Defect in ionic solids

Real crystals are imperfect. Every lattice site is not occupied. There can be a vacancy at a site called a Schottky defect created by moving an atom from its lattice site in the interior to the surface of the crystal. Another kind of defect is a Frenkel defect, in which an anion is transferred from its site to an interstitial position. The number of Schottky vacancies present is dependent on the temperature. It can be shown form statistical mechanics that the fraction of sites f vacant at a given temperature is proportional to

F = exp – E / k T

Where E is the defect formation energy, that is, the energy needed to move an atom from its lattice site to t he exterior of the crystal. The question we are interested in examining is how nanosizing affects the fraction of defects in the material, which means that we have to examine the effect of nanosizing on the defect formation energy. For a crystal consisting of nonpolarizable constituents, the defect formation energy is the lattice energy per atom U. in the case of ionic solids U is an overestimate of the defect formation energy because it neglects the polarization energy produced by the vacancy. when an ion is removed from its site, the ions around the site experience an electric field and are polarized. The potential ϕ from t his polarization must be included I determining the defect formation energy, which is then

E = U – e ϕ / 2

We consider simple model for the potential ϕ. When a positive ion is removed from its site, the neighbouring ions experience and electric filed e/r^2, where r is the distance of the ion from the vacancy. the neighbouring ions become polarized having a dipole

μ + = a + e/ r^2 for positive ions and μ- = a – e/r^2 for negative ions,

where a is the polarizability of the ion, which will differ between Na+ ions and Cl- ions. The potential produced by a dipole is proportional to μ /r2 and depends on the orientation of the dipole. Thus the induced dipole on the ions produces a potential ϕ at the vacancy given by

ϕ = ∑ a+e /r j^ 4 + ∑ a- e/ r k ^4

where rj is the distance of positive ions from the vacancy and rk is the distance of negative ions form the vacancy.

to illustrate the effect of nanosizing on the energy to form a vacancy, let us calculate the defect formation energy to remove a positive ion in the middle of our linear NaCl lattice discussed above. The energy will be e ϕ /2. because the l / r^4 term decreases rapidly with r at most, we need only go to next nearest neighbours in the summation to calculate the polarization contribution to the formation energy. In this case the energy will be

E = 2a + e / R^4 [ ½^4 ¼^4] + 2 a = e ^2 / R^4 [ 1 + 1/3 ^4]

Frank Owen, Charles Poole, The physics and Chemistry of Nanosolids, Wiley, 2008