Analytical calculation of compressibilities of highly aymmetrical particles

Abstract

We investigate the size-dependence of the compressibility of cuboctahedra and icosahedra. For particles of inert-gas atoms these structures play a n important role as small clusters of optimum stability (icosahedra), as well as greater crystallites corresponding to the fcc-lattice (cuboctahedra). Similar clusters of various transition metal atoms have been observed too.The compressibility, a t zero temperature, is defined asThe interaction energy E is given by means of pair potentials Uif,where i and j denote a pair of atoms. I n order t o derive analytical formulae, valid as well for microscopic as for macroscopic particles, a topological simplification is introduced : -I n eq. ( 2 ) we consider only the contributions of nearest neighbours.-The clusters are assumed to have equal bond lengths R. (The difference in bond By these suppositions ( 2 ) may be written as lengths of 5 % in icosahedra is neglected.)where U ( R) denotes the interaction energy of a nearest-neighbouring pair of atoms, and C N ( i ) is the coordination number of the i-th atom. The total number of atoms N can be expressed by the number of cluster "shells" n, n = 1 , 2 , 3, ...Then, it follows (for details see VOGELSBERGER et al.) with15 for icosahedra 18 for cuboctahedra k = (The atoms are supposed to be spheres with diameter R touching each other. Now, we define the surface of the cluster as the "equimolecular dividing surface" (see VOGELS-BERGER et al.). Thus, the surface involves all the atoms (spheres) of the cluster com-