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Using a plane-wave expansion method, we have solved Maxwell’s equations for the propagation of electromagnetic waves in a periodic lattice of dielectric spheres (dielectric constant ε a) in a uniform dielectric background (ε b). Contrary to experiment, we find that fcc dielectric structures do not have a ‘‘photonic band gap’’that extends throughout the Brillouin zone. However, we have determined that dielectric spheres arranged in the diamond structure do possess a full photonic band gap. This gap exists for refractive-index contrasts as low as 2.