The thermodynamic properties of small clusters of rare-gas atoms are determined from a biased random-walk Monte Carlo procedure. The cluster sizes studied range from $N=3\mathrm{}$ to $N=13\mathrm{}$ atoms. The internal energy is calculated as a function of temperature. Each cluster exhibits an abrupt liquid-gas phase transition at a temperature much less than for the bulk material. An abrupt solid-liquid transition is also observed for $N=11\mathrm{} \mathrm{and} 13$. The entropy of fusion is $S\simeq 1.76$ e.u. For smaller clusters, the solid-liquid transition is more gradual. The bond-length distributions are calculated for each cluster at various temperatures. The fluctuations of bond lengths from their average values increase gradually with temperature until the melting temperature is reached, at which point there is a large, abrupt increase. The indications are that a harmonic normal-mode analysis is applicable only at temperatures substantially below the melting point and even there, the frequency spectrum is temperature dependent. For $N=3\mathrm{}$, the equilibrium structure is an equilateral triangle. An $N=5\mathrm{}$ cluster forms a trigonal bipyramid and for $N=7\mathrm{}$ the form is a pentagonal bipyramid. The nine-particle structure has symmetry $C_{1h}$ and is a pentagonal bipyramid with two additional atoms, and the $N=13\mathrm{}$ cluster forms an icosahedron.

• Received 3 December 1973

DOI:https://doi.org/10.1103/PhysRevA.11.1068