SCALING RANGE AND CUTOFFS IN EMPIRICAL FRACTALS

Fractal structures appear in a vast range of physical systems. A liter ature survey including all experimental papers on fractals which appea red in the six Physical Review journals (A-E and Letters) during the 1 990s shows that experimental reports of fractal behavior are typically based on a scaling range Delta that spans only 0.5-2 decades. This ra nge is limited by upper and lower cutoffs either because further data are not accessible or due to crossover bends. Focusing on spatial frac tals, a classification is proposed into (a) aggregation, (b) porous me dia, (c) surfaces and fronts, (d) fracture, and (e) critical phenomena . Most of these systems [except for class (e)] involve processes far f rom thermal equilibrium. The fact that for self-similar fractals [in c ontrast to the self-affine fractals of class (c)] there are hardly any exceptions to the finding of Delta less than or equal to 2 decades, r aises the possibility that the cutoffs are due to intrinsic properties of the measured systems rather than the specific experimental conditi ons and apparatus. To examine the origin of the limited range we focus on a class of aggregation systems. In these systems a molecular beam is deposited on a surface, giving rise to nucleation and growth of dif fusion-limited-aggregation-like clusters. Scaling arguments are used t o show that the required duration of the deposition experiment increas es exponentially with h. Furthermore, using realistic parameters for s urfaces such as Al(111) it is shown that these considerations limit th e range of fractal behavior to less than two decades in agreement with the experimental findings. It is conjectured that related kinetic mec hanisms that limit the scaling range are common in other nonequilibriu m processes that generate spatial fractals.