Shear Modulus of a Coulomb Crystal of Ions: Effects of Ion Motion and Electron Background Polarization

Abstract

Shear modulus of a body-centered cubic Coulomb crystal of ions is calculated by thermodynamic perturbation theory taking into account ion motion. Classic and quantum regimes of ion motion are considered. The calculations in the classic range of high temperatures agree well with previous Monte Carlo simulations. In this case, the shear modulus is given by a sum of a positive contribution due to the static lattice and a negative ∝ T contribution due to the ion motion. In the quantum regime of low temperatures, the contribution to the shear modulus due to the ion motion saturates at a negative constant value, determined by zero-point ion vibrations.The correction to the static lattice shear modulus due to the polarization of the electron background is also calculated. The background polarization is described by a dielectric function in the linear response formalism. We compare two dielectric functions (simple Thomas-Fermi and more realistic Jancovici) and find dependence of the correction to the shear modulus on the ion charge number and the electron degree of relativity.The numerical results are approximated by simple analytical formulas. They can be applied to matter in white dwarf cores and neutron star crusts for precise modeling of oscillations of these astrophysical objects.

Electron screening effects on crustal torsional oscillations

“…Furthermore, one might consider the phonon contribution in the shear modulus. But, since such a contribution is much smaller than that coming from a static lattice [42], one can neglect it. Thus, we will calculate the frequencies of torsional oscillations in the crust region with Eqs.…”

Screening corrections to the Coulomb crystal elastic moduli

“…The electron polarization correction to the static Coulomb crystal effective shear modulus was calculated by Baiko (2012). In this paper, screening was described in the linear response formalism.…”

“…At the same time, neither corrections to the individual elastic coefficients nor details of the calculations were reported. Based on the numerical results of Baiko (2012), Kobyakov & Pethick (2013) produced a fit for the effective shear modulus screening correction in the Thomas-Fermi model.…”

“…The goal of the present work is to extend the work of Baiko (2012) as well as to provide necessary details of these calculations. In particular, we report on the screening contributions to all second order elastic coefficients of the static crystal with the bcc lattice.…”

Elastic properties of polycrystalline dense matter

“…In recent years the elastic properties of neutron stars have attracted particular attention because vibrations of the crust could provide a mechanism for quasiperiodic oscillations observed in X-ray burst sources (Duncan 1998;Strohmayer & Watts 2006). The elastic properties of single crystals of dense matter have been calculated in a variety of works (see, e.g., Haensel, Potekhin & Yakovlev (2007); Baiko (2011Baiko ( , 2012). However, because of initial variations in the local temperature and composition, as well as the presence of gravitational and magnetic fields, it appears unlikely that the solid part of a star is one giant single crystal, and the standard assumption in recent work is that matter is polycrystalline, with a random distribution of crystal orientations (Ogata & Ichimaru 1990;Baiko 2011Baiko , 2012.…”