Point defect equilibria

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or 

Whenever electrons and holes are generated in a solid through ionization, another equilibrium is set up 0 = c +h. The corresponding intrinsic equilibrium constant is 

In the above equations necessary simplifications such as [Vx] 1, [V,] [Xx] = 1 etc. have been introduced. Although it is possible in principle to obtain solutions for these equations once the various K values are known, the form of equation (5.9) makes it very tedious. Brouwer made the approximation that for large negative deviations from stoichiometry, [Vm] and [p] become negligible  

We thus have [Vx] = KJR from equation (5.3) and [n] = [V ] = 7 vx[ x] = K /xiRJRV from equation (5.6). Since the Schottky equilibrium constants are known for ionized defects, it is profitable to express in terms of using the relation 

At large positive deviations from stoichiometry, we can use the approximation 

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Antisite defect equilibria can be treated in the same way as the other point defect equilibria. The creation of a complementary pair of antisite defects consisting of an A atom on a B atom site, AB, and a B atom on an A atom site, BA, can then be written ... 

Some earlier thermodynamic studies on rutile reported expressions involving simple idealized quasi-chemical equilibrium constants for point defect equilibria (see, e.g., Kofstad 1972) by correlating the composition x in TiOx with a function of AGm (O2), which is the partial molar free energy of oxygen. However, the structural effects were not accounted for in these considerations. Careful measurements of AGm (O2) in the TiOjc system (Bursill and Hyde 1971) have indicated that complete equilibrium is rarely achieved in non-stoichiometric rutile. 

Solid-solid reactions are as a rule exothermic, and the driving force of the reaction is the difference between the free energies of formation of the products and the reactants. A quantitative understanding of the mechanism of solid-solid reactions is possible only if reactions are studied under well-defined conditions, keeping the number of variables to a minimum. This requires single-crystal reactants and careful control of the chemical potential of the components. In addition, a knowledge of point-defect equilibria in the product phase would be useful. 

Transport plays the overwhelming role in solid state kinetics. Nevertheless, homogeneous reactions occur as well and they are indispensable to establishing point defect equilibria. Defect relaxation in the (p-n) junction, as discussed in the previous section, illustrates this point, and similar defect relaxation processes occur, for example, in diffusion zones during interdiffusion [G. Kutsche, H. Schmalzried (1990)]. 

Up to this point we have assumed implicitly that each defect responsible for the atomic motion has an infinite lifetime. In real crystals, however, this lifetime is finite because of the dynamic nature of the point defect equilibria. This means that only m consecutive jumps are correlated (corresponding to the defect lifetime). It has been shown [R. Kutner (1985)] that under these conditions... 

The rare earth elements are different from other elements because the optical transitions between levels of the fn configuration are inherently very sharp-lined and have well-resolved structure characteristic of the local crystal fields around the ion. In minerals, this characteristic provides an excellent probe of the local structure at the atomic level. Examples will be shown from our work of how site selective laser spectroscopy can be used to determine the thermal history of a sample, the point defect equilibria that are important, the presence of coupled ion substitution, the determination of multiple phases, and stoichiometry of the phase. The paper will also emphasize the fact that the usefulness and the interpretation of the rare earth luminescence is complicated by the presence of quenching and disorder in mineral samples. One in fact needs to know a great deal about a sample before the wealth of information contained in the site selective luminescence spectrum can be understood. 

These basic techniques can be used in a number of ways to get detailed information about point defect equilibria and dynamics. 

Shear Plane-Point Defect Equilibria.—The question of the existence of point defects in compounds where extended defects are known to occur has been controversial. Indeed, it has occasionally been claimed that point defects cannot form in such phases and that they will always be eliminated with the formation of extended structures. We reject these latter arguments as thermodynamically unsound. From a thermodynamic standpoint, the formation of extended defects can be viewed as a special mode of point defect aggregation as such, shear planes will be in equilibrium with point defects, with the position of the equilibrium depending on both temperature and the extent of the deviation from stoicheiometry. Thus, if we assume, as is suggested by our calculations, that anion vacancies are the predominant point defects in reduced rutile (a further point of controversy as mentioned above) then there will exist an equilibrium of the type... 

Equations for oxygen transport can be derived from the point defect equilibria discussed in Section 10.6.2.2. This provides us with some general insight... 

Wright, J.C., 1985, Laser Spectroscopy of Point Defect Equilibria in Insulators Transitions of Insulators to Superionic State, in Proc. Int. Conf. on Defects in Insulating Crystals, Cryst. Lattice Defects and Amorphous Mat., Vol. 12, ed. F. Liithi (Gordon and Breach, New York) pp. 505. 

For the thermodynamic treatmerrt of the point defect equilibria, one has to take into accoimt the electroneutrality condition... 

In ternary compounds of the type (A,B)X, the point defect concentrations and hence the deviation from stoichiometry depend on the concentrations of the two cations. Since in general the equilibrium constants of the different point defect equilibria in ternary compounds are composition dependent, their thermodynamic treatment is mote difficult, and usually much more experimental data are required than in the case of the binary compounds. The complexity of the ternary compounds is the reason why point defect chemistry has been worked out more or less quantitatively only for some oxide and fluoride systems, using the concepts of Kroger and Vink, Kroger, Schmalzried and Wagner, and Schmalzried. ... 

Equations for oxygen transport can be derived from the point defect equilibria discussed in Section V.B.2. This provides ns with some general insight into the transport behavior of oxygen-deficient perovskites. Strictly speaking, the equations presented below are valid at low defect concentrations only, i.e., assuming oxygen defects to be randomly distributed. 

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