Coexistence of Anderson-Bogoliubov phonon and quadrupole cluster vibration in the inner crust of neutron stars

Background: Low-lying collective excitations of the inner crust matter in neutron stars are expected to affect observables such as the quasiperiodic oscillation in giant flares or cooling of the inner crust in transient phenomena. The coupling between the Anderson-Bogoliubov (AB) superfluid phonon in superfluid neutron gas and collective excitations of nuclear clusters is crucially important.

Purpose: Our aim is to describe the characteristics of the low-lying excitation modes of the superfluid inner crust matter, focusing on quadrupole excitations around a spherical nuclear cluster. We studied the effect of the inhomogeneous structure of the inner crust matter on the AB phonon of neutron superfluid and on the possible quadrupole shape vibration of clusters.

Methods: The coordinate-space Hartree-Fock-Bogoliubov method and the quasiparticle random phase approximation formulated in a spherical Wigner-Seitz cell were used to describe the neutron superfluidity and low-lying collective excitations. We performed systematic numerical calculations for the quadrupole excitations by varying the neutron chemical potential and the number of protons in the cell.

Results: The calculated results indicate the appearance of both AB phonon and quadrupole shape vibration of the cluster with small mixing between the two collective modes. The quadruple AB phonon is similar to that in the uniform superfluid, apart from the small admixture of the shape vibration of cluster. The excitation energy and the collectivity of the cluster vibration mode show a strong and oscillatory dependence on the neutron chemical potential (the neutron gas density), resulting in softening and instability under certain conditions. This is due to the resonance shell effect, where unbound, but resonant single-particle states of neutrons are crucially important.

Conclusions: The AB phonon of the superfluid neutron gas and the quadrupole shape vibration of the nuclear cluster coexist in the inner crust. The coupling between the AB phonon and the quadrupole shape vibration is weak. The collectivity of the quadrupole shape vibration is controlled by the resonance shell effect, which suggests that the appearance of deformed nuclear clusters is possible in any layer of the inner crust.

Figure 1

(a) Strength function $S(Q_n;E)$ for the neutron quadrupole operator, (b) $S(Q_p;E)$ for the proton quadrupole operator, (c) $S(P_{\mathrm{add}};E)$ for the neutron pair-addition operator, and (d) $S(P_{\mathrm{rm}};E)$ for the neutron pair-removal operator, calculated for a $Z=28$ system at ${\lambda }_n=4.0$ MeV, plotted as a function of the excitation energy $E$. The arrows indicate the threshold energy $2{\mathrm{\Delta }}_{\mathrm{ext}}$ and the smearing parameter is $\epsilon =10$ keV. The dashed curve in (b) represents the proton quadrupole strength function $S(Q_p;E)$ obtained with $\epsilon =500$ keV (with the scale on the right axis).

Figure 2

Transition densities of the low-lying quadrupole collective modes with excitation energies of (a) $E=1.57$ MeV, (b) 2.06 MeV, and (c) 3.03 MeV, obtained for the $Z=28$ system at ${\lambda }_n=4.0$ MeV. The red, gray, green, and blue curves represent $\delta {\rho }_{\mathrm{ph}}^{\nu }\left(r\right)$, $\delta {\rho }_{\mathrm{ph}}^{\pi }\left(r\right)$, $\delta {\tilde{\rho }}_{\mathrm{hh}}^{\nu }\left(r\right)$, and $\delta {\tilde{\rho }}_{\mathrm{pp}}^{\nu }\left(r\right)$, respectively. The arrows indicate the surface position of the cluster [26]. See text for details.

Figure 3

Effects of the residual interactions for the system $Z=28$ at ${\lambda }_n=4.0$ MeV: (a) strength function $S(Q_n;E)$, (b) $S(Q_p;E)$, and (c) $S(P_{\mathrm{add}};E)$ calculated by assuming full residual interaction (red curve), residual pairing interaction only (green dashed line), and residual particle-hole interpretation only (blue long-dashed line). The unperturbed results (magenta long-dashed dotted line) are also plotted. Transition densities of the low-lying collective mode obtained in calculations assuming (d) residual pairing interaction only and (e) the residual particle-hole interaction only. The red, gray, green, and blue curves represent $\delta {\rho }_{\mathrm{ph}\left(\mathrm{r}\right)}^{\nu }$, $\delta {\rho }_{\mathrm{ph}}^{\pi }\left(r\right)$, $\delta {\tilde{\rho }}_{\mathrm{hh}}^{\nu }\left(r\right)$, and $\delta {\tilde{\rho }}_{\mathrm{pp}}^{\nu }\left(r\right)$, respectively. Please refer to captions of Figs. 1 and 2 as well.

