distributions in the slab phase as functions of $z$ at beta equilibrium. The results of TF approximation (${\lambda }_W=1/36$) are shown by dashed lines. Panel (a) shows those at $n_B=0.08{\mathrm{fm}}^{-3}$, while panel (b) shows those at $n_B=0.02{\mathrm{fm}}^{-3}$. The left end ($z=0$) corresponds to the center of a slab. The obtained lattice constants are $a=20.0\mathrm{fm}$ for both the band and TF calculations for panel (a), while they are $a=108.8\mathrm{fm}$ (band) and $101.4\mathrm{fm}$ (TF) for (b).

Figure 6

Calculated nuclear energy per nucleon $E_B/A$ as functions of baryon density $n_B$, for the proton ratio $Y_p=0.5$, $Y_p=0.25$, and $Y_p=0.1$. The results of the TF calculation are shown by blue diamonds (${\lambda }_W=1/36$) and by green triangles (${\lambda }_W=1/144$), while those of the band calculation are by red circles. The solid lines show those of the uniform matter.

Figure 7

The same as Fig. 5, but at $n_B=0.1{\mathrm{fm}}^{-3}$ and $Y_p=0.25$.

Figure 8

(a) Calculated chemical potential for protons, (b) single-particle energies for protons, and (c) nuclear energy per nucleon, as functions of lattice constant $a$, for the case of $Y_p=0.1$ and $n_B=0.08{\mathrm{fm}}^{-3}$. In panel (b), each bundle of lines, which correspond to a single band index $\alpha$, contains 16 lines with different Bloch wave numbers $k_z$.

Figure 9

Neutron band structure at beta equilibrium as a function of Bloch wave number $k_z>0$; (a) $n_B=0.04{\mathrm{fm}}^{-3}$, (b) $n_B=0.05{\mathrm{fm}}^{-3}$, (c) $n_B=0.06{\mathrm{fm}}^{-3}$, and (d) $n_B=0.07{\mathrm{fm}}^{-3}$. The lattice constants and neutron chemical potentials are given as (a) $a=56.4\mathrm{fm}$, ${\mu }_n=8.45$ MeV, (b) 42.2 fm, 9.55 MeV, (c) 30.8 fm, 10.71 MeV, (d) 26.4 fm, and 11.93 MeV, respectively. In each panel, the right end corresponds to the end point of the Brillouin zone $k_z=\pi /a$.

Figure 10

Neutron effective mass ${(1/m^{*})}_{\alpha ,k_z}^{zz}$ as a function of $k_z$, calculated at beta equilibrium with $n_B=0.04{\mathrm{fm}}^{-3}$. The circled numbers next to the lines represent the band index $\alpha =1–10$. The band with $\alpha >10$ indicates similar behavior to that of $\alpha =10$, except that ${(1/m^{*})}_{\alpha ,k_z}^{zz}$ with odd $\alpha$ diverge at $k_z=0$ and $\pi /a$ with the sign opposite to that of $\alpha =10$. The colors correspond to those in Fig. 9.

Figure 11

The neutron density profile ${\rho }_n\left(z\right)$ and the difference in the free neutron densities, ${\tilde{\rho }}_n^f\left(z\right)-{\overline{\rho }}_n^f\left(z\right)$, as a function of $z$. Panels (a) and (b) correspond to $n_B=0.04{\mathrm{fm}}^{-3}$ and $0.06{\mathrm{fm}}^{-3}$, respectively.

Figure 12

Neutron mobility coefficients ${\mathcal{K}}^{zz}$ calculated at beta equilibrium as a function of baryon density. The circles show those obtained with the self-consistent band calculation, while those for the uniform matter are shown by the solid line. See text for explanation of dashed and dash-dotted lines.

Figure 13

Neutron effective masses $m_z^{*}$, ${\overline{m}}_z^{*}$, and ${\tilde{m}}_z^{*}$ calculated at beta equilibrium as functions of density.

Figure 14

Nuclear energy per nucleon $E_B/A$ as functions of density $n_B$ for different values of $Y_p$. The dashed lines indicate $E_B/A$ of the slab nuclei, while the solid lines correspond to the uniform matter EOS.

Figure 15

Lattice constant $a$ as a function of density $n_B$. Different lines correspond to different values of proton ratio $Y_p$.