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This supershell structure was explained by an appeal to a semiclassical theory of Balian and Bloch . The analysis is based on the correspondence between classical and quantum physics in the limit of large quantum numbers.
Metallics – Nanoparticles – J-Blackman – Handbook of Metal Physics
The motion of a particle in a quantum state is related to classical motion in a closed orbit. The closed orbits of a particle moving in a spherical cavity square well potential are polygons free particle motion with bouncing off the walls. Experimental evidence of the supershell effect has been claimed in sodium , lithium  and gallium.
It is clear from Figure 2. For other materials, the geometric shell model is usually the starting point for studies of stability and associated magic numbers and is the dominant mechanism even for quite small clusters. The prototype materials for close-packing are the rare gases , for which the bonding is via the non-directional van der Waals interaction. They adopt an icosahedral geometry for sizes up to many thousands of atoms. This is a common motif for close-packed structures although, for metals, there is generally a transition to structures other than icosahedral at a much smaller size than for rare gas clusters.
We consider now some of the geometries based on close-packing. We begin by recalling the macroscopic approach to the stability of metallic particles based on the Wulff construction . Using this procedure for an fcc metal produces the polyhedron comprising eight. The simplest model for the surface energy assumes pairwise interactions denoted by f between nearest se broken bond model.
An atom on the. This is illustrated in Figure 2. There are a large number of ways in which a symmetric cluster can be constructed from a fcc close-packed lattice. A basic shape is apoostila octahedron Figure 2. The total number of atoms in a cluster containing K octahedral shells is.
A cluster with an odd value of K has a single atom at its centre, while one with an even value of K is built round an elementary octahedron cartuxhos six atoms. Polyhedral structure based on the Wulff construction.
Reproduced with the permission of Elsevier Science from Marks . A cuboctahedron, an octahedron truncated by a cube, cartucos be produced in two ways. Perhaps the simplest is that shown in Figure 2. The total number of atoms in a cluster comprising K shells is. This polyhedron has a central atom irrespective of the size of the cluster and we can consider it built of.
A atom truncated octahedron with square and hexagonal faces. An alternative cartuchps is the one with hexagonal and square faces already shown in Figure 2. Like the octahedron itself, the cluster is built round a central atom or a central octahedron of atoms depending on whether the shell number is odd or even.
Unlike the polyhedron in Figure 2.
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The number of atoms in this cluster is given by and the magic numbers are therefore 1, 38,y. The total surface energy can be minimised if all the faces of the polyhedron are 1 planes and the surface area can be kept at a minimum. This can be accomplished with an icosahedral structure with 20 triangular faces [32,3], as in Figure 2. The Mackay icosahedron can be regarded as being made up of 20 tetrahedra with slight distortion as shown in Figure 2. The icosahedra can be constructed by arranging 12 neighbours of a central atom at the corners of an icosahedron.
Successively larger clusters can be. The total number of atoms in an icosahedron of K shells is identical to that of the cuboctahedron with triangular faces Eq. Because of the distortion, the packing density of the icosahedron is lower 0.
However, it has been shown  that, compared with the close-packed cuboctahedral structure, the reduction in surface energy achieved by the icosahedron more than compensates for the extra strain energy resulting from the distortion, and even up to quite large clusters the icosahedral packing is energetically favourable.
The icosahedron, like the cuboctahedron with triangular faces, has 12 vertices. Mackay  has shown how the cuboctahedron can be transformed into the icosahedron by a continuous rotation of its triangular faces.
The reason for the assignment of magic numbers to the geometric shell structure for large sodium clusters is clear. The positions of the minima in the mass spectra in Figure 2. The vertex O is at the centre. The angles subtended at O are u. The number of atoms in a decahedron of K shells is. Decahedra with odd values of K are atom-centred and those with even K have a seven-atom decahedron at the centre.
Like the icosahedron, the surfaces are close-packed, but in this case the total surface area is very large. This can be reduced  by truncation Figure 2.
The total number of atoms in a truncated decahedron with square faces is given by the same expression as for the icosahedron and cuboctahedron with triangular faces Eq. Further truncation can be performed to expose re-entrant 1 faces in addition to the faces [31,35] as is shown in Figure 2. The rhombic dodecahedron is the primary cluster shape based on the bcc lattice, as shown in Figure 2. It is made up of 12 identical faces whose diagonals have lengths with the ratio 1: The number of atoms for a cluster of K shells is given by.
Parte 7 de 7 Figure 2. Shell Models of Isolated Clusters34 and maxima near and Geometric shell structure 2. Close-packing It is clear from Figure 2. Wulff construction We begin by recalling the macroscopic approach to the stability of metallic particles based on the Wulff construction .
Using this procedure for an fcc metal produces the polyhedron comprising eight 2. Geometric shell structure Tags Metallics – Nanoparticles Sapatas- Original Teoria sapatas original teoria. Gold Nanoparticles Gold Nanoparticles.