Crystalline solids unit cells and

Crystalline Solids Unit Cells and Basic Structures 520... 

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

Most solids are crystalline solids. The atoms and molecules in crystalline solids are arranged in definite geometric patterns. Each geometric pattern piece is called a unit cell. Unit cells repeat over and over in the solid. Think of each unit cell as a building block. Within each crystalline solid, the exact organization of the atoms... 

In order to discuss the selection rules for crystalline lattices it is necessary to consider elementary theory of solid vibrations. The treatment essentially follows that of Mitra (47). A crystal can be regarded as a mechanical system of nN particles, where n is the number of particles (atoms) per unit cell and N is the number of primitive cells contained in the crystal. Since N is very large, a crystal has a huge number of vibrations. However, the observed spectrum is relatively simple because, as shown later, only where equivalent atoms in primitive unit cells are moving in phase as they are observed in the IR or Raman spectrum. In order to describe the vibrational spectrum of such a solid, a frequency distribution or a distribution relationship is necessary. The development that follows is for a simple one-dimensional crystalline diatomic linear lattice. See also Turrell (48). 

At an earlier point (p. 185) the unit cell of a crystalline structure was described as one of a large number of identical prisms, which, when oriented in the same way and stacked together in three dimensions, form a perfect crystal. The corners of an array of unit cells put together in this way are said to be the points of a space lattice the surroundings about each point of the lattice must be identical to the surroundings about every other point. Additional lattice points may sometimes be put at the face centers or at the body centers of the unit cells in crystalline sodium chloride (Fig. 12-2), for example, chloride ions are located both at the corners and the face centers of the unit cell, and an observer at a corner would have the same surroundings as one at a face center. The description of the structure of a crystalline solid is then a description of the size and shape of the unit cell and of the locations of the atoms within it. 

Polymorphism refers to the situation in which materials of the same chemical composition possess different sub-cell packings in the solid state (Small, 1986). Figure 7.3 compares the level of structure for the unit cell and sub-cell in crystalline fats. Polymorphism arises from both variations in the tilt of TAG molecules in a bilayer and from variations in the hydrocarbon chain packing (Larsson, 1994). 

In the case of crystaUine sohds, more than one equivalent structural unit may be present in the primitive cell. This results in sphttings of the fundamental vibrational modes of these units. In the case of many crystalline solid materials covalent units (e.g. oxo-anions for oxo-salts) are present, together with other groups bonded by ionic bonds (e.g. the cations in the oxo-salts). According to the above group approximation, the internal vibrations of the covalent units can be considered separately from their external vibrations hindered rotations and translations of the group that finally contribute to the lattice vibrations and to the acoustic modes of the unit cell) and those of the other units. The presence of a number of covalent structural units in the primitive cell, causes their internal modes to spHt... 

The main consequences of each step of the preparation procedures on the physicochemical properties of the solids (unit cell parameter, X-ray crystallinity, etc.) have already been described in ref. (18). However for the sake of clarity, the most important characteristics of the solids are given again in Tables I and II together with the dealumination operating conditions. 

Crystalline solids consist of particles tightly packed into a regular array called a crystal lattice. The unit cell is the simplest portion of the crystal that, when repeated, gives the crystal. Many substances crystallize in one of three types of cubic unit cells, which differ in the arrangement of the particles and, therefore, in the number of particles per unit cell and how efficiently they are packed. 

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)... 

Band Structures. - Conceptually the simplest approximation amounts to assuming that the crystalline solid is infinite and periodic in all three dimensions. Then, when expanding the solutions to the Kohn-Sham (or Hartree-Fock) single-particle equations in a set of atom-centered basis functions [cf. Eq. (13)], the translational symmetry can be utilized in constructing symmetry-adapted basis functions from the identical ones of different unit cells, i.e.,... 

The regular arrangement of the components of a crystalline solid at the microscopic level produces the beautiful, characteristic shapes of crystals, such as those shown in Fig. 10.8. The positions of the components in a crystalline solid are usually represented by a lattice, a three-dimensional system of points designating the positions of the components (atoms, ions, or molecules) that make up the substance. The smallest repeating unit of the lattice is called the unit cell. Thus a particular lattice can be generated by repeating the unit cell in all three dimensions to form the extended structure. Three common unit cells and their lattices are shown in Fig. 10.9. Note from Fig. 10.9 that the extended structure in each case can be viewed as a series of repeating unit cells that share common faces in the interior of the solid. 

In a crystalline solid there is a relatively small repeating unit, called a unit cell, that is made up of a unique arrangement of atoms and embodies the structure of the solid. The structure of the crystal can be built by stacking this unit over and over in all three dimensions. Thus, the structure of a crystalline solid is defined by (a) the size and shape of the unit cell and (b) the locations of atoms within the unit cell. 

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. 

---