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We analyse the rigidity of discrete energies where at least nearest andnext-to-nearest neighbour interactions are taken into account. Our purpose isto show that interactions beyond nearest neighbours have the role of penalisingchanges of orientation and, to some extent, they may replace thepositive-determinant constraint that is usually required when only nearestneighbours are accounted for.} In a discrete to continuum setting, we prove acompactness result for a surface-scaled energy and we give bounds on itspossible Gamma-limit.  In the second part of the paper we follow the approach developed in the firstpart to study a discrete model for (possibly heterogeneous) nanowires. In theheterogeneous case, by applying the compactness result shown in the first partof the paper, we obtain an estimate on the minimal energy spent to matchdifferent equilibria. This gives insight into the nucleation of dislocations inepitaxially grown heterostructured nanowires.

Interactions beyond nearest neighbours and rigidity of discrete energies: a compactness result and an application to dimension reduction