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 et al. (Ann Inst H Poincaré Anal Non Linéaire 18(6):687–760, 2001). 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.