Electrostatic energy of Coulomb crystals with polarized electron background

Outer crusts of neutron stars and interiors of cool white dwarfs consist of bare atomic nuclei, arranged in a crystal lattice and immersed in a Fermi gas of degenerate electrons. We study electrostatic properties of such Coulomb crystals, taking into account the polarizability of the electron gas and considering different structures, which can form the ground state: body-centered-cubic (bcc), face-centered-cubic (fcc), hexagonal close-packed (hcp), and MgB_{2}-like lattices. At zero temperature the electrostatic energy provides a fundamental contribution to the total energy of the classical Coulomb crystal, which allows us to study structural transitions in the neutron-star crusts and crystallized white-dwarf interiors. To take the electron background polarization into account, we use the linear response theory with the electron dielectric function given either by the Thomas-Fermi approximation or by the random-phase approximation (RPA). We compare the widely used nonrelativistic (Lindhard) version of the RPA with the more general, relativistic (Jancovici) version. The results of the different approximations are compared to assess the importance of going beyond the Thomas-Fermi or Lindhard approximations. We also include the contribution of zero-point vibrations of ions into the ground-state energy. We show that the bcc lattice forms the ground state for any charge number Z of the atomic nuclei at the densities where the electrons are relativistic (ρ≳10^{6}g cm^{-3}), while at the nonrelativistic densities (ρ≲10^{6}g cm^{-3}) the fcc and hcp lattices can form the ground state. The MgB_{2}-like lattice never forms the ground state at realistic densities in the crystallized regions of degenerate stars. The RPA corrections strongly affect the boundaries between the phases. As a result, transitions between different ground-state structures depend on Z in a nontrivial way. The relativistic and quantum corrections produce less dramatic effects, moderately shifting the phase boundaries.