  Whereas many different models exist in the mathematical and physics
literature for ground states of non-relativistic crystals, the relativistic
case has been much less studied and we are not aware of any mathematical result
on a fully relativistic treatment of crystals. In this paper, we introduce a
mean-field relativistic energy for crystals in terms of periodic density
matrices. This model is inspired both from a recent definition of the
Dirac-Fock ground state for atoms and molecules, due to one of us, and from the
non-relativistic Hartree-Fock model for crystals. We prove the existence of a
ground state when the number of electrons per cell is not too large.