Figure 4

(a),(b) Transition densities of the AB phonon dominant mode and the cluster vibration dominant mode obtained in the weak pairing case with $V_{\mathrm{pair}}=-422$ MeV ${\mathrm{fm}}^{-3}$ for the system with $Z=28$ and ${\lambda }_n=4$ MeV. (c),(d) The same but in the strong pairing case with $V_{\mathrm{pair}}=-505$ MeV ${\mathrm{fm}}^{-3}$.

Figure 5

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

Figure 6

Transition densities $\delta {\rho }_{\mathrm{ph}}^{\nu }\left(r\right)$, $\delta {\rho }_{\mathrm{ph}}^{\pi }\left(r\right)$, $\delta {\tilde{\rho }}_{\mathrm{hh}}^{\nu }\left(r\right)$, and $\delta {\tilde{\rho }}_{\mathrm{pp}}^{\nu }\left(r\right)$ of the AB phonon dominant mode for $Z=28$ systems at ${\lambda }_n=$ (a) 1.0 MeV, (b) 2.0 MeV, (c) 3.0 MeV, (d) 5.0 MeV, and (e) 6.0 MeV.

Figure 7

Excitation energy $E_{\mathrm{AB}}$ of the lowest AB phonon dominant mode (open circles) and the energy $E_{\mathrm{cl}}$ of the cluster vibration dominant mode (filled diamonds) for $Z=28$ systems, plotted as a function of the neutron chemical potential ${\lambda }_n$. The blue dashed curve represents the threshold energy $2{\mathrm{\Delta }}_{\mathrm{ext}}$ and the red dots connected by line indicate the proton quadrupole strengths $B\left(Q_p\right)$ of the cluster vibration dominant mode. The results for near-drip-line nuclei of ${}^{80,82,...,88}\mathrm{Ni}$ are also plotted. The thin black line represents the hydrodynamic estimate, $\hslash {\omega }_{\mathrm{hyd}}$, of the quadrupole AB phonon mode in uniform superfluid neutron gas in the same Wigner-Seitz cell.

Figure 8

Excitation energy $E_{\mathrm{AB}}$ of the lowest AB phonon dominant mode (open circles) and the energy $E_{\mathrm{cl}}$ of the cluster vibration dominant mode (filled diamonds) for $Z=20$, 40, and 50 systems, plotted as functions of the neutron chemical potential ${\lambda }_n$. The blue dashed curve represents the threshold energy $2{\mathrm{\Delta }}_{\mathrm{ext}}$ and the red dots connected by line indicate the proton quadrupole strengths $B\left(Q_p\right)$ of the cluster vibration dominant mode. The thin black line represents the hydrodynamic estimate, $\hslash {\omega }_{\mathrm{hyd}}$, of the quadrupole AB phonon mode in uniform superfluid neutron gas in the same Wigner-Seitz cell.

Figure 9

(a) Neutron single-particle energies for $Z=$ 28 systems at ${\lambda }_n=1.0,2.0,...,6.0$ MeV. The thick black bars connected by dashed lines represent the resonant single-particle states with quantum numbers $1h_{11/2}$, $1h_{9/2}$, $1i_{13/2}$, and $1i_{11/2}$. The bound high-$j$ orbits are also connected by thin dashed lines for reference. The red horizontal bars connected by a dashed line indicate the neutron Fermi energy (chemical potential) ${\lambda }_n$ and the blue bars indicate the depth of the neutron HF potential, $U_{\mathrm{ext}}$, in the region of the neutron gas. (b) Proton single-particle energies for $Z=$ 28 systems at ${\lambda }_n=1.0,2.0,...,6.0$ MeV, with the same colors as those in pane (a). Numbers 28, 50, and 82 indicated in the levels are spherical magic numbers.