  We have studied phonon and thermodynamic properties of a body-centered cubic
(bcc) Coulomb crystal of ions with weakly polarized electron background in a
uniform magnetic field ${\bf B}$. At $B=0$ the difference between phonon
moments calculated using the Thomas-Fermi (TF) and random phase (RPA)
approximations is always less than one percent and for description of phonon
properties of a crystal TF formalism was used. This formalism was successfully
applied to investigate thermodynamic properties of magnetized Coulomb crystals.
It was shown that the influence of the polarization of the electron background
is significant only at $\kappa_{\rm TF}a > 0.1$ and $T \ll T_{\rm
p}(1+h^2)^{-1/2}$, where $\kappa_{\rm TF}$ is the Thomas-Fermi wavenumber, $a$
is the ion sphere radius, $T_{\rm p}\equiv\hbar \omega_{\rm p}$ is the ion
plasma temperature, $h\equiv \omega_B/\omega_{\rm p}$, $\omega_B$ is the ion
cyclotron frequency and $\omega_{\rm p}$ is the ion plasma frequency.
