  We investigate the local and global optimality of the triangular, square,
simple cubic, face-centred-cubic (FCC), body-centred-cubic (BCC) lattices and
the hexagonal-close-packing (HCP) structure for a potential energy per point
generated by a Morse potential with parameters $(\alpha,r_0)$. The optimality
of the triangular lattice is proved in dimension 2 in an interval of densities.
Furthermore, a complete numerical investigation is performed for the
minimization of the energy with respect to the density. In dimension 3, the
local optimality of the simple cubic, FCC and BCC lattices is numerically
studied. We also show that the square, triangular, cubic, FCC and BCC lattices
are the only Bravais lattices in dimensions 2 and 3 being critical points of a
large class of lattice energies (including the one studied in this paper) in
some open intervals of densities, as we observe for the Lennard-Jones and the
Morse potential lattice energies. Finally, we state a conjecture about the
global minimizer for the 3d energy that exhibits a surprising transition from
BCC, FCC to HCP as $\alpha$ increases. Moreover, we compare the values of
$\alpha$ found experimentally for metals and rare-gas crystals with the
expected lattice ground-state structure given by our numerical
investigation/conjecture. Only in a few cases does the known ground-state
crystal structure match the minimizer we find for the expected value of
$\alpha$. Our conclusion is that the pairwise interaction model with Morse
potential and fixed $\alpha$ is not adapted to describe metals and rare-gas
crystals if we want to take into consideration that the lattice structure we
find in nature is the ground-state of the associated potential energy.
