  We study the local optimality of Simple Cubic, Body-Centred-Cubic and
Face-Centred-Cubic lattices among Bravais lattices of fixed density for some
finite energy per point. Following the work of Ennola [Math. Proc. Cambridge,
60:855--875, 1964], we prove that these lattices are critical points of all the
energies, we write the second derivatives in a simple way and we investigate
the local optimality of these lattices for the theta function and the
Lennard-Jones-type energies. In particular, we prove the local minimality of
the FCC lattice (resp. BCC lattice) for large enough (resp. small enough)
values of its scaling parameter and we also prove the fact that the simple
cubic lattice is a saddle point of the energy. Furthermore, we prove the local
minimality of the FCC and the BCC lattices at high density (with an optimal
explicit bound) and its local maximality at low density in the
Lennard-Jones-type potential case. We then show the local minimality of FCC and
BCC lattices among all the Bravais lattices (without a density constraint). The
largest possible open interval of density's values where the Simple Cubic
lattice is a local minimizer is also computed.
