Crystal, defect, point packing

It should be clear that the presence of line defects in a crystal lattice leads to a disruption of the continuity of the lattice just as the presence of point defects affects the packing of a given lattice. The line defect. 

At all temperatures above 0°K Schottky, Frenkel, and antisite point defects are present in thermodynamic equilibrium, and it will not be possible to remove them by annealing or other thermal treatments. Unfortunately, it is not possible to predict, from knowledge of crystal structure alone, which defect type will be present in any crystal. However, it is possible to say that rather close-packed compounds, such as those with the NaCl structure, tend to contain Schottky defects. The important exceptions are the silver halides. More open structures, on the other hand, will be more receptive to the presence of Frenkel defects. Semiconductor crystals are more amenable to antisite defects. 

In retrospect, one can understand why solid state chemists, who were familiar with crystallographic concepts, found it so difficult to imagine and visualize the mobility of the atomic structure elements of a crystal. Indeed, there is no mobility of these particles in a perfect crystal, just as there is no mobility of an individual car on a densely packed parking lot. It was only after the emergence of the concept of disorder and point defects in crystals at the turn of this century, and later in the twenties and thirties when the thermodynamics of defects was understood, that the idea... 

Since the state of a crystal in equilibrium is uniquely defined, the kind and number of its SE s are fully determined. It is therefore the aim of crystal thermodynamics, and particularly of point defect thermodynamics, to calculate the kind and number of all SE s as a function of the chosen independent thermodynamic variables. Several questions arise. Since SE s are not equivalent to the chemical components of a crystalline system, is it expedient to introduce virtual chemical potentials, and how are they related to the component potentials If immobile SE s exist (e.g., the oxygen ions in dense packed oxides), can their virtual chemical potentials be defined only on the basis of local equilibration of the other mobile SE s Since mobile SE s can move in a crystal, what are the internal forces that act upon them to make them drift if thermodynamic potential differences are applied externally Can one use the gradients of the virtual chemical potentials of the SE s for this purpose ... 

From the fact that the structure of hexamethylethane is close-packed, we may infer that the point defects responsible for the translational motion of the molecules are monovacancies. The data indicate an overexponential increase in diffusivity with temperature near Tm, analogous to some observations made on inorganic crystals. Various explanations could be given for this increase such as anharmonicity of the... 

A major difference between crystals and fluids refers to the necessity of distinguishing between different sites. So the autoprotolysis in water could, just from a mass balance point of view, also be considered e.g. as a formation of a OH vacancy and a IT vacancy. In solids such a disorder is called Schottky disorder (S) and has to be well discerned from the Frenkel disorder (F). In the densely packed alkali metal halides in which the cations are not as polarizable as the Ag+, the formation of interstitial defects requires an unrealistically high energy and the dominating disorder is thus the Schottky reaction... 

The concept of a zero-dimensional intrinsic point defect was first introduced in 1926 by the Russian physicist Jacov Il ich Frenkel (1894-1952), who postulated the existence of vacancies, or unoccupied lattice sites, in alkali-halide crystals (Frenkel, 1926). Vacancies are predominant in ionic solids when the anions and cations are similar in size, and in metals when there is very little room to accommodate interstitial atoms, as in closed packed stmctures. The interstitial is the second type of point defect. Interstitial sites are the small voids between lattice sites. These are more likely to be occupied by small atoms, or, if there is a pronounced polarization, to the lattice. In this way, there is little dismption to the stmcture. Another type of intrinsic point defect is the anti-site atom (an atom residing on the wrong sublattice). 

We have discussed the packing of ions in terms of coordination polyhedra. When we create defects in a crystal we can create new polyhedra that are not found in the perfect crystal. Pauling s rules were developed for perfect crystals, but the principles still apply when we examine defects. One complication is that as we introduce grain boundaries, for example, new sites are produced that depend on the detailed nature of the grain boundary. Amorphous materials present a new challenge when describing point defects. Two amorphous materials can have different structures that depend on the processing history even if the chemistry is the same. 

Zero-dimensional defects or point defects conclude the list of defect types with Fig. 5.87. Interstitial electrons, electron holes, and excitons (hole-electron combinations of increased energy) are involved in the electrical conduction mechanisms of materials, including conducting polymers. Vacancies and interstitial motifs, of major importance for the explanation of diffusivity and chemical reactivity in ionic crystals, can also be found in copolymers and on co-crystallization with small molecules. Of special importance for the crystal of linear macromolecules is, however, the chain disorder listed in Fig. 5.86 (compare also with Fig. 2.98). The ideal chain packing (a) is only rarely continued along the whole molecule (fuUy extended-chain crystals, see the example of Fig. 5.78). A most common defect is the chain fold (b). Often collected into fold surfaces, but also possible as a larger defect in the crystal interior. Twists, jogs, kinks, and ends are other polymer point defects of interest. 

The defects which disrupt the regular patterns of crystals, can be classified into point defects (zero-dimensional), line defects (1-dimensional), planar (2-dimensional) and bulk defects (3-dimensional). Point defects are imperfections of the crystal lattice having dimensions of the order of the atomic size. The formation of point defects in solids was predicted by Frenkel [40], At high temperatures, the thermal motion of atoms becomes more intensive and some of atoms obtain energies sufficient to leave their lattice sites and occupy interstitial positions. In this case, a vacancy and an interstitial atom, the so-called Frenkel pair, appear simultaneously. A way to create only vacancies has been shown later by Wagner and Schottky [41] atoms leave their lattice sites and occupy free positions on the surface or at internal imperfections of the crystal (voids, grain boundaries, dislocations). Such vacancies are often called Schottky defects (Fig. 6.3). This mechanism dominates in solids with close-packed lattices where the formation of vacancies requires considerably smaller energies than that of interstitials. In ionic compounds also there are defects of two types, Frenkel and Schottky disorder. In the first case there are equal numbers of cation vacancies... 

Our first example of a reaction in inhomogeneous metallic systems will be the equilibration of vacancies. In close-packed elemental crystals such as Cu or Al, essentially the only point defects are vacancies - as long as the metal is sufficiently pure. According to Gibbs, the... 

As pointed out in the previous section, there are voids between the double-twist cylindas when they are packed in 3-D space. Liquid crystal must fill the void. Because of the boundary condition imposed by the cylinders, the hquid crystal director is not uniform in this space and forms a defect... 

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