Metallic clusters magic numbers

Reproduction of metal-cluster magic numbers using a -deformed, 3-dimensional, harmonic oscillator model... 

As holds for other cluster systems, certain magic cluster electron counts exist, which indicates for a certain cluster-halide ratio and interstitial present the filling of all bonding molecular orbitals and therefore the thermodynamically most stable situation. For main group interstitial atoms these are 14 cluster-based electrons whereas for transition-metal interstitials the magic number is 18 [1, 10-12]. All of these phases are synthesized by high-temperature solid-state chemical methods. A remarkable variety of different structure types has been... 

Slovokhotov, Yu.L. and Struchkov, Yu.T (1984) X-ray crystal structure of a distorted tetrahedral cluster in the salt [(Ph P)4Au4N] BF4 . Geometrical indication of stable electronic configurations in post-transition metal complexes and the magic number 18-e in centred gold clusters. Journal of Organometallic Chemistry, 177, 143-146. 

The metal nuclei of the so-called full-shell clusters, also referred to as magic number clusters, generally exhibit hexagonal or cubic close packing (ccp). Figure 6 shows the stepwise formation of the first magic-number clusters built by 13, 55, 147, and 309 atoms. 

There are similarities and differences between the Hirsch rule and those developed by physicists, originally to rationalize the magic numbers observed experimentally for simple metal clusters. The reader is referred to the excellent review by de Heer for more information [71]. The magic numbers for sodium clusters, 8, 20,... 

Aggregation of the atoms or microclusters may give metal nuclei. The micro-cluster itself may work as the nucleus. Although the size of microcluster or nucleus is not clear, the nucleus may consist of 13 atoms, which is the smallest magic number, This idea may be supported by the structural analysis of PVP-stabilized Pt nanoparticles (64) and other systems. In fact, a particle of 13 atoms is considered an elemental duster. In the case of preparation of PVP-stabilized Rh nanoparticle dispersions by alcohol reduction, formation of very tiny particles, the average diameter of which is estimated to be 0.8 nm, was observed (66). These tiny particles in the metastable state may consist of 13 atoms each and easily increase in size to the rather nanoparticles with average diameter of 1.4 nm, i.e., the particles composed of 55 atoms. This observation again supports the idea that the elemental cluster of 13 atoms is the nucleus. 

The jellium model of the free-electron gas can account for the increased abundance of alkali metal clusters of a certain size which are observed in mass spectroscopy experiments. This occurrence of so-called magic numbers is related directly to the electronic shell structure of the atomic clusters. Rather than solving the Schrodinger equation self-consistently for jellium clusters, we first consider the two simpler problems of a free-electron gas that is confined either within a sphere of radius, R, or within a cubic box of edge length, L (cf. problem 28 of Sutton (1993)). This corresponds to imposing hard-wall boundary conditions on the electrons, namely... 

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