A number of different point defects can form in all ceramics, but their concentration and distributions are interrelated. In the event of the production of a vacancy by the displacement of a lattice atom, this released atom can be either contained within the crystal lattice as an interstitial species (forming a Frenkel pair), or it can migrate to the surface to form part of a new crystal layer (resulting in a Schottky reaction). Figure 1 represents a Frenkel pair: both cations and anions can undergo this type of disor­der reaction, resulting in cation Frenkel and anion Frenkel pairs, respectively. In ceramic materials, both the vacancy and interstitial defects are usually charged, but the overall reaction is charge-neutral. The energy necessary for this reaction to proceed is the energy to create one vacancy by removing an ion from the crystal to infinity plus the energy to create one interstitial ion by taking an ion from infinity and placing it into the crystal. The implication of remov­ing and taking ions from infinity implies that the two species are infinitely separated in the crystal (unlike the two species shown in Figure 1). As separated species, these are defects at infinite dilution, a

well-defined thermodynamic limit. As the two defects are charged, they will interact if not infinitely sepa­rated, a point we will return to later.

As with the Frenkel reaction, the Schottky disorder reaction must be charge-neutral. Here, only vacancies are created, but in a stoichiometric ratio. Thus, for a material of stoichiometry AB, one A vacancy and one B vacancy are created. The displaced ions are removed to create a new piece of lattice. It is impor­tant to realize that we are dealing with an equilibrium process for the whole crystal. Thus, as the tem­perature changes, many thousands of vacancies are created/destroyed and new material containing many thousands of ions is formed. Thus, it is not simply that one new molecule is formed, but there is also an increase in the volume of the crystal, which is why the lattice energy is part of the Schottky reaction.

The energy for the Schottky disorder reaction to proceed in an AB material is the energy to create one A-site vacancy by removing an ion from the crystal to infinity plus the energy to create one B-site vacancy by removing an ion from the crystal to infinity, plus the lattice energy associated with one unit of the AB compound. For example in Al2O3, the energy would be that associated with the sum of two Al vacancies plus three O vacancies plus the lattice energy of one Al2O3 formula unit. Again, the vacancy species are assumed to be effectively infinitely separated.

In a crystalline material with more than one type of atom, each species usually occupies its own sub­lattice. If two different species are swapped, this pro­duces an antisite pair (see Figure 2). For example, in an AB compound, one A atom is swapped with the B atom. While this would be of high energy for an AO compound, where A and O are of opposite charge (e. g., Mg2+ and O2—), in an ABO3 material where A and B may have similar or even identical positive charges, antisite energies can be

chemistry terms, a lattice

site means a position in the crystal that an ion will usually occupy in that crystal structure), the number of ways, OA, of arranging n A-site vacancies is

N!

n!(N — n)!

As we have n B-site vacancies to distribute over N, B-lattice sites, the total number of configurations is the product of OA and OA:

N! ‘

n!(N — n)!

( N! ‘

n!(N — n)!

where N and n are large, as they are when dealing with crystals, we can invoke Stirling’s approxima­tion, which states that ln(MZ) = M ln(M) — M. Thus,

ASc = 2k[N ln(N) — (N — n)ln(N — n) — nln(n)]

Therefore,

AG=nAgf — 2kT [N ln(N) — (N—n)ln(N—n) — nln(n)]

=nAgf — 2kT

To find the equilibrium number of defects, we need to find the minimum of AG with respect to n (see Figure 3). That is

0=Agf — 2kT ln

Assuming that the number of defects is small in comparison to the number of available lattice sites, then N n N:

n Ag Ah As

N =exp — Ш =exp — rn exp 2k

Usually, we assume that the energy associated with the change in vibrational entropy is negligible so that the concentration of defects (n/N) is dominated by the enthalpy of reaction:

M=N=“p(-2T)

However, this is not always a valid assumption and care must be taken. When defect concentrations are measured experimentally, they are presented on an Arrhenius plot of ln(concentration) versus 1/ T which yields straight lines with slopes that are pro­portional to the disorder enthalpy (see Figure 4).