  We use the phase field crystal model to study nucleation of edge dislocations
in two dimensions under an applied stress field. A dislocation dipole nucleates
under the applied stress, consistent with Burgers vector conservation. The
phase field correctly accounts for elastic energy storage prior to nucleation,
and for dissipative relaxation during the nucleation event. We show that a
lattice incompatibility field is a sensitive diagnostic of the location of the
nucleation event, and of the Burgers vector and slip direction of the
dislocations that will be nucleated above threshold. A direct calculation of
the phase field energy accurately correlates with the nucleation event, as
signaled by the lattice incompatibility field. We show that a Schmid-like
criterion concerning the resolved stress at the nucleation site correctly
predicts the critical nucleation stress. Finally, we present preliminary
results for a three-dimensional, body-centered cubic lattice. The phase field
allows a direct computation of the lattice incompatibility tensor for both
dislocation lines and loops.
