Vacancies interaction energy

Let us consider a crystal similar to that discussed in Sections 1,3.3 and 1.3.4, which, in this case, shows a larger deviation from stoichiometry. It is appropriate to assume that there are no interstitial atoms in this case, because the Frenkel type defect has a tendency to decrease deviation. Consider a crystal in which M occupies sites in N lattice points and X occupies sites in N lattice points. It is necessary to take the vacancy-vacancy interaction energy into consideration, because the concentration of vacancies is higher. The method of calculation of free energy (enthalpy) related to is shown in Fig. 1.12. The total free energy of the crystal may be written... 

In the limit that nearest-neighbor contributions dominate, such a one-center nb automatically leads to an approximate 21/2-fold increase in overlap and two-fold increase in interaction energy, compared with a two-center 7tb donor. A corresponding enhancement results when the pi-acceptor is reduced from two-center (7tb ) to one-center (nb ) form, i.e., a valence p-type vacancy. Unlike the intrinsically bidirectional character of conjugation between two pi bonds (7ta->-7tb, 7tb 7ta ), the interactions of a pi bond with a nonbonding center are intrinsically mono-directional and lead to uncompensated transfer of pi charge from one moiety to the other. 

When divalent cation impurities (e.g. Cd, Sr ) are present in an ionic solid of the type MX consisting of monovalent ions, the negatively charged cation vacancies (created by the divalent ions) are bound to the impurity ions at low temperatures. Similarly, the oppositely charged cation and anion vacancies tend to form neutral pairs. Such neutral vacancy pairs are of importance in diffusion, but do not participate in electrical conduction. The interaction energy of vacancy pairs or impurity-vacancy pairs decreases with the increase in distance between the two oppositely charged units. 

It is noted that by increasing <) to some extent the interaction energy between vacancies plays an important role in non-stoichiometric compounds, as mentioned below. 

Consider a crystal Mj X which contains both metal vacancies and interstitial metal atoms in low concentration, i.e. M occupies lattice points in N lattice sites of metal, X occupies N in N lattice sites (generally, N, N, but in this calculation we assume = Nf and, moreover, interstitial M occupies in Na, where a is a constant which is fixed by crystal structure. If the conditions N N — N ), (N — N ), N are satisfied, it is not necessary to take the interaction energy between defects, as mentioned below, into consideration. The free energy of the crystal may be written as... 

Fig. 1.12 Interaction energy, s , between metal vacancies and its calculation, (a) denotes the interaction energy (enthalpy) between the first nearest vacancies, (b) A metal vacancy has metal sites as first nearest neighbours (labelled 1 ). The probability of being a vacancy in a metal site equals (N — Aml/A. The interaction energy between a metal vacancy and its first nearest neighbour vacancies is The total...

In Fig. 1.14, the dotted lines for each curve show the activity of the coexisting phases at chemical equilibrium. Similarly in Fig. 1.16 the dotted line BDF shows the activity of the coexisting phases (5 = 0.185 and 0.815). The coexisting phases, which have the same structure, differ in the concentration of vacancies. This phenomenon is generally called phase separation or spinodal decomposition (it is observed not only in the solid phases but also in the liquid phases), and originates from the sign of the interaction energy... 

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. 

For an anisotropic defect, like crowdions or di-atomic quasi-molecules (H and Vic centres), the problem becomes much more complicated and often permits only a numerical solution (e.g., [72]). For example, an estimate of the interaction energy of a crowdion with a vacancy in Cu in the direction perpendicular to the crowdion axis is 0.1 eV at the relative distance /2ao (ao is a lattice constant) if both are in the same plane, but this energy becomes 0.02 eV only for a distance twice as large (ip = 0 in Fig. 4.8(a)). Increase of the angle

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The coordination numbers of the lanthanide cation in these higher oxides depend on the type and numbers of the modules assembled in the phase. When one module is stacked upon another, then a cation located on the interface would be in both modules. The type of stacked module, as shown in Figure 25, determines the coordination number of the cation. The separation distance between the two oxygen vacancies dominates the Coulomb interaction energy of the system the largest separation should be favorable since the shorter one has a higher Coulomb interaction energy. 

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. 

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