  In this work, we derive some analytical properties of Berry's phase in
one-dimensional quantum and classical crystals, also named Zak's phase, when
computed with a Fourier basis. We show that Zak's phase can be divided in two
terms: a global phase required to make the Bloch wave periodic in the Brillouin
zone and an internal phase which measures the relative delay of the different
Fourier terms within the Brillouin zone. While the former phase is dependent on
the origin of coordinates of the unit cell, the latter is independent of it, so
that it can be interpreted as an internal property of the band itself. We show
that this internal phase is always zero for a symmetric crystal while it can
take any value when this symmetry is broken, showing therefore that it can be
interpreted as a measure of the assymetry of the band. Since for a symmetric
crystal Zak's phase is entirely determined by the global part, we show that
this can be easily calculated by means of the parity of the Fourier terms at
the center and edge of the Brillouin zone, being therefore unnecessary the
integration of the modes through the unit cell and the entire Brillouin zone.
We provide numerical examples analyzing the internal part for both electronic
and classical waves (acoustic or photonic). We analyze the weakest electronic
potential capable of presenting asymmetry, as well as the double-Dirac delta
potential, and in both examples it is found that the internal phase varies
continuously as a function of a symmetry-control parameter, but it is zero when
the crystal is symmetric. For classical waves, the layered material is
analyzed. Although Zak's phase has been mainly studied in connection with the
existence of edge states in finite crystals, we consider that the study of the
internal phase can be more relevant to understand bulk properties of quantum
and classical crystals.
