Schottky equilibria

The approximations to use depend upon the pressure regime and the values of the equilibrium constants. This oxide is an insulator under normal conditions, and so, in the middle region of the diagram, Schottky equilibrium is dominant, that is, > Ke and the electroneutrality equation is approximated by... 

Derive the polynomial expressions for the concentration dependence of electrons, metal vacancies, and anion vacancies for a compound MX showing Schottky equilibrium (Section 7.9.4). Insert values of the equilibrium constants to obtain polynomial expressions for the cases where electronic defects dominate. 

Insertion of the equilibrium constants will result in equations that can be solved for [h ] as a function of px2, which, in turn, can be used to determine the values of the other defects as a function of pXl (Section 7.10.2). The diagrams for donor and acceptor doping of MX in which electronic defects dominate over Schottky equilibrium (Fig. 8.2a and 8.2c) can be compared to that for undoped material (Fig. 7.11), redrawn here (Fig. 8.2b). 

Consider a Schottky equilibrium taking place in a forsterite crystal and involving Mg cations and anions, displacing them from their lattice positions and transporting them to the surface of the crystal ... 

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... 

Figure 2-2. a) Fraction /Vp of defect pairs (e.g., [B.V] in Schottky-disordered AX) as a function of the normalized temperature (R/ AGp )-T for various dopant concentrations Nb) NP as a function of at given T. The parameter Ks denotes the Schottky equilibrium constant FVv-A )-... 

At moderate partial pressures of X2, the assumption made above is replaced by the assumption that the crystal is an insulator and Schottky equilibrium is dominant. In this case ... 

In other words, the oxygen atoms from the gas phase are incorporated into the crystal by filling the oxygen vacancies present, which naturally decreases their concentration. This, in turn, must increase the cation vacancy concentration to maintain the Schottky equilibrium, i.e.. Eq. (6.9). 

Combining this result with the expression for D n reveals the observed behavior. The physics of the situation can be summed up as follows Increasing the oxygen partial pressure (i.e., going from left to right in Fig. 7.4) decreases the concentration of oxygen vacancies which, in order to maintain the Schottky equilibrium, results in an increase in [V n] and a concomitant increase in Divin-... 

Ks = KJLVa ] and in NiO 0 V, + Vo = MKo]. Schottky equilibrium leads to vacancy pairs and this type of disorder dominates in alkaline halides and in oxides with a halite (NaCl) structure. 

Another kind of disorder equilibrium exists between A vacancies and B vacancies in a binary AB crystal—the so-called Schottky equilibrium. In this case the exchange of particles between A and B vacancies and the crystal is considered. This means that either A or B particles are transferred to the surface of the crystal, which is thus enlarged, while A or B vacancies are generated, or vice versa. For example, the A vacancies may be singly negative and the B vacancies singly positive. When the particles are brought to the surface, two of the particles there will be... 

In the Schottky types of disorder, a pair of ions moves from the normal sites to the surface of the solid [2—4]. For Lil the Schottky equilibrium can be expressed by... 

In this oxygen excess region, the formation of cation vacancies will have the effect of strongly decreasing the oxygen vacancy concentration through the Schottky equilibrium. 

Although such equiUbria for undoped oxides are, as already said, necessarily a basis for dissolution of protons in the doped cases, their thermodynamics—a combination of hydration of oxygen vacancies and Schottky equilibrium—is little studied. 

The first equilibrium is certainly the Schottky equilibrium for the neutral vacancies and is thus written as ... 

Here it is the Schottky equilibrium with respect to the SrO component that is meant and, hence, the formation of Sr + and 0 " vacancies with the formation of an SrO deficiency with respect to Ti02- On account of the high vapour pressure of PbO, the same applies significantly at relatively low temperatures to the Pb titanates or zirconates, which are important as actuators. 

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