Statistical description of systems on the basis of the Mandelbrot law: discontinuous metal films on dielectric substrates

Abstract

The inverse power law has been used to describe the structures and the statistical distributions of discontinuous films of Au, Cu, and Mn on dielectric substrates. To this end, the rank of an island (k) was connected with its area (X) for the films with different coverage coefficients (metal contents) p. The dependencies of the island areas on the rank orders in a double-logarithmic plot are straight lines according to the Mandelbrot law. The slope of the straight line, determined as -1/µ, characterizes the distributions of the islands. One can distinguish three ranges of the critical index µ. For µ>2, the moments m1, m2 are finite; if the number of islands is large, the distribution of island areas tends towards a Gaussian or log-normal one. The films structures exhibit Euclidean character. When 1<µ<2, the moment m1 is finite, m2 is infinite; the islands become irregular in shape, and their areas may be described using Poisson distributions. For µ<1, the `hierarchization' of X is more pronounced, the moments m1, m2 are infinite, and the island areas are distributed according to a Lévy stable distribution (for example, Pareto's distribution). At the percolation threshold (phase transition), the infinite cluster appears and the inverse power law is no longer valid.