Migration, vacancy

Thus, the vacancies migrate along with the interstitial atoms (ions) equalfy... 

In this book and in general, the concentration, d, is used to represent the number of mobile species, ions or vacancies, per unit volume whereas is always the fraction of the crystallographically equivalent sites that are occupied by ions. Often if > 0.5 we talk of ion migration and c refers to ion concentration, whereas if < 0.5 we talk of vacancy migration and c refers to the concentration of vacancies. However, this is simply a convenient way of thinking about ion transport in solids by focusing on the minority species. It is equally possible to describe conduction always in terms of ions or of vacancies provided account is taken of the fact that both the concentration of mobile species and sites to which they may migrate are important. The importance of c and (1 — c) is emphasised and placed within a unified framework in Chapter 3. The concentration of ions Ci is related to the occupancy by = cJC, where C is the concentration of the sites. 

Let us now consider the crystal MO. If the diffusion takes place by migration of cationic vacancies, the number of atoms that undergo the process depends on the vacancy concentration [Vm] and the thermal state of single atoms M (the jump takes place only whenever atom M in the neighborhood of the vacancy has sufficient energy to perform it). The diffusion coefficient associated with the vacancy migration process is given by... 

Vm = migration enthalpy of cationic vacancy = migration enthalpy of anionic vacancy /7m = migration enthalpy of cationic interstitial //xj = migration enthalpy of anionic interstitial... 

Basically, whenever isotopic exchanges occur between different phases (i.e., heterogeneous equilibria), isotopic fractionations are more appropriately described in terms of differential reaction rates. Simple diffusion laws are nevertheless appropriate in discussions of compositional gradients within a single phase— induced, for instance, by vacancy migration mechanisms, such as those treated in section 4.10—or whenever the isotopic exchange process does not affect the extrinsic stability of the phase. 

Figure 4.40 Illustration of diffusion mechanisms in alloys and ionic solids (a) interchange (exchange) (b) ring rotation (rare) (c) interstitial migration and (d) vacancy migration. From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

The extrinsic case applies at low temperatures or large doping levels. The site fraction of cation vacancies is equal to the solute-atom site-fraction and is therefore temperature independent. In the extrinsic regime, no thermal defect formation is necessary for cation self-diffusion and the activation energy consists only of the activation energy for cation vacancy migration. 

The mobility, of the ions is governed by the potential barrier between interstitial site in the case of interstitial ion migration or between lattice sites in the case of vacancy migration. 

Schematic illustration of several mechanisms proposed for substitutional diffusion. (A) Ring interchange, (B) simple interchange, and (C) vacancy migration. 

Ernest Kirkendall s doctoral thesis on the interdififiision between copper and brass disproved the almost universally accepted concept that diffusion in substitutional solutions occurs by interchange of atoms and led to the conclusion that diffusion was the result of vacancy migration. His results were so startling to the leading metallurgists of the day that publication of his work was held up by reviewers who doubted his results. 

In the absence of an electric field the charged vacancy migrates randomly, and its mobility depends on temperature since this determines the ease with which the Na+ surmounts the energy barrier to movement. Because the crystal is highly ionic in character the barrier is electrostatic in origin, and the ion in its normal lattice position is in an electrostatic potential energy well (Fig. 2.17). 

Ref.205) this is not the case for a vacancy migrating in an electrical experiment, nor for an isotope in a tracer experiment moving according to an interstitial mechanism (Fig. 42 bottom). The relation between D and Dq then reads... 

Williford RE, Weber WJ, Devanathan R, Cormack AN (1999) Native vacancy migrations in zircon. J Nucl Mater 273 164-170... 

---