Neutron star inner crust: reduction of shear modulus by nuclei finite size effect

Abstract

The elasticity of neutron star crust is important for adequate interpretation of observations. To describe elastic properties one should rely on theoretical models. The most widely used is Coulomb crystal model (system of point-like charges on neutralizing uniform background), in some works it is corrected for electron screening. These models neglect finite size of nuclei. This approximation is well justified except for the innermost crustal layers, where nuclei size becomes comparable with the inter-nuclear spacing. Still, even in those dense layers it seems reasonable to apply the Coulomb crystal result, if one assumes that nuclei are spherically symmetric: Coulomb interaction between them should be the same as interaction between point-like charges. This argument is indeed correct, however, as we point here, shear of crustal lattice generates (microscopic) quadrupole electrostatic potential in a vicinity of lattice cites, which induces deformation on the nuclei. We analyze this problem analytically within compressible liquid drop model. In particular, for ground state crust composition the effective shear modulus is reduced for a factor of 1 − u5/3/(2 + 3 u − 4 u1/3), where u is the ratio of the nuclei volume to the volume of the cell. This result is universal, i.e., it does not depend on the applied nucleon interaction model within applied approach. For the innermost layers of inner crust u ∼ 0.2 leading to reduction of the shear modulus by $\sim 25\%$, which can be important for correct interpretation of quasi-periodic oscillations in the tails of magnetar flares.

The Elasticity of the Neutron Star Mantle: The Improved Compressible Liquid Drop Model for Cylindrical Phases

“…For a typical filling factor of the cylindrical phases, u ≈ 0.2 ÷ 0.3 (e.g., [2,13]), the cluster deformation evolves from ε p ≈ 0.3ε to the value, which is close to ε (ε p ≈ 0.85ε). For comparison, the dotted line indicates a similar ratio for spherical clusters, which was obtained by the authors in [47]. One can see that this ratio depends on the shape of the clusters.…”

“…To calculate the elastic constant C, we proceed in the same way as in [47]. First, we note that a straightforward consideration of deformation for a system of cylindrical clusters, ordered into a perfect static two-dimensional hexagonal lattice (as was carried out in [38]) is equivalent to the deformation of one unit cell with periodic boundary conditions.…”

“…Now, it is easy to write down equations for internal variables by minimizing the energy density at fixed n b and ε. As in the case of spherical nuclei, considered in [47], the majority of internal variables (n ni , n pi , n no , r c , r p ) can be derived at ε = 0 up to corrections of order of ε 2 , 3 and only ε p ∝ ε:…”

“…In this paper, we elaborate on CLDM for the elasticity of the mantle. Specifically, we use our recent work [47], where we have demonstrated that shear deformation of a neutron star crust induces quadrupole deformation for the initially spherical nuclei, and this effect reduces the effective shear modulus. Here, we generalize the approach of [47] for cylindrical pasta phases and consider transversal shear, i.e., shear deformations in the plane perpendicular to the symmetry axis of the cylinders.…”

“…Specifically, we use our recent work [47], where we have demonstrated that shear deformation of a neutron star crust induces quadrupole deformation for the initially spherical nuclei, and this effect reduces the effective shear modulus. Here, we generalize the approach of [47] for cylindrical pasta phases and consider transversal shear, i.e., shear deformations in the plane perpendicular to the symmetry axis of the cylinders. Regarding spherical nuclei, we demonstrate that the respective elastic constant (coefficient C in notations of [38]) can be analytically calculated within a simplified Wigner-Seitz approximation, and the result agrees well with accurate lattice-based calculations [38], if the same assumption (enforced circular shape of the cluster cross-section) is applied.…”