Nearest-neighbor clustering

In the CVM, the free energy of a given alloy is approximated in terms of probabihties for a selected set of finite clusters. The largest cluster explicitly considered in the free energy functional specifies the level of the approximation. The common practice for an fcc-based system is the tetrahedron approximation [26] in which nearest neighbor tetrahedron cluster is taken as the largest cluster. Hence, within the tetrahedron approximation, the free energy expression, F,is symbolically expressed as... 

Impurity-cluster size nearest-neighbor approximation... 

Table I. The electrostatic energies of the central silicon site and one nearest-neighbor oxygen site for a 977 point-ion cluster, and the Madelung energies for silicon and oxygen in alpha-quartz...

Recently we reported EXAFS results on bimetallic clusters of iridium and rhodium, supported on silica and on alumina (15). The components of this system both possess the fee structure in Efie metallic state, as do the components of the platinum-iridium system. The nearest neighbor interatomic distances in metallic iridium and rhodium are not very different (2.714A vs. 2.690A). From the results of the EXAFS measurements, we concluded that the interatomic distances corresponding to the various atomic pairs (i.e., iridium-iridium, rhodium-rhodium, and iridium-rhodium) in the clusters supported on either silica or alumina were equal within experimental error. Since the Interatomic distances of the pure metals differ by only 0.024A, the conclusion is not surprising. 

There appears to be concentration of rhodium in the surface of the iridium-rhodium clusters, on the basis that the total number of nearest neighbor atoms about rhodium atoms was found to be smaller than the nunber about iridium atoms in both catalysts investigated. This conclusion agrees with that of other workers (35) based on different types of measurements. The results on the average compositions of the first coordination shells of atoms about iridium and rhodium atoms in either catalyst Indicate that rhodium atoms are also incorporated extensively in the interiors of the clusters. In this respect the iridium-rhodium system differs markedly from a system such as ruthenium-copper (8), in which the copper appears to be present exclusively at the surface. 

The charge distribution determined within clusters by CNDO has been reported for only a few cases. Let us consider only one cluster, the 13-atom fee cluster with only two geometrically different types of atom. There is a center atom with 12 nearest neighbors, and there are 12 surface atoms each with 4 nearest neighbors. At the equilibrium bond length (0.34 nm) the center atom has a net positive charge, but this situation is reversed at the bulk experimental distance (0.288 nm). 

The I U) characteristic of the arrays showed a linear behavior over a broad voltage range. If each cluster is assumed to have six nearest neighbors and a cluster-to-cluster capacitance of 2 x 10 F is implied, the total dot capacitance will be 1.2 x 10 F. A corresponding charging energy can thus be approximated to 11 meV, which is only about half of the characteristic thermal energy at room temperature. This excludes a development of a Coulomb gap at room temperature. 

The algorithm originally proposed by Moody and Darken (1989) uses >means clustering to determine the centers of the clusters. The hypersphere around each cluster center is then determined to ensure sufficient overlap between the clusters for a smooth fit by criteria such as the P-nearest neighbor heuristic,... 

The objective functions for both k-means clustering and the F-nearest neighbor heuristic given by Eqs. (20) and (21) use information only from the inputs. Because of this capacity to cluster data, local methods are particularly useful for data interpretation when the clusters can be assigned labels. 

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