Schottky defect equilibrium population

It is possible to create a population of Schottky defects that is much higher than the equilibrium population that is based on Eq. (7.32). If a crystal is heated to high temperature, lattice vibrations become more pronounced, and eventually ions begin to migrate from their lattice sites. If the crystal is quickly cooled, the extent of the motion of ions decreases rapidly so that ions that have moved from their lattice sites cannot return. As a result, the crystal will contain a population of Schottky defects that is much higher than the equilibrium population at the lower temperature. If a crystal of KC1 is prepared so that it contains some CaCl2 as an impurity, incorporating a Ca2+ ion in the crystal at a K+ site... 

A Schottky defect in a crystal consists of a cation and anion vacancy combination that ensures overall electroneutrality in the crystal (Section 1.9). The estimation of the configurational entropy change in creating a population of Schottky defects in a crystal can be obtained in the same way as that of a population of vacancies in a monatomic crystal. The method follows that given in Section 2.1 for the equilibrium concentration of vacancies in a monatomic crystal and is set out in detail in Supplementary Material S4. 

Because Schottky defects are present as equilibrium species, the defect population can be treated as a chemical equilibrium. For a crystal of composition MX ... 

The estimation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects by estimating the configurational entropy (Supplementary Material S4). This approach confirms that Frenkel defects are thermodynamically stable intrinsic defects that cannot be removed by thermal treatment. Because of this, the defect population can be treated as a chemical equilibrium. For a crystal of composition MX, the appropriate chemical equilibrium for Frenkel defects on the cation sublattice is... 

Self-diffusion of Ag cations in the silver halides involves Frenkel defects (equal numbers of vacancies and interstitials as seen in Fig. 8.116). In a manner similar to the Schottky defects, their equilibrium population density appears in the diffusivity. Both types of sites in the Frenkel complex—vacancy and interstitial— may contribute to the diffusion. However, for AgBr, experimental data indicate that cation diffusion by the interstitialcy mechanism is dominant [4]. The cation Frenkel pair formation reaction is... 

A population of vacancies on one subset of atoms created by displacing some atoms into normally unoccupied interstitial sites constitute a second arrangement of paired point defects, termed Frenkel defects (Figure 2(b), (c)). Because one species of atom or ion is simply being redistributed in the crystal, charge balance is not an issue. A Frenkel defect in a crystal of formula MX consists of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. Equally, a Frenkel defect in a crystal of formula MX2 can consist of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. As with the other point defects, it is found that the free energy of a crystal is lowered by the presence of Frenkel defects and so a popnlation of these intrinsic defects is to be expected at temperatures above 0 K. The calculation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects. The appropriate chemical equilibrium for cation defects is ... 

Under equilibrium conditions, the Gibbs energy of a crystal, G, is lower if it contains a small population of Schottky defects, similar to the situation shown in Figure 3.12. This means that Schottky defects will always be present in crystals at temperatures above 0 K, and hence Schottky defects are intrinsic defects. The approximate number of Schottky defects, ns, in crystal with a formula MX, at equilibrium, is given by ... 

We give some experimental values for the enthalpy of formation of Schottky defects in Table 11.4. We can use these numbers to calculate equilibrium defect concentrations as we have for NaCl in Table 11.5. The population of point defects is very low, but it is clear from Eq. Box 11.1 that vacancies are stable in the crystal at any temperature above absolute zero. Because energies for point defect formation in stoichiometric oxides such as... 

An intrinsic defect is one that is in thermodynamic equilibrium in the crystal. This means that a population of these defects cannot be removed by any forms of physical or chemical processing. Schottky, Frenkel, and antisite defects are the best characterized intrinsic defects. A totally defect-free crystal, if warmed to a temperature that allows a certain degree of atom movement, will adjust to allow for the generation of intrinsic defects. The type of intrinsic defects that form will depend upon the relative formation energies of all of the possibilities. The defect with the lowest formation energy will be present in the greatest numbers. This can change with temperature. 

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