Capture numbers in rate equations and scaling laws for epitaxial growth

In this paper, we present a detailed exposition of the functional form of capture numbers that we found using an extended-island model. Our results suggest that the assumption sigma(s) = sigma(1) for all s is only valid up to a time that scales like O(R-1/2). After this time, a better approximation is sigma(s) = as + b + small correction and we show that in the limit R --> infinity, sigma(s)-->as + b. We link the functional form to the amount of nucleation of new islands on the surface and explain the differences between what is obtained with our extended-island model to what is obtained with a point-island model. Finally, we use our results to derive scaling laws for the adatom and total number densities. We found that the scaling in R remains unchanged, but that the time evolution is influenced by the functional form of the capture numbers.