  We study the influence of the space and time dispersion on the frequency
dependence of the wave vectors of electromagnetic waves propagating in
three-dimensional photonic crystals. Two types of structures are considered:
media with weak periodic modulation of the permittivity, and photonic crystals
composed of the periodically arranged identical resonant dielectric particles.
It is shown that in these systems, in contrast to electrons in solid crystals,
different types of excitations exist. For example, a peculiar kind of
polaritons arises in the photonic crystals due to the interaction of the
electromagnetic field, eigenoscillations of the dielectric medium, and Debye
resonance. The widths of the transparency zones and of the band gaps have been
calculated as functions of the frequency and of the parameters of the media. It
is shown that in the photonic crystals with dispersion, the number of
transparency bands is larger than in non-dispersive systems, and the width of
the gaps in the frequency spectrum of photons depends on the wave vector. The
interaction of different types of waves deforms the Brillouin zone, so that it
may not have a plane boundary (for example, a sphere), in which case the
classical Bragg condition does not hold.
