Spatiotemporal Distribution of Nucleation Events during Crystal Growth

We consider irreversible second-layer nucleation that occurs when two adatoms on a terrace meet. We solve the problem analytically in one dimension for zero and infinite step-edge barriers, and numerically for any value of the barriers in one and two dimensions. For large barriers, the spatial distribution of nucleation events strongly differs from 2, where is the stationary adatom density in the presence of a constant flux. Theories of the nucleation rate based on the assumption that it is proportional to 2 are shown to overestimate by a factor proportional to the number of times an adatom diffusing on the terrace visits an already visited lattice site.