We have studied phonon modes of a body-centered cubic (bcc) Coulomb crystal of ions in the presence of a uniform magnetic field B taking into account the polarizability of the electron background (electron screening) described by the Thomas-Fermi formalism. For kκTF (k and κTF are the phonon wavevector and Thomas-Fermi wavenumber, respectively), electron polarizability is not important. At kκTF, the electron response results in a pronounced effect. One of the three available modes is acoustic. For orthogonal propagation (kB), its frequency Ω is independent of B and κTF. For kB, Ω1/κTF and is independent of B. Another mode is quadratic. Its frequency is 1/(BκTF) for orthogonal propagation and 1/B and independent of κTF for the parallel case. The third mode is optic with ΩωB (ωB is the ion cyclotron frequency). A general expression is derived for the dynamic matrix of a Coulomb crystal with a polarizable background and more than one ion in the primitive cell. It is employed for a study of a magnetized hexagonal close-packed Coulomb crystal. We have also presented an analysis of phonon polarization vectors in a magnetized bcc crystal with or without screening. The results obtained can be used for realistic calculations of electron-phonon scattering rates and electron thermal and electrical conductivities in neutron star crusts.

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