Neutron star crust in Voigt approximation: general symmetry of the stress-strain tensor and an universal estimate for the effective shear modulus

Abstract

I discuss elastic properties of neutron star crust in the framework of static Coulomb solid model when atomic nuclei are treated as non-vibrating point charges; electron screening is neglected. The results are also applicable for solidified white dwarf cores and other materials, which can be modeled as Coulomb solids (dusty plasma, trapped ions, etc.). I demonstrate that the Coulomb part of the stress-strain tensor has additional symmetry: contraction $B_{ijil}=0$. It does not depend on the structure (crystalline or amorphous) and composition. I show as a result of this symmetry the effective (Voigt averaged) shear modulus of the polycrystalline or amorphous matter to be equal to $-2/15$ of the Coulomb (Madelung) energy density at undeformed state. This result is general and exact within the model applied. Since the linear mixing rule and the ion sphere model are used, I can suggest a simple universal estimate for the effective shear modulus: ${\sum }_{Z}0.12n_Z{Z}^{5/3}{e}^2/a_{\mathrm{e}}$. Here summation is taken over ion species, $n_Z$ is number density of ions with charge $Ze$. Finally $a_{\mathrm{e}}=(4\pi n_{\mathrm{e}}/3{)}^{-1/3}$ is electron sphere radius. Quasineutrality condition $n_{\mathrm{e}}={\sum }_{Z}Z{n}_Z$ is assumed.

Comments: 5 pages, accepted for publication in MNRAS:Letters