  We study the periodic Hartree-Fock model used for the description of
electrons in a crystal. The existence of a minimizer was previously shown by
Catto, Le Bris and Lions (Ann. Inst. H. Poincare Anal. Non Lineaire} 18 (2001),
no.6, 687--760). We prove in this paper that any minimizer is necessarily a
projector and that it solves a certain nonlinear equation, similarly to the
atomic case. In particular we show that the Fermi level is either empty or
totally filled.
