Crystalline solids sphere packing

This opening chapter has introduced many of the principles and ideas that lie behind a discussion of the crystalline solid state. We have discussed in detail the structure of a number of important ionic crystal structures and shown how they can be linked to a simple view of ions as hard spheres that pack together as closely as possible, but can also be viewed as the linking of octahedra or tetrahedra in various ways. Taking these ideas further, we have investigated the size of these ions in terms of their radii, and... 

Unit Cells and the Packing of Spheres in Crystalline Solids... 

The surface of a crystalline solid is strongly correlated to its bulk structure. The atoms of a crystal are arranged in a periodical sequence forming the crystal lattice. Most frequently metals and metal alloys tend to form close-packed sphere (cps) arrangements reflecting the isotropy of the forces of atomic interaction. 

Describe the three types of cubic unit cells and explain how to find the number of particles in each and how packing of spheres gives rise to each calculate the atomic radius of an element from its density and crystal structure distinguish the types of crystalline solids explain how the electron-sea model and band theory account for the properties of metals and how the size of the energy gap explains the conductivity of substances ( 12.6) (SP 12.4) (EPs 12.57-12.75)... 

Atoms and ions can be considered to be spheres which pack together in special and reproducible patterns to form solid state crystalline materials. The patterns of the spheres repeat in all three directions. The simplest, basic repeating unit in a crystalline solid is called the unit cell. It is of importance to realize that all crystalline solids, no matter how complex, are described by unit cells. The unit cell must be consistent with the chemical formula of the solid, must indicate the coordination number and geometry of each type of atom or ion, and must generate the crystal structure by simple translation or displacement of the unit cell in three dimensions. 

The structures adopted by crystalline solids are those that bring particles in closest contact to maximize the attractive forces between them. In many cases the particles that make up the solids are spherical or approximately so. Such is the case for atoms in metallic solids. It is therefore instructive to consider how equal-sized spheres can pack most efficiently (that is, with the minimum amount of empty space). 

There are two classes of solids that are not crystalline, that is, p(r) is not periodic. The more familiar one is a glass, for which there are again two models, which may be called the random network and tlie random packing of hard spheres. An example of the first is silica glass or fiised quartz. It consists of tetrahedral SiO groups that are linked at their vertices by Si-O-Si bonds, but, unlike the various crystalline phases of Si02, there is no systematic relation between... 

When we determined the crystalline structure of solids in Chapter 4, we noted that most transitional metals form crystals with atoms in a close-packed hexagonal structure, face-centered cubic structure, or body-centered cubic arrangement. In the body-centered cubic structure, the spheres take up almost as much space as in the close-packed hexagonal structure. Many of the metals used to make alloys used for jewelry, such as nickel, copper, zinc, silver, gold, platinum, and lead, have face-centered cubic crystalline structures. Perhaps their similar crystalline structures promote an ease in forming alloys. In sterling silver, an atom of copper can fit nicely beside an atom of silver in the crystalline structure. 

When all the rotations are possible in the solid state the symmetry increases to hexagonal. This form corresponds to the close packing of spheres or cylinders and the molecule is in a rotational crystalline state, characterized by rigorous order in the arrangement of the center (axes) of the molecules and by disordered azimuthal rotations [118]. If the chain molecules are azimuthally chaotic (they rotate freely around their axes), their average cross sections are circular and, for this reason, they choose hexagonal packing. The ease of rotation of molecules in the crystal depends merely on the molecular shape, as in molecules of an almost spherical shape like methane and ethane derivatives with small substituents, or molecules of a shape close to that of a cylinder (e.g., paraffin-like molecules). 

Both Fade approximant " and the more powerful Tova approximant" predictions of the tails of the virial series are consistent with the location of the first-order pole at the crystal close-packed density, as required by the closure for the virial series. In fact, Baram and Luban" give this as a conclusion of their work. The known virial coefficients in the soft-sphere, inverse twelfth power models also imply that the virial series contains information on the crystalline phase at very high pressure, but is unrelated to the freezing transition, the glass transition, or the amorphous solid equation of state. °... 

One of the driving forces for crystallization is the maximum occupancy of space. This is particularly true for metallic solids, whose crystalline structures can often be considered as the packing of identically sized spheres. Metallic bonding is considered to be nondirectional. Unlike the nonmetals, which can fill their valence shells by sharing only a few pairs of electrons, the metals require much larger coordination numbers in order to satisfy their outermost valence shells. As a result, the metals... 

By now, you should have some appreciation for the different types of crystalline lattices and have some idea of the difficulty in predicting the exact structure that a compound will form. There are a variety of factors involved in the process of crystallization, including (but not limited to) closest packing of spheres, closest packing of polyhedra, electrostatic interactions, hydrogen bonding, hybridization, crystal field effects, and the degree of polarization. When it comes to the solid state, simple stoichiometries do not necessarily imply simple structures. 

The geometrical structure of pores is of great concern, but the three-dimensional description of pores is not established in less-crystalline porous solids. Only intrinsic crystalline intra-particle pores offer a good description of the structure. The hysteresis analysis of molecular adsorption isotherms and electron microscopic observation estimate the pore geometry such as cylinder (cylinder closed at one end or cylinder open at both ends), cone shape, slit shape, interstice between closed-packing spheres and inkbottle. However, these models concern with only the unit structures. The higher order structure of these unit pores such as the network structure should be taken into accoimt. The simplest classification of the higher order structures is one-, two- and three-dimensional pores. Some zeolites and aluminophosphates have one-dimensional pores and activated carbons have basically two-dimensional slit-shaped pores with complicated network structures [95]. 

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