Melting of clusters and melting

A model is presented for the melting and freezing behavior of finite clusters. The model has the property—and suggests that some real clusters may so behave—that the rigid, solidlike form is the only thermodynamically stable form below a "freezing" temperature $T_f$, that the solidlike form may coexist with a nonrigid, liquidlike form within a sharply-bounded range of temperatures between $T_f$ and a higher "melting" temperature $T_m$, and that above $T_m$, only the liquidlike form of cluster is thermodynamically stable. The temperatures $T_f$ and $T_m$ are functions of $N$, the number of particles in the cluster; it is suggested that the traditional melting point is the common temperature to which $T_f\mathrm{}\left(N\right)\mathrm{}$ and $T_m\mathrm{}\left(N\right)\mathrm{}$ converge as $N\to \infty$.