  We investigate one-dimensional periodic chains of alternate type of particles
interacting through mirror symmetric potentials. The optimality of the
equidistant configuration at fixed density -- also called crystallization -- is
shown in various settings. In particular, we prove the crystallization at any
scale for neutral and non-neutral systems with inverse power laws interactions,
including the three-dimensional Coulomb potential. We also show the minimality
of the equidistant configuration at high density for systems involving inverse
power laws and repulsion at the origin. Furthermore, we derive a necessary
condition for crystallization at high density based on the positivity of the
Fourier transform of the interaction potentials sum.
