Correlated walk model of the melting transition in small clusters

Abstract

A novel criterion to locate the melting temperature, T_m, in clusters is proposed. Based on the characteristics of the configuration space of the clusters we identify a class of clusters that: (1) have the global minimum of the potential-energy surface well detached by an energy gap from all other local minima; (2) present a large number of local minima above the gap that can be accessed by the system before evaporation takes place and (3) have a mean energy spacing between the local minima that is very small compared to the gap. For this class of clusters, one trajectory in phase space split into short time intervals can be mapped onto one state of a one-dimensional walker that steps on the various minima in configuration space. The average number of accessed minima above the gap, f, is obtained in a closed form. This quantity has a sigmoid shape as a function of temperature, i.e. it changes fairly rapid from zero at low temperatures to one at high temperatures. Thus f(T) is identified as the signature of the melting transition, and we define T_m as the temperature at which f(T) reaches the value of ½. This phenomenological model is supported by a comparison with a molecular-dynamics simulation of 12-, 13- and 14-atom Lennard-Jones clusters. Values of the parameters pertaining to the theory are extracted from the simulation and a comparison to Lindemann's criterion for melting is provided.