Anderson-Bogoliubov phonons in the inner crust of neutron stars: Dipole excitation in a spherical Wigner-Seitz cell

Background: The Anderson-Bogoliubov (AB) phonon, called also the superfluid phonon, has attracted attentions since it may influence the thermal conductivity and other properties of the inner crust of neutron stars. However, there are a limited number of microscopic studies of the AB phonon where the presence of clusters is explicitly taken into account.

Purpose: We intend to clarify how the presence of clusters affects the AB phonon in order to obtain microscopic information relevant to the coupling between the AB phonon and the lattice phonon.

Methods: The Hartree-Fock-Bogoliubov model and the quasiparticle random-phase approximation formulated in a spherical Wigner-Seitz cell are adopted to describe neutron superfluidity and associated collective excitations. We perform systematic numerical calculations for dipole excitation by varying the neutron chemical potential and the number of protons in a cell.

Results: The model predicts systematic emergence of the dipole AB phonon mode, which, however, exhibits strong suppression of phonon amplitude inside the cluster. We find also that the phonon amplitude around the cluster surface varies with the neutron density. At higher neutron densities ($\gtrsim 0.006$ ${\mathrm{fm}}^{-3}$) the AB phonon mode exhibits behavior similar to the pygmy dipole resonance in neutron-rich nuclei.

Conclusions: The dipole AB phonon mode does not penetrate into the clusters. This suggests that the coupling between the AB phonon and the lattice phonon may be weak.

Figure 1

(a) Calculated neutron densities (thick lines) and proton densities (thin lines) for $Z=28$ system with ${\lambda }_n=$ 1.0–6.0 MeV (long-dashed red, dashed brown, short-dashed yellowish green, dotted blue green, dashed dotted blue, and long-dashed dotted purple in order). (b) The same but for $Z=$ 20 (solid red), 28 (short-dashed light green), 40 (dashed cyan), and 50 (long-dashed blue) systems with fixed ${\lambda }_n=$ 5.0 MeV. (c),(d) Calculated neutron pair densities and (e),(f) neutron pair potentials for the same systems as (a),(b), respectively. For comparison, densities and pair potentials of drip-line nucleus ${}^{88}\mathrm{Ni}$ are also plotted with solid gray lines in (a), (c), and (e).

Figure 2

Relation between neutron pair gap ${\mathrm{\Delta }}_{\mathrm{ext}}$ and neutron density ${\rho }_{\mathrm{ext}}$ in the external neutron superfluid region for $Z=$ 20 ($▽$), 28 ($△$), 40 ($\square$), and 50 ($\diamond$) systems with ${\lambda }_n$ varied up to 10 MeV. The calculated pair gap of pure neutron matter is also plotted with circles for comparison.

Figure 3

(a) Strength function $S(D_n;E)$ for neutron dipole operator, (b) $S(D_p;E)$ for proton dipole operator, (c) $S(P_{\mathrm{add}};E)$ for neutron pair-addition operator, and (d) $S(P_{\mathrm{rm}};E)$ for neutron pair-removal operator, calculated for the $Z=28$ system with ${\lambda }_n=$ 2.0 MeV, plotted as a function of the excitation energy $E$. Arrows indicate the threshold energy $2{\mathrm{\Delta }}_{\mathrm{ext}}$. The smearing parameter is $\epsilon =$ 10 keV.

Figure 4

(a) Unperturbed strength function $S(D_n;E)$ for neutron dipole operator and (b) $S(P_{\mathrm{add}};E)$ for neutron pair-addition operator, calculated for the $Z=28$ system with ${\lambda }_n=$ 2.0 MeV. The scale of the vertical axis of $S(P_{\mathrm{add}};E)$ is ten times larger than that of Fig. 3.

Figure 5

(a) Transition densities of the AB phonon mode at $E=1.71$ MeV obtained for the $Z=$ 28 system with ${\lambda }_n=$ 2.0 MeV. Solid red, short-dashed gray, dashed green, and long-dashed blue lines correspond to $\delta {\rho }_{\mathrm{ph}}^{\nu }$, $\delta {\rho }_{\mathrm{ph}}^{\pi }$, $\delta {\tilde{\rho }}_{\mathrm{pp}}^{\nu }$, and $\delta {\tilde{\rho }}_{\mathrm{hh}}^{\nu }$, respectively. (b) The same but for the displacement motion of the cluster at $E=0$. The scale is in arbitrary units. (c) The same but for the GDR-like mode, calculated at $E=12.2$ MeV and employing a large smearing width $\epsilon =500$ keV.

