This paper reports on the photonic band structure of periodic dielectric structures, specifically fcc and diamond lattices. Using a plane-wave expansion method, the authors solve Maxwell's equations for electromagnetic wave propagation in periodic dielectric spheres. They find that fcc structures do not have a full photonic band gap that extends throughout the Brillouin zone, but diamond structures do possess a full photonic band gap. The diamond structure can have a photonic band gap even with low refractive-index contrasts. The authors compare their calculations with experimental data and find that while the experimental data and theory agree well over most of the Brillouin zone, there are two symmetry points (W and U) where the experiment indicates a gap while calculations show propagating modes. They conclude that the fcc structure exhibits a pseudogap rather than a full photonic band gap. In contrast, the diamond structure does have a full photonic band gap. The authors also discuss the importance of considering the vector nature of the electromagnetic field in calculations. They show that the scalar-wave approximation is inadequate for accurately predicting photonic band gaps. The results indicate that the diamond structure can have a full photonic band gap with low refractive-index contrasts, which is promising for future experimental verification. The study highlights the importance of symmetry and the vector nature of the electromagnetic field in determining photonic band gaps.This paper reports on the photonic band structure of periodic dielectric structures, specifically fcc and diamond lattices. Using a plane-wave expansion method, the authors solve Maxwell's equations for electromagnetic wave propagation in periodic dielectric spheres. They find that fcc structures do not have a full photonic band gap that extends throughout the Brillouin zone, but diamond structures do possess a full photonic band gap. The diamond structure can have a photonic band gap even with low refractive-index contrasts. The authors compare their calculations with experimental data and find that while the experimental data and theory agree well over most of the Brillouin zone, there are two symmetry points (W and U) where the experiment indicates a gap while calculations show propagating modes. They conclude that the fcc structure exhibits a pseudogap rather than a full photonic band gap. In contrast, the diamond structure does have a full photonic band gap. The authors also discuss the importance of considering the vector nature of the electromagnetic field in calculations. They show that the scalar-wave approximation is inadequate for accurately predicting photonic band gaps. The results indicate that the diamond structure can have a full photonic band gap with low refractive-index contrasts, which is promising for future experimental verification. The study highlights the importance of symmetry and the vector nature of the electromagnetic field in determining photonic band gaps.