Nucleus-nucleus interactions in the inner crust of neutron stars

The interaction between nuclei in the inner crust of neutron stars consists of two contributions: the so-called “direct” interaction and an “induced” one due to density changes in the neutron fluid. For large nuclear separations $r$ the contributions from nuclear forces to each of these terms are shown to be nonzero. In the static limit they are equal in magnitude but have opposite signs and they cancel exactly. We analyze earlier results on effective interactions in light of this finding. We consider the properties of long-wavelength collective modes and, in particular, calculate the degree of mixing between the lattice phonons and the phonons in the neutron superfluid. Using microscopic theory, we calculate the net non-Coulombic contribution to the nucleus-nucleus interaction and show that, for large $r$, the leading term is due to exchange of two phonons and varies as $1/r^7$; it is an analog of the Casimir-Polder interaction between neutral atoms.

Figure 1

Mode velocities in the inner crust. The solid lines represent the results of the full calculations based on Eq. (27) and Table 1. The lines correspond to the results for a neutron superfluid density equal to the density of neutrons between nuclei with hybridization (full lines) and in the absence of hybridization $(v_{np}=0$, dashed lines). The dash-dotted line shows the velocity of the transverse elastic wave, which is the same at that in Ref. [3]. The crosses show results for Chamel's values for the superfluid density [19].

Figure 2

Diagrams representing the simplest contributions to the energy of a single impurity (a). and to the interaction energy of two impurities in a normal Fermi gas (b).

Figure 3

Diagrams representing the contribution to the interaction energy between two static impurities in a quantum liquid due to one-phonon exchange (a) and two-phonon exchange (b). A cross denotes an impurity and the wavy line a phonon propagator in the quantum liquid.

Figure 4

Diagrams representing contributions to the vertex for coupling of a nucleus (solid lines) to two phonons (wavy lines).