  The critical separation of clusters, corresponding to the maximum density of
clusters, affects growth characteristics during physical vapor deposition
(PVD). In particular, this separation can affect surface smoothness in growing
single-crystalline films, grain size distribution in growing polycrystalline
films, and diameter in growing nanorods. This Letter reports a theoretical
expression of the critical separation as a function of deposition conditions
and accompanying verifications using lattice kinetic Monte Carlo (MC)
simulations of PVD on Cu{111}. In contrast to existing theories, the
theoretical expression in this Letter is (1) closed-form, (2) in better
agreement with the MC simulations than the lattice approximation and (3) in
better agreement with the MC simulations than the mean field approximation when
the critical separation is large- larger than 25 nm for PVD on Cu{111}.
