Mean field and pairing properties in the crust of neutron stars

Properties of the matter in the inner crust of a neutron star are investigated in a Hartree-Fock plus BCS approximation employing schematic effective forces of the type of the Skyrme forces. Special attention is paid to differences between a homogenous and inhomogeneous description of the matter distribution. For that purpose self-consistent Hartree-Fock calculations are performed in a spherical Wigner-Seitz cell. The results are compared to predictions of corresponding Thomas-Fermi calculations. The influence of the shell structure on the formation of pairing correlations in inhomogeneous matter are discussed.

Figure 1

(Color online) The local density for homogeneous matter $(\rho =0.06{\mathrm{fm}}^{-3})$ calculated according to Eq. (5) using Basis 1 (2), Basis 2 (6), and the mixed basis as described in the text.

Figure 2

(Color online) Energy per nucleon for homogenous matter in $\beta$ equilibrium. The results for infinite matter are compared to those obtained in spherical boxes of different radii. The Skyrme I has been used for the NN interaction.

Figure 3

(Color online) Proton abundance $Y_p$ for homogenous matter in $\beta$ equilibrium. For further comments, see Fig. 2.

Figure 4

(Color online) Density distributions derived from HF calculations for nuclear matter in $\beta$ equilibrium.

Figure 5

(Color online) Energy per nucleon of $\beta$-stable nuclear matter in the homogenous phase (symbols connected by dashed line) and inhomogeneous phase (symbols connected by solid line).

Figure 6

(Color online) Proton abundances of $\beta$-stable nuclear matter in the homogenous case of homogenous infinite matter (solid line), inhomogeneous matter determined in the Thomas-Fermi approximation, and in HF calculations employing Wigner-Seitz cells of different radii.

Figure 7

(Color online) Local gap function $\Delta \left(r\right)$ as defined in Eq. (15) for $\beta$-stable matter of density $\rho =0.02{\mathrm{fm}}^{-3}$. The results are displayed for the plane wave plus BCS approach (dashed line) and the HF plus BCS approach (solid line). While the upper panel exhibits the results obtained in Basis 1 [see Eq. (2)], the lower part of the figure displays the results obtained for the mixed basis.

Figure 8

(Color online) The results for the state-dependent pairing gaps ${\Delta }_{nlj}$ defined in Eq. (16) for $\beta$-stable matter of density $\rho =0.02{\mathrm{fm}}^{-3}$ as derived from plane wave plus BCS calculation. These gaps are displayed as a function of the corresponding ${\epsilon }_{nlj}$. Each gap is presented by a symbol, and gaps belonging to states with identical orbital angular momentum $l$ and angular momentum $j$ are connected by a line.

Figure 9

(Color online) Results for the state-dependent pairing gaps ${\Delta }_{nlj}$ as derived from a HF plus BCS calculation. For further details, see Fig. 8.