  In the context of nanowire heterostructures we perform a discrete to
continuum limit of the corresponding free energy by means of
$\Gamma$-convergence techniques. Nearest neighbours are identified by employing
the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the
nanowire is done in such a way that we perform not only a continuum limit but a
dimension reduction simultaneously. The main part of the proof is a discrete
geometric rigidity result that we announced in an earlier work and show here in
detail for a variety of three-dimensional lattices. We perform the passage from
discrete to continuum twice: once for a system that compensates a lattice
mismatch between two parts of the heterogeneous nanowire without defects and
once for a system that creates dislocations. It turns out that we can verify
the experimentally observed fact that the nanowires show dislocations when the
radius of the specimen is large.