Figure 6

Upper panels: (a) Strength function $S(D_n;E)$ and (b) $S(P_{\mathrm{add}};E)$ calculated with full QRPA residual interaction (red), only residual pairing interaction (green dashed), and only residual particle-hole interaction (blue long-dashed). The system is for $Z=28$ and ${\lambda }_n=2.0$ MeV. Lower panels: Transition densities of the largest peak in calculations with (c) only residual pairing interaction, (d) only residual particle-hole interaction, and (e) modified pairing interaction. Solid red, short-dashed gray, dashed green, and long-dashed blue lines correspond to $\delta {\rho }_{\mathrm{ph}}^{\nu },\delta {\rho }_{\mathrm{ph}}^{\pi }$, $\delta {\tilde{\rho }}_{\mathrm{pp}}^{\nu }$, and $\delta {\tilde{\rho }}_{\mathrm{hh}}^{\nu }$, respectively. See text for details.

Figure 7

Strength functions (a) $S(D_n;E)$, (b) $S(P_{\mathrm{add}};E)$, and (c) $S(P_{\mathrm{rm}};E)$ for the $Z=28$ system with ${\lambda }_n=$ 1.0 MeV (red), 2.0 MeV (brown), 3.0 MeV (yellow green), 4.0 MeV (blue green), 5.0 MeV (blue), and 6.0 MeV (purple). Colored arrows indicate the threshold energy $2{\mathrm{\Delta }}_{\mathrm{ext}}$ for corresponding ${\lambda }_n$. See text for details.

Figure 8

Transition densities $\delta {\rho }_{\mathrm{ph}}^{\nu },\delta {\rho }_{\mathrm{ph}}^{\pi }$, $\delta {\tilde{\rho }}_{\mathrm{pp}}^{\nu }$, and $\delta {\tilde{\rho }}_{\mathrm{hh}}^{\nu }$ of the dipole AB phonon mode for the $Z=28$ system with ${\lambda }_n=$ (a) 1.0 MeV, (b) 3.0 MeV, (c) 4.0 MeV, (d) 5.0 MeV, and (e) 6.0 MeV. See text for details.

Figure 9

Excitation energy of the dipole AB phonon mode for $Z=$ 20 ($▽$), 28 ($△$), 40 ($\square$), and 50 ($\diamond$) plotted as a function of neutron chemical potential ${\lambda }_n$. Blue symbols are the threshold energy $2{\mathrm{\Delta }}_{\mathrm{ext}}$. The hydrodynamic estimate of the excitation energy is also plotted with a solid line. See text for details.

Figure 10

Upper panels: Strength functions (a) $S(D_n;E)$ and (b) $S(P_{\mathrm{add}};E)$ for $Z=$ 20 (solid red), 28 (short-dashed light green), 40 (dashed cyan), and 50 (long-dashed blue) systems with fixed ${\lambda }_n=$ 4.0 MeV. Colored arrows indicate the threshold energy $2{\mathrm{\Delta }}_{\mathrm{ext}}$ for corresponding $Z$. Lower panels: Transition densities $\delta {\rho }_{\mathrm{ph}}^{\nu }$, $\delta {\rho }_{\mathrm{ph}}^{\pi }$, $\delta {\tilde{\rho }}_{\mathrm{pp}}^{\nu }$, and $\delta {\tilde{\rho }}_{\mathrm{hh}}^{\nu }$ of the dipole AB phonon for (c) $Z=20$, (d) $Z=40$, and (e) $Z=50$ systems. See text for details.

Figure 11

Transition densities $\delta {\rho }_{\mathrm{ph}}^{\nu }$, $\delta {\rho }_{\mathrm{ph}}^{\pi }$, $\delta {\tilde{\rho }}_{\mathrm{pp}}^{\nu }$, and $\delta {\tilde{\rho }}_{\mathrm{hh}}^{\nu }$ of the second AB phonon state: (a) 2.16 MeV state for $Z=28$ with ${\lambda }_n=$ 2.0 MeV and (b) 3.82 MeV state for $Z=28$ with ${\lambda }_n=$ 5.0 MeV.