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

Abstract

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 modelled as Coulomb solids (dusty plasma, trapped ions, etc.). I demonstrate that the Coulomb part of the stress–strain tensor has additional symmetry: contraction Bijil = 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: $\sum _Z 0.12\, n_Z Z^{5/3}e^2 /a_\mathrm{e}$. Here summation is taken over ion species, nZ is number density of ions with charge Ze. Finally, ae = (4πne/3)−1/3 is electron sphere radius. Quasi-neutrality condition ne = ∑ZZnZ is assumed.