Page 117 - Nanotechnology an introduction

The above three equations can be used to derive some more:

(10.2a)

(10.2b)

(10.2c)
To each of these there corresponds a mass action law, i.e., for the last three, writing x for the mole fraction, identified by its subscript:

(10.3a)

(10.3b)

(10.3c)
where the K are the equilibrium constants. The defect concentrations are supposed to be sufficiently small so that the chemical potentials μ depend
on the logarithm of x; e.g., for vacancies:

(10.4)
where E is the energy to annihilate one mole of vacancies.
□
Of course the usual electroneutrality condition holds:

(10.5)
Impurities
This formalism can be powerfully applied to the effect of impurities on defects. Examples:

(10.6)
(silver chloride doped with sodium chloride): here there would be practically no change in defect concentration as a result of the doping; but if it is
with divalent cadmium, there would evidently be a significant increase in silver vacancies and a decrease in interstitial silver ions:

(10.7)
and

(10.8)
In these mixed phases, the electroneutrality condition becomes

(10.9)
where y is the lattice concentration of the dopant. This relation, together with the three (10.3a), (10.3b) and (10.3c), can be used to derive explicit
relations for the concentration dependence of each of the four defect types in equation (10.8) on y.
Surfaces
In very small particles, a significant fraction of the atoms of the particle may be actually surface atoms. Because of their different bonding, the
surface atoms may be sources or sinks of defects, thereby perturbing the equilibria (10.1a). The adsorption of impurities at the surface of a
+
nanoparticle will generate defects. For example, adsorption of two monovalent Ag onto the surface of a ZnS nanoparticle will require the formation
of a zinc vacancy in the cation sublattice. The environment of the nanoparticle (the matrix) thus plays a crucial role in determining its properties.
10.4. Spacial Distribution of Defects
If p is the probability that an atom is substituted by an impurity, or the probability of a defect, then the probability of exactly k impurities or defects
among n atoms is

(10.10)
where q = 1 − p. If the product np = λ is of moderate size (~1), the distribution can be simplified to:

(10.11)

the Poisson approximation to the binomial distribution. Figure 10.2 shows an example. Hence, the smaller the device, the higher the probability that
it will be defect-free. The relative advantage of replacing one large device by m devices each 1/mth of the original size is , assuming
that the nanification does not itself introduce new impurities.