Vacancy point defects equilibrium concentration

In other cases, however, and in particular when sublattices are occupied by rather immobile components, the point defect concentrations may not be in local equilibrium during transport and reaction. For example, in ternary oxide solutions, component transport (at high temperatures) occurs almost exclusively in the cation sublattices. It is mediated by the predominant point defects, which are cation vacancies. The nearly perfect oxygen sublattice, by contrast, serves as a rigid matrix. These oxides can thus be regarded as models for closed or partially closed systems. These characteristic features make an AO-BO (or rather A, O-B, a 0) interdiffusion experiment a critical test for possible deviations from local point defect equilibrium. We therefore develop the concept and quantitative analysis using this inhomogeneous model solid solution. 

The number of interstitial atoms Np in the Frenkel type and the number of vacancies TYj in the Schottky type at thermal equilibrium can be obtained, following a similar calculation to that for the concentration of point defects of elements mentioned in Section 1.3.1, as... 

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 resulting equilibrium concentrations of these point defects (vacancies and interstitials) are the consequence of a compromise between the ordering interaction energy and the entropy contribution of disorder (point defects, in this case). To be sure, the importance of Frenkel s basic work for the further development of solid state kinetics can hardly be overstated. From here on one knew that, in a crystal, the concentration of irregular structure elements (in thermal equilibrium) is a function of state. Therefore the conductivity of an ionic crystal, for example, which is caused by mobile, point defects, is a well defined physical property. However, contributions to the conductivity due to dislocations, grain boundaries, and other non-equilibrium defects can sometimes be quite significant. 

Let us now discuss some details of practical relevance. From the Gibbs phase rule, it is evident that crystals consisting of only one component (A) become nonvariant by the predetermination of two thermodynamic variables, which for practical reasons are chosen to be Pand T. In these one-component systems, it is easy to recognize the (isobanc) concentration dependence of the point defects on temperature. From the definition of the vacancy chemical potential for sufficiently small vacancy mole fractions Nv, namely //v = /A (P, T) + RT- In Vv, together with the condition of equilibrium with the crystal s inerL surroundings (gas, vacuum), one directly finds... 

Interstitial point defects involving normally substitutional atoms will always exist (although typically at very low concentration) at equilibrium in a crystal at finite temperatures because, as in the case of vacancies described above, their enthalpy of formation can always be compensated by a configurational entropy increase. 

The current majority opinion is that both types of point defects are important. Thermal equilibrium concentrations of point defects at the melting point are orders of magnitude lower in Si than in metals. Therefore, a direct determination of their nature by Simmons-Balluffi-type experiments (26) has not been possible. The accuracy of calculated enthalpies of formation and migration is within 1 eV, and the calculations do not help in distinguishing between the dominance of vacancies or interstitials in diffusion. The interpretation of low-temperature experiments on the migration of irradiation-induced point defects is complicated by the occurrence of radiation-induced migration of self-interstitials (27, 28). 

Point Defect Models of Diffusion in Silicon. Under conditions of thermal equilibrium, a Si crystal contains a certain equilibrium concentration of vacancies, C v°, and a certain equilibrium concentration of Si self-interstitials, Cz°. For diffusion models based on the vacancy, Cv° Cf and the coefficients of dopant diffusion and self-diffusion can be described by equation 27 (15)... 

Calculation of the oxygen vacancy concentration at the interconnector surface On the basis of the point defect theory, the oxygen vacancy concentration (mole fraction) 8 on the fuel and air side surfaces of the interconnector are calculated [34], In an equilibrium state, the formation of the oxygen vacancy can be described as follows using Kroger-Vink notation [35] ... 

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

All real crystals above 0 K contain point defects which are thermodynamically inherent [21,22]. In a monatomic crystal, the simplest defects are the vacancy, a lattice site that is empty, and the interstitial atom, an atom on an interstitial site in the lattice. The equilibrium concentration of these defects is thermally controlled and has an exponential dependence on temperature. For example, the site fraction of vacancies, c in a pure monatomic crystal is given by ... 

Note that via the potent combination of microscopic evaluation of relevant material parameters such as the vacancy formation energy and statistical mechanical reasoning to treat the entropic effects of the presence of vacancies we have arrived at a prediction for the equilibrium concentration of point defects. 

Surfaces are heterogeneous on the atomic scale. Atoms appear in flat terraces, at steps, and at kinks. There are also surface point defects, vacancies, and adatoms. These various surface sites achieve their equilibrium surface concentrations through an atom-transport process along the surface that we call surface diffusion. Adsorbed atoms and molecules reach their equilibrium distribution on the surface in the same way. This view of surface diffusion as a site-to-site hopping process leads to the random-walk picture, in which the mean-square displacement of the adsorbed particle along the. r-component of the coordinate is given by... 

Figure 11.15b also shows an interesting way of extrapolating the equilibrium hole fraction h T,P = 0) to lower temperatures. When considering the vacancies of the S-S lattice as Schottky point defects, their equilibrium concentration heq may be expressed by the Schottky equation. 

Figure 11.3 is a plot of nAE, AS, and AG. From this plot you can see that introducing vacancies lowers the free energy of the crystal until an equilibrium concentration is reached adding too many vacancies increases G again. At higher temperatures the equilibrium number of vacancies increases. The implications are important. In pure crystals we expect to hnd point defects at all temperatures above OK. Since these defects are in thermodynamic equilibrium, annealing or other thermal treatments cannot remove them. 

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... 

Intrinsic vacancies are much more numerous in metals. For example, in a 1-cm crystal of aluminum at room temperature there are about 9 billion vacancies. In a crystal of silicon in equilibrium at room temperature there are only about 1 x 10 intrinsic vacancies per cubic centimeter. This is considerably less than typical concentrations of extrinsic point defects (dopants) in silicon—about 0.0001% another fortunate fact. 

If a crystal is annealed at a sufficiently high temperature and for long enough, then the equilibrium concentration of vacancies will increase. Abrupt quenching of the material can freeze in unusually high concentrations of point defects. 

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