Vacant lattice sites

A crystalline solid is never perfect in that all of tire lattice sites are occupied in a regular manner, except, possibly, at the absolute zero of temperature in a perfect crystal. Point defects occur at temperatures above zero, of which the principal two forms are a vacant lattice site, and an interstitial atom which... 

The smallest imperfections in metal crystals are point defects, in particular vacant lattice sites (vacancies) and interstitial atoms. As illustrated in Fig. 20.21a, a vacancy occurs where an atom is missing from the crystal structure... 

The electrochemical intercalation/insertion is not a special property of graphite. It is apparent also with many other host/guest pairs, provided that the host lattice is a thermodynamically or kinetically stable system of interconnected vacant lattice sites for transport and location of guest species. Particularly useful are host lattices of inorganic oxides and sulphides with layer or chain-type structures. Figure 5.30 presents an example of the cathodic insertion of Li+ into the TiS2 host lattice, which is practically important in lithium batteries. 

The notion of point defects in an otherwise perfect crystal dates from the classical papers by Frenkel88 and by Schottky and Wagner.75 86 The perfect lattice is thermodynamically unstable with respect to a lattice in which a certain number of atoms are removed from normal lattice sites to the surface (vacancy disorder) or in which a certain number of atoms are transferred from the surface to interstitial positions inside the crystal (interstitial disorder). These forms of disorder can occur in many elemental solids and compounds. The formation of equal numbers of vacant lattice sites in both M and X sublattices of a compound M0Xft is called Schottky disorder. In compounds in which M and X occupy different sublattices in the perfect crystal there is also the possibility of antistructure disorder in which small numbers of M and X atoms are interchanged. These three sorts of disorder can be combined to give three hybrid types of disorder in crystalline compounds. The most important of these is Frenkel disorder, in which equal numbers of vacancies and interstitials of the same kind of atom are formed in a compound. The possibility of Schottky-antistructure disorder (in which a vacancy is formed by... 

We turn now to the microscopic description of an imperfect crystal. The various defects in any imperfect crystal can be imagined to be formed from a corresponding perfect crystal by one or more of the following processes (a) remove an atom of species Os from the crystal leaving a vacant lattice site, (b) remove an atom of species Os from the crystal and replace it by an atom of a different species (either Oi or at), (c) add to the crystal an atom of any species to a site on a sublattice unoccupied in the perfect crystal. We refer to the latter as atoms in interstitial positions. Let B be a set of numbers such that Br is the number of sites on sublattice number r in the perfect crystal, and let be the number of sublattices in the crystal (including interstitial sublattices not occupied in the perfect crystal). The total number of sites of all kinds in the perfect crystal is then... 

Fig. 3-6. Ions on the surface and in the interior of solids 0=occupied or vacant lattice site - surface kink site (S> = surface adsorption site = surface lattice vacancy, (f) = step plane = terrace ai = unitary level of occupied or vacant lattice site ions a = unitary level of surface kink site ions.

Since the concentration of kink sites ( k 10 to 10 in the molar fraction) on the metal surface is much greater than the concentration of vacant lattice sites (xy = 10 to lO in the molar fraction) in the interior, the metal ion level of a metal phase is close to the unitary level of surface metal ions a . as shown in Eqn. 3-12 and in Fig. 3-6 ... 

Similarly, we obtain the ion level in the interior of the semiconductor as a function of both the energy level of holes in the interior Ey and the concentration of vacant lattice sites in the same way as Eqn. 3-10 as shown in Eqn. 3-16 ... 

States due to different biographical structural defects existing on any real surface and playing the part of local disturbances in the strictly periodic structure of the surface (Sec. IX,A). These include vacant lattice sites in the surface layer of the lattice, atoms or ions of the lattice ejected onto the surface, and foreign atomic inclusions in the surface of the lattice (surface impurities). 

It must be realized that actually for each oxygen ion built into the lattice, according to (i) a vacant lattice site must be created in the sublattice of nickel ions. This is due to the geometrical impossibility of accommodating excess oxygen in the lattice. Excess oxygen really means nickel deficiency. More complex notations than the notation used here are necessary to deal with this situation (51) but for our purpose we need not go into this. If now the ionization equilibrium... 

In addition to stress, the other important influence on solid state kinetics (again differing from fluids) stems from the periodicity found within crystals. Crystallography defines positions in a crystal, which may be occupied by atoms (molecules) or not. If they are not occupied, they are called vacancies. In this way, a new species is defined which has attributes of the other familiar chemical species of which the crystal is composed. In normal unoccupied sublattices (properly defined interstitial lattices), the fraction of vacant sites is close to one. The motion of the atomic structure elements and the vacant lattice sites of the crystal are complementary (as is the motion of electrons and electron holes in the valence band of a semiconducting crystal). 

Chemical solid state processes are dependent upon the mobility of the individual atomic structure elements. In a solid which is in thermal equilibrium, this mobility is normally attained by the exchange of atoms (ions) with vacant lattice sites (i.e., vacancies). Vacancies are point defects which exist in well defined concentrations in thermal equilibrium, as do other kinds of point defects such as interstitial atoms. We refer to them as irregular structure elements. Kinetic parameters such as rate constants and transport coefficients are thus directly related to the number and kind of irregular structure elements (point defects) or, in more general terms, to atomic disorder. A quantitative kinetic theory therefore requires a quantitative understanding of the behavior of point defects as a function of the (local) thermodynamic parameters of the system (such as T, P, and composition, i.e., the fraction of chemical components). This understanding is provided by statistical thermodynamics and has been cast in a useful form for application to solid state chemical kinetics as the so-called point defect thermodynamics. 

The model of Marchetti et al. is based on the compressible lattice theory which Sanchez and Lacombe developed to apply to polymer-solvent systems which have variable levels of free volume [138-141], This theory is a ternary version of classic Flory-Huggins theory, with the third component in the polymer-solvent system being vacant lattice sites or holes . The key parameters in this theory which affect the polymer-solvent phase diagram are ... 

Defects which have extent of only about an atomic diameter also exist in crystals—the point defects. Vacant lattice sites may occur—vacancies. Extra atoms—interstitials—may be inserted between regular crystal atoms. Atoms of the wrong chemical species—impurities—also may be present. 

On the basis of thermodynamic considerations, some of the lattice sites in the crystal are vacant, and the number of vacant lattice sites generally is a function of temperature. The movement of a lattice atom into an adjacent vacant site is called vacancy diffusion. In addition to occupying lattice sites, atoms can reside in interstitial sites, the spaces between the lattice sites. These interstitial atoms can readily move to adjacent interstitial sites without displacing the lattice atoms. This process is called interstitial diffusion. The interstitial atoms may be impurity atoms or atoms of the host lattice, but in either case, interstitial atoms are generally present only in very dilute amounts. However, these atoms can be highly mobile, and in certain cases, interstitial diffusion is the dominant diffusion mechanism. 

A common lattice defect is a vacant lattice site or vacancy. The presence of a vacancy increases the enthalpy by Ahf and the entropy by As/. Because the free energy of a system is lowered by increased entropy, there is an equilibrium fraction of vacant lattice, xv, which increases with temperature. At equilibrium, A gf = Ahf — T A Sf = 0, so A hf = —T Asf. From statistical mechanics, ASf = —klnx , so... 

As is well known, in thermal equilibrium, every crystal, at a temperature above absolute zero encloses a certain number of vacant lattice sites, where the probability, P, of finding a vacancy in a solid at equilibrium at an absolute temperature, T, is given by... 

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