Nearest neighbour clustering

The new data matrix using nearest neighbour clustering is presented in Table 4.20, with the new values shaded. Remember that there are many similarity measures and methods for linking, so this table represents only one possible way of handling the information. 

Table 4.20 Nearest neighbour cluster analysis, using correlation coef-ficients for similarity measures, and data in Table 4.16. ...

Figure 14 ORTEP representation of two nearest neighbour clusters with ethanol ligands on the outer Mn(II) ions. H atoms are omitted for clarity. Dotted lines represent the shortest O-H - N contacts (2.74 A) between adjacent clusters [73],...

Atoms in rare gas solids arrange in a close-packed structure, each atom being surrounded by 12 nearest neighbours. Clusters of rare gases are expected to... 

The LDOS have been calculated using 10 exact levels in the continued fraction expansion of the Green functions. For clean surfaces the quantities A Vi are the same for all atoms in the same plane they have been determined up to p = 2, 4, 6 for the (110), (100) and (111) surfaces, respectively, and neglected beyond. The cluster C includes the atoms located at the site occupied by the impurity and at al the neighbouring sites up to the fourth nearest neighbour. 

In the first example (46), hydrogen adsorption on different sites of a nickel crystal was studied by considering the interaction of a hydrogen atom with a limited number of metal atoms. The nickel crystal was truncated to obtain the clusters shown in Fig. 31, containing 13,10, 9 and 8 nickel atoms, and representing models for the bulk crystal and for the (111), (100) and (110) surfaces respectively. The nearest-neighbour distance in all clusters was... 

The CPA has proved to be an enormously successful tool in the study of alloys, and has been implemented within various frameworks, such as the TB, linear muffin-tin orbital and Korringa-Kohn-Rostoker (Kumar et al 1992, Turek et al 1996), and is still considered to be the most satisfactory single-site approximation. Efforts to do better than the single-site CPA have focused on multi-site (or cluster) CPA s (see, e.g., Gonis et al 1984, Turek et al 1996), in which a central site and its set of nearest neighbours are embedded in an effective medium. Still, for present purposes, the single-site version of the CPA suffices, and we derive the necessary equations here, within the framework of the TB model. 

One of the characteristics of weak flocculation is that the system is reversible. At low volume fractions the system will form some clusters and some single particles. The clusters can be easily disrupted by gentle shaking. As the concentration is increased the system will reach a percolation threshold . The number of nearest neighbours around any test particle reaches 3 at about (p — 0.25 and the attractive forces between... 

In between these extremes lie a large number of CVM treatments which use combinations of different cluster sizes. The early treatment of Bethe (1935) used a pair approximation (i.e., a two-atom cluster), but this cluster size is insufficient to deal with fhistration effects or when next-nearest neighbours play a significant role (Inden and Pitsch 1991). A four-atom (tetrahedral) cluster is theoretically the minimum requirement for an f.c.c. lattice, but clusters of 13-14 atoms have been used by de Fontaine (1979, 1994) (Fig. 7.2b). However, since a comprehensive treatment for an [n]-member cluster should include the effect of all the component smaller (n — 1, n — 2...) units, there is a marked increase in computing time with cluster size. Several approximations have been made in order to circumvent this problem. 

Recall that, if y > 1, increase in T results in decrease in yt. Thus the left-hand side of the above equation becomes negative and hence Hf is positive - endothermic. In a similar way one may find that, if y, < 1, then the solution process is exothermic. If the solution process in an i-j binary system is endothermic, the i-i and j-j attractions are greater than the i-j attraction, i atoms attempt to have only i atoms as nearest neighbours - tendency toward phase separation or clustering. 

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