Vacancy energy formation

Table 4 Tight-binding vacancy formation energies compared to first-principles calculations and experiment. Energies were computed using a 108 atom supercell. The experimental column shows a range of energies if several experiments have been tabulated. Otherwise the estimated error in the experiment is given.

Vacancy Formation Energy (eV) Element Tight-Binding Experiment Fixed Relaxed... 

This has important consequences for the energetics of defects in metals. For example, the vacancy formation energy within a nearest neighbour pair... 

This disagrees with experiment where the vacancy formation energy in metals is typically only about one-half the cohesive energy. 

Suppose now that such a source is present in a crystal that is rapidly quenched from a temperature Tq to a temperature Ta to produce supersaturated vacancies. Find an expression for the critical value of the quenching temperature, Tq, which must be used to produce sufficient supersaturation to activate the source so that it will be able to create dislocations loops capable of destroying the supersaturated vacancies by climb. The vacancy formation energy is Ey and the segment length is L. 

By definition, the rate at which the tracer atom is displaced by a surface vacancy is the product of the vacancy density at the site next to the tracer times the rate at which vacancies exchange with the tracer atom. For the case where the interaction between the tracer atom and the vacancy is negligible, the activation energy obtained from the temperature dependence of the total displacement rate equals the sum of the vacancy formation energy EF and the vacancy diffusion barrier ED. When the measurements are performed with finite temporal resolution and if there is an interaction present between the vacancy and the indium atom, this simple picture changes. 

The importance of the measurements that we have presented so far for the diffusion of embedded tracer atoms becomes evident when we now use these measurements and the model discussed in Section 3 to evaluate the invisible mobility of the Cu atoms in a Cu(00 1) terrace. The results presented in Section 2 imply that not just the tracer atom, but all atoms in the surface are continuously moving. From the tracer diffusion measurements of In/Cu(0 0 1) we have established that the sum of the vacancy formation energy and the vacancy diffusion barrier in the clean Cu(0 01) surface is equal to 717 meV. For the case of self-diffusion in the Cu(0 01) surface we can use this number with the simplest model that we discussed in Section 3.2, i.e. all atoms are equal and no interaction between the vacancy and the tracer atom. In doing so we find a room temperature hop rate for the self-diffusion of Cu atoms in a Cu(00 1) terrace of v = 0.48 s-1. In other words, every terrace Cu atom is displaced by a vacancy, on average, about once per two seconds at room temperature and about 200times/sec at 100 °C. We illustrate this motion by plotting the calculated average displacement rate of Cu terrace atoms vs. 1 /kT in Fig. 14. 

The energetics of non-stoichiometric surfaces can be characterized in two different ways. One consists in defining the vacancy formation energy Evf. It is the energy required to extract one neutral oxygen atom from the... 

An alternative perspective on the subject of point defects to the continuum analysis advanced above is offered by atomic-level analysis. Perhaps the simplest microscopic model of point defect formation is that of the formation energy for vacancies within a pair potential description of the total energy. This calculation is revealing in two respects first, it illustrates the conceptual basis for evaluating the vacancy formation energy, even within schemes that are energetically more accurate. Secondly, it reveals additional conceptual shortcomings associated with... 

If a vacancy is created in an otherwise perfect lattice, the implication is that an additional atom must now occupy the surface of the crystal. The basic idea behind the vacancy formation energy is that it is a measure of the difference in energy between two states one being the perfect crystal and the other being that in which an atom has been plucked from the bulk of the crystal and attached to the surface. From a computational perspective, the vacancy formation energy may be defined as... 

What is remarkable about this result is that we have found that the vacancy formation energy is with opposite sign equal to the cohesive energy per atom. An assessment of the accuracy of this conclusion is given in table 7.1. We note... 

Table 7.1. Cohesive energies and measured vacancy formation energies (in eV)for several representative metals (adapted from Carls son (1990)).

In this expression, we have made the simplifying assumption that the model is strictly reckoned on the basis of near-neighbor interactions. The outcome of the calculation is that the vacancy formation energy is given by... 

This result illustrates the amendment to the vacancy formation energy due to the embedding term. For Johnson s simple analytic model discussed in chap. 4, recall that the embedding function is given by... 

For the particular choice of parameters adopted by Johnson in his treatment of Cu, this result implies a vacancy formation energy of e ac 1.3 eV. Note that this result refers to the unrelaxed vacancy formation energy, and hence we should anticipate a further reduction in the energy to form such a vacancy once the atoms in the vicinity of the vacancy are allowed to adjust their positions. 

Fig. 7.13. Schematic illustrating the geometry of a supercell of the type used to evaluate the vacancy formation energy (adapted from Payne et at. (1992)). Dotted lines denote the limits of the computational cell which is then repeated periodically.

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