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

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.