Crystalline solid simple cubic

Amorphous solid Crystalline solid Simple cubic... 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

We have shown the least complicated one which turns out to be the simple cubic lattice. Such bands are called "Brilluoin" zones and, as we have said, are the allowed energy bands of electrons in any given crystalline latttice. A number of metals and simple compounds have heen studied and their Brilluoin structure determined. However, when one gives a representation of the energy bands in a solid, a "band-model is usually presented. The following diagram shows three band models ... 

Under the influence of such a potential, the spheres easily settle into a crystalline form when the temperature is lowered or the pressure is raised. The solid phase can be either a face-centered cubic (fee) lattice or a body-centered cubic (bee) lattice, depending on the detailed nature of the potential. In the case of argon, the crystal freezes into an fee lattice, while liquid sodium freezes into a bee lattice. The transformation of simple spherical molecules (such as argon and sodium) into the crystalline solid phase is now rather well understood, largely because of the extensive use of computer simulation studies, accompanied by theoretical analysis using methods of statistical mechanics. 

A unit cell is the basic repeating structural unit of a crystalline solid. Figure 11.14 shows a unit cell and its extension in three dimensions. Each sphere represents an atom, ion, or molecule and is called a lattice point. In many crystals, the lattice point does not actually contain such a particle. Rather, there may be several atoms, ions, or molecules idaitically arranged about each lattice point. For simplicity, however, we can assume that each lattice point is occupied by an atom. This is certainly the case with most metals. Every aystalline solid can be described in terms of one of the seven types of unit cells shown in Hgure 11.15. The geometry of the cubic unit cell is particularly simple because aU sides and aU angles are equal Any of the unit cells, when repeated in space in all three dimensions, forms the lattice structure characteristic of a crystalline solid. 

Fig. 5 Diagram of states for a tethered chain, destaibed by the bond fluctuation model on the simple cubic lattice, for chain length N = 64, in the plane of variables bulk coupling fii, and surface coupling 5. The solid lines show the locatimi of well-deflned maxima in the fluctuation of surface contacts or bead-bead contacts, respectively broken and dotted lines show the locations of less well-pronounced anomalies in these fluctuations. In the hatched region, the precise behavior is still uncertain. The states that compete with each other are desorbed expanded (DE, i.e., a three-dimensional mushroom) adsorbed expanded (AE, i.e., a d = 2 SAW), desorbed collapsed (DC, as in Fig. lb, but tethered to the grafting plane) adsorbed collapsed (AC, a compact but disordered structure with many surface contacts) and various crystalline layered structures (IS). Reprinted with permission from [5]. Copyright 2008, American Institute of Physics...

Figure 15.1 shows the three ways the atoms of a crystalline solid can be arranged. As a molecule goes from a simple cubic structure to a face-centered cubic structure, the density increases. The less space between the atoms, the more tightly packed the entire molecule, and the harder and less flexible. Unlike amorphous solids, a lattice structure provides for predictable breaks along set lines. This is the reason why diamonds and gemstones can be cut into facets. The round, oval, pear, emerald cut, and diamond-shaped cuts used in jewelry can be cut by dilferent gem cutters all over the world due to their characteristic lattice structures. 

The reader would be familiar with the packing of atoms in crystalline solids to produce regular, repeating, three-dimensional patterns such as the simple cubic, body-centered cubic, face-centered cubic, and hexagonal close-packed structures. The packing density and coordination number of these crystal structures for a pure metal are listed in Table 6.2. 

Section 11.7 In a crystalline soUd, particles are arranged in a regularly repeating pattern. An amorphous solid is one whose particles show no such order. The essential structural features of a crystalline solid can be represented by its unit cell, toe smallest part of toe crystal that can, by simple displacement, reproduce the three-dimensional structure. The three-dimensional structures of a crystal can also be represented by its crystal lattice. The points in a crystal lattice represent positions in toe structure where there are identical environments. The simplest unit cells are cubic. There are three kinds of cubic unit cells primitive cubic, body-centered cubic, and face-centered cubic. 

Every crystalline solid can be described in terms of one of the seven types of unit cells shown in Figure 12.15. The geometry of the cubic unit cell is particularly simple because all sides and all angles are equal. Arty of the imit cells, when repeated in space in all three dimensions, forms the lattice stmcture characteristic of a crystalline solid. 

Solids can be crystalline or amorphous. A crystalline solid has an ordered arrangement of structural units placed at crystal lattice points. We may think of a crystal as constructed from unit celb. Cubic unit cells are of three kinds simple cubic, body-centered cubic, and face-centered cubic. One of the most important ways of determining the structure of a crystalline solid is by x-ray diffraction. 

The crystalline form of interest in Zr-based ceramic compounds is the cubic fluorite structure based on the mineral CaF2. In this structure, consisting of interpenetrating face-centered-cubic and simple cublic arrays of cations (Zr ) and anions (O ), respectively, oxygen ion conductivity is enhanced by replacing zirconium (Zr ) ions on the cation lattice with soluble dopant cations having a valence less than 4, typically divalent (Mg, Ca ) and trivalent (Y, Yb , Sc ) cations. These dopants, which are in solid solution, are incorporated into the zirconia structure by the following types of defect reaction ... 

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