Packings, structured geometry

The octahedral holes in a close-packed structure are much bigger than the tetrahedral holes—they are surrounded by six atoms instead of four. It is a matter of simple geometry to calculate that the radius of a sphere that will just fit in an... 

Although the (111) face of fee metals is of the lowest surface free energy — a fact which may explain the reconstructions of the (100) and (110) faces —, the (111) face itself may also reconstruct Au(l 11) is normally reconstructed with a structure that may nevertheless still involve the hexagonally close-packed layer geometry (since extra sites of hexagonally arranged spots appear in LEED), but with a lattice constant different from that of the bulk . ... 

We can, therefore, understand the origin of the four-atom structural trend from tetrahedron rhombus -> linear chain -> square as a function of the electron count for the case of a = The presence of three-membered rings in a given geometry skews the eigenspectrum asymmetrically downwards. Hence, the close-packed structures, the tetrahedron and the rhombus, are stabilized for fractional electron occupancies less than one-half but destabilized for fractional electron counts more than one half On the other hand, the open structures, the square and the linear chain, have symmetric... 

The sequence and spacings given for close packing are not artificial descriptions or approximations, as these are determined by geometry. The PTOT system is the most detailed and definitive treatment presented for close-packed structures, and many other structures can be described in this system. 

There are commonly void spaces (holes) in a crystal that can sometimes admit foreign particles of a smaller size than the hole. An understanding of the geometry of these holes becomes an important consideration as characteristics of the crystal will be affected when a foreign substance is introduced. In the cubic close-packed structure, the two major types of holes are the tetrahedral and the octahedral holes. In Fig. 10-1(h), tetrahedral holes are in the centers of the indicated minicubes of side a/2. Each tetrahedral hole has four nearest-neighbor occupied sites. The octahedral holes are in the body center and on the centers of the edges of the indicated unit cell. Each octahedral hole has six nearest-neighbor occupied sites. 

Ionic crystals can also be described in terms of the interstices, or holes, in the structures. Figure 7-5 shows the location of tetrahedral and octahedral holes in close-packed structures. Whenever an atom is placed in a new layer over a close-packed layer, it creates a tetrahedral hole surrounded by three atoms in the first layer and one in the second (CN = 4). When additional atoms are added to the second layer, they create tetrahedral holes surrounded by one atom in the one layer and three in the other. In addition, there are octahedral holes (CN = 6) surrounded by three atoms in each layer. Overall, close-packed structures have two tetrahedral holes and one octahedral hole per atom. These holes can be filled by smaller ions, the tetrahedral holes by ions with radius 0.225r, where r is the radius of the larger ions, and the octahedral holes by ions with radius 0.414r. In more complex crystals, even if the ions are not in contact with each other, the geometry is described in the same terminology. For example, NaCl has chloride ions in a cubic close-packed array, with sodium ions (also in a ccp array) in the... 

A different measure of atomic size is the volume occupied by a mole of atoms of the element in the solid phase. Figure 5.23 shows the pronounced periodicity of the molar volume, with maxima occurring for the alkali metals. Two factors affect the experimentally measured molar volume the size of the atoms, and the geometry of the bonding that connects them. The large molar volumes of the alkali metals stem both from the large size of the atoms and the fact that they are organized in a rather open, loosely packed structure in the solid. 

The differences in the stable geometries of the AM and AE clusters have been investigated from the electronic structure view point. Ekhardt and Penzar, using a self-consistent jellium model, reported a more stable prolate structure than the spherical one for Na4 (25). The model placed four valence electrons of the Na4 cluster into a spherical potential. Two electrons occupy the I5 shell in the spherical potential and the other two electrons are accommodated in the p shell. Prolate distortion splits the I/7 levels and then the lowered Ip level is filled with two electrons. Therefore, the Na4 cluster prefers the prolate deformation. Using a molecular orbital method, Rao and Jena came to a conclusion which is consistent with the jellium results (13). The Li4 cluster adopts a planar structure while the Be4 cluster has a close packed structure since the latter cluster has eight valence electrons and the molecular orbitals corresponding to the p shell for the Jellium model are completely filled with the electrons. 

The formation of surfaee aggregates of surfaetants and adsorbed micelles is a challenging area of experimental research. A relatively recent summary has been edited by Sharma [51], The details of how surfactants pack when aggregated on surfaces, with respect to the atomic level and with respect to mesoscale structure (geometry, shape etc ), are less well understood than for micelles free in solution. Various models have been considered for surface surfactant aggregates, but most of these models have been adopted without firm experimental support. 

The adsorption of anions on metal electrodes has been one of the major topics in surface electrochemistry. Specific adsorption of anions occurs when the anion loses aU or part of its solvation shell and forms a direct chemical bond with the substrate. In this situation the surface coverage by anions can be high and the adlayer tends to form a close-packed structure that depends critically on the surface atomic geometry of the underlying substrate and the balance between the anion-metal and anion-anion interaction energies. The structures of halide anions adsorbed onto Au(Jtkl), Ag(hkl), and Pt(hkl) low-index surfaces have been the most widely studied systems by SXS, and a comprehensive review of ordered anion adlayers on metal electrodes is given by Magnussen [57]. 

It should also be mentioned that different chainpacking arrangements have been derived theoretically by Kitaigorodskii (1957) from the geometry of extended chains. The results are in good agreement with the experimentally determined chain-packing structures described below. 

---