Long-wavelength phonons in the crystalline and pasta phases of neutron-star crusts

We study the long-wavelength excitations of the inner crust of neutron stars, considering three phases: cubic crystals at low densities, and rods and plates near the core-crust transition. To describe the phonons, we write an effective Lagrangian density in terms of the coarse-grained phase of the neutron superfluid gap and of the average displacement field of the clusters. The kinetic energy, including the entrainment of the neutron gas by the clusters, is obtained within a superfluid hydrodynamics approach. The potential energy is determined from a model where clusters and neutron gas are considered in phase coexistence, augmented by the elasticity of the lattice due to Coulomb and surface effects. All three phases show strong anisotropy, i.e., angle dependence of the phonon velocities. Consequences for the specific heat at low temperature are discussed.

Figure 1

Illustration of different types of deformations leading to a change of Coulomb and surface energies in the lasagna phase. The small arrows indicate the displacement field $\mathbf{\xi }$. (a) Spatially varying shear in the $xz$ plane [cf. Eq. (25)], (b) compression by displacing protons inside the lasagne plates $[B_{11}$ term in Eq. (20)], and (c) compression by changing the distance between the plates $[B_{33}$ term in Eq. (20)].

Figure 2

Angle dependence of the velocities of the three phonons in the lasagna phase (parameters see Table 1).

Figure 3

Temperature dependence of the contributions to the specific heat of the lasagna phase corresponding to the three phonons shown in Fig. 2. For comparison, the electron contribution $c_{v,e}={\mu }_e^{2}T/3$ is also shown.

Figure 4

Angle dependence of the velocities of the four phonons in the spaghetti phase. The parameters are given in Table 1.

Figure 5

Temperature dependence of the contributions to the specific heat of the spaghetti phase corresponding to the four phonons shown in Fig. 4.

Figure 6

Dependence of the velocities of the four phonons in the bcc crystal on the azimuthal angle $\varphi$ for five different values of the polar angle: $\theta =0, {30}^{\circ }, {45}^{\circ }, 54.7^{\circ }$, and ${90}^{\circ }$, corresponding to $cos\theta =1, \sqrt{3}/2, 1/\sqrt{2}, 1/\sqrt{3}$, and 0, respectively. The speeds of sound are indicated by the radial distance of the curves from the origin. Note that the scales are different for the transverse (upper scale) and longitudinal (lower scale) modes. The parameters are given in Table 1.

Figure 7

Density dependence of the angle-averaged velocities of the four phonons in the crystalline phase: (a) longitudinal waves and (b) transverse waves. The solid lines ($\delta =1$) correspond to the superfluid hydrodynamic model for the entrainment, Eq. (2), while the dashed lines ($\delta =0.5$, 0) take into account a possible reduction of the superfluid density inside the clusters [cf. Eq. (50)]. The density dependence of the parameters ($R,R_{\mathrm{WS}},n_{n,i},n_{p,2}$) of the crust were taken from the ETF calculation of Ref. [8]. The points are the results of Ref. [26].