  The emergence of long-range order at low temperatures in atomistic systems
with continuous symmetry is a fundamental, yet poorly understood phenomenon in
Physics. To address this challenge we study a discrete microscopic model for an
elastic crystal with dislocations in three dimensions, previously introduced by
Ariza and Ortiz. The model is rich enough to support some realistic features of
three-dimensional dislocation theory, most notably grains and the Read-Shockley
law for grain boundaries, which we rigorously derive in a simple, explicit,
geometry. We analyze the model at positive temperatures, in terms of a Gibbs
distribution with energy function given by the Ariza-Ortiz Hamiltonian plus a
contribution from the dislocation cores. Our main result is that the model
exhibits long range positional order at low temperatures. The proof is based on
the tools of discrete exterior calculus, together with cluster expansion
techniques.
