  The goal of this paper is twofold. First we prove a rigidity estimate, which
generalises the theorem on geometric rigidity of Friesecke, James and M\"uller
to 1-forms with non-vanishing exterior derivative.
  Second we use this estimate to prove a kind of spontaneous breaking of
rotational symmetry for some models of crystals, which allow almost all kinds
of defects, including unbounded defects as well as edge, screw and mixed
dislocations, i.e. defects with Burgers vectors.
