  We study the light dispersion relation in a periodic ensemble of atoms at
fixed positions in the Fano-Hopfield model (the atomic dipole being modeled
with harmonic oscillators). Compared to earlier works, we do not restrict to
cubic lattices, and we do not regularize the theory by hand but we renormalize
it in a systematic way using a Gaussian cut-off in momentum space. Whereas no
omnidirectional spectral gap is known for light in a Bravais atomic lattice, we
find that, for a wide range of parameters, an omnidirectional gap occurs in a
diamond atomic lattice, which may be realized in an experiment with ultra-cold
atoms. The long-wavelength limit of the theory also provides a Lorentz-Lorenz
(or Clausius-Mossotti) relation for an arbitrary lattice.
