Scaling in the equilibrium size distribution of one-dimensional surface clusters

The length distribution of the one-dimensional atomic rows self-assembled on a crystal surface in thermal equilibrium at low coverage is calculated both in the framework of an analytical theory and with the use of the Monte Carlo simulations. It has been shown that in the case of interatomic interactions restricted to nearest-neighbor atoms the distribution exhibits the scaling properties formally similar to those found in the theories of irreversible growth. The difference consisted in the behavior of the scaling functions which in the equilibrium case was monotonously decreasing while in the case of irreversible growth exhibits a monomodal character. We found that our scaled distribution described without any fitting parameters the monotonous distributions recently observed in the growth of Ga rows on Si(001) by Albao [Phys. Rev. B 72, 035426 (2005)]. The implications of the scaling for the experimental definition of interatomic interaction parameters is briefly discussed.