  The goal of this work is to investigate the optimality of the $d$-dimensional
rock-salt structure, i.e., the cubic lattice $V^{1/d}\mathbb{Z}^d$ of volume
$V$ with an alternation of charges $\pm 1$ at lattice points, among periodic
distribution of charges and lattice structures. We assume that the charges are
interacting through two types of radially symmetric interaction potentials,
according to their signs. We first restrict our study to the class of
orthorhombic lattices. We prove that, for our energy model, the $d$-dimensional
rock-salt structure is always a critical point among periodic structures of
fixed density. This holds for a large class of potentials. We then investigate
the minimization problem among orthorhombic lattices with an alternation of
charges for inverse power laws and Gaussian interaction potentials. High
density minimality results and low-density non-optimality results are derived
for both types of potentials. Numerically, we investigate several particular
cases in dimensions $2$, $3$ and $8$. The numerics support the conjecture that
the rock-salt structure is the global optimum among all lattices and periodic
charges, satisfying some natural constraints. For $d=2$, we observe a phase
transition of the type 'triangular-rhombic-square-rectangular' for the
minimizer's shape as the density decreases.
