  We use a hybrid Monte Carlo algorithm to simulate the shaking of spheres at
different vibrational amplitudes, and find that spontaneous crystallisation
occurs in specific dynamical regimes. Several crystallising transitions are
typically observed, leading to end states which can be fully or partially
ordered, depending on the shaking amplitude, which we investigate using metrics
of global and local order. At the lowest amplitudes, crystallisation is
incomplete, at least for our times of observation. For amplitude ranges where
crystallisation is complete, there is typically a competition between hexagonal
close packed (hcp) or face-centered cubic (fcc) ordering. It is seen that fcc
ordering typically predominates; in fact for an optimal range of amplitudes,
spontaneous crystallisation into a pure fcc state is observed. An interesting
feature is the breakdown of global order when there is juxtaposition of fully
developed hcp and fcc order locally: we suggest that this is due to the
interfaces between the different domains of order, which play the same role as
dislocations.
