Diffusion mechanism vacancy

Consider the following diagram, given as 4.7.1. on the next page, showing two types of self-diffusion. Self diffusion can occur by at least two mechanisms, vacancy and interstitial. Both are "hopping" motions, as described above. 

One type of diffusion mechanism is known as the interstitial mechanism because it involves movement of a lattice member from one interstitial position to another. When diffusion involves the motion of a particle from a regular lattice site into a vacancy, the vacancy then is located where the site was vacated by the moving species. Therefore, the vacancy moves in the opposite direction to that of the moving lattice member. This type of diffusion is referred to as the vacancy mechanism. In some instances, it is possible for a lattice member to vacate a lattice site and for that site to be filled simultaneously by another unit. In effect, there is a "rotation" of two lattice members, so this mechanism is referred to as the rotation mechanism of diffusion. 

In the case of interstitials—self-interstitials, impurities, or dopants—two diffusion mechanisms can be envisaged. In the simplest case, an interstitial can jump to a neighboring interstitial position (Fig. 5.8a). This is called interstitial diffusion and is sometimes referred to as direct diffusion to distinguish it from vacancy diffusion (indirect diffusion). 

The vacancy will follow a random-walk diffusion route, while the diffusion of the tracer by a vacancy diffusion mechanism will be constrained. When these processes are considered over many jumps, the mean square displacement of the tracer will be less than that of the vacancy, even though both have taken the same number of jumps. Therefore, it is expected that the observed diffusion coefficient of the tracer will be less than that of the vacancy. In these circumstances, the random-walk diffusion equations need to be modified for the tracer. This is done by ascribing a different probability to each of the various jumps that the tracer may make. The result is that the random-walk diffusion expression must be multiplied by a correlation factor, / which takes the diffusion mechanism into account. 

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 climb of mixed dislocations possessing some screw character can proceed by basically the same jog-diffusion mechanism as that for the pure edge dislocation.10 On the other hand, a pure screw dislocation can climb if the excess vacancies convert it into a helix, as in Fig. 11.10. Here the turns of the helical dislocation possess... 

On the basis of thermodynamic considerations, some of the lattice sites in the crystal are vacant, and the number of vacant lattice sites generally is a function of temperature. The movement of a lattice atom into an adjacent vacant site is called vacancy diffusion. In addition to occupying lattice sites, atoms can reside in interstitial sites, the spaces between the lattice sites. These interstitial atoms can readily move to adjacent interstitial sites without displacing the lattice atoms. This process is called interstitial diffusion. The interstitial atoms may be impurity atoms or atoms of the host lattice, but in either case, interstitial atoms are generally present only in very dilute amounts. However, these atoms can be highly mobile, and in certain cases, interstitial diffusion is the dominant diffusion mechanism. 

When applying this relationship, one must be aware of (z) all diffusion mechanisms operative in a non-growing compound, (zz) the concentration of vacancies of a given component in this compound and (zzz) the value of its self-diffusion coefficient associated with the vacancy mechanism. In view of the lack of specially planned experiments aimed at obtaining all necessary data for the same compound, including reaction- and self-diffusion coefficients of its components, at present only calculations based on the results compiled from several works are possible. 

As explained in Chapter 5, the transport mechanism in dense crystalline materials is generally made up of incessant displacements of mobile atoms because of the so-called vacancy or interstitial mechanisms. In this sense, the solution-diffusion mechanism is the most commonly used physical model to describe gas transport through dense membranes. The solution-diffusion separation mechanism is based on both solubility and mobility of one species in an effective solid barrier [23-25], This mechanism can be described as follows first, a gas molecule is adsorbed, and in some cases dissociated, on the surface of one side of the membrane, it then dissolves in the membrane material, and thereafter diffuses through the membrane. Finally, in some cases it is associated and desorbs, and in other cases, it only desorbs on the other side of the membrane. For example, for hydrogen transport through a dense metal such as Pd, the H2 molecule has to split up after adsorption, and, thereafter, recombine after diffusing through the membrane on the other side (see Section 5.6.1). 

The first qualitative observation of vacancy-induced motion of embedded atoms was published in 1997 by Flores et al. [20], Using STM, an unusual, low mobility of embedded Mn atoms in Cu(0 0 1) was observed. Flores et al. argued that this could only be consistent with a vacancy-mediated diffusion mechanism. Upper and lower limits for the jump rate were established in the low-coverage limit and reasonable agreement was obtained between the experimentally observed diffusion coefficient and a theoretical estimate based on vacancy-mediated diffusion. That same year it was proposed that the diffusion of vacancies is the dominant mechanism in the decay of adatom islands on Cu(00 1) [36], which was also backed up by ab initio calculations [37]. After that, studies were performed on the vacancy-mediated diffusion of embedded In atoms [21-23] and Pd atoms [24] in the same surface. The deployment of a high-speed variable temperature STM in the case of embedded In and an atom-tracker STM in the case of Pd, allowed for a detailed quantitative investigation of the vacancy-mediated diffusion process by examining in detail both the jump frequency as well as the displacement statistics. Experimental details of both setups have been published elsewhere [34,35]. A review of the quantitative results from these studies is presented in the next subsections. 

In Section 3 we derive that for the vacancy-mediated diffusion mechanism, one expects the shape of the jump length distribution to be that of a modified Bessel function of order zero. Both distributions can be fit very well with the modified Bessel function, again confirming the vacancy-mediated diffusion mechanism for both cases. The only free parameter used in the fits is the probability prec for vacancies to recombine at steps, between subsequent encounters with the same embedded atom [33]. This probability is directly related to the average terrace width and variations in this number can be ascribed to the proximity of steps. The effect of steps will be discussed in more detail in Section 4. 

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