Distribution of cluster sizes from evaporation to total multifragmentation

A model for studying fragmentation phenomena is proposed and developed. The model leads to a single, simple, and exact expression for the cluster size distribution function. Various limits of this distribution function show: (1) evaporation-like behavior, (2) scale-invariant power law behavior, (3) a broad region with a dependence which is linear growth in small clusters and exponential falloff of large clusters and, finally, (4) total multifragmentation with an exponential-like falloff of all clusters except the monomer or unit element. The cluster size distribution function in any region is given by various limits of one expression: ${\mathit{Y}}_{\mathit{A}}$(k,x)={A!/[k!(A-k)!]}xB(x+A-k,k). Here, the size k is the number of elements in a cluster taken from a fixed total number of A elements, x is an evolutionary tuning parameter which determines the various regions, and B(x+A-k,k) is a beta function. Cellular rules and a particular choice of weight function lead to self-similar behavior on Young’s triangular lattice. A scale invariant hyperbolic power law emerges in a row by row evolution of the lattice. A counterclockwise rotation of Ferrer’s block diagram of partitions shows a pictorial resemblance of the present model with recent work on self-organized critical states, and a comparison is made. The cumulative mass distribution at a critical point of the model is a staircase function whose continuous limit is analogous to that of a uniform bar. The uniform bar may then be hammered into various shapes which will be discussed. Some observations on the form of x are given by comparing the multifragmentation limit of the model with the law of mass action or Saha equation. The evaporation limit of the model is discussed and evaporation barriers are shown to evolve into binding energy enhancement factors in the Saha equation.